Skip to main content

Full text of "An easy introduction to the mathematics : in which the theory and practice are laid down and familiarly explained"

See other formats


Google 


This  is  a  digital  copy  of  a  book  that  was  preserved  for  generations  on  library  shelves  before  it  was  carefully  scanned  by  Google  as  part  of  a  project 

to  make  the  world's  books  discoverable  online. 

It  has  survived  long  enough  for  the  copyright  to  expire  and  the  book  to  enter  the  public  domain.  A  public  domain  book  is  one  that  was  never  subject 

to  copyright  or  whose  legal  copyright  term  has  expired.  Whether  a  book  is  in  the  public  domain  may  vary  country  to  country.  Public  domain  books 

are  our  gateways  to  the  past,  representing  a  wealth  of  history,  culture  and  knowledge  that's  often  difficult  to  discover. 

Marks,  notations  and  other  maiginalia  present  in  the  original  volume  will  appear  in  this  file  -  a  reminder  of  this  book's  long  journey  from  the 

publisher  to  a  library  and  finally  to  you. 

Usage  guidelines 

Google  is  proud  to  partner  with  libraries  to  digitize  public  domain  materials  and  make  them  widely  accessible.  Public  domain  books  belong  to  the 
public  and  we  are  merely  their  custodians.  Nevertheless,  this  work  is  expensive,  so  in  order  to  keep  providing  tliis  resource,  we  liave  taken  steps  to 
prevent  abuse  by  commercial  parties,  including  placing  technical  restrictions  on  automated  querying. 
We  also  ask  that  you: 

+  Make  non-commercial  use  of  the  files  We  designed  Google  Book  Search  for  use  by  individuals,  and  we  request  that  you  use  these  files  for 
personal,  non-commercial  purposes. 

+  Refrain  fivm  automated  querying  Do  not  send  automated  queries  of  any  sort  to  Google's  system:  If  you  are  conducting  research  on  machine 
translation,  optical  character  recognition  or  other  areas  where  access  to  a  large  amount  of  text  is  helpful,  please  contact  us.  We  encourage  the 
use  of  public  domain  materials  for  these  purposes  and  may  be  able  to  help. 

+  Maintain  attributionTht  GoogXt  "watermark"  you  see  on  each  file  is  essential  for  in  forming  people  about  this  project  and  helping  them  find 
additional  materials  through  Google  Book  Search.  Please  do  not  remove  it. 

+  Keep  it  legal  Whatever  your  use,  remember  that  you  are  responsible  for  ensuring  that  what  you  are  doing  is  legal.  Do  not  assume  that  just 
because  we  believe  a  book  is  in  the  public  domain  for  users  in  the  United  States,  that  the  work  is  also  in  the  public  domain  for  users  in  other 
countries.  Whether  a  book  is  still  in  copyright  varies  from  country  to  country,  and  we  can't  offer  guidance  on  whether  any  specific  use  of 
any  specific  book  is  allowed.  Please  do  not  assume  that  a  book's  appearance  in  Google  Book  Search  means  it  can  be  used  in  any  manner 
anywhere  in  the  world.  Copyright  infringement  liabili^  can  be  quite  severe. 

About  Google  Book  Search 

Google's  mission  is  to  organize  the  world's  information  and  to  make  it  universally  accessible  and  useful.   Google  Book  Search  helps  readers 
discover  the  world's  books  while  helping  authors  and  publishers  reach  new  audiences.  You  can  search  through  the  full  text  of  this  book  on  the  web 

at|http: //books  .google  .com/I 


mr^ 


.!        »t.'.  7f  .•    .  ,.-.1    :.. 


*f   ,< 


'*>  >- ' ."  »i 


w 

Wf^m 

s 

^^ 

iffi^' 

Ep- '^^ 

^-^  «Sl^'S^H*^ 

r,"5^"^v%wi 

iS 

'•^*li 

iiSrp^'7>  ^ 

^^^in^bdEwS^ 

^W 

WJ>3-- 


iiiiiiiiiniinniiimiiiiiniyiimnn|ff 


THE  GIFT  OF 

Prof  .William  H.Eutta 


niiiiiiii!i:iitiiiiiiniiiiiiii: 


fA        --  ■       -  !       7 


i 


-*   t» 


v'         -             ' 

1 

^ 

A 

,i 

•  r* 

I 


H 


i  ... 

if 


AN 

EASY   INTRODUCTION 

TO    THE 

MATHEMATICS; 

IN  WHICH 

THE  THEORY  AND  PRACTICE 

ARE  LAID  DOWN  AND  FAMILIARLY  EXPLAINED. 

To  each  subject  are  prefixed, 

A  BRIEF  POPULAR  HISTORY  OF  ITS  RISE  AND  PROGRESS,  CONCISE  MEMOIRS 

OF  NOTED  MATHEMATICAL  AUTHORS  ANCIENT  AND  MODERN, 

AND  SOME  ACCOUNT  OF  THEIR  WORKS. 

The  whole  forming 

A  COMPLETE  AND  EASY  SYSTEM 

or 

ELEMENTARY    INSTRUCTION 

IN   THE 

LEADING  BRANCHES  OF  THE  MATHEMATICS; 

DESIGNED  TO  FURNISH  STUDENTS  WITH  THE  MEANS  OF  ACQUIRING  CONSIDERABLE 
PROFICIENCY^   WITHOUT   THE  NECESSITY   OF  VERBAL  ASSISTANCE. 

Adapted  to  the  use  of 

SCHOOLS,  JUNIOR  STUDENTS  AT  THE  UNIVERSITIES,  AND  PRIVATE 

LEARNERS, 

B8FECIALLT  THOSE  WHO   STUDY  WITHOUT  A  TUTOR. 


IN  TWO  VOLUMES. 


BY  CHARLES  BUTLER. 


n^PT  irh  S|f  t^m,  Uv  ftii  r«y  ^m 

fiuyy  ns,  h  ir^rt  r»t$  ^nrwfumt.     «»IARMnN. 


Shake  off  your  ease,  and  send  your  name  rJfA  tv 

To  immortality  and  fame,  /   "^^CLx^l^ 

By  ev'ry  hour  that  flies.  Watts.  I  O^st^j^y,  ^^ 

VOL.  II. 


OXFORD: 

PRINTED  BY  BARTLETT  AND  NEWMAN; 

AND   SOLD  BT  LONGMAN,   HURST,    RE£S,   ORME,  AND  BROWN,  PATERNOSTER  ROWj 

LONDON;   PARKER,   OXFORD;  AND   DEIOHTON,    CAMBRIDGE, 


0 


18H. 


^, 


o 


(a^^u^.  i*/-^^--  /4./:iint?- 


CONTENTS. 


ALGEBRA. 

PAQB 

GsNERAL  Problems.  Their  Nature  and  Properties  explained  1 
Method  of  registering  the  Steps  of  an  Operation .    .    17 

AaiTUMETICAL  PROGRESSION. 

Its  Rules  A^ebraically  investigated 36 

and  applied     .     .    39 
Problems  exercising  Arithmetical  Progression     .     .    40 

Permutations 49 

Combinations 43 

Simple  Interest,  its  Rules  invest^ted  and  applied     •    .    45 

'^       Discount^  its  Rules  investigated  and  applied 48 

^i       Tbe  Doctrine  of  Ratios '    •    *    .    ^ 

^  Continued  Fractions .••«••    58 

Proportion^  Direct •    •    .    .    62 

/V3  Inverse^  or  Reciprocal  Proportion 69 

\  Harmonical  Proportion      .    r 70 

^  Contra-harmpnical  Proportion 73 

t       Comparison  of  variable  and  dbfenobnt  Quantities  .    74 
^      Geometrical  Progression. 

Its  Rules  investigated 89 

and  applied 87 

Problems  in  GeometricaLProgression  .....  89 
Compound  Interest^  its  Rules  investigated  and  applied  .  91 
Properties  of  Numbers^  an  Investigation  of  those  ivhich 

*      are  most  generally  useful     . 93 

SauATioNs  of  several  Dimension?*  • 

A  general  View  of  the  Nature,  Formation^  Roots,  &c. 

of  Equations Ill 

Generation  of  tbe  higher  Equations     .    .    ,    «    .113 

Depression  of  Equations 117 

Transformation  of  Equations •  118 

To  find  the  Limits  of  the  Roots      ^ i^e 

aS 


iv  CONTENTS. 

«  PAOB 

To  find  the  possible  Roots  of  an  Equation     .    •    .  129 

By  Newton*8  Method  of  Divisors  132 

Recurring  Equations 134 

Cubic  EctUATioNS,  Cardan's  Rule 138 

BiauADRATic  EauATioNs,  Des  Cartes'  Rule 143 

Euler'sRule 146 

Simpson's  Rule 147 

Afpboximation. 

To  revolve  Equations  by  the  simplest  Method     .     .150 

By  Simpson's  Rule 153 

By  Bernoulli's  Rule 155 

Exponential  Equations 159 

•  Dr.  Button's  Rule  for  extracting  the  Roots  of  Num- 

bers by  Approximation 162 

Problems  producing  Equations  of  three  or  morb 

Dimensions »    .     .     .  163 

Indeterminate  Analysis 165 

Solution  of  Indeterminate  Problems 173 

Diophantine  Problems 176 

Infinite  Series^  their  Nature,  &c 181 

To  reduce  Fractions  to  Infinite  Series 182 

To  reduce  compound  quadratic  Surds  to  Infinite 

Series       184 

Newton's  Binomial  Theorem 185 

To  find  the  Orders  of  Diffiirences 190 

To  find  any  Term  of  a  Series 191 

To  interpolate  a  Series 199 

To  revert  a  Series 195 

To  find  the  Sum  of  a  Series 197 

The  Investigation  and  Construction  of  Logarithms, 

both  hyperbolical  and  common 1204 

GEOMETRt. 

•  Historical  Introduction       .'211 

On  the  Usefulness  of  Geometry  .     .     .    .-    .    .     .241 
Description  of  Mathematical  Instruments      .     .     .  242 
Of  Geometry  considered  as  the  Science  of  Demon- 
stration, "with  some  Account  of  the  Principles  of 
Reasoning,  as  introductory  to  the  Study  of  Budid  250 

Observations  on  some  Farts  of  the  first  Book  of 
£uclid*s  Elements %S9 


CONTENTS.  V 

PAOB 

On  Euclid's  second  Book 291 

On  Euclid's  third  Book 297 

On  Euclid's  fourth  Book 301 

On  Euclid's  fifth  Book 304 

On  Euclid's  sixth  Book 308 

An  Appendix  to  the  above  six  Books  of  Euclid   .     .314 
Pbactical  Geometry^  exemplifying  and  applying  Euclid's 
Theory;   the  Use  of  the  Mathematical  Instru- 
ments, &c 327 

Methods  of  constructing  Scales  of  equal  Parts     .     .  343 
To  construct  Scales  of  Chords,  Sines^  Tangents,  Se- 
cants, &c 344 

The  Mensuration  of  a  great  variety  of  plane  and 
solid  Figures,  Land,  Planks,  Timber,  Stone,  &c.     346 

TfilGONOMBTRY. 

Historical  Introduction 359 

On  the  new  (French)  Division  of  the  Quadrant  (note)  367 
Definitions  and  Principles  of  Plane  Trigonometry     371 

Variation  of  the  Algebraic  Signs 375 

Introductory  Propositions 380 

Investigation  of  Formula 389 

Method  of  constructing  Tables  of  natural  Sines, 

Tangents,  &c 394 

Method   of  constructing  Tables  of  Logarithmical 

Sines,  Tangents,  &c 399 

The  fundamental  Theorems  of  Plane  Trigonometry  400 

Solution  of  right  angled  Triangles 407 

Solution  of  oblique  angled  Triangles 413 

Mensuration  of  inaccessible  Heights  and  Distances  423 

Description  of  the  Quadrant ibid. 

Theodolite 426 

Mariner^s  Compass ibid. 

Perambulator 429 

Guntei's  Chain ibid. 

Measuring  Tapes,  Rod,  &c.    .     .  430 

Problems     .     .     . ' ibid. 

Conic  Sections.       ' 

Historical  Introduction *  .     .     .441 

The  Parabola    . 417 

The  Ellipse 458 

The  Hyperbola 478 

List  of  Subscribers 501 


ERRATA. 


7  To  the  note  at  the  bottom  of  the  page  add>  ''  The  sign  *.* 

denotes  therefore** 
18  Last  line^  for  ss^Ae  difference,  read  d^the  difference. 
44  Line  8^  for  n— 1  read  n— 1. 


Line  20,  for  n.n — l.n— 2.n— 3.n— 4^  read 
n.n— l.n»9^— 3.fi— 4. 
64  Line  3,  for  Fo  since,  read  For  etnce  ad, 
^6  Art.  68,  after  the  word  convbrtxndo^  add,  Euclid  pr.  £. 

.      Books, 
71  The  note  at  the  bottom  is  useless  here>  as  it  occurs  in  the 
latter  part  of  T?ie  Froperties  of  Numbers,  pp,  108, 109* 

97  Last  line,  for«6=1.9>  read  &-f  l.g. 

123  Dele  the  third  and  five  following  lines. 

^52  Art.  15.  line  9,  dele  *<  or  simple:* 

320  The  three  lines  JG,  BD,  and  EC  in  the  figure,  should 
intersect  in  the  point  Fon  the  circumference.  Two  or 
three  of  the  figures  in  Part  X.  are  very  indifferently   * 
cut^  but  it  is  hoped  that  there  is  nothing  which  can 
possibly  mislead>  or  affect  the  demonstrations. 


AN 

EASY    INTRODUCTION 


TO  TKX 


MATHEMATICS,  &c- 


PART   IV. 


ALGEBRA. 


OENEfeAL  PROBLEMS^ 

ART.  1. 

«/jlLGEBRA  is  divided  into  two  kinds^  numeral  and  literal, 
both  depending  on  the  same  principles  and  employing  the 
same  operations. 

^»  Numeral  algebra '  is  that  chiefly  used  in  the  solution  of 
numeral  problems,  in  which  all  the  given  quantities  are  ex- 
pressed by  numbers^  the  unknown  quantities  only  bei^g  de- 
noted by  letters  or  other  convenient  symbols.  This  kind  of 
fdgebra  has  been  largely  treated  of  in  the  preceding  volume. 

3.  Literal  or  specious  algebra  ^  is  that  in  which  all  the  quan- 


•  Numeial  algebra  is  that  part  of  the  science,  which  thcc  Earafeaos  received 
from  the  Arabs,  about  the  siddie  of  the  15th  cfoHiry.  It  doe*  oot  appear  thai 
the  latter  people,  or  even  Diophaotns,  (who  is  the  only  Oitek  writer  oa  the 
subject  at  present  known,)  nnderstood  any  thing  of  the  general  methods' now 
in  use ;  accordingly  we  find  but  little  attempted  bcyoad  the  solution  <^  nuaie* 
ijcal  problems,  in  the  writibgs  of  liucas  de  Bnrgo,  Cardan,  Drophantus,  Tar- 
talea,  BombeUi,  f^eletarios,  Stevinus,  Reoorde,  or  any  other  of 'the  early  au- 
thors who  treated  on  algebra. 

>>  Vieta,  the  great  hnpiover  of  ahnMt  every  branch^  of  the  Mathiwaatics 

YOIi.  II.  B 


S  ALOSBRA.  Fakt  IV. 

titksj  both  kaown  and  unknoim,  are  lepreaented  by  letteiB  and 
other  general  ebaracten.  This  general  mode  of  designation  is 
of  the  greaitest  use ;  as  efery  conclusion,  and  indeed  evety  step  by 
which  it  IS'  obtained,  becomes  an  universal  rule  Ibr  performing' 
every  possible  operation  of  tite  kind* 

4.  In  literal  algebra,  the  initial  letten  a,  6,  c,  d,  &c.  are  usuaBy 
employed  to  represent  known  or  ^ven  quantities,  and  the  final 
letters  x,  y,  z,  to,  v,  &c.  to  represent  unloiown  quantities,  whose 
values  are  required  to  be  found. 

5.  A  general  algebraic  problem  is  that  in  which  all  the  quan- 
tities concerned^  both  known  and  unknown,  are  represented  by 
letters  or  other  general  characters.  Not  only  such  problems  as 
have  their  conditions  pn^osed-  in  general  terms,  are  here  im* 
plied,  every  particular  numeral  problem  may  be  made  general, 
by  substituting  letters  for  the  known  quantities  concerned  in  it : 
when  this  is  done,  the  problem  which  was  originally  proposed  in 
a  particular  form,  is  now  become  a  general  problem. 

6.  Every  problem  consists  of  two  parts,  the  data,  and  the 
qtuBsita';  the  data  Include  all  the  conditions  and  quantities 
given,  and  the  qusesita  the  quantities  sought. 

7*  The  process  by  which  the  quaesita  are  obtained  by  means 
of  the  data,  that  is,  by  which  the  values  of  the  unknown  quan- 
tities are  found,  is  called  the  analysis  \  or  the. analytical 


■rr- 


known  in  his  time,  is  considered  as  the  first  who  introdaced  the  literal  aota* 
tion  of  given  quantities  into  genera!  practice,  about  the  year  1600.  Cardan 
had  indeed  given  specimens  of  such  an  improvement,  in  his  algebra,  as 
early  as  1545 ;  but  as  the  advantages  of  a  general  mode  of  notation  were  thea 
in  all  probability  not  sulBcienUy  understood,  the  method  was  not  adopted  wtil 
about  the  time  we  h«fe  mentioned.  The  impioTement  of  Viet*  was  forthor 
i^vanced  and  applied  by  Thomas  Harriot,  the  fathcar  of  modern  algebra,  abont 
1620;  likewise  by  Onghtred  in  1631,  Des  Cartes  in  1637,  and  afterwards  by 
Wallis,  Newton,  Leibnits,  the  Bemoallis,  Baker,  Raphson,  Sterling,  £uler,&ie. 
and  is  Justly  peilierred  by  all  modem  algebraists,  on  account  of  the  universality 
of  its  application.  The  letters  of.  the  alphabet  are  called  by  Vieta,  tpeciesf- 
whence  algefara  has  been  named  oritAmeiicu  spedata:  reasoning  in  species,  as 
applied  to  the  solotion  of  mathematical  problem%  appears  to  have  been  bor* 
rowed  from  the  Civiliaiis,  who  determine  cases  at  law  between  imaginary  per- 
sons, representing  them  abstractedly  by  A  and  9;  these  tliey  call  •cases  te 
a^ecUi!  this  is  the  more  probable,  as^^^ta  hunself  was  a  lawyer. 

«  The  MTord  data  means  tbtngs  given,  and  puuiia  things  sought. 

*  Thawofd  analysis^  (from  the  Greek  mmOiw*  c^m^i}  i»  lU  geneval  sense. 


Pakt  IV.  GENERAL  VROELEMB.  t 

iNTSSTIGATtDBTl  it  18  alsO  lUUned  the  SOI.UTI<IN>Or  KIBoiUTlON 

of  the  proyem. 

S.  When  the  values  of  the  unknown  qdanlhies  are  fbund  and 
express^  in  known  termsj  the  subetituttng  these  values^  each 
for  its  respective  unknown  quantity  in  the  given  equations;  that 
18^  by  reasoning  in  an  aider  the  convesse  of  anal)sis9  and  there- 
by ultimately  proving  that  the  quantities  thus  assumed  have  the 
properties  described  in  the  problem^  is  called  the  synthesis  % 
or  SYNTHETICAL  OEMOMsxaATioKof  theprobliMayandfiequentlf 
the  coirposiTioN. 

9.  When  the  value  of  any  quantity^  which  was  at  fifBt  un« 
knowa>  is  found  and  expressed  in  known  terms,  the  translate 
ittg  of  this  value  out  of  algebraic  into  oommon  language,  whece« 
In  the  relation  of  the  quantities- concerned  is  simply  declared,  is 
called  deducing  a  theorem  ^5  but  if  the  tianslation  be  exhibited 
in  the  form  oi9i  precept,  it  is  called  a  canon  <j  or  rulb. 


implies  the  reaolvii^  of  any  thing  which  is  compounded,  into  its  constituent  si9<* 
pie  elements :  thus  in  algvbra,  several  quantities,  known  and  unknown,  being 
tomponnded  together, analysis  is  the  disentangling  of  them;  by  its  opera- 
tion, each  of  the  quantities  included  in  the  composition  is  disengaged  from  the 
rest,  and  its  value  found  in  terms  of  the  kitown  quantities  concerned.  This  being 
the  proper  business  of  algebra,  the  science  itself  on  that  account  is  frequently 
termed  analysis,  which  name  however  implies  other  brandies  besides  algetoi. 

^  Synthesis  (from  the  Greek  rvy^irif,  compotUia)  is  the  converse  of  analysis. 
By  analysis,  as  we  hate  shewn,  compound  quantUies  are  decompounded  ;  hj 
synfliesis,  the  quantities  disentangled  and  brought  out  by  the  analysis,  are 
again  compounded,  by  which  op^iM^oo  the  original  compoijnd  quantity  it  re- 
produced ;  hence  synthesis  is  colkd. (Ae  method  of  dgmtmniraiunt^  mni  analgia 
the  metifid  ef  investtgifUiQH,. 

<*  A  theorem  (from  the  Greek  ^t^fftifMh  a  epecukuioni)  .is  a  proposittoa  ter* 
minatittg  in  theory,  in  which  something  is  simply  itiSrmed  or  denied.  Theorems, 
as  we  have  observed  before,  are.  initestigated  or  discovered  by  anaJ^sis^  and 
their  truth  demonstrated  by  syntbesi««  ^ 

s  A  caaoa  (froin  the  Greek  »mmf)  cf  role  (from  the  Latia  nguim)  is  •• 
system  of  precepts  difectiog^wiiat  operations  mu^  be  perfoimed^  in  ordea  ta, 
produce  any  pr<^osed  result^  such^  are  the  rules  of  eonmon  arithmetiq*  U  is . 
noticed  f^bove,  that  a  theorem,  ^d  a  canon,  are  of  nearly  the  same  iiaport, . 
differing  only  in  the  form-of  words  in  which  tl^ey  are  laid  down  ;  the  distiae* 
tion  may  appear  trifling,  but  it  is  observed  by  writer|>  whose  skiU  and  judg- 
ment are  nn^estiviii^i^^x  and  on  that  iiccoant  we  tb$nght  pioper  aot 
tirdy  to  omit  it. 

b2 


4  ALQESSA.  Part  IV 

VO.  A  coKoirLABT  **  10  a  truth  obtwrtfd  intonawBiitriy,  umI 
by  the  bye;  an addiUooaltnith, over aod above  wbat the prahl^a 
yipopteed  to  aeareh  out,  or  prore. 

11.  A  ftCHouvM  ia  a  remark  or  eaplaDatory  ofcaorvalioiiy  io^ 
tended  to  illuatnite  80inetbui§^  preoediiig'. 

19.  To  make  what  ha»  been  delivered  perlbctly  pfattn»  to  the 
analytical  investigation  of  several  of  the  following  proUems^  is 
added  the  synthetical  demonstration ;  instances  are  given  of  de* 
dooir^  theorems  and  of  deriving  canons  or  rules  from  the  analy-* 
sis ;  examples  are  likewise  proposed,  where  necessary,  to  shew  the 
method  Of  applying  the  gehend  condnsions  to  particular  cases ; 
and  finally,  tbe  manner  of  converting  any  porticukir  numerical 
problem  into  a  general  form,  and  of  substitttting  and  deriving 
expressions  for  the  unknown  quantities,  in  a  great  variety  of 
ways,  are  shewn  and  explained. 

PROBLBM  1 '.  Given  the  sum  and  difference  of  two  magni- 
tudes, to  find  the  magnitudes* 

Analysis.  Lei  x=:the  greater  magnitudey  y^the  less,  i= 
the  given  sunh  d=stke  given  difference. 

Then  by  the  problem  ^r-f  yas*. 

And    x— ysrrf.  ^  • 

»-fd 


Whence  by  addition  2j;sx«4-<f,  or  xs 


2 


^  Tbe  t«nii  cofollaiy  ir  derived  from  the  Latin  oonMty.^ometkimg'  given  over 
etnd  above  f  and  teiiolinm  fiKim  rx*yjm9  a  ekvrt  comment, 

■  Sereral  of  the  problems  here  given,  with  others  of  the  kind,  may  be  found 
in  Sannderifoa'*  Elemento  of  A4j^bm»  2  vot  4to.  1740.  in  the  Abri%ment  of 
the  Mme,  and  in'  Ludhun't  Rudiments  of  Mathematics. 

^  In  the  lechnieal  bmgnage^  the  mathematicians,  Q.  E.  f.  denotes,  quod 
erai  investigandom*  which  woe  to  be  imiettigaied ;  Q.  £.  D.  quod  erat  de- 
Bionstimmlum,  iViAicA  wot  to  he  demmatraUd ;  and  Q.  E.  F.  quod  emt  facir 
toAwa^^'Wkichwa$tohed$ne*  Tbe  iirst  is  subjoined  to  analytical  investiga- 
tioni,  the  seeottd  to  synthetical  demonstrations,  and  the  third  to  the  proof  t)^at 
a  proposed  ptaetical  operation  is  actuaUy  performed  and  done.  We  hare 
adapted  the  distinctions  of  anafyeU,  tynihesis,  thmremy  camm,  &c.  and  like- 
wise  tbe  above  abbrtfviations  in*  a  few  instances,  to  assist  this  learner  in  a  knowr 
ledge  of  their  use,  wheb  any  boeh  eontaining  the»  may  happen  te  flOl  into  his 
hands^ 


Pabt  IV.  GENERAL  PROBLEMS.  .5 

'    -STNTHifiB.  Bemwte  hf  ihe  prMem  x^^fttis,  «nd  iX'^t^zad, 

if  the  valuet  ftmnd  6jr  efte  analysis  he  really  equwalent  to  x  ami  f 

reepecthely,  then  those  values  being  euhetituted  for  x  and  p  m  the 

gwen  equations^  and  the  latter  value  added  to  the  former  in  ihe 

fipst  equation,  and  subtracted  from  it  in  the  secomdj  the  results  will 

be  s  and  d.    Let  us  make  the  expemnent 

^      s-^d     «— d     2^ 

First  — - — I — -— xs— a^,  .tMch  atuwers  the  firet  ixmtftfion, 

namely  that  x-^ysxs^ 

Seax&dUf  — == — ssd^  which  answers  the  second  con^ 

^2  «        3  . 

diijum^  namely  that  x^^y^es^d;  wherefore  the  values  of  x  and  y 

J<mnd  by  the  <malysis,  jure  those  which  the  problem  requires. 

TiifBQftigt^  1.  If  the  differenoe  of  any  two  magnitudes  be 
«dded  to  their  sum,  half  the  result  will  be  the  greater  magni- 
.titde;  bnt  if  the  difference  ht  Miiatracted  from  thQ  spn,  half 
the-reeuH  will  be  the  less.  . 

Scholium,    llie  form  of  any  general  algebraic  expression 

may  be  changed  at  pleasure,  provided  its  value  be  not  altered 

thereby :  by  this  means  ^  theorem  may  sometimes  be  laid  down 

in  a  more  convenient  form  than  thai  derived  immediately  from 

s-\-d 
the  analysis.  The  value  of  x  found  idx>ve,  viz.  -——may  he  thue 

s       d  f^— d  s       d 

expre0ed^7;4-— j  and  the  value  of  y,  viz.-—-— ,thi|s, —— — : 

,  hence  we  obtain  the  above  theorem  in  a  pioife  convenient  form* 

■ 

viz. 

Theorem  %.  Half  tlie  differenoe  of  two  magnitudes  being 
added  to  half  their  sum,  the  result  will  be  the  greater  3  and 
half  the  difierente  being  subtracted  "from  half  the  sum;  the  re- 
sult will  be  the  less. 

Corollary*  Hence  it  appears,  that  theorenis  ^^ip4  canons 
may  be  derived  from  uny  general  algebi^ic  investigadQn,  which 
will  solve  every  perticular  c£»e  subject  to  the  same  conditions 
with  the  general  problem^  to  which  that  investigation  belongs. 

Cam  ON  I.  (From  theqran  1.)  Add  the  difference  of  any  tinfo 
mUgnitudes  to  their  sum,  and  divide  the  result  Vy  ^»  ^^^  ^lotieat 

93 


6  ALGSMLL  Past  W. 

^vffl  be  the  greater  magnitude.  SuMraet  the  diftffwme  from  the 
mm,  and  divide  the  result  by  %  the  quotient  wiH  be  the  kas. 

Canon  3.  (from  theorem  2.)  Add  half  the  differenoe  of  anj 
two  magnitudes  to  half  their  8um«  and  the  Tegult  will  be  the 
greater  magnitude.  Subtract  half  the  difference  from  half  the 
sum^  and  the  Ksult  will  be  the  leas. 

SxAuPLEs.-^l.  Giv^i  the  sum  of  two  numbeiB  20>and  their 

difference  12,  to  find  the  numbers. 

-^  30-f  12     32 

By  canon  1.  — - —  ^ ---=16  =z  the  greater  number, 

20-12      8 

^'-— - — =r«~s43xlfte  leu  nmmber. 

2  2 

^  20     12 

By  cowan  2.  ~+— =:10+6=16=«Ae  greater  number. 

20     12 

"5 — =ia— 6=r4=/^  k8B  number,  as  before. 

2.  If  the  sum  of  two  numbers  be  dl>  and  theur  difference 

14^  what  are  the  numbers  ? 

^  31  +  14     45    ^^.      ,, 

By  canon  1.  — - — ^—=9fl^=  the  greater. 

31—14     17     «       ,,    , 

— - — i=z-^=S\^the  Ubs, 
2  <  2> 

'  31     14 
By  ctmon  2.  -3-+— sxl5i+ 7=92^8=*^  freoesr. 
2        2 

y — Y=16*— 7=«8f  satte  Isst,  «»  6^efe. 

S«  The  sum  of  two  numbers  is  16^  and  their  difference  6,  to 
find  the  numbers  ?    Am.  11  and  5. 

4.  Given  the  sum  109>  and  the  difference  51,  to  find  the 
numbers  ?    Jne.  754^  and  244-. 

5.  Given  the  sum  of  two  numbers  44.,  and  their  diflference  I4., 
to  find  the  numbers  ?     Jns.  244-  and  l^.    . 

6.  Given  the  sum  123>  and  difference  104>  to  find  the 
numbers  ? 

Problem  2.  What  magnitude  is  that,  to  which  a  given  mag- 
nitude being  added,  and  from  it  the  same  given  magnitude 
b^ng  subtracted,  the  sum  shall  be  to  the  remainder  in  a  ^iven 
ratio?  • 


Paut  ly.  GENERAL  PROBLEMS.  7 

A«rAi.7Sis.    Let  xssihe  magmiude  reqmred,  a^zthe  gk)€n 

magnitude  to  be  added  and  subtracted;  r  and  s  the  tern^i  of  the 

gwen  nafta;  then  by  thefirohkm,  x+a  ;  f — a  ::  r  :  s,\'  rx-^ar 

ar-^tig    r+f 
5=sj74-a*,  •.•  rx-^sxzszarA-aSy  (tnd  x= = a,  the  mag- 


nitude  required  \  Q.  E.  I. 

_.       ar+ai  ar-^as+ar^as     ^ar 

Synthesis.    First, \-ass = , 

r— «  r — s  r—s 


^        „     ar-^-as  ar'\'as-^ar'\-as    %ae 

Secondly,  — a= ■"     = 

r—s  r—s  r — * 

2ar      ^ag         2a  2a  ^  ^   ^ 

Xr  : X «  : :  r  :  f.    Q,  E.  D. 


m       • 


r— «       r—s  r—s 

Examples. — 1.  What  number  is  tbat^  which  with  3  added  to 
it^  and  also  subtracted  from  it,  the  sum  is  to  t|ie  remainder  as 
9  to  7  ? 

Here  a^S,  r=9,  «=7.  and  a?=-i^x3=---x  3=8x3 

=24.  - 

2.  Required  a  number^  which  being  increased  and  4eQreased 

1 

by 'T^,  the  sum  is  tQ  the  remainder  as  3  to  1  ? 

tiere  as=-— ,  r=s3,  s^sl,  \' x^^- — r  X  T-r=-:r  X  t:::— :r: 
12  3—1     12^     2      12     24 

3.  If  10  be  added  to,  and  subtracted  from,  a  certain  number, 
the  sum  will  be  to  the  remainder  as  11  to  9 }  what  is  the  num* 
ber?     Ans,  100. 

4.  If  -^  be  added  to,  and  subtracted  from^  a  required  number^ 
the  results  will  be  as  15  to  13 ;  what  is  the  number  ? 

1  Here  it  is  plain  if  r==s,  then  s  +  a^x-^a,  consequently  a=o,  whence  any 
iBagnitode  taken  at  pleasure  for  x  will  satisfy  the  conditions  of  the  problem. 

li  r  y*  (the  qoantity  —-^  a,  or)  the  Talne  of  x  will  be  affirmatii^e  ;  bat  if 

r— # 

r^  #9  the  Talne  of  x  will  be  negative :  in  the  former  case  the  ratio  is  that  of 
the  greater  tfi«gva/tfy,  but  in  the  latter,  it  is  the  ratio  of  the  ietser  inequmliiy, 
and  the  given  problem  is  changed  into  the  following  ;  **  To  find  a  magnitude, 
from  and  to  which  a  given  magnitude  being  subtracted  and  added,  the  remainder 
•ball  be  to  the  sum  as  r  to  s*' 

B  4 


S  ALOBBRA*  Past  IV. 

DMgakude  into  two  psHi  in  m 


giTeniatio. 

Analysis.  Let  aatihe  given  magttUude,  x^aneoftheparU, 
then  will  a^Tszthe  other  part;  also,  let  r  and  s  represent  the 
terms  of  the  given  ratio. 

Then  6y  the  problem  x  :  a-^x  ::  r  :  s,\'  sx^ar^rx,  and  rx 

,                            ^^         J  «*"       ar^as—ar       as 

-^sxssar,  •.!  x= ,  and  a-^x^za' 


Q.  £.  /. 

c  ^         or         as      ar-^-as     r-^-sa 

Synthesis.  First, 1 = — - — ^=  ~ — =a. 

r+»     r+j       r+#       r-^-s 

Secondly,  ——  :  -~—  ::  ar  :  as  :i  r  :  s.   Q.  E.  D. 

ExAMFLM. — 1.  Divide  the  number  32  into  two  parts>  in  the 
ratio  of  9  to  7. 

JEferea==32,  r=9,«==7,  a»dx==--— -=2x9s:l«,  and  a 

9  +  7 

-JC=(-^=)  32-- 18=14. 

3  2  4 

2.  Dinde  —  into  two  parts,  in  the  ratio  of  —  to  — . 
7  "^  5        9 

oi, ^^  ^_  3  2  4  3       2       2       4       6 

Here  «=:---,  ^=-^-.  *="r'*  «»rf  J^= — X — i = — 

7  5  9'  75       6^9      35 

38      6      45      3       9       27  ^  3        27 

'*-4T-35^  38  =y^  15=153"'    ''"^    "-"^^^  7^133"= 

399—189      210        30 


931  931        133' 

3.  Divide  60  into  two  parts,  in  the  ratio  of  1  to  3.  Ans,  15 
and  45. 

4.  Divid€f  5  into  two  parts,  in  the  ratio  of  20  to  19, 

Problem  4.  To  divide  a  given  number  into  two  parts,  such, 
that  certain  proposed  multiples  of  the  parts  being  taken,  their 
sum  shall  equal  another  given  number  ? 

Analysis.  Let  ais^the  given  number  to  be  divided,  x  and  »=» 
the  parts  respectively,  r=zthe  multiplier  of  x,  sz=the  multiplier  of 
y,  and  bv^the  sum  of  the  multiples  of  ts  and  y;  then  by  the  pro*- 
blem,  x^y=za,  and  rx-^sy=zb.    From  the  first  of  these  equations, 

Vie  ftave  y^a-^x;  and  from  the  lalt^r^  ysz ;  •.•  a— xa; 


Pahi^  IV.  GENERAL  PROBLEMS.  9 

,  .        b'-as     ar'^aS'-b-\'as     ar—b    -.    _  , 

=  (a— X=:)  a = a:- :.  <?.  E.  I. 

r-^s  r — 8  r— -« 

^  ^       6—05     ar—b     ar-^as     r-^$.a 

Synthesis.  First, 1 = = =«. 

r— «       r— #       r— *        r— » 

^        „     6— a»  ar— 6  6r — a$r     asr — bs 

Secondly, x  r-\ X5=( 1 ■ 

r—s  r— «  r — s         r— « 


br—bs    .r—s.b     ,    ^   ^   ^ 

= =) s6.  0.  £.  D. 

r— «        r— 5 

Examples. — 1.  Let  100  be  divided  into^two  parts>  so  that 
foor  times  one  part  beilig  added  to  three  times  the  other^  the 

sum  will  be  355. 

6— <i# 

Here   a=ziOO,   r=4,   5=3,  and  6=355:    •.•  x= ao 

r— 5 

355—100x3     355—300       ^         ,        ar— 6     100x4—365 

— = =55,  and  »= = =ss 

4—3  1  *         ^      r— »  4—3 

400—355 


=45. 


1 

2.  To  divide  13  into  two  p^rts,  so  that  three  times  one  part, 
added  to  five  times  the  other^  will  make  47. 

47—13x5 


Here  a=13,  r=3,  5=5,  and  6=47?    '.• 


3-5 


472:6B_  — 18_  13x8— 4739— 47     -8_ 

—2    ""—2""  '^"         3—5      ■"    -2    "^—2""  * 
3.  To  divide  23  into  two  parts,  so  that  the  Bum  of  9  times  the 

first  part,  added  to  7  times  the  second,  may  make  199. 

^ROBLEM  5.  Given  the  sum  and  quotient  of  two  numbers,  to 
findt  them. 

Analysis.    Let  s=:the  given  sum,  qszthe  given  quotienty  x 
and  yzs: the  numbers  required;  then  by  the  problem,  x^ysss,  and 

X 

' — =9.    From  thejirst  x^zs—y,  and  from  the  second  x=^qy,  •.• 

5  _        ,        .    05      ^  „  - 

qy^s-^yyorqy-^-y^s,  •.•  y=— --,a«dx=(9y=)--^.  Q,EJ. 

q-r^  9  +  1 

^          05            5         qs-\-s     0+1.5 
Synthesis.  First, -^ 1 =•= =2 =«. 

9+1     9+1      9+1     9+1 

Secondly,  -?i*-^— i~=-l=n.   Q,  E.  D. 
9+1     9+1      1 


10  ALGKBRA.  PaktIY. 

ExAMPLss.— 1.  The  sum  of  two  numbers  is  54,  and  tlieif  pa- 
tient 8,  to  find  tbe  numbers  ? 

rr  .-.  «  9'        8x54     433     ^„         J 

'    ^  9  +  1      8+1        9 


*  — -H— '^^-fi 


9+1     8+1      9 

2.  Given  the  sum  3,  and  quotient  11,  of  two  numbers,  to  find 

them? 

33        3  3       1 

Heres^S,  g=ll,  •.'x=j-=2-~.a»<iy=  -  =— . 

• 

3.  If  the  sum  be  144,  and  quotient  %^,  what  ase  the  nlim' 
bers  ?     Ans.  100  and  44. 

4.  Let  the  sum   be   91,  and  quotient    65    required  the 
numbers  ? 

Problem  6.  The  sum  of  two  numbers  and  the  difference  of 
their  squares  being  g^ven,  to  find  the  numbers  ? 

AifALTsis.  JjCt  sssthe  given  sum,  b^the  given  difference  of 
their  squares^  x  and  y^the  required  numbers :  then  bjf  the  problem, 
j?+y=s,  and  a^'-y'^sszb.    From  the  first,  x^ss^^y;   this  value 

being  substituted  for  x  in  the  second,  it  becomes  («— yl* — y«=«*— 

^2 5 

2«y+y*— y*=)  »*— 25y=6,  v  2«y=«*-.6,  and  ws= ■; 

^s 

whence x:=:^(s^y=:)s — —=z — ="2^.   «.  E  I. 

Synthesis.  First,  ^-^ — : — = — =». 

'    ^s  ^  ^s       2s 


Secondly,  — JL— 


2     ««— fe>     54^.255^4.5. 


2« 


45 


a 


54_25a5^fc«      4,95  ^    ^   _ 

4  «*  4  5* 

Examples* — 1.  Given  the  sum  14,  and  the  difierence  of  the 
squares  28,  of  two  numbers,  to  find  them  ? 


"■  When  Tfi  "^  b,  j^  will  be  negative,  and  the  first  given  equatiMi  it 
changed  into  s—y^s,  bat  the  second  remains  the  same ;  for  the  sign  of  y* 
is  not  altered  by  changing  the  sign  of  y.  Tbe  problem  by  this  change  becomes 
the  following ;  Given  the  difference,  and  the  difference  of  tlw  sqnares,  to  find 
the  numbers.    See  Ludlom,  p.  150. 


FA»nr.  GENERAL  PROBLEMS.  U 


Here    «s=14,   6=28,  •.•   x=  ^  ^  -  =■*— =8,  a«(i  ysa 

2x14       28  '^ 

14^— 28168_ 
2  X  14  "■  28  ""   ' 

2.  If  the  sum  be  4,  and  the  difference  of  the  squares  likewise 
4f  what  are  the  numbers  ? 

Here  »3x4,  6=4,  •.*  xx=24,  yas  I4.. 

3.  The  sum  is  101,  and  the  difference  of  the  squares  100, 
what  are  the  numbers  ? 

Problem  7*  Ghren  the  product  and  quotieot  of  two  numbers, 
to  find  the  numbers  ? 

Analysis.  Let  psz the  given  product,  qssthe  given  quotient, 
X  and  y^the  required  numbers  respectively  ;  then  by  the  problem, 

X 

xy=p,  and  — =59  ;from  the  latter^  x=:qy ;  tJus  substituted  for  x 

y 

P  P 

in  the  former,  gives  qy'^:=p  \'  y*=-^,  and  yc=:^-=--;  •/  x=qys: 

9Vj-V~  =  VP9'   Q'E.I. 

p  p*<7 

Synthesis.  First,  VP9  X  v'-^-s  y'i—i  =  ^p«=ap. 

Secondly,  ^pq-^  v^— =  ^pq  X  v"— ==  V— = 

q  P  P 

^^^ssq.    Q.  E.D. 

Examples. — 1.  Given  the  prodoct  196,  and  quotient  4,  to 
find  the  numbers  ? 


Here    p=:196,    9=4, '.v^^^  X  4=^784=28=1:;    and 
196 

V-4-=V49=7=y. 

2.  The  in*oduct  is  — ,  and  the  qi^otient  I4 ;  required  the 

numbers  ? 

„  55  26      5^  42 

flerep=-,  9=-,  •/  x=  ^3^=-^,  and  V^Vj^^' 

3.  If  the  product  be  605,  and  the  quotient  5,  what  are  the 
anmbers? 


IS  ALGEKU.  Paxt  IV. 

Problbm  8.  Given  the  imn  and  |»nodiict  of  two  nuaiben.  to 
find  them? 

AvALTHs.  Let  iz^the  gwen  tum,pssihe  given  product,  x 
amd  ff=zthe  numbers  required.  Then  by  the  problem,  x-^-yszs,  and 
xy^p;from  the  first  jr^t— x;  this  vabie  substituted  for  y  in  the 
second,  it  becomes  sx—a/^ssp,  •/  j*  -  sx=  —p  ;  complete  the  square* 

and  x«-»+_=--^p=^-_r,   .,.   x__.=: -|,  ^__^=:  + 
_ ^,  ..j.= ^and  jf=(«^^5=)« — =^ T 


2 ;  2 2 

Secondly, ^ x T"^^' 

Q.  E.  D. 

Examples. — 1.  Given  the  sum  17,  and  product  72>  to  find  tlie 
numben  ? 


K^-.     1^         iro              17+^/289-288     17+1     ^ 
J5ferei=17,p=72,  v  j:= — — ^      = — =^^0 


or 


8,  a»4f  jf= 3 — - — =:8  or  9;  whence,  y  31^9, 

thenysiSf  but  ifs^szS,  thenyssg. 

2.  If  the  sum  be  — ,  and  product  -—,  what  are  the  nurabeis  ? 

i«  o 

w  11  1  2  1 

Here  #= — ^,  p= — ,  *= — ^.  v= — . 

12"^     6 '         3    '^      4 

3.  liet  the  sum  be  21,  and  product  90,  required  the  numbers  ? 

PaoBLEif  9.  The  sum  of  two  numbers,  and  the  sum  of 
their  squares  being  given,  to  find  the  numbers  ? 

Analysis.  Let  s::sthe  sum,  a:sithe  sum  of  the  squares,  x  and 
yssthe  numbers  sought.  Then  hy  the  problem,  x+y^s,  and  aj»+ 
y^=za;  now  from  thefcrsty:ss^x,  vy*=*a^2*r+JC«;  thisvalue 
substituted  for  y«  in  the  second  equation,  it  becomes  j«+««— 9«r+ 

«•  stf  ;  that  is,  2x*— 2  Jtrasa— «*,  v  x^-^sx^ — — ,  v  jr«— «r+ 
£ _  a^l»    js  _  3a--j« 
4*"^    2    ■*'4"^^      T"' 


Past  IV.  GENERAL  PROBLEMS.  13 

fVheri  ike  nquare  u  completed,  the  procea  ma^  he  simpiyied 
by  substituting  a  more  convenient  expression  for  the  known  side  of 

the  equation ;  thus,  in  the  above  equation,  instead  of ,   let 

R^  *«      JR«  * 

—  ^  be  substituted,  and  it  will  become  a^-^sx-\ — = —  :  whence 

by  evolution,  x---^^±^—=z±j;  v  ar=(-^±~=)-|-, 

2              2  2 

Synthesis.  First,  -^t=-  4 = — =». 

#»HP2^«+/J«     2*«+2B«     *«+il«      .  .         r         ,        ^ 
=  — _ SB —  — SI  {by  restoring    the  value  of 

^  2 

Examples. — I.   Let  the  sum  9>  andj|;i^  sum  of  the  s^iyaret 
45,  be  proposed,  to  find  the  nimibere  ? 

Here  sss9,  a=45,  then  B=  J^aZIil^sx  ^90—81=  ^9=3^ 
-       9-f-3     12     ^        .        9—3      6 

2.  Let  the  sum  2.25,  snd  tke  sum  of  the  squares  2.5625»  he 
given. 

Hct-c  *=:2.25,  a=2.5<525^  1?S5.25,  ar=l.25,  yss:!. 

3.  Given  the  sum  15,  and  sum  of  the  sqoaies  137>  to  find  the 
numbers  ? 

Problem  10.  Given  the  product,  and  the  sum^of  the  squares 
of  two  numbers,  to  find  them  ? 

ANALYSIS.   Let  p^the  product,  asathe  sum  of  the  squares, 
X  and  yiathe  required  numbers  •;  then  by  the  problem,  xy=:p,  and 


•  Since  9a^fi»Iti,  it  follow*,  that  if  ^  ^  2a,  the  probtem  wiU  be  impoisU 
ble ;  because  R^  will  be  negativQ  inlbat  case,<aiid  consequentlj  will  btve  od 
square  root. 

•  Let    x=the    greater  of   two   numbers,   y^the    less,   «=:  their    sum, 
S3  jdM    difference,  p  *stke    product,  q  «&  tbe    quotient,   a  =»  the    sum  tf 


14  ALGEBRA.  Past  IV. 

jt*+y'=a.     Prow   the  first,  y=— ,  z  y*=^;   iub^iUuie   thii 

value  for  y*  in  <&«  second^  and  a*+^=:a,  •.•  i:*4-p*s=ar",  or  a?* 

tf*       a*             a*  "^4  p* 
'-^ajfizs—p^^  •.•  X*— ax*H —  =( — — p«=s i-,  lo/iicA  6y  «<6- 

,  a+R      ,  p     J  a+U    . 

and  *=+  ^-=-  ;  ai$o  y^{^sz)  p-«.-h  ^-=-  i  but,  m  order 

^0  o6<ai]i  the  value  of  y  in  terms  of  a  and  R,  toe  must  tuiatituie 

for  p  Us  equal  ^ ,  {which  is  derived  from  the  above  equa-- 

tf«— 4p«     |i«  a  +  R 

iian ss-j-,)   wherefore  f/^p^±^/-^»   be€omes=z± 

Sywthesis.    First,    ±  ^  -=^  X  +  V  -3—  =  v^ — 7 — » 

{yoKvch  by  restoring  the  value  of  B^,  viz,  «•— 4p^)=s  ^ ^ 

4p« 

Q.  £.  D. 

£xAMrLE8.-^L  If  the  prtiduct  be  ^4,  and  the  sum  of  the 
squares  52,  what  are  the  numbers  ? 

Here  p=24,  a=52, 11=^^34^^-:)  2o,*=r  ^51±^=3 

o^     ^  52-20        82 

2.  Given  the  product  I.32«  and  the  siim  of  the  sqiiarie^  2.65, 
to  find  the  numbers  ? 

£ferep=1.32,  11=2.65,  li=.23>  a?=:1.2,  y=:l.l. 


ibe  sqnaresy  ft  =3 the  differenee  of  tb«  squares ;  any  UK>  of  thesie  eight  (jty  y,  s, 
dtp*qt^  9XiA  h)  beiDg  giTeOy  the  remiunios  six  may  thence  be  foDod,  as  was 
lint  ifaewn  by  Dr.  Pell,  in  his  Additions  to  Rhonius's  Algebra,  1688.  Tbefe  pro- 
blemt  ma/ be  fouod  wrought  oat  at  length  in  ff^ar^M  Yoi^ng  Mathematician's 
Guide,  8th  cdHioay  London,  1724. 


Past  IV.  GENERAL  FBOBLEMS.  15 

3.  Given  tke  product  Uf,  and  the  sum  of  the  squaits  ^50, 
to  find  the  numbers  ? 

Problem  11.  A  vintner  makes  a  mixture  of  100  gallons, 

with  wine  at  6  shillings  a  gallon,  and  wine  at  10  shillings  a 

gallon :  what  quantity  of  each  sort  must  he  put  in^  so  as  to  afford 

to  sell  the  compound  at  7  shillings  a  gallon  without  loss  ? 

Analysis.    Lei  a=i6,  6s:  10^  sszlOO,  ib=75  xssthe  quajt" 

iitff  at  6  shillings  J  y^the  quantity  at  10  shilMngs*     Then  by  the 

problem,  x-i-y=«,  and  ax'^by^ms;from  ike  first,  x=zs^y',from 

^,  ,         ms — by  ms—by 

the  second,  jt= ,  •.•  »— 1^= ,  or  as'^ay:=fns^by,  or 


a-^m 


ay-^byz^as-^ms;  that  is,  a — b.y:=ia^m,s,  \*  yt:z -.*, '.'xsi 

a — 6 

.        a— OT       as—bs     as-^ms     ms — bs    m—b 

a—b         a^b        a—b         a— 6        a-^b 
Q.  E.  I. 

tit— 6       a— m       a*— & 

Stnthbsis.    First,    r^H t-.«= ^.s=«.    Likewise 

a^b       a—b        a^b 

m — b       ,     a— w       am—ah       ab'—bm       am — bm       a  -^b 

ax i-<+6x  — i-«= ^-'^^ r  •*= i— •*==  — i 

a—b  a—b  a—b  a—h  a-^b         a — b 

V 

.  ms=ims,   Q,  E.  D. 

m.'—h 

The  above  problem  resolved  in  numbers,  gives  j?= 7.*= 

^ XlOO= — Xl00= — X  100=75  gallons  at  6  shillings  s^ 

6 — 10  —4  4 

and  tf=^5^^.*=-^  X 100=^  X  100=---x  100=25  gaZ^ons 
^     a— 6        6—10  —4  4  ** 

al  10  sfullings  a  gallon, 

Pboblem  12.  Towards  the  expense  of  building  a  bridge,  A 
paid  1000/.  more  than  B,  and  2000/.  more  than  C,  and  the 
square  of  A*s  payment  equalled  the  sum  of  the  squares  of  the 
other  two  3  what  sum  did  each  contribute  > 

Analysis.    Let  aszlOM,  then  2a=2000,  also  let  x^zCs 
payment,  then  willx-^a=:B*s  payment,  and  x4-2  a=zA*s  payment; 
whence  by  the  problem  x-h2a)*=x+a)*4-x®;  that  m,  x*■f4xa-f- 
4a*=a?*+2aa^-a*4-x*,  or  3a*=x*— 2j:a;  that  w,  x*— 2ar= 

3  a*,  •/  a?*— 2flaf+a*=4a*,  •/  «— a=-jh^4a*=+^^'  ^""^  *=^ 
3a=3000=C«  share,  \'  jr4-a3r4a=s40b0s=:Fj  share,  and  x-J- 
^  a=5  fl=5000=if  #  share,    Q,  E,  L 


X6  AIX3£BBA.  Part  IV. 

Stutubsis.  ra\'^ss{$quare^  jripafmentzs)  4al*-fs3*=s 
(sum  of  the  squares  of  Bs  and  C«=)  25  a«.  Moreover  At  pof- 
ment  (Sa)  exceeded  Es  (4  a)  hy  a,  and  Cs  (3  a)  Ay  2  a.    Q.  E.  D. 

Problbm  13.  It  18  required  to  divide  11  into  two  such  parts, 

tbat  the  product  of  their  squares  may  be  784. 

Analysis,  ^t  aszllybzs7S4,xa^dysa Sports  required s 

then  by  the  problem,  x-^ysza,  and  $f^^^b;fr(m  the  first,  y=«— 

X;  the  square  of  this  value  substituted  for  y*  in  the  second,  gives 

a — ^^  X  x*=6,  whence  by  evtdution  a— xucs  ^b;  that  is,  or— x*= 

a'       a^  a9_^   /A 

^b,orx^^ax=s^  ^b,  •/  j?«-ajc-».— =(--— y'^^Z — 1^=) 

R^  a  R^         R        ^       a+R 


4'   • 


•  •  X- 


a ==—=:--_.    Q.  E,  I. 


a-^R     a-^-R     2a  ^      a4-R 

Synthesis.  Fwst,  -^=— -f._s= — =a.     rftat    — 


«  +  /? 


2      •      3         2        '  2 


X 


2  "1  ^ 4 == 16 =(*^ 

restoring  the  value  of  R*=za*^4^b=s) 
a4_2«*-f8a»^fe+a*— 8a»v^6+16  6     165 

Te ^16=*-  <?^0- 

The  solution  of  the  problem  in  numbers,  is  x=z^-—^=: 

2 

a-h  ^/a' —4 ^/b     ll+^121-4v'784     ^       ^        a-^R 
^ = -^ =7,  and  y=— -=4. 

Phoblem  14.  Given  the  sum  of  two  numbers  24,  and  the 
product  equal  thirty-five  times  their  diflference,  to  find  ihe 
numbers  ? 

Analysis.  Let  x  and  y  be  the  numbers  required,  #=s24,  mi= 
85  J  tl^n  by  the  problem,  x+y=s,  and  xy==(i».i^=:)  mx^my. 
From  the  first,  y=s—x;  this  value  substituted  in  the  second, 
git7e«*x--r»  =  (wix— »w4-»ix=)2  wix— »w,  or  x'  +2  m— *.x3asw5,- 

whence  (putting  a=2wi— *)  x»+ax=m5,  •/  x^+ax-f  ?l=(m*+ 

a'       Ams^'     R'  a         R  ^-R^a 

T""^      4 — ^  T'  '*'  ^"^T"^- 2"'  ""^  ^"^  "  2      *  **^^"^ 


TAkr  IV.  GENERAL  PROBLEMS.  ¥7 

Synthesis*  First, 1 = — =s, 

2  2  2 

^      '   ,      R-a     2ff--Jl+a    2«JR— JJ»4-2aft— 2m-o« 
Secondly.  __X  — — -=x ^ 4      .  ,      ■     ■• 


— —  =  (since  a+s=z2m) 


4 

■■■  ;  (to AtcA>  because  4nw+a"=it*,)=3 

4mA— '4ma — Ams        '  R—a        2*— -R+a 

4  -  2  2 

Q.  E.  D. 

R— a 
The  answer  to  this  problem  in  numbers  is^  xs  — — =s 

74—46     28     ,  ^        2i-R  +  a     48—74+46     20     , 

= — =14i  and  »=———= —  ss — ^10. 

22'^  2  2  2 

TO  REGISTER  THE  STEPS  OF  AN  ALGEBRAIC 

OPERATION. 

The  register  p  is  a  method  whereby  the  place  from  whence  any 
«tep  is  derived,  and  the  operation  by  which  it  is  produced,  are 
dearly  pointed  out^  by  means  of  symbols  placed  opposite  the  said 
stept  in  the  margin. 

The  symbols  employed  are  +  for  addition,  —  for  subtraction, 
X  for  multiplication,  h-  for  division,  ^  for  involution,  *m  for 
jevolution,  a  for  completing  the  square,  =  for  equality,  and 
ir.  fbr  transposition. 

When  the  regbter  is  used  in  the  solution  of  any  problem,  it 
reqiures  three  columns ;  the  right  hand  column  contains  the  alg^- 


9  The  re^ster  will  be  fonnd  to  be  a  rery  coDTenieiit  mode  of  reference,  where 
mn  ample  detail  of  the  work  U  required ;  bat  at  modern  algebraists  prefer  noting 
down  results,  and  omit  as  much  as  possible  particularising  those  intermediate 
steps  which  are  in  a  great  degree  evident,  the  register  is  now  less  in  use  than 
formerly.  We  are  indebted  to  Dr.  John  Pell,  an  eminent  English  mathematician, 
for  the  invention :  it  was  first  published  in  Rhonius's  Algebra,  translated  out  of 
the  High  Dutch  into  English  by  Thomas  Brancker,  altered  and  augmented  by 
Dr.  Pell,  4to.  London,  1688.  The  learner  will  be  enabled,  by  the  specimen 
here  given,  to  apply  the  method  to  other  cases  if  he  thinks  proper ;  at  least  he 
•bonld  understand  its  use,  as  it  is  employed  in  the  writings  of  Emerson, 
Ward,  Carr,  and  some  other  books  which  are  still  read^ 

VOL.  II.  Q 


la 


ALGEDSA. 


Pam  iy« 


braic  operation,  in  the  n^ct  the  steps  ate  numbered,  and  in  the 
left  hand  column  opposite  to  eaoh  step  are  placed,  first  the  num« 
ber  of  Hie  $t^  from  whence  it  is  derived,  aiul  then  the  symbcd 
denoting  the  operation  by  which  it  is  obtained.  And  here  it 
must  be  noted,  that  the  numbers  1,  2«  3,  &e.  in  the  register 
column,  always  denote -the  numbers  of  the  steps,  as  fli'st,  secotd; 
third,  &c.  but  when  a  figure  has  a  dash  over  it,  as  3,  it  denoUft 
a  number  concerned  in  the  operation. 

In  the  following  estample  an  additional  column  is  placed  Cfx 
tbe  left,  fOff  th^  purpose  of  exphdnl^  the  process, 


15,  Given  ---+~ss7,  and  ^-—=±3,  to  find  x  and  y. 
6      2  ^16  •  ^ 

j^t  ^2dS,  dxx^,  mitt7^  caslS, 


£j?p/aisatioii. 


/»  equatum  1.  $uhtr acting  ^. 


Multiplying  eq.  3.  tn/o  6. 
Multiplying  eq.  5.  iiilo  ir. 

V 

2>tvu2ti^  eqiMiott  6.  ^  ^. 


E^ua^m^  ^Ae4M  and  7th  steps, 

Multiplffkng  kq.  S,  h^  d. 

Multiplying  e^.  9.  info  y. 
Transposing  in  eqt^tttkm  10. 

Dividi$ig  equation  11.^6. 
Camp,  the  square,  4rc.  ineqA2. 


Register. 


Giveh 


1 


U 


d 


3x6 


2^ 
Sxc 


4=57 
8xd 

10  tr. 

12to*^. 


Evolving  the  root  of  eq,  13.  "  13  *m  ' 

^  -  -.     dm  ijni 

Jddvng  —  to  eq.  14.  14-^  -^ 

*     ,  2 

/^rom  //te  7^A  and  15^^  e^.      7  .  • .  15 


By  restitution  in  the  \^th  eq. 
By  restitution  in  the  I6th  eq. 


ibrestit. 
\6restit. 


—=18,  or  24, 
Wherefore  if  y=8,  then  isslS}  but  if  y-se,  then  »s=24. 


No, 
1 

2 

> 

3 
4 
5 


•6  ^pjjr^cn*. 

4      y 
cSh* 


7J 
8 


9 
10 


frdj»iy^ijr*'5=ciitt», 

11  I^T— Wmyas— cd»V. 

cdn* 
y*»-dmy=s r-. 

.      ^         d«w« 
y*— dmy-f ss 


12 

131 


14 
15 
16 


17 
18 


Operation, 
X      y 

^  c 


07= dm- 


*f 


i' 


sy 


( 


d'm'     cdn' 


«) 


4  h 

bd*m*'^4cdn'     R' 


4b 
•    dm+JR 


2 


xsscn'  X 
2cn' 


dnt±|{ 


dm+£' 

2x7+2    ^      ^ 

x=5l6?<9x-— r— • 

2x7+2 

2 
cl44x— ,  or  144X 


2 


80 


ALGMfiRA. 


PAIT  IV, 


16.  Giyen  the  difference  9,  and  quotieat  4,  of  two  nmnbers, 

to  find  them  ? 

Let  x^  the  greater,  y^theUu,  d=s9,  9=4. 


Register 
Given 


I 


2xy 

1+y 

3s4 

5-y 
6-1-9—1 

1+7 


I 

3 

4 
5 
6 

7. 
8 


OpercUion. 
x^ysid, 

X 

—=9. 

y 

x=ct+y. 
9ysd+y. 

9y-y=d. 

d_ 


17.  Given  the  sum  of  the  squares  of  two  numbere  61,  and 
the  difference  of  their  squares  11^  to  find  the  numbers  ? 

Let  xssihe greater,  yisthe  le$s,  a=:6l,  (ssU. 


{ 


.Roister 
Given 
l+« 
3-h2 

4  *M 

1-2 
6-»-2 


7 


No, 

1 
2 
3 


5 
6 

7 


OperaHan. 

2ar»=a+6. 
T' 


•  « 


X'sz- 


2 


=6. 


2y»=:a—6. 


y=sv- 


=5. 


PittW. 


GENERAL  FBDttLEMB. 


fl 


18.  The  ^fierefeoe  of  two  nunabere  eaneeds  their  quotient  bj 
Sj'and  their  product  exceeds  their  suih  by  dO:  what  lare  Hul' 

numbers? 

Let  xs=itfie  greater,  y^the  lea,  a=S,  h^^O. 


Registers 
Given  J 

Ixy 
3  if. 

4ss5 
6-y 

70and 
iuhst 


No. 


Operation* 


8  AAI* 

9<r. 
10. 


1    — =sx-^^— (t, 

4  xy-^xrsy*+ay. 
y'-fa'-lyssk 


3 


6 

7 


8 


9 
10 

11 

12 


y'+a— i.y-f 


«*+ 


r::? 


y+ 


4  ""4 

g— 1     jft 

fi— a+l 


y-^1 


il— a+l 


2 


=r8. 


19.  The  square  of  the  greater  of  two  numbers  oniltiplied 
into  the  less,  produces  75  i  and  the  square  of  the  less  tnultiplied 
into  the  greater,  45:  what  are  the  numbers? 

Let  xszthe  greater,  y:=sithe  less,  az=75,  6=45  3  <^^^  x'yss 
c«  and  xy'^=b,  Inf  the  problem  j  divide  the  first  by  the  second,  and 

— = —  •.•a?=-T-;  substitute  this  value  for  x  in  the  second,, 

y     b  b  ■  ■ 

an*  b'  b'  '        av 

md^ssb,  oray*:£zb%  vy^ss— ,  cmd  jr=V^— 3,  V2r:t:(-^ 

=)5. 

90,  To  divide  100  into  two  piirts,  sueh,  that  their  product 
may  equal  the  difference  of  their  squares^ 

Let  x=ithe  greater  part,  yssthe  less,  atnlOO;  then  by  the 
fTohlem,  T-f  y=a,  and  xytrt*  ^y^  ;from  the  first  x^a—y  $  this 
iubstituted  in-  the  $,econd,:ii  becomes  ay— ft*sst(a?»— y»a=)  a«-«- 

c3 


9t  AlCBBHA.  IktolV. 

38.1966011^5  -/  x=(a^y=  a-^^±^-^*^)^^±*-/*=. 

61.80339888. 

21.  What  tv«ro  nunrfMri  tti&  thoee,  whose  diflfereoce  is  4,  and 
the  product  of  their  cubes  ^f^\  ? 

Lei  d=4,  p=9261,  txi'=zthe  less,  then^  x^d:=tke  greater  ^ 

fohence  by  the  problem,  (op*  xx+^f  5cp,  •.'  »x«-hd='v'P»  '^^ 

J      •  •      d*  <J'      . 

M,  x»  +  dar=3  ^p,  •/  x'-^dx-i — -=«  ^pH — .  j  iftu  resohed,  gives 

4  4 


28.  The  greater  of  two  numbers  is  to  the  leas  as  3  to  2«  and 

the  sum  of  their  squares  is'  906  5  required  the  i^umbers } 

Lei  assS,  bzs^  c=:208,  xszihe  greater  number,  then  (a  :  b 

bx  b*x^  a'^ 

: :  OP :  ) — ^thelesS',  v  by  theprab,x*4- sac.  vxas^/ 

a  ^       r  ^  a'        '  ^  a* -ft* 

bx    ^x 
=sl2,  and  — = — s=8. 
a      3 

2&.  t>ivide  thentwiber  35  into  two  safib  paits^  that  the  sum 
of  their  square  roots  may  be  7. 

Let  azs^S,  6=7>  x=z(me  part,  then  a-^xssthe  oihpr,  \\by 
theprob,  >/x4-  ^a— «^6;  square  both  sides,  and  x-^-2^ax— x* 

6*— a 

+a-r«=r6s  •/  ^aJp— a?»a-— -J  ag^tin  square  both  sides,  oife^ 


«^-:t.=:^i^'.  „hich  resolved,  gives  x^^^  ^a^^fc^^^ 

16,  or  9,  and  a— »=9,  0^  16. 

214.  What  number  is  t^t^  to  which  its  biqi^viiiate  bei;^ 
ailded,  and  from  the  sum  twice  its  cube  subtracted^  the  remaqi- 
der  will  be  1722  ? 


9  Here  we,  moat  endently  take  the  negative  valve  of +  0  y^S,  otherwise  y 
would  come  oat  greater  than  100,  and  consequently  s  would  be  negative  j 
which  is  contrary  to  what  was  proposed. 

»  Ilcielhe<«ffirm^tiv«  Kslve  of  4*^«  V5Wil«t  (etdten. 


Fi»Y  m  GENfiilAL  l^dfiLEAtS.  «§ 


d;+.5==a^  6y  the  ftroh.  wA!l?Aoe^-^l.Sir>  +  .(li95s±d^  •/  ^  cem 

I  11    ■  ■■ 

pleting  the  square,  and  Tedxlction,  *=a  ^^.75+ ^a-j-.^Sar'^.S,  •.* 

25.  To  fiad  two  idumbere/such^  that  their  sUtn^,  ptdduct,  ^d 
the  difference  of  their  squares,  may  he  etfiaX  to  each  other  i 

Letx^s.  the  greater,  y=i  the  less,  then  by  theprob.  x-^y=sxyjand 

m-^yxssi^^ffi;  divide thelatterhyX'\'y,undl^f;M*^iftPrstxeii'^yt 
iubstiiute  this  value  for  x  in  the  first,  and  l-k-Stys^y^y*,  whence 

y^—^  V^^=ai^^^=:L6180339887,  «fC.  cOid  *=*(!  +|f=«)-|- 

4  Tb 

26.  The  product  of  two  numbers  Is  1944?  and  the  sUftoUd 

iroot  of  the  greater  is  to  the  cube  it)ot  of  the  1^^  as  li  to  1 1 

%rhfit  at«  the  numbets  ? 

X       ^tx        %x\^ 
Let  x^^.the  ^eatet,  then,  I4. 1 1  ; :  ;r  :  (77^)—-,  *.'  -^  = 

83^  -        8 

—  :szthe  less.     Let  <^=5ri  p=«1944i  then  hy  the  problem  (r*X 

W:s&)  C3^^p^',*X:=M^ j^^^^ ^^h6\:^^^\' x^ii^tA^^thigfeatet^ 

o 

8 
<md  csr^isi(— ><3»te)8S5*^  [est. 
27 

^. .  tto  fufii^  4nd  product  ^f  two  imp^fs  ^H  ^ml«  nod  if 
to  either  sum  or  product  the  9mp  of  the  square  besdded^  th? 
|P9$ult  will  he  \%  y  what  aire  the  numbers  ? 

Let  ^  £tad'  y  repr^emi  the  numkertf,  d^t^,  ih^  ^+yM4|^ 
Md  x-^y^x^  -fy'a'WK  iy  theprob.  Take  twice  $hefifHjr^m  ih0 

•  ^       ^u^     \.  \        4Ut4.1 

hy  completing  iht  s^strei»^f  ^-^st^y^-^^^-^-^^t)—^  j 

1  4  fl  +1       ■4-  ^4  (1 4- 1 

*••  fry  eHolutvon  x-^^y^—^ssst^  ^ — — »==  -^  i--^- — —f  •»•  i+yad 

■T-'    -^ ' =x4,  iffhence  also  xy±±   •^■^' — r-  =a4*    From  ^/^ 

i^uafe  of  the  last  i0i  onig,  take  fim  Hmtiif  fh^  tagiytmM  if*-^^  xy 
4.y*si^5  %•  hy  «o«wfkm,i^y±i:3(J,  0nd  x^y,  \*  :fy=***«fc|r*aDi4/ 
*t*xi^^/andft±% 

c4 


ALOSBIU.  r^tTm 

98.  G^ven  the  product  p(^li^)  and  Ibe  $um  of  the  ftxirtb 
{K^were  f  (=»337>)  of  two  mimbers^  to  find  them  ? 

Le<  ar=s:^^  greater,  if^the  leu,  then  xjfvp,  OMd  ir^+y^stf 
add  twice  the  square  of  the  first  to,  and  subtract  itfrdm,  the  second, 
and  extract  the  square  root  of  the  sum  and  difference^  and  there 
will  arise  x*  +y*  =  ^«+2p*,  and  x»— y*  =  ^«— 2p* :  taketh^ 
sum  and  difference  of  these  two  equations,  and  extract  the  square 

root  from  each,  and  t^  ^-j-^*-h2p*  +  V«— *P*  =*>  «'»^  y=* 

* 

«  -     '      -•  I 


29.  The  sum  of  two  numbers  is  &,  ftnd  if  they  he  divided 
alternately  by  each  other>  the  sum  ^  the -quotients  w91  be  4{-i 
required  the  numbers  ?  * 

Let  a=:25^  b^z4^,  xzsthe  greater  number,  then  a-^x^^the 

X  fl*'"*X 

less,  and  by  the  problem 1 =6 ;  whence  :t'  +  d»  —2  aX 

a— d?        X 

4 x»=flto— &r«,  or  2j?*  +  6ct'— 2<ur— fl&xss— a»  ;  <toit»>  2+6 

.a?*  —2+6.00?=:    -a* ;  divide  this  eguation  6y  2+6i  and  x»— ittsae 

-— r — rj  w^6»(  '  by  completing  the  square,  and  extracting  the  roof, 

2  +  6 

tr         a»       a*  ^  a  i-     a*      a* 


^c.  .:=^+  ^-.  -^^=:20,  or  5  j  and  a-x^-T  ^/•5~2+ 6 

=5,  or20.  "  . 

30.  6ivetf  th6  sum  of  two  numbers  9,  and  the  sum  of  their 

e^jibes  189>  to  find  the  numbers  ? 

Let  2«=:9>  6=sl89,  ^xtnthe  difference  of  the  required 

numbers  f  then  {by  proh,  1.)  «+2rs=t^  greater,  and  s-^xssthe  less, 

and  by   the  proh,    (*+j1^+«— j?]'s=^+3t»ir+S«P*+a:*+*»— > 

3«'ar+3air»— x'rs)  2«»+6«r'=s6,  or  6*e«ss6— 2*»;  \- »*» 

6*— £«*  6*-2«'      1  '91 

g^    >  g»d  *±S;/— - — ac— ,  tp^nce  t+«=K~+— =p5ya»d 

9       1' 
*^j:s=;-;-— .--=4. 

'22 

> 

^  31.  Given  the  simy  e,  and  the  sum  of  the  biquadrat^  272»  of 
two  numbek^;  to  find  them  ? 

Let  ^s^S,  2xs£^e  difference  of  the  required  numbers,  6.s«.. 
27.2 ;  th$n,^as  in  the  preceding  problem,  s-^xssthe  greater,  and  s^ 
xsithe  less,  whence  *+xl*+«— a7*=6;  ic^ic^  tnvo^r^d  and  re* 


.    ; 


IfARTiV;  G£N£BAIi  PR09L£MS.  « 

duced,  we  Aarc2«*+12«*x'+2x*=6^  or  j:*  +  6«*x»=— 6— «*j 
V  6y  completing  the  «^tkir«,  jc*+6«*x«+9f*32---fc+8**|'.' 6y 

evolution,  x'  +3*»  =  +  v'"S"*+8**,  x*  =s— 3  «•  +  V^  6-I-8**, 


4mc2ar3=;^^— 3**Hh  V''^^+8«*=1;  whence  «-fa:=4,  aiwf  *— 

39.  Given  the  sum  \0,  and  the  sum  of  the  fifth  powers  17050» 
of  two  numbers,  to  find  them  ? 

^  Let  ^ssslO^  6=17050^  2  j=f^e  difference  of  the  numbers 
required ;  then  S'\'X::=ithe  greater^  and  s^x-^^the  less,  and  by  pro* 
eeeding  as  in  problems  30  and  31>  we  have^s^-^-Ws'x*  -h  10  sx* 

acA,  whence  xs:t  ^ ^ 1 s*  s=2,  •.'  «-f-x=s7>  ond  s — «s=3. 

33.  Given  the  product  p,  and  the  sum  of  the  nth  powers  s, 
of  two  numbers,  to  find  them  ? 

Let  X  and  y  represent  the  numbers,  then  by  the  problem  ar"H- 

y^sss,  and  xyszp ;  from  the  second  equation  yssJ^f  thisvalue^ub^ 

X 

t 

stituted  for  y  in  the  first,  gives  x"-fi— sat,  or  a?**+p"as««",  or 

s^     s* 

X*"— «r"s:— p")  hence,  completing  the  square,  x^'-'Sac^-] s=— 

4       4 

5  7^  s         s* 

—p"  J  whence  ^— -g-=  ±  >/-^-/>'>  ^—'^±  V^ — P'>  ««<i  *== 


,t 


«          •'        -     «    «4-a/«'— 4p'  .         p 

V-5-± 'v^X'^*^'*  V"^ '    ami  ya-^*=p-^• 


* «  * 


34.  Given  the  product  p,  and  the  difference  of  the  nth  powers 
d^  of  two  numbers,  to  find  them  ? 

Let  X  and  y  be  the  numbers,  then  xy=sp,  and  x*— -y"=d ; 
whence     by    proceeding    ai    in    the  foregoing    problem,    xsw 

^2 


«  ALOKBRA.  ^AiT  lV^ 


36.  Required  the  values  of  x  and  y  in  the 
lis.  V**x  ^y'atsSyS  and  12V*— ^y^s^^ 

te«  ass*  V*,  »=x  V'y,  lAca  it'sBX,  aad  ifi^f;  v  fAe  gteeM 
equaiumt  become  u*^szZ  t*,  and  19  u^zss^  ;  divuie  tAe  last  bui 

one  by^t*,  and  zae — ;  ^Aif  equatiim  added  to  the  preceding,  gvees 

a* 
12  uaB«+— >  or  »'  -«-24  axs— 44  j  Ito  tgaa^ioa  retobed,  girel 

tt=:2,  •••  «=(^»)  2,  «« (tt'=)  8,  CMd  f:xM{z'  =.  >4. 

36.  If  18  oxen  in  5  weeks  can  eat  6  acres  of  grass,  and  4d 
oxen  in  9  weeks  eat  21  acres  of  the  same^  how  many  must  there 
be  to  eat  38  acres  in  19  weeks,  the  grass  being  allowed  to  grow 
uniformly  ? 

Let  ais:lS,  6=s5,  c=56,  ds45,  msa2U  ^^9,  rs38,  trrlf, 
Irs  the  quantity  eaten  by  an  ox  in  a  week,  w=:the  quantity  on  an 
acre  at  first,  xszthe  weekly  increase  on  an  acre  after  the  first  5 
weeks,  x^sthe  number  of  oxen  required,  p25(«— 6^)4,  l=i 
(t— -fcsrr)  14 ;  then  will  rw=:the  grass  on  r  acres  ai first,  and  riz=5 
the  inermae  on  r  acres  m  t  weeks  ;  the  mm  of  these,  by  theproblemg 
equcUs  the  qu€Mtity  x  oxen  ate  in  s  weeks,  that  is,  ixs=fir+r<2; 
again,  mwt^the  grass  on  m  acres  at  first,  and  mpx^sthe  increase  of 
the  safne  in  p  weeks ;  the  sum  of  these  two  equals  what  d  oxen  ate 
in  n  weeks,  that  is,  mw'\*mpx=idn;  also  cw^(the  grass  on  C  acres 
at  first  y^  the  quantity  a  oxen  can  eat  in  b  weeks,  thai  is,  cw=sabf 

whence  wssz — •  to  mp  times  the  first  equation,  add  rt  times  the 

second,  and  mpsX'\*mriw-\-mprtzszdnrt'^mprw^n^tz,  or  mp^ 

szdnrt^mprw-^mrtw  ;  for  w  in  this  equation,  substitute  its  equal 

a6         ,     -  ^.       ,  _  _    abntpr     abmrt 

— ,  and  the  equation  becomes   mpsxxdnrt+  - — ^  — ,  of 

c  c  c 

,    .       -  ,  ,  cdnt-^abmp-^abmt 

cmpsxsz  cdnrt -f-  abmpT'^abmrt ;  whence  «3C ^  -« X 

cmps 


cdnt-j-^abmxp-^t  34020+ 1890  x —10     „^     ^^    ,. 

rss ■^ ^ xrrr    ■     .     '      ^"  ^ XS8aB:0Oy   the 

cmps  9576 


answer. 


37*  A  waterman,  who  can  row  11  miles  an  hour  wifh  the  tide^ 
and  2  miles  an  hour  i^nst  it,  rows  5  miles  op  a  nver  atodbacli 


Paut  m  OENIiKAIi  PROBLEMS.  «P 

ag^  iii  3  hmr^^  now  8M;i^0Bi9g  the  Ucto  t»  wii  uttiflnteilx  tlie 

Let  mssll^  »=s2,  ps5j  rs3»  v:=ithe  velocittf  refiiifed*  oiul 
irs&fAe  /im^  Ae  rowed  with  the  tide,  then  will  r^x^the  time  he 

towed  agaimt  it;  whence  {x  :  p  ::  I  hour  :  )—=:hi8  velocity  with 

X 

the  tide,  and  (r—x  :  p  i:  1  hour  :  )  =sAtf  velocity  againtt 

(ifje;  now  .since  the  tide  assists  himssv  when  he  goes  with  it,  it 
Tilust  evidently  retard  himszv  when  he  goes  against  it^  whence 

P 
^vsAthe  difference  of  his  velocity  with,  and  against  tide,  •/  —  — 

•  X 

P  P 

■^va^v,  ov  t7S£X-.^p^«.,ii^  .  9I4M9  because  his-vehn^ty  with,  i$ 


r— a?  2a?     f2r— 2a? 

to  his  velocity  against,  tide,  as  m  to  n -,  so  his  time  of  rowing  with, 

if  to  hie  time  Of  towing  c^in^,  tide,  as  n  to  m,  since,  the  time  if 

nr 
inversely  as  the  velocity ;  wherefore  x  ;  r^x  ; :  n  :  m,  •/  x= 

M-f-fl 

6  7 

isj^ofdn  hour  ss  the  time  he  rowed  with  tide,  and  r— jr=s2  — 

hours:=:the  time  he  rowed  against  it }  for  x  substitute  its  value  — 

P  P 

in  the  equation  above  derived.  And  it  becomes  v=s  (-C  —  — i- — 3=) 
^       ^2x    2r-^Six    ^ 

n  ^        12     05^    (Rl     3510     ,  19     .,       .     ^ 

pH---—pH-2r— ---;=: --■--~3Es----=s4  —-  miles  per  hour=sthe 
'^     13    ^  13     12     66      793        44  ^ 

lulocity  of  the  tide. 

38.  The  ages  of  five  persona,  A»  B,  C,  D«  and  B,  Bve  mi^» 
that  the  sum  of  the  first  four  is  95^  that  of  the  three  first  and 
l^st  97>  that  of  the  two  first  and  t\vo  last  103,  that  of  the  first 
and  three  last  106>  and  that  of  the  four  last  107 )  required  the 
age  of  each  ? 

Let  a=z95,  6=97,  c=103,  (f=sl06,  e==l07,  s=the  sum  of 
all  their  ages,  and  let  x,  y,  z,  v,  w,  be  put  for  their  ages  retpec- 
tively ;  then  wiU  s—wsza,  «— «=6,  »— zapc,  s — y:std,.and  «— * 

'  Velocity  ^fmm  tlM  hmtk  mh^,  ft«ift»>  it  ttM  aftntiMi^f  mHiieD,  wbatcbf 
»  BlMii%  hsdy-  fuam  <wwr  •  cntoisk  tpM^  ia a  ontaifi  time;  pr  in  tammoa 
btngtiage,  it  it  tbe  degree  of  twiftaet^  with  wbicb  a  body  moves :  it  is  liUcwiiM* 
mMMd  eekvit9s  (mi^tlie  l«Ciii  ««Ai^  Mft  orvfin^lAt. 


tt  AtX^EBRA;  pARtlV. 

tse;  aid  the»e  fine  eqwUUms  together,  md  the  sum  is  (5«— 
X— y— 2— »— io=:5«— »=)    4*=a+6+c+d+e;    whenee  ssaf 

',  now  if  this  value  he  substituted  for  s  in  the  five 

preceding  equations,  we  shall  thence  obtain  the  required  numbers, 
viz.  10=32,  ©=30,  2=24,  y=s21,  and  x=20,  being  the  ages  of 
E,  D,  C,  B,  and  A,  respectively.  \ 

39.  To  find  a  point  in  the  straight  line  which  joins  two  lumi- 
naries, or  in  the  line  produced,  which  is  equally  enlightened  by^ 
both  •. 

Let  asstheir  distance  apart,  x^the  distance  of  the  least  of 
them  from  the  required  point,  then  a^x^zrzthe  distance  of  the 
other  :  lei  the  quantity  of  Ught  emitted  by  the  first  in  a  given  time 
be  to  that  emitted  by  the  second  in  the  same  time,  asm  ton;  then 

fgjiU  —  : be  the  ratio  of  the  effects  they  produce,  supposing. 

^'     a±x\ 

i»=»,  and  -J  :  will  be  the  ratio,  supposing  m  and  n  un^ 

tn 

tqual:  but  these  effects  are  by  hypothesis  equal;  whence  — = 

X 

n  — •   ■  . 

-,  •/  iiia'-j-2a;n4:4'WW?*=siw:*,orm— n«r*+2awu:= 


a'+2aj:-f  a? 


2am             ma'                 2am          am  I* 
-^ma',  '/  X*  A a:=  — ,  •/ar'H x-^-  1   = 


■  III  —— ^^-^— i 


am  \'      ma*                am                am 
— ^,  '/  x± =»±  v/ 


TO— »       m — n         "  m-^n     —  '  m — n 


*      ma' 


m — n 


and  x: 


^--  am         ,  am  1*      ma^      -f  am+ -/m»a*    ^-{-a.m-^  Vmna* 

(-f +>/ =3 -=*-2r 3») =» 

^    m—n^     m—w     m—n  wi— »  m — n 

s=  the  distance  required. 

40.  The  weight  w,  and  the  specific  gravity  of  a  mixture,  and 
the  specific  gravities  a  and  b,  of  the  two  simples  which  compose 
it,  being  given,  to  find  the  qua,ntity  of  each  «  ? 


•  A'lnmiaary,  (from  the  Latin  hiinen,  light,)  is  a  body  that  gires  light,  as  the 
fan,  moon,  a  plan^,  star,  &c. 

*  The  double  sign  serves  both  cases,  tIx.  a^x  when  the  point  tvqoired  is-, 
beyQud  the  smaller  luminary,  and  a-^x  when  it  is  beti9«;eii  them ;  als9  in  the 
answer,  the  upper  sign  —  applies  to  the  ficst  case,  and  the  lower  sign  -^  iO'^lke  > 
second. 

■  The  gravity  of  a  body,  (from  the  Latin  gramSf  heavy,)  is  tta  weighty  < 


Part  IV.  GENSRAL  PROBLEMS.  129 

JUt  xm^  iDe^fhi  of  the  simple,  whose  «peci/Sc  gravHy  is  the 
great€»t,  then  w—xazthe  weight  of  the  other. 

X 

a 
v)—x I    the  magnitude  of  the    I 

'   '  '  ""       hody,  whose  weight  is 


w 
s 


w 


x-      w— J?      w 
Whence — h — ; — = — ,    or   bsx-^-asw^asxssabw,'.'  bsx-^ 
a         b         s 

,  abw—asw     b — smw 

mxssabw — asw,  or  a?= — ; =         — . 

bs--as         f,^a^ 

'  41.  Suppose  two  bodies,  A  and  B,  to  move  in  c^ipoeite  direc- 
tions towards  the  same  point  with  given  velocities,  the  distance 
of  the  places  from  whence  they  set  out,  and  the'difierence  of  the 
times  in  which  they  beghi  to  move,  being  likewise  given,  thence 
to  determine  the  point  where  they  meet } 

Let  d^sithe  distance  from  A  to  B  at  the  time  of  setting  out, 
SO-srAs  distance  from  the  point  of  meeting,  then  d — x=Rs  distance 
from  the  point  of  meeting;  let  t=:the  difference  between  the  times 
of  their  beginning  to  move,  and  suppose  A  moves  through  the  space 
a  in  the  time  n,  and  B  through  the  space  b  in  the  time  m,  then 

nx 
(a:n::x  :)  — sithe  time  of  As  motion,  and  (6  :  m  : :  d— x  :  ) 

d^""  x^m  nx 

. — r — ^ithe  time  of  B*s  motion;  whence  by  the  problemj^ —  — 

d — x.m  bt-\'dm 

— T — =*,  VX5=-- .a. 

o  on-^am 

^juid  the  specific  grarity  is  its  weight  compared  with  that  of  a  body  of  equal 
hulk,  hot  of  a  difflereDt  kind :  thus,  a  cabic  £D0t  of  oommon  water  weighs  1000 
oances  avoirdapois,  and  a  cubic  inch  of  each  of  the  following  substances  weighs 
•8  follows ;  Tix.  fine  gold,  ]9640os.  fine  silver,  11091  ox.  cork,  240  ox.  new 
falXen  snow,  86  ox.  oommon  air,  1.232  ox.  &c.  &c.  these  numbers,  then,  repre- 
sent the  specific  gravities  of  the  aboTe-mentioned  substances  respectiTcly,  com- 
.fared  with*co«imoa  water. — ^Tables  of  the  spedfie  grarity  of  a  great  variety  of 
bodies,  both  solid  and  fluid,  may  be  found  in  the  writings  of  Mersenne,  Muf- 
chenbroeck,  Ward,  Cotes,  Emerson,  Hntton,  Vyse,  Martin,  &c.  and  are  useftil 
^or  computing  the  weight  of  such  bodies  as  are  too  large  and  unwieldy  to  be 
inoved ;'  by  means  of  their  kind  and  dimensions,  which  must  be  prerionsl^ 
known. 


N 


W  AL&SAIIA.  PAftf  tV. 

SxAicnms^*^!.  A  tett  out  from  London  to>M«Rk  Diiriiam 
dteUnt  257  miles,  md  lAweli  II  tsfleB  iA4liOtfi$  B«ils<nit 
from  Durham  8  boon  later,  and  travels  towards  London  at  the 
rate  of  10  miles  in  S  hours  :  whereabouts  on  the  road  wHl  they 
meet? 

Here  d==257>  t^B,  #sfell,  msl4,  6s^10»  ms3.* 

Then  x=— ^J!±?^?^  X  ll  =  m  ^  lailef  from  Xo«A«. 
10X4+11x3  73  -^ 

.    2*  Supposing  Africa  to  be  9QfiOO  miles  round,  and  a.  ship  to 

iail  from  the  Isthmus  of  Suez  down  the  Bed  Siea,  with  int^at 

to  coast  it  round  that  vast  conthient,  sailing  on  an  average  ^ 

miles  an  hour } — a  week  after  anethief  ship  sails  from  the  opposite 

side  of  the  same  Isthmus  with  the  santt  intent*  and  piling  the 

traits  of  Gibmltar,  sails  at  the  mte  of  3^  miles  an  Ixniff;*** 

Aear  what  place*  on  tlie  coast  will  they  raedt  ? 

4$.  If  two  bodies,  A  and  B,  move  in  (he  same  direction  and 
in  the  same  straight  line,  their  velocities,  distance  at  setting  out, 
land  the  interval  between  the  times  of  their  beginning  to  move, 
being  given,  thenCe  to  determuie  the  point  where  they  will  come 
together. 

Let  A  ie  the  farthest  from  the  required  point,  ct=<^  distattce 
froth  A  to  B,  xzsAts  distance  from  the  point,  then  wiU  x — d^B's 
distoHde;  tiko  let  tmihe  Mtfvmlof  tsmt  Aehmeoi  ihtir  wetting  xmt, 
and  let  A  move  through  the  space  a  in  the  time  r,  and  B  thret^ 

tx 

the  space  h  in  the  time  <;  then  U>iU  {a  :r  n  x  : ) — ^the  time  of 

As  motion,  and{h:szi  x^d  :  )  -Il-ie^Ae  time  of  Bs  motion; 

nohmee  bf  the  prohlem,  !!i--iZLjft±f,  *.•  x^J^^:a,  when  A 

«  o  hT'^as 

sets  out  first;  and  — r =^  •/  x=  — ^-.o^  when  B  ^ets 

b         a  as^hr  ^ 

cut  first. 

EKAMPLBS.---1.  A  ship  sbiIb  from  the  D«wnS)  east,  toii^aitfe 
Petersburg,  at  the  rate  of  54  mites  in  «8  htmt ;  «4  hwurs  aftelr 
another  ship  saib  frbm  Lisbon,  distant  ttom  the  l>owns  660 
miles  west,  in  pursuit  of  her,  and  goes  at  the  rate  of  8  miles  aa 
hour :  whereabouts  will  the  latter  ship  overtake  the  former^ 


PlkT IV. 


GENERAL  PROBLEMS. 


91 


Heted^&SO,  ^ssM,  assS,  rzl,  b^M,  Jrcstt;  0nd4i^au$e 


B  sets  out  first,   therefore  xsz 


54x344-23x560 


xSss 


8x23—54x1 

858.21538,  8(c.  miles  from  Lisbon,  or  (858.21538,  ^.--560^) 
308.21538,  SiC  miles  from  the  Downs. 

2.  Suppose  the  skip  from  Lisbon  sets  sail  24  hourii  before  tbe 
other  ? 

-,.  54  X  24--23  X  550  ■         ^^«w,«^    ,         zt     a^ 

Then  x=i  -—■ — ; — 7r''7z;r-  x  8=<>98.triS8,  ?rc.  mites  from. 
!    .  54x1—8x23 

Lishm,  or  .(698.7138,  ^c--550=d)  14$i7138»  ic.  miles  fron^ 

the  Downs, 

3.  A  is  trOO  tniles  south  of  London,  and  sets  otit  on  a  journey 
north^^Fard,  travelling  37  miles  etery  ^4  hours ;  B  from  London 
pursues  the  same  roiit,  selling  out  49  hours  lafter  A,  and  tra- 
velling at  the  rate  of  ll  miles  every  8  hours :  where  will  they 
be  together  ? 

43.  Qiven  the  forcfs  of  several  agents  «  separately,  to  deter- 
mine  their  Joint  force  ? 

Let  A,  B,  C,  D,  ^c.  be  the  agents,  and  suppose 


A  -1 
B 

"I 


b  \n 

can  produce  an  effect,  ^   c   >  times,  in  the  time  ^  r 


Uc 


CaU  the  gtMn  ^eet  1,  aitkd  hi  ^stihe  iime  in  whidi  theif  can- 
produce  ity  all  operating  together  : 

Thsn  will 


ax'\ 


m.{time) :  a,{effect)  : :  x  A  time  :  — * 

m 

bx 


n 


X 


^c. 


d 


*c. 


n 

'BX 

dx 
s 


The  effect  produced 
in  the  time  x,  b^ 


A 
B 
C 
D 


>  An  agent,  (in  Latin  agent,  from  »ym  to  drire,)  is  that  by  which  any 
thing  - 18  done  or  effected,  niilosophert  call  that  the  agent,  which  is  the 
iinmediate  cause  of  any  effect,  and  that  on  which  the  effect  is  produced  they 


38  ALGEBRA.  PaatIT. 

.  Bui  the  ^m  of  thne  effects  it  equtd  to  the  gkfen  efeei  I,  pro- 

dttced  by  thejomt  apemttum  of  aU  the  agents^  m  th^  time  x ;  whenct 

ax     hx     ex     dx  a       b   ,   c   ,    d  . 

—+ h h—  *c,=l,  or  X.--+ h— +— *c.5=l,vx= 

ffififf  m      n      r      $ 


a    .  b       c      d    ^ 
m      n       r      s 


Examples. — 1.  A  can  reap  5  acres  of  wheat  in  8  days,  B  caa 
reap  4  acres  in  7  days,  and  C  6  acres  in  9  days  ;  how  lon^  will 
they  require  to  reap  a  field  of  SO  acres,  all  working  together  > 

Here  m=8,  a=s5,  n=s7>  6=4,  r=9,  c=s6. 

1  1  168 

Thenx=z __X30=-- j  x30=-yt  xSOss 

a       b       c  5       4       6  dlo 


m      n       r  8       7      ^ 


32 


2.  A  vessel  has  three  cocks.  A,  B.  and  C  -,  A  can  fill  it  twice 
in  3  hoiurs,  B  3  times  in  4  hours,  and  C  4  times  in  5  hours  -,  in 
what  time  will  it  be  filled  with  the  three  tocks  all  open 
together  ? 

44.  If  two  agents,  A  and  B,  can  jointly  produce  an  effect  ia 
the  time  m,  A  and  C  in  the  time  n,  and  B  ^nd  C  in  the  time  r ; 
in  what  time  will  each  alone  produce  the  same  effect  ? 

Let  Jyy^sthe  time  <  B>  would  require  to  produce  the 
given  effect;  and  let  the  effect  be  called  1. 


call  the  patient ;  the  effect,  as  communicated  by  the  agent,  they  call  an 
eeticn ;  but  as  reeeived  by  the  jiatient,  a  pauiou :  a  smith  striking  oa  an 
anvil  has  been  frequently  proposed  as  a  proper  example ;  thus  the  smith  is  the 
superior  agent^  the  hammer  with  which  he  strikes  is  the  it^itrufr  agents  the 
blow  he  strikes  is  the  adtofi,  the  anvil  is  the  patient^  and  the  blow  it  receives, 
the  pasnon. 


Part  IV. 


GENERAL  PROBLEMS. 


ds 


0?  {time)  :  1  {effect) 


Then  is 
TO  Umie)  :  — 

X 


X 


1 
1 
1 
1 
1 

TO        TO 


TO 


n 


ft 


TO 
7 

n 


I 


z 
r 

y 

r 


II 


J 


A  in  the  time  m 

B TO 

A n 

C n 

B r 

C r 


fflience—+'^=:l,  or  (1)  —4—=—. 
a?       y  ^       :p       y      TO 

— +— =1,  or  (2)  —+_=—. 
r       r  ^      X       z       n 

*•       *•       ,         ,ov   1       1        1 

— 1---=1,  or  (3)  —+—=--., 

y       z  ^      y       z      r 

Add  equations  I,  %  and  3  together,  and  the  sum  will  be 

1 


1 


I 


— 4— -f  —  X2=— +— +— ,  or  (4)  —+—+—=— -4 
j:yz  TOMr  x       y       z       2to     2n 

4-3-  ;  /roTO  eq.  4' subtract  eq,  \,  2,  an^  3  severally,  and  the  re^ 

mainders  are 


JL—  ^  J  ^  1_ 

z      ^m  2n  ^r  to 

1  _  1     ^  JL  JL 

y'~2w  «n  2r  n 

1  _  1  JL  J,    1 

a?  ""2  TO  2n  2r      r  ^ 


2TOnr 


►  whence  < 


Xrs- 


TOf-f-mn — nr 
2TOnr 

nr+mn-^mr- 
2TOnr 

»r4-rar — win  * 


Examples. — 1.  A  and  B  can  unload  a  waggon  in  3  hours^  B 
and  C  in  2^^  hours,  and  A  and  C  in  2^^  hours  j  how  long  will 
each  be  in  doing  the  same  by  himself? 

Here  m:Kz3,  71=24,  »*=^t>  *=a 7i ^^^^    ^  = 

'  ^  24x2iH-3x2i— 3x2^ 


37.126 


4.6875 
VOL.  II. 


=7.92  /lour^. 


34  ALG£BRA.  Past  IV. 

2x3x24X^  37.125 


24  X  2^+3  X  24-3  x2i    7.6875 


=:4.82926889  hours. 


2  X  3  X  24  x^i  37125     ^  ^ , «^^.^e  L 

2=:-: — — -^ ^ = =4.21276595  hours. 

3  x2i+3x  24-^x24    8.8125 

2.  A  quantity  of  provfeions  will  serve  A  and  B  8  mcmtlis, 
A  and  C  9  months^  and  B  and  C  10  months ;  how  long  would 
the  same  quantity  serve  each  person  singly  ?    . 

Ans.    A  14  fit.  20$4  days,  B  17  m.  16|f  days,  C  33  m.  ^ff 
daySf  reckoning  30  days  to  a  month. 

45.  It  is  required  to  divide  the  number  22  into  three  such 
parts^  that  once  the  first,  twice  the  second,  and  thrice  the  third 
being  added  together,  the  sum  will  be  47»  and  the  sum  of  the 
squares  of  the  parts  166  ? 

Let  X,  y,  and  z,  denote  the  three  parts  respectively,  a=22j  h 
=47>  c=166j  thenify  the  problem  x+y-^-zssa,  x+2y-|-3z=6> 
and  j?*+y*-f-z*=c;  subtract  the  first  from  the  second,  and  y-\-2z 
szb^a,  whence  y:=^b — a-^2z;  subtract  double  the  first  from  the 
second,  and  z— a:=5— 2fl,  whence  xssz+2a— 6^  let  f^b^a, 
^=6— 2  a;  these  values  being  substituted  in  the  two  latter  equa- 
tions, they  become  yszf-^^z,  and  xzs^z^g;  svhstitute  these  values 
for  y  and  x  in  the  third  given  equation,  and  it  will  become  z*-^2g9 

+g*+/*-4/z+4z«+z«=:c,  or  z^^^I±MzJ'''^''^.^  puth^ 

^  o 

2f4-jT  c f* fi* 

-^^^j  and  the  latter  equation  becomes  z^^hzss — "^      ^  ,m 

which  by  completis^  the  square,  8fC.  it  becomes  zss — + 

V g H-j  (f/ohich^  by  restoring  the  values  of  c,f,  g,  and 

k,  viz,  c=166,  /=r6-a=47-22=26,  gs=5>-gtfs=47— 44= 

J  1.     2/-f  5f     50-1-3       63^     53         166—625-9  .  2809 

3,  and  A=-=^^ — S= =  — )= \- ^ \ 

3        -3  3^6^^  6  ^36 

=9,  whence x=z{z^g:=i)  6,  and  yas  (/— 2«=s)  7. 

46.  Required  the  values  of  x  and  y  in  the  following  equations, 
viz.  a^+3!«y-ha?*y«+a:y'+y*=211=ii,-  and  a«+«*'9*+«*y*+ 
t?y-hy«=U605=6? 

Divide  the  second  by  the  first,  add  the  quotient  to,  and  sub* 
tract  it  from  the  first,  and  the  results  will  be  (2:r^+2xV+2^:s 


Paat  IV.     GENERAL  PROSLEMS.  i5 


=— fl— — ,  i^^  ttJiW   fAe  two  ei]u<xiion9,  o^t^tf  derit^rf,  6^oin« 

spxn,  •/  p= — i  iAis  being  squared,  and  the  nquare  added  to  «•— 

s 


w'                                            in         111* 
p*=w,  gi«c5  «*=mH — ^,  or  «♦— iiw*=«%  •••  5=^—4-  ^ f-n* 

n 
=13,  andp=( — =)  6.     ^ow  since  (*=?)  a?*+y*=:13,  and  (p=) 

xy:=:6,  if  the  square  root  of  the  sum  and  difference,  of  the  former 
and  double  the  latter  be  taken,  we  shall  thence  obtain  x=i3,  and 

'    47.  Given  the  sum =5,  and  the  product  =p,  of  any  two  num- 
bers, to  find  the  sum  of  their  nth  powers  ? 

Let  X  and  y  represent  the  ttco  numbers,  then  will  x4-y=5, 
and  xpssp.  First,  {x+yl'^sa)  j^-f^ipy-f  y«32#«,  and  ^xyrm^p; 
subtract  the  latter  from  the  former,  and  ^fi-^-y^ss^-^QpnstkesHm 
(ff  the  squares.  Secondly,  x'^ + y^jx -j- y:=^s^^%pjSy  or  x^  +xy.x + y 
-f-y'=s'— 2sp,  which  (by  substituting  sp  for  its  equal  ^y^x-^-y) 
becomes  a:^-fsp+y^=«^— 2fip,  •/  3^-\-y^z=::,^'Ssprzthe  sum  of 
the  cubes.  Thirdly,  a?3+y'.a?4-y=«3— 3«p.s,  or  a?*-ha?y.a:*+y«-|- 
y*=:s*-^35*p,  which  {ky  a^stiiuUng  p^'^—^p  for  its  equal 
xy.x^'^y^)  becomes  a:*4-p,«*— 2p4-y*=:«*— 3«^,  •/  :t*+y*=(**— 
3s*p— p.«*— 2p=)  *^— 4s''p-f  2p'=/^e  sum  of  the  biquadrates. 

In  like  manner  it  may  be  shewn,  that  s^'^5^p'\-bsp''ssthe 
sum  of  the Jifth  powers;  s^— 6y*p+9s'p*— 2p^=*/»e  sum  of  the 
sixth  powers,  ^c. 

By  comparing  together  these  several  results,  the  law  of  con- 
tinuation will  be  manifest;  for  it  appears  from  the  foregoing  pro- 
cess,  that 

The  sum  of  any  powers  is  found  by  multiplying  the  sum 
of  the  next  preceding  powers  by  s,  and  from  this  product 
subti*acting  the  sum  of  the  powers  next  preceding  those  multi« 
plied  hyp. 

D  2 


36  AL6BBRA.  Part  IV. 


Thus,  the  sum  of  the  4th  fHwensss  x  sum  of  the  cuhes^p  x 
sum  of  the  squares. 

The  sum  of  the  ^th  powers^s  x  sttm  of  the  4th  powers^p  x 
sum  of  the  cubes. 

The  sum  of  the  6th  powerszss  X  sum  of  the  ^th  powers—p  x 
sum  of  the  4th  powers,  Stc  ^c. 

Hence  the  sum  of  the  nth  powers  of  x  and  y  will  be  as  follows  ;. 
n— 2  n— 3  n— 4  n— 4«— 5  n— 6 

.  ft— 5n— 6n— 7  w— 8    . 

13.  To  investigate  the  rules  of  arithmetical  progressioa. 
Let  a^ihe  Ua»t  term  I  ^^^^^^  ^^^^  ^^  ^^^^^ 
z=itJie  greatest     -> 
n=zthe  number  of  terms 
d=zthe  common  difference  of  the  terms 
8=:  the  sum  of  all  the  terms, ' 


.  Then  will  a+a+<i+a+2(2+a+3d+^  SfC.  io  a-f  ft— l.cE 
he  an  increasing  series  of  terms  in  arithmetical  progression;. 

And  24-z— d+«— 2rf-f-« — 3rf+,  8(C.  to  z-^n — l,d  will  he 
a  decreasing  series  in  arithmetical  progression. 


14.  Now  since  in  the  increasing  series  a -^n^ I, d=: the  greatest 

term,  and  z:=:  the  greatest  term  by  the  notation,  therefore  z^:^ a •\' 

n— l.d  (theorem  1.)  JVhence  by  transposition,  8sc,  assz — n— l.d 

z—a  Z'^^a 

(theor,  2.)  d= (theor.  3.)  and  nzx — ; — 1-1  (theor.4.) 

^    •  n— 1 ^  '  d  ^ 

Whence,  of  the  first  ttrm,  last  term,  number  of  terms,  and  difference, 
any  three  being  given,  the  fourth  may  be  found  by  one  of  these  four 
theorems, 

15.  Next^  in  order  to  find  «,  and  to  introduce  it  into  the  fore- 
going theorems^  let  either  of  the  above  series,  and  the  same  series 
inverted  be  added  together ;  and  since  the  sum  of  each  series  is:=:  . 
s  by  the  above  notation,  the  sunt  of  both  added  together,  will  6t'i-  ^ 
dently  be  2  s.    Thus, 

The  series a-|-a4-d+a+2d+a-i-3d+^c.=«^ 

The  series  inverted  a-^-S  d+a-\'2  d-\'a'\'d'{-a  ,  ,  ,  =«. 


i.t  ■«  ■  I- 1 


Their  sum 2a+3d-h2a  +  3d-f  2a-f  3d+2a-f  3<f=^.2« 


Part  IV.      ARITHMETICAL  PROGRESSION.  37 


That  is  (2  a-f  3(2.11,  or  a-i-a+3d.»,  or^  since  a+3  rfsrz) 

'        '  ■'  ft  JL,  T  ft  ««— — »  n  • 

a+z.n=s2«,  whence  *=( — ^— =)  <»+«--^  (theok.  5.)  From  this 

equatumare  deriveda=: z  (theok.  6.)  z=^^ — ra  (t^eor. 7.) 

and  n= (theor.  8.)  Also  by  equating  the  vaiues  of  z  in 


2«        .  ...  s 


theorems  1  and  7>  (»w^  o+w — 1.4=-- — -a.)  we  obtain  a=r 

n         '  n 

——.a  (theor.  9.)  fl=(       ■ ;=s)  — . -(theor.  10.) 

2  n.»— jl      «— I         »  «— 1 


5=—n.2a+7i—l.d  (theor.  ll.)andn=- 

(theor.  12.) 

16.  In  like  mannevy  by  equating  the  values  of  a  in  theorems 

3   and  6,   (viz,  z^n — l.d= z,)  loc  derive  z= 1 .d 

9  n         2 


2  wz — ^^ 


(theor.   13.)    d= — . (theor.   14.)  sr= — n.Sz— n-'l.d 

^  '  n   n— 1    ^  2 


id4~z^  A/ld-l-z^— 2  d* 

(theor.  15.)  andn= ^^~i (theor.  16.)  and 

a 

z-'-a 
equating  the  values  of  nin  theorems  4  and  8,  we  have  — — J- 1= 


2* 

—-7,    whence  z=  ./a— 4^d)*4-2  d«— i^   (theor.   I7.)   a= 

. z-f-a.z— a 

^z+T^*— 2d5+4.d  (theor.  18.)  d=r-—:;; (theor.   19.) 

z— a+ci  z-j-a  , 
#=  — -^ —  •—^^  (theor.  20.) 

17.  Hence  any  three  of  the  five  quantities  a,  z,  d,  n,  s,  being 
given,  the  other  two  may  be  found :  also  if  the  first  term  a=zo, 
any  theorem  containing  it  may  be  expressed  in  a  simpler  manner. 

IS.  The  following  is  a  synopsis  of  the  whole  doctrine  of 
arithmetical  progression,  wherein  all  the  theorems  above  de- 
rived are  brought  into  one  view* 


d3 


58 


ALGEBRA. 


Pait] 


PTheor.  I  Given  |  Req.|      Solutiog  when  a^o. 


I. 
XI. 

m. 

V. 
IV. 

XX. 

VII. 

X. 

VIII. 

XIX. 

XVII. 
XII. 

II. 

XV. 
IX. 

xm. 

VI. 

XIV. 
XVIII. 

XVI. 


a,  d,  n-l 


2=:a-i-n— l.d 


Theor. 


a,  d,z 


I. 


a,  n,  8 


2,  d. 


i: 


z,  d,  s 


5=4..n.2a-|-n— I.d 


d= 


n 

■  I  ..  ■ 


2— <t 

71=-— -+i 


5= 


z — a+d  a-f« 


^8 

'zz=z a 

n 


d= 


^  s^na 


n   n — I 


W=" 


a+2 


XXI. 


2=11— I. d 


XXII. 


f=4-n.n— l.d 


XXIII. 


d=- 


XXiV. 


n 

5  =  2.-- 

2 


XXV. 


a 


XXVI. 


XXVII. 


2^ 


XXVIII. 


^at— 


XXIX. 


d= 


2+a.z— a 
2«— a — 2 


2=  ^a— 4J*+2<f*— 4^ 


n= 


^— «4-  >v/-W— 3«4.2(i« 


a=2— n — l.d 


ai^Bi^vMriiH 


5=4-n.?2— »— -l.d 


^aa^^aMte* 


«       w— 1    , 

a= :r— .d 

w         2 


«         2 


2s 

a= 2 

n 


2  n2-*» 
""  n  *n— I 


«=  >v/2-R31*— 2  ds+4-d 


n 


XXX. 


XXXI. 


XXXII. 


When  a:=zQ. 


n— I 


*=• 


2  +  cZ    2 

T*"2 


2* 

n 


2 


»   n— 1 


ff= 


2 


d=- 


2«— 2 


2=v'id«+2dM 


«= 


id+vI^M^ 


S  8  8 

When  d=iOj  then  azsz^:—^ }  8:=znazznz  ;  n= — =  — 


n 


a 


Pabt  IV.      ARITHMETICAL  PROGRESSION.  d9 

ExAMPLBfl. — I.  la  an  arithmetical  ftogretsioa,  the  first  term 
is  3,  the  number  of  terms  60,  and  the  common  difference  S : 
what  is  the  last  term^  and  the  sum  of  the  series  ? 

Here  a^S,  n^bO,  d=x^. 

Whence,  tJum.  1.  z=3+50— 1x^=2 101  =si^  last  term. 


And,    theor.  2.    «=4-x 50x3 x 3  +  50- lx2=2600=f^ 
sum. 

2.  Given  the  first  term  3^  the  last  term  101^  and  the  number 
of  terms  50  5  to  find  the  common  di£ference  and  the  sum  of  the 
series  ? 

Here  ascS,  2=101,  n=50. 

Whence,  theor.  3.  ^=*r --=2=^^e  common  difference. 


50 


And  theor.  5.  «=3  +  iOl  x  --=3600=s  the  sum. 

3.  The  first  term  is  S,  the  common  difference  %  and  the  last 
term  101 5  required  the  number  of  terms,  and  the  sum  ? 

Here  0=3,  d=:2,  z=101.  • 

101—3 

Wherefore,   theor,  4.   »= ^.l^zzBO^ithe  number  of 

'terms. 

^  J  ^L       ^^         101-3+2     101+3    ^^^      - 
And,  theor.  20.  «= ^         X      ^      =2600==th£  sum. 

4.  The  first  term  is  3,  the  number  of  terms  50,  and  the  sum 
of  the  series  2600,  to  find  the  last  term,  and  £fference  ? 

Here  a=3, 11=  50,  s=^600. 

2  X  2600 

Then,  theor.  7.  2= — 3=101=*Ae  last  term. 

'50 

..  ^     .X         ,        ,     ^      2600—50x3     ^     ^. 

j#wf,  <*ew.  10.   <l3s:--x ; 3=3=3:  iAe  cmmuw 

50   ^      50—1 

difference. 

b:  Given  the  first  term  5,  the  last  term  41,  and  the  sum  of 

the  series  299,  to  find  the  number  of  terms,  and  the  common 

differenced    Ans.  6y  theor*  9.  »3=13,  and  by  theor.  19.  ds^iS. 

6.  Given  the  first  term  4,  the  common  deference  7>  and  the 
turn  355,  to  find  the  last  terra,  and  number  of  terms?  Ans.  by 
theor.  17.  zsxejf  and  by  theor.  12.  fi=10. 

7.  Tte  last  terra  is  67>  the  difierence  7,  md  tht  number  of 

D  4 


40  ALGEBRA.  Paet  IV. 

terms  10>  being  given,  to  find  the  first  term  and  sum  ?   Jng.  by 
thear.  2.  asz4,  and  hy  theor,  15.  f  =5355. 

8.  Let  the  common  di£ference  3,  the  number  of  terms  13> 
and  the  sum  299  be  given,  to  find  the  first  and  last  terms  ? 
Ans.  by  theor.  9.  a=5,  and  by  theor,  13.  z=41. 

9.  Let  the  last  term  67,  the  number  of  terms  10,  and  the 
sum  355,  be  given,  to  find  the  first  term  and  difference  ?  Am. 
by  theor.  6.  a =4,  and  by  theor.  14.  dsT. 

10.  If  the  last  term  be  9>'the  difiference  1,  and  the  sum  44, 
required  the  first  term,  and  number  of  terms  ?  Ans,  by  theor, 
18.  a=5,  and  by  theor.  16.  n=8. 

11.  The  first  term  O,  the  last  term  15,  and  the  number  of 
terms  6,  being  given,  to  determine  the  di£ference  and  sum  ? 
Ans.  by  theor.  23.  d=3,  and  by  theor,  24.  «=45. 

12.  Bought  100  rabbits,  and  gave  for  the  first  6d.  and  for  the 
last  34d.  what  did  they  cost  ?     Ans.  SL  6s.  Sd. 

13.  A  labourer  earned  3d.  the  first  day,  8d.  the  second,  ISd. 
the  third,  and  so  on,  till  on  the  last  day  he  earned  4s.  lOd.  how 
long  didHie  work  ?     Ans.  1*2  days, 

14.  There  are  8  eqdidifierent  numbers,  the  least  is  4,  and  the 
greatest  32  -,  tvhat  are  the  numbers  ?  Ans,  4,  8,  12,  16,  20, 
24,  28,  and  32. 

15.  A  man  paid  1000^.  at  12  equidifiercnt  payments,  the  first 
was  10/. — ^what  was  the  second,  and  the  last  ?  Ans.  the  second 
23;.  6s.  8d.  the  last  1661.  I3s.  4d. 

16.  A  trader  cleared  502.  the  first  year,  and  for  20  years  he 
cleared  regularly  every  year  bl.  more  than  he  did  the  preceding; 
•what  did  he  gain  in  the  last  year,  and  what  was  the  sum  of  his 
gains? 

17.  The  sum  of  a  series,  consisting  gf  lOQ  terms,  and  be- 
ginning with  a  cipher,  is  120  5  required  the  conunon  difference^ 
and  last  term  ? 

19.  PROBLEMS  EXERCISING  ARITHMETICAL 

PROGRESSION. 

1.  To  6nd  three  numbers  in  Arithmetical  Progression,  the 
common  difference  of  which  is  6,  and  product  35 } 

Let  the  three  numbers  be  x---6,  x,  and  j;+6  respectwehf. 
Then  by  the  problem,  (x— 6.x4;+6=)  «'-.-36a?5=35,  orx^— 36x 
—35=^05  this  equation  divided  by  Xrf  1^  give»  (x^-^o;-- 35=0,  or) 


Pabx  IV.       AMTHMETICAL  PROGRESSION.  4i 

«*— jp=:35  5  whiah  resolved,  we  h(me  a?=^35.25+.5,  whence 
a:--6=:^35.25-f  5.5,  c/nd  .a:+ 6=^^^5.25 +6.5  :  the  numbers 
therefore  are  .43717,  6AS7l7,(ind  12.43717,  nearly. 

2.  An  artist  proposed  to  work  as  many  days  at  3  shillings  per 
day,  as  he  had  shillings  in  his  pocket;  at  the  end  of  the  time 
having  received  his  hire,  and  spent  nothing,  he  finds  himself 
worth  44  shillings  j  what  sum  did  he  begin  with  ?. 

Let  x=his  number  of  shillings  at  first,  whence  also  x=:the 
number  of  days  he  worked :  we  Jiave  therefore  here  given  the  first 
term  x,  the  common  difference  3,  and  the  number  of  terms  x-^-l, 
in  an  arithmetical  progression,  to  find  the  last  term}  now  by 
theor.  1.  (z=a+n--l.d,  or)  44=af-f  a:+l  — 1  x3,  that  is,  4x= 
44,  whence  a:=ll  shillings  =z  the  sum  he  began  with, 

3.  To  find  three  numbers  in  arithmetical  progression,  such« 
that  their  sum  may  be  12,  and  the  sum  of  their  squares  56  ? 

Let  x^zthe  common  difference,  3  5=(12)  the  s^um,  then  wUl 

s=^the  middle  number,  s — x=^the  less  extreme,  and  s+x=:the 

greater  extreme,  also  let  fl=56j  then  by  the  problem,  («— x)*-fr 

5«-|-7+il«i=)    3««-|-2a;«=a,  whence  2:r2=a— 3«S  and    xss 

a— 3««         56-48  ,       ,  ^  ", 

V — 5 — =  V ; — =^  3  therefore s=i4,  s— x=2,  and  s+xrz6, 

%  2 

that  is,  2,  4,  and  6,  are  the  numbers  required. 

4.  To  find  four  numbers  in  arithmetical  progression,  whereof 
the  product  of  the  extremes  is  52,  and  that  of  the  means  70  ? 

Let  xzzithe  less  extreme^  y=the  common  difference ;  then  will 

X,  x-i-y,  x-^^y,  and  x-^-Sy,  represent  the  progression.     Let  a= 

52,  Z>=70,  then  by  the  problem  (a?.^+3y=)  a?*+3xy=:a,  and* 

(j;-|-y.a?+2y=)  a?*+3a:y+2y*=6;/roTO<^  latter  equation  sub' 

b — a 
tract  the  former,  and  2y*=6— a,  whence  y=^— — =35  suo- 

stitute  this  value  for  y  in  the  first  equation,  and  it  becomes  a^+dx 

81      9 

=a ;  completing  the  square,  Sfc.  we  obtain  a:=  a/^H =4 : 

4       2 

wherefore  4,  7>  10,  and  13,  are  the  numbers  required. 

5^  The  sum  of  six  numbers  in  arithmetical  progression  is  48, 
and  if  the  common  difference  d  be  multiplied  into  the  less  ex- 
treme, the  product  equals  the  number  of  terms  -,  required  tbQ 
terms  of  the  progression  ? 


ALGSMU.  Past  IV. 

Let  a^s-the first  term,  then  da=6,  and  a^s^—i  also, since s^s 


(•i^fi.da+n — l.ds)  mH — '--^—-d  by  theor,  11.  we  have  bff  sub* 

stitutum,  48=6  a+---— .d^  <^<  is,  6  a+ 15  d=:48  -,  whence  2  a-f 

2 

5d=:16^or  (ptt/^iii^— /or  a)  5  <?+ 12=16  d,  or  <P— — d=  — 
18 

•7-  i  whence  fry  completing  the  square,  4rc.  if =?^  therefore  azc 
o 

6 

(— =)  3;  coiuegiMiUZy  the  numbers  are  S,  B,  7,9,11,  and  13. 

6.  The  continual  product  of  four  numbers  in  arithmeticil 
progression  is  880^  and  the  sum  of  their  squares  214 ;  what  are 
the  numbers  ? 

Let  p=:880>  «=214,  2x=<fte  common  difference^  y^Sxav 
the  less  extreme;  then  will  y— 3x,  y— x,  y-h^Pt  ond  y4-3x=*^ 
<eni»  of  the  progression  s  wherefore  by  the  problem,  y— 3x.y — x. 
y+x.y"+3x=p,afirf  y— 3xl*-f  y— xp+y+xl«-f  y+3xl*=«;  these 
equations  reduced,  become  y*— 10y*x*+9x*=p,  and  4y*-h20x* 

s=#;  /rom  tAe  2a^^€r  of  these  y*= 5x*,  therefore  y** — — 

4  16 

Ssjfi 

'-^+25x*;  tf  <toe  values  Be  substituted  for  their  equals  in  the 

s^     5  sx^  5  <x^ 

former,  we  have  — ~ — I-  25  x* h  50x* + 9x*=:p,  whence 

16         2  2 

-T — =-! — ! -.  or   (mUtttue  ass  —.  and  — =-^ 


— — --')x*— «x*= — 5  then  by  completing  the  square,  ^c,  x^ 
J±^=ily  rutariag  the  values  of  a  and  R)  H,  »A.«ce  y= 


(^-- — 5x*=i:)  64-:  therefore  y— 3x=2,  y— x=:5,  y-f  x=:8,  and 

y  -|-  3  x=  1 1  ^  /A6  numbers  required, 

20.  To  find  the  number  of  permutattons^  which  can  be  made 
with  any  number  of  given  quantities. 

Defs  The  permutations  of  quantities  are  the  diflerent  orders 
in  which  they  can  be  arranged. 


Pabt  1Y.  PERMDTATIONS.  43 

Let  a  arid  b  he  two  quantitiisj  thete  wUl  evidently  admii  of 
two  permutations,  viz.  ab  and  bo,  whkh  number  of  pemmtaiUme 
may  be  thus  expressed,  1x2. 

Let  a,  by  and  c,  be  three  quantities  j  these  admit  of  six  pemm* 
tations,  abc,  bac,  cah  acb,  6ca,  and  cba,  viz.  1x2x3. 

Let  a,  b,  c,  and  d,  be  four  quantities)  these  admit  of  24  per* 
mutations;  thus,  abed       bacd      cabd      dabc 

abdc  bade  cadb  dacb 
aebd  bccid  cbad  dbac 
acdb  bcda  cbda  dbca 
adbc  bdac  cdab  dcab 
adcb       bdca      cdba      dcha 

That  is, 4  things  admit  o/ 1x2x3x4 permutations. 

In  like  manner, 

5  tilings  admit  qflx2x3x4x5  ^ 

0 1x2x3x4x5x6        >permuiations. 

7 Ix2x3x4x5x6x7j 

S;c 5fc. 

jind  therefore  n  things  admit  of  1  x  2  x  3j  8sc.  to  n^ 
permutations. 

Examples. — 1.  How  maay  ways  can  the  musical  notes  uty  re, 
mi,  fa,  sol,  la^  be  sung ?     Ans.  Ix2x3x4x5x 6:s720  ways. 

2.  How  many  changes  can  be  rung  on  12  bells  J  Answer, 
479001000. 

3.  How  many  permutations  can  be  made  with  the  24  letters 
of  the  alphabet  ? 

.    21.  To  find  the  number  of  combinations  that  can  be  mad« 
out  of  any  given  number  of  quantities. 

Def.  The  comUnations  of  quantities^  or  things^  is  the  takii^ 
a  leas  collection  out  oi  a  greater  as  often  as  it  .can  be  done> 
without  regarding  the  order  in  which  the  quantities  so  taken 
are  surranged. 

Thus,  if  a,  b,  and  c,  be  three  quantities,  then  ab,  ac,  and  be, 
are  the  combinations  of  these  quantities,  taken  two  and  two :  and 
here  it  is  necessary  to  remark,  that  although  ab  and  ba  form  two 
different  permutations,  yet  they  form  but  one  combination;  in  the 
same  manner  ac  and  ca  make  but  one  combination,  as  also 
be  and  cb. 

Let  there  be  n  things  given,  namely  a,  b,  c,  d,  S;c.  (to  n  terms,) 
then  if  a  be  placed  before  each  of  the  rest,  n— 1  permutatUmf 


^ 


44  ALGEBRA.  Pabt  IV. 


ioiU  be  formed;  if  h  be  placed  before  each  of  the  rest,  n — 1  pet' 
mutations  will  in  like  tnanner  be  formed;  and  if  c,  d,  e,  8;c.  be 
placed  respectively  before  each  of  the  rest,  n— 1  permutations  in 
each  case  will  arise;  consequently,  if  each  of  the  n  things  be 
placed  before  all  the  resty  there  will  be  formed  in  the  whole  n.n — 1 
permutations;  that  is,  there  can  ».n— 1  permutations  be  formed  of 
n  things  taken  two  at  a  time. 

Hence,  if  instead  of  nwe  suppose  n —  1  things,  b,  c,  d,  e,  8(C, 
the  number  of  permutations  which  these  afford  of  the  quantities 
taken  twS  and  two,  will  (by  what  has  been  shewn)  be  n — l.n — 2 } 
now  if  a  he  prefixed  to  each  of  these  permutations^  there  will  be 
n—  1^ — 2  permutations  in  which  a  stands  first;  in  the  same  man^ 
ner  it  appears,  that  there  will  be  fi--l.n— 3  permutations  in  each 
case  when  b,  c,  d,  e,  dtc  respectively  stand  first ;  and  therefore 
when  each  of  the  n  things  have  stood  first,  there  wUl  be  formed  in 
the  whole  n.n — l.n— 2  permutations  of  n  things  taken  three  and 
three.  .  By  similar  reasoning  it  appears  that  n  things  taken 

4  at  a  time  afford  n.n— l.n— 2.n— 3  •% 

5  at  a  time tt.n— l.n— 2.n— 3.n— 4  ^'^ 

r  at  a  time .  . . 


„.n-l.n-g.n-3.n--4        (tations. 

n.n—  l.n-r2.n— 3.n— 4  . . .  n— r + 1-' 


This  being  premised,  we  may  readily  obtain  the  number  of 
combinations,  each  consisting  of  ^,  3y  4,  B,  8fC.  to  r  things,  which 
can  be  made  out  of  any  given  number  n ;  for  it  appears  by  the  pre- 
ceding problem,  that  2  things  admit  of  2  permutations,  but  by  the 
definition  they  admit  of  but  1  combination ;  and  therefore  any^ 
number  of  things  taken  .2  at  a  time,  admit  of  half  as  many 
combinations  as  there  are  permutations;  but  the  number  of 
permutations  in  n  things,  taken  two  and  two,  has  been  shewn 
to  be  n.n — l-,  therefore  the  number  of  combinations  in  n  thirds, 

taken  two  and  two,  will  be  — ^ .  or  which  is  the  same  — ^ — -— . 

2  1.2 

If  three  things  be  taken  at  a  time,  then  6  permutations  will 
arise  from  every  3  things  so  taken,  and  but  1  combination ;  and 
therefore  any  number  of  things  taken  3  at  a  time,  admit  of  one 
sixth  as  many  combinations,  as  there  are  permutations ;  but  the 
number  of  permutations  in  n  things  taken  3  at  a  time,  has  been 
shewn  to  be  n.n— l.n— 2  5  and  therefore  the  number  of  combina^ 


Pakt  IV. 


COMBINATIONS. 


45 


tions  in  n  things,  taken  ^  at  a  time,  will  be 


n.n — l.n — 2 


or 


n.n— In— 2 


1.2.3 
JBjf  similar  reasoning  it  mat^  be  shewn,  that  the  number  of 
combinations  in  n  things,  taken 


4 
5 


n.n— l.n — ^2.n— 3 
1.2.3.4 


>  at  a  time  will  be  < 


n.n— l.n— 2.n— '3.n— 4 
1.2.3.4.5 


n.n-^  l.n— 2.n— 3.n— 4.n — 5 
1.2.3.4.5.6 


n.n— l.n— 2.n— 3,  5rc.  to  n— r+l 


1.2.3.4,  4c.  to  r 

Examples. — 1.  How  many  combinations  can  be  made  of  2. 
letters,  out  of  10  ? 

rr  ,^1.        n.n^     10X9     ^^     . 

Here  n=10.  whence = — - — =45.  Ans» 

1.2  2 

2.  How  many  combinations  of  5  letters  can  be  made  out  of 

the  24  letters  of  the  alphabet  ? 


Here  n=24,  then 


njn^  1  .n— 2.n— 3.n— 4 


=  10626.  Ans, 


1.23  4.5 

3.  In  a  ship  of  war  there  are  40  officers,  and  the  captain  in- 
tends to  invite  6  of  them  to  dine  with  him  every  day  ;  how  many 
parties  is  it  -possible  to  make,  so  that  the  same  6  persons  may 
not  meet  at  his  table  twice  ? 

22.  To  investigate  the  rules  of  simple  interest. 

Def.  1 .  The  sum  lent  is  called  the  prvnci'pal, 

2.  The  money  paid  by  the  borrower  to  the  lender  for  the  use 
of  the  principal,  is  called  interest. 

3,  The  interest  (or  quantity  of  money  to  be  paid)  is  previ- 
ously agreed  upon ;  that  is,  at  a  certain  sum  for  the  use  of 
lOOZ.  for  a  year :  this  is  called  the  rate  per  cent,  per  annum  '. 


y  Per  cent,  means  by  the  hundrefi,  and  per  annam,  by  the  year  ;  the  term 
5  per  cent,  per  annum ,  means  5  pounds  paid  for  the  use  of  100/.  lent  daring 
the  space  of  a  year,  &c. 

VarioDS  rates  of  interest  have  been  i^iven  in  this  country  for  the  use  of 


46 


ALGrEBEA. 


Part  IV. 


4.  The  principal  and  interest  being  added  together^  the  sum 
is  called  the  amount. 

Let  pxzthe  principal  lent,  r=ithe  interest  of  I  pound  for  a 
year,  t=zthe  time  during  which  the  principal  has  been  lent,  i^ 
the  interest  of  p  pounds  for  t  years,  a=^the  amount;  then  toiU 
1  (pound)  :  r  {interest)  : :  p  (pounds)  :  pr^the  interest  of  p  pounds 
for  a  year:  and  1  (year)  :  pr  (interest)  :  :  t  (years)  ;  ptr=zi 
(thbob.  l.)zzthe  interest  of  p  pounds  for  t  years,  or  t  parts  of  a 

•  m  • 

year:  hence p^ — ,  *= — ,andr=z — .      If  to  this  interest  the 
^  "^     tr         pr  P^ 

principal  be  added,  we  shall  have  ptr^^pssa  (thbor.  2.)  hence 

by  transposition,  ^c.  p= (theor.  3.)  t=z ^  (theor.  4.) 

'^     If-fl  ^  ^         pr    ^ 

and  ras — -±  (theor.  5.)    The  following  is  a  synopsis  of  the  whole 
doctrine  of  simple  interest. 


Theor.  Given. {Req.j  Solution. 


.,«,r.{* 


irzzptr. 
aszptr-^-p, 
a 


mency,  at  different  periods,  from  5  to  50  per  cent,  but  the  law  at  present  is, 
that  not  more  than  5  per  cent,  per  annum  can  be  taken  here,  although  the 
legal  rate  of  interest  is  much  higher  in  some  of  our  colonies. 
The  interest  of  money  is  oompntod  as  follows ; 

In  the  courts  of  law in  years,  quarters,  and  days. 

On  South  Sea  and  India  bonds calendar  months  and  days. 

On  Exchequer  bills  ....  quarters  of  a  year  and  days. 
Brokerage,  or  commission,  is  an  allowance  made  to  brokers  and  agents  in 
foreign,  or  other  distant  pfaioes,  for  buying  and  selling  goods,  and  perform- 
ing other  money  transactions,  on  my  account ;  it  is  reckoned  at  so  much  per 
cent,  on  the  money  which  passes  through  their  hands,  and  is  calculated  hj  the 
rules  of  simpU  interest,  the  time  being  always  considered  as  1.  The  same 
Yules  senre  for  finding  the  value  of  any  quantity  of  stock  to  be  bought  or  Mid, 
and  likewise  iot  finding  the  price  of  insurance  on  hovses,  ships,  goods,  Ac 


PjiitIV.  simplb  intbbjest.  4r 

£xAMPi.B8.*-l.  Required  the  simple  interest  of  7^/.  lOf.  for 
4  years^  at  5  per  cent,  per  annum  ? 

Herep^{76Bl  10*.=)  765.5.  t=:4.  r=(— =).05. 

Tbm  i=zptr  (tkeor,  I .)  =^765.6  x  4  x  .05=  153.1  =:  1631.  2*. 
Anst^er. 

2.  What  is  the  amount  of  752.  10«.  6d.  for  S^  jetm,  at  44 
per  cent,  per  annum  ? 

Here  p=(75i.  10«.  6(f.=)  75.525,  ^=(84.=)  8.5,  r=(ii=) 

.0475:  whence  {theor.  2.)  prr+p=:75.525X&5Xi0475 4-7^.585 
5=106.01821875=  106i.  0«.  4d:^.49=a,  (he  amount. 

S.  What  sum  of  money  being  put  out  at  3  per  oent.  simple 
interest,  will  amount  to  4022.  10s.  in  5  years  ? 

Here  a=(4022.  10y.=)  402.5,  f=5,  r=(— =).03:  vjhere- 
r      i.x.        ox      «  402.5         402.5     „,  , 

4.  In  what  time  will  3502.  amount  to  4022.  IO9.  at  3  per  cent, 
per  annum  ? 

f/cr«  p=350,  a=402.5,  r=.03. 

nn.      /.I.        ^x«— P     402.5—3.50     52.5     ^  ^     ^ 

Then  (theor,  4.) i-= = — -3:5  years:=ti.  the 

^  '    pr         350X.O3       10.5        ^ 

answer. 

5.  At  what  rate  per  cent,  will  752.  amount  to  772.  Ss.  l^xL  in 
1^  3'ear,  ? 

Her€j)=75,  fl=:(772.  Ss.  Hd.=)  77.40625,  2=  (14-=)  1.5. 

r.^      .^       .X        «— P     77.40625—75     ^,.^    ^ 

T?iew  (theor.B.)  rar i.=i^^ r-^ =.021 38 s=2-iV per 

pt  75x1.5  '^ 

otsmp.  neathfyssir,  the  answer. 

6.  What  is  the  interest  of  2542.  17*.  3d.  for  24-  years,  at  4 
per  cent,  per  annum  ?     Ans.  252.  9s.  S^d. 

7.  What  is  the  amount  of  2502.  in  7  years,  at  3  per  oent  per 
annum  ?     Ans,  3022.  lOs.  Od. 

K 

8.  What  sum  being  lent  for  4  of  a  year,  will  amount  to  15«. 
C^d-  at  5  per  cent  ?     Ans*  15  shillings. 

9.  In  what  time  will  252.  amount  to  252.  1  Is.  3(2.  at  4^-  per 
cent,  per  annum  ?     Ans.  half  a  year. 

10.  At  what  rate  per  cent,  fer  annum  will  7962. 1^  ■ttotii^ 
to  9762.  Os«  4^(2.  in  5  years  ?     Ans.  44-  per  cent. 


48  ALGEBRA.  Part  IV. 

• 

11.  Required  the  interest  of  140L  lOf.  6d.  for  ^^  yeais^at  5 
per  cent,  per  annum  ? 

1^.  To  find  the  amount  of  2002.  in  8  years^  at  44  per  cent, 
per  annum  ? 

13.  Suppose  a  sum^  which  has  been  lent  for  120  days  at  4  per 
cent,  per  annum,  amounts  to  243/.  3^.  l-^d,  what  is  the  sum  ? 

14.  In  what  time  will  7252.  15«.  amount  to  7312.  25.  8^.  at 
4  per  cent,  per  annum } 

15.  At  what  rate  per  cent,  per  annum  will  5592.  45.  Od. 
amount  to  7352.  7*.  Od.  in  7  years  ? 

23.  To  investigate  the  rules  of  discount. 

Def.  1 .  When  a  debt  which  by  agreement  between  debtor  and 
creditor  should  be  paid  some  time  hence,  is  paid  imni^diately,  it  is 
usual  and  just  to  make  an  allowance  for  the  early  payment  3  this 
allowance  is  called  the  discount. 

2.  The  sum  actually  paid  (that  is^  the  remainder,  after  the 
discount  has  been  subtracted  from  the  debt,)  is  called  the 
present  worth. 

3.  The  debt  is  considered  as  the  amount  of  the  present  worth, 
put  out  at  simple  interest,  at  the  given  rate^  and  for  the  given 
time  *. 

Let  p::=:the  given  debt,  r=zthe  interest  of  1  pound  for  a  year, 

tzzztlie  time  the  debt  is  paid  before  it  is  due,  in  years  or  parts  of  a 

year;  then  will  l-{-tr.:=^the  amount  of  1  pound  at  the  rate  r,  and 

for  the  time  t:  {Art.  22.  theor.  2.)  then  also  will  the  amount  of  1 

pound  be  to  1  pound,  {or  its  present  worth,)  as  the  given  debt,  to 

its  present  worth ;  also  the  amount  of  1  pound,  is  to  the  interest  of 

1  pound,  as  the  given  debt,  to  the  discount ;  that  1*5,  1  +  ^r  :  1  : :  p  : 

P 
l  +  ^r 


:=.the  present  worth  of  p  pounds  paid  t  time  before  d«e>  at  r 


Tptr 
per  cent,  interest:  also  l-\-tr  ;  tr  ::  p  :  -^ — =2/tc  discount  aU 
^  ^     H-2r 

lowed  on  p  pounds,  at  the  said  rate,  and  for  the  said  time. 

Examples. — 1.  What  is  the  discount^  and  present  worth  of 
2502.  paid  2  years  and  75  days  before  it  falls  due,  at  5  per  cent. 
per  annum  simple  interest  ? 


■  In  Smart's  Tables  of  Interest,  there  is  inserted  a  table  of  discounts,  by 
wbich  tb«  diaooant  of  aoy  snm  of  money  may  be  calculated  with  ease  and 
cz|>edition. 


Part  IV.  DISCOUNT.  49 

Here  p==950^  r=s:.05,  <s=(«  y.75d=)  2.^548  years, 
^  ^  ^«50X8.80548X  .05^87^685  ^3,33^5^ 
l  +  tr       H-2.20548x.05       1.110274 
242.  ld«.  7d^=the  discount 

P  250  250 

1  +  *r      1  +  2.20548  X  .05      1.1 10274 
2252.  35.  A\d.:=the  present  worth, 

2.  Required  the  present  worthy  and  discount,  of  4872.  I2s. 
due  6  months  hence^  at  3  per  cent,  per  annum  ?  Ans.  pr.  worth 
480/.  7*.  lO^d.  disc.  7l  4*.  l^d. 

3.  Sold  goods  for  8752. 5s.  6d.  to  be  paid  for  5  months  hence } 
ivhat  are  the  present  worth  and  discount  at  44-  per  cent,  per 
annum  ?     Ans.  pr.  worth  8592.  Ss.  Z\d.  disc.  162.  2«.  2^d. 

4.  What  is  the  present  worth  of  1502.  payable  as  follows ;  viz. 
one  third  at  4  months^  one  third  at  8  months^  and  one  third  at 
12  months  ^  at  5  per  cent,  per  annum  discount  ? 

5.  How  much  present  money  can  I  have  for  a  note  of  352* 
15s.  8(2.  due  13  months  hence,  at  4^  per  cent,  per  annum 
discount  ? 

OF  RATIOS. 

24.  Ratio  •  is  the  relation  which  one  quantity  bears  to  another 
in  magnitude,  the  comparison  being  made  by  considering  how 
often  one  of  the  quantities  contains,  or  is  contimQed  in,  the 
other. 

Thus,  if  l^  be  compared  with  3,  we  observe  that  it  has  a 

certain  relative  magnitude  with  respect  to  3,  it  is  4  times  as  great 

%  as  S,  or  contains  3/otfr  times;  but  in  comparing  it  with  6,  ire 

discover  that  it  has  a  different  relative  magnitude  with  respect  to 

6,  for  it  contains  6  but  twice. 


■  Ratio  is  a  Latin  word  implying  comparison. 

The  stodent  must  be  carefiil  not  to  oonfoond  the  idea  of  ratio  with  that  of 
proportion,  as  some  thronf^  inattention  have  done :  he  mnst  bear  in  mind, 
that  ratio  is  simplj  the  C9mp€iri8on  ^  one  quantity  to  another,  both  being 
quantities  of  the  san^e  kind ;  whereas  proportion  is  the  equality  of  two  ratios  : 
the  former  requires  two  qaaotittes  of  the  same  kind  to  express  it,  the  latter 
requires  at  least  three  quantities,  which  must  be  all  of  the  same  kind  ;  or  four 
quantities,  whereof  the  two  first  must  be  of  a  kind,  and  the  two  last  likewise  of 
a  kind.    See  the  note  on  Art:  53,  and  the  note  on  Art.  137'  Part  I .  Vol.  1. 

VOL.  11.  £ 


50  AX^BBRA.  Paxt  !▼. 

25.  The  ratio  of  iw6  quantifies  il  usuaDf  expre»«d  by  inter- 
podng  two  dots^  placed  vertically^  betw^n  them. 

Thus  the  rating  of  a  i^h^  cuhI  o^  5  ^d  4,  ore  vntxtieni  a  :  h, 
and  5  :  4. 

96.  The  former  quiuntity  is  cdkd  the  mUeedentj  and  tiiie  kit* 
ter  the  consequent. 

Thus  in  the  above  ratios,  a  and  5  are  the  antecedents,  and  h 
and  4  the  consequents. 

The  antecedent  and  consequent  are  Called  terms  of  the  ratio. 

37*  To  determine  what  multiple^^  part^  or  parts  the  antece* 
dent  is  of  the  consequent,  (that  is»  to  find  how  often  it  eontains 
or  is  contained  in  the  consequent,)  the  former  must  be  divided 
by  the  latter  j  and  this  division  is  expressed  by  placing  the  con- 
sequent below  the  antecedent  like  a  fraction. 

Thus  the  ratio  of  a  to  h,  or  a  i  h,  is  likewise  prcfAr^  ex- 
pressed thus  — ,  and  5  :  4  fhus  -~. 
o  4 

28.  Hence^  two  ratios  are  equal,  when  the  antecedent  of  the 
first  ratio  is  the  same  multiple,  part>  or  parts  of  its  consequent, 
that  the  antecedent  of  the  other  ratio  is  df  its  consequent  5  or  in 
other  words,  when  the  fttM^iofi  made  by  the  terms  of  the  former 
ratio  (Art.  27«)  is  equal  to  the  iractioa  made  by  the  tenm  of 
the  latter. 

Thus  the  ratio  of  6  :S  is  equal  to  the  ratio  of  3  :  4>/^ 

6 3^ 

8  ~4' 

29.  Hence,  if  both  terms  of  any  ratio  be  ttiultifdi^  or  di** 

tided  by  the  same  quantity,  the  ratio  h  Hot  alfefed. 

3 
Thus  if  the  terms  ofS:4or  —  be  both  multiplied  by  any 

number,  suppose  6,  the  result = — ,  which  fractv&n  is  e©i* 


dmtly  equal  to  the  givt/n  frattitm  4  5  that  u^t  x  4  k  tke 

as  18  **  24  >  in  like  manner ^  if  the  terms  of  the  taiia  a  i  b,  or 

xp  be  both  multipUed  by  my  qtiantzty  n,  the  resuhing  fatio  an  t 
b 

bn,  or  -j^  is  the  same  as  a  :  b,  or  -fr- ;  and  the  same  in  general, 
on  b 


FitfrlV.  lUflOK  61 


30.  Ilettee>  oaei  rMo  i§  g^^after  than  another,  when  the  tinte- 
cedept  of  the  fbriOQer  ratio  is  a  greater  multiple)  part>  or  parts 
ii$  its-eonseqiient^  thati  the  antecedent  of  the  latter  ratio  is  of 
its  consequent ;  or>  when  the  fraction  constituted  by  the  teroit 
mi  the  fin»t  rtttto>  is  ^reatcnr  thsin  that  conttitnted  by  the  termi 
of  the  latter. 

J!^  tf :  S  »  greater  than  8  :  4^  for  6  contains^  thrte  Hme$, 

p  ft 

whereas  8  contains  4  but  twice,  or  ---  U  greater  than  — . 

31.  Having  two  or  more  ratios  given^  to  determine  which  is  the 
greater.  ^ 

Rule.  Having  expressed  the  given  ratios  in  the  form  of  frac- 
tions, (Art.  S7-)  reduce  these  fractions  to  other  equivalent  ones 
having  a  common  denominator,  (Vol.  1.  P.  1.  Art.  180.)  The  lat- 
ter will  expr^  the  given  ratios  h^mag  a  common  c6nse(pieiiti 
wherefore  the  numerators  will  express  the  relative  magnitudes 
6t  the  ratios  respectively. 

£xAMFi.£S.-*-l.  ^Vhich  is  the  greater  ratio,  7  :  4^  or  8  :  5  ? 

7         d 
These  raths  expressed  «» fortn  qf  fra^ti^n^i^ar^  —  ^  ---^ 

'  "         '4  5 

whence  7x5=85,  and  8x4=32,  these  are  the  new  numerators; 

tik^4xB=:M,^  common  denominator. 

««      /.       7     35    '      8      32 

I««r^arc —•=—;„  and -—=--- 3  and  the  former  q£  these 
4'     2Cr         5     20  ^        .    ^ 

heing  the  greater,  shews  that  the  raiiq  of  7  i4,is  greater  thim  ihe 
roHo  ofSiB. 

.12.  Whkb  is  tiie  greaier  rMicb  tteat  of  6:  Iti  of  ^t  df 
23:fe> 

8  2^    '"' 

These  raH^  epDpreseed  Wee  fraetiens^  are  --  and  — ,  wfiich 

reduced  to  other  equivalent  fractions  with  a  common  denofninator, 

256  253 

become  ~~,  and  --3  retpeetwely  s  ^^  former^of  these  being  ^  the 

greater,  shews  thatidJie  ratuy  S :  11,  is  greater  than  the  ratio 
23:32.  . 

3.  Which  is  greatest/  th/^  »(ip  of  18 :  25>  or  that  of  19  :  27  ^ 
4n»s  the  format, 

4.  Whiehi  is  \h»  greatest,  and  whieh  «h^  least,  of  the  ration 
9 :  10,  37 :  41,  and  75  :  83 1 


59  ALGEBBA.  Pabt  IV. 

39.  When  the  antecedent  of  a  ratio  is  greaiter  than  its  poase- 
quent,  the  ratio  is  called  a  ratio  of  thegreaUr  inequaliUf. 

Thus  b  :  3,  II  :  7,  and  2  :  I,  are  ratios  of  the  greater  m^ 
equaUty. 

33  \  When  the  antecedent  is  less  than  its  consequent,  tkt 
ratio  Is  called  a  ratio  of  the  lesser  inequality. 

Thus  3  :  5,  7 :  11>  ^d  I  :%  are  ratios  of  the  lesser  in* 
equality. 

34.  And  when  the  antecedent  is  equal  to  its  consequent,  the 
ratio  is  called  a  ratio  of  equality, 

'  Thus  5  :  5,  1  :  1,  and  a  :  a,  are  ratios  of  equality, 

35.  A  ratio  of  the  greater  inequality  is  diminished  by  adding 
a  common  quantity  to  both  its  terms. 

ThuSi  if  I  be  added  to  both  terms  of  the  ratio  5:3,  it  6e- 

5      90        ,6      18     ,    ,  ^     .....  . 

comes  6:4}  out  -—=:—,  and  — =--,  the  latter  of  which  {hemg 

the  ratio  arising  from  the  addition  of  1  to  the  terms  of  the  given 

ratio)  is  the  least,  and  therefore  the  given  ratio  is  diminished  : 

and  in  general,  if  x  be  added  to  both  terms  of  the  ratio  3:9,  it 

3         •   •    3+JC 
^comef  3+« :  9-f-r,  that  is  -—■  becomes ;  these  fractions  re* 

ducti  to  m  common  denominator  a$  before,  become  ^±^  and 

,  4+2x 


^    ■■  I        tH       II ».,»■. 


^  Wketk  tbe  aotecedent  is  a  mitltiple  of  its  coiMeqnent,  the  ratio  \g  named  a 
multiple  ratio ;  but  when  the  antecedent  is  an  aliqnot  part  of  its  conse^nenty 
tiie  ffalia  is  naned  a  tubmuiUple  ntio.  U  tha  antecedent  aoataias  the 
aonseqnent  ^ 

twice,  as-         V2 :  61  fdnple,        1 

thrice,  asF         12  :'4  >it  is  eaU^  a|:trit>le»>        }k  ration 
fbnr  times,  as  13  :  3  J  (^  quadruple,  J 

&c.  &C. 

U  the  antecedent  be  contained  in^  tbe  consequent 

twice,  as-         S-i  \9\  Tsubdapl^        T 

thrice, as         4  :  13  >it  is  called  a^  subtriple,         ^ration- 
four  times^  as  3  :  12  J  (,sabqnadn\ple,cj 
Sec.  &c. 

There  is  a  great  variety  of  denominations  applied  to  different  ratios  by  tfie 

early  writers,  whUsb  is  Mcessary  to  be^  nUdei^tood  by  those  who  read  the  works 

either  of  the  ancient  mathematicians,  or  of  their  commentators,  and  nmy  ba- 

seen  in  Chambers'  and  Hatton*»  Dictionary :  at  present  it  ia  uioal  ta  nalne 

ratios  by  tbe  least  numbers  that  will  express  thea». 


Part  IV.  RATIOS.  63 

■      ■  •  respectively ;  and  since  the  latter  is  evidentlif  the  least,  it 

^   r*  *  Sf 

follows  that  the  given  ratio  is  Mmintshed  by  the  addition  of  sif  to 
'  each  of  its  terms. 

36.  A  ratio  of  the  lesser  inequality  is  increased  by  the  addt- 
•tion  of  a  common  quantit}i  to  each  of  its  terms. 

Thus  if  I  he  added  to  both  termsofthe  ratio  3  it},  it  becomes 

4 : 6,  but  -7-=rr>  and  —•=—-,  the  latter  of  which  being  the 
6      30  6      30  *^ 

greater,  shews  that  the  given  ratio  is  increased :  in  general,  let 

2 :  3  have  any  quantity  x  added  to  both  its  terms,  then  the  ratio 

becomes  2-|-x :  S+x,  that  is  —  becomes :  these  reduced  to  a 

3  3+x' 

6+2a?  64-3jc 

common  denominator,  become  ,and ,  of  which  the 

9+3  Jp  2+3  a?'  -^ 

latter  being  the  greater,  it  shews  that  the  given  ratio  is'  increased^ 
37*  Hence,  a  ratio  of  the  greater  inequality  is  increased  by 
taking  fr«m  each  of  its  terms  a  common  quantity  less  than 
either.  • 

Thus  by  takvi^  1  from  the  terms  of  4:3,  it  beeomes  3 : 2, 

A  Q  ^  O 

but  — — ~,  and  -—=—-,  the  latter  being  the  greater,  shews  that 

the  given  ratio  is  increased, 

38.  And  a  ratio  of  the  lesser  inequaUty  is  diminished  by  tak* 
iDg  from  each  of  its  lerms  a  common  quantity  less  than  either. 

Thus  by  taking  2  from  the  terms  of  3  1 4,  it  becomes  I :  % 

hut  —=---,  and  ---=---,  the  latter  being  the  leasts  shews  that  the 

given  roUio  is  dimunshed, 

39.  Hence,  a  ratio  of  equality  is  not  altered  by  adding  tOi  or 
subtracting  from,  both  its  terms  any  common  quantity. 

40.  If  the  terms  of  one  ratio  be  multiplied  by  the  terms  of 
another  respectively,  namely  antecedent  by  antecedent,  and  con- 
sequent by  consequent,  the  products  will  constitute  a  new  ratio, 
which  is  said  to  be  compounded  of  the  two  fonnerj  this  compo- 
sition is  sometimes  called  addition  of  ratios. 

Thus,  if  the  ratio  3i4  be  compounded  with  the  ratio  2  :  3, 
the  resulting  ratio  (3x2:4x3,  or)  6  :  12  is  the  ratio  com^ 
pounded  of  the  two  given  ratios  3  :  4  and  2  : 3,  or  the  sum  of  the 
ratios  3  ;  4  and  2  ;  3. 

e3 


54  AWEBHA.  Tmt  it. 

41.  If  the  ratio aihhe compounded  with  itself^  the  refultlog 
ratio  a^ib^w  the  ratio  of  the  squares  of  a  and  b,  and  is  said  to 
)^  double  the  i^^tio  a :  bf  and  the  mtio  a :  6  is  mi  to  be  ha^  the 
ratio  cfiib^;  in  like  manner  the  ratio  a'  :  6^  is  spill  to.be  triple 
.the  ratio  a :  b,  and  a :  6  one  third  the  ratio  a^  :V}  also  «*  :  £i*  is 
said  to  be  n  ^tmef  the  ratio  of  a :  (^  ^d  ai :  bi  om  ii*^of  ihe  ra- 
tio o£  a:b, 

41. B.  Let  a :  1  be  a  given  ratio^  then  ^r,  l,a^  :l,  €^  :  I, 
a^:l,  are  twice,  thrice,  four  times,  n  times  the  giv^n  r^tio,  where  n 
shews  what  multiple  or  pail  of  the  ratio  |t" :  1  ihe  |^en  xdti» 
<i :  1  is ;  hence  tbie  indices  I,  2^  3, 4« , . . «,  are  caUed  ik»  mea* 
sures  of  the  ratios  of  a,  a^,  a^,  o^,  ...  a*  to  1  r^p^ctive]y>  of 
the  logarithms  of  the  quantities  a,  o^  c^,  aS  •  •  .  a** 

49.  If  there  be  several  ratios,  so  that  the  consequent  of  thf 
first  ratio  be  the  antecedent  of  the  second  3  the  consequent  of  the 
second,  the  antecedent  of  the  third  $  the  consequent  of  the  third, 
the  antecedent  of  the  fourth,  &e.  then  wfll  the  ratio  compounded 
of  all  these  ratios,  be  that  of  the  fiiBt  antecedent  to  th%  last  con- 
sequent. 

For  letaih,b:€,e:4,die,'8fe»he  any  number  of  given  ra* 

tios ;  the$e  compounded  by  Art,  4Q.  pUl  be  Qixb:f^cxd:hxc^4 

dxbxcXd     a 

X  e,  or)  , —  = — ,oraie,  the  ratio  of  ihfifir^mtecedmt 

bv^cxdxe      e  ^     •  ^     .  -r 

a  to  the  last  consequent  e. 

4S.  Hence,  in  any  series  of  quantities  of  the  same  Und,  4ht 
tot  wfll  have  to  the  last,  the^ratio  compounded  of  the  ratios  of 
the  fim  to  the.second^  of  the  secoQd  to  the  thir4»  of  the  thJini 
to  the  fourth,  &c.  to  the  last  quantity. 

44.  If  two  ratios  of  the  greater  inequafity  be  tompoundei 
together,  each  ratio  is  increased. 

Thus,, let  4:S  be  compounded  with  B  i^,ihe resulting  ratio 

^  4         5 

(4  X  5  ;  3  X  2  or)  —-  w  greater  thga  either  -^ytr  -^,m  ^Vf^f^^kn 

reducing  thesefractions  to  a  common  denominator.  Art,  31. 

45.  If  two  ratios  of  the  lesser  inequality  be  compounded  to- 
gether, each  ratio  is  diminished. 

Thus,  let  S:  4  be  compounded  unth  2  :  5,  the  resulting  ratio 

(3  X 2  :  4  X  5  or)  — ,  w  less  than  either  of  the  givi^  r^w—  or 
— ,  as  appears  by  reducing  thes^  fractions  as  before* 


40.  If  a  ratfa)  of  the  greater  inequality  be  compounded  with 
a  ratio  of  the  leaa,  the  former  will  be  diminished^  and  the  latter 
increased. 

Thus,  let  4:3  be  compaunied  mth  ^:&,  the  r^nMng  r9tio 

3  4 

(4xS:3x5  4>r)  :^,ii  Usg  than  the  nUw--, but  greater  than  the 

15  S 

ratio  -r. 
6 

47.  From  the  composition  of  ratios,  the  method  of  their  de- 
coBoposition  evideatly  £d11owb;  for  since  ratios  may  be  repre- 
sented like  fractions,  and  the  sum  of  two  ratios  is  found  by  mul- 
tiplying these  fractions  representing  them  together,  it  is  plain 
that  in  order  to  take  one  ratio  from  another,  we  have  only  to 
divide  Ihe  fraetion  r^resenting  the  formerly  that  representhig 

o 

the  latter.    Hence^  if  the  ratio  of  (3:4  or)  —  be  compounded 

5 
with  the  ratio  of  (5 : 7  or)  —^  w^  obtaux  the  ratio  of  (15 :  28  or) 

—  i  DOW  if  'from'  this  raObfo^ve  decompound  the  fahatr  of  tl^e 

givea  nHoB,  naibely  ---,  the  tvsult  wfflbe  (— x  --as—a)  — ^, 

which  is  the  latter  of  the  given  ratios  -,  and  if  from  the  com- 

15  5 

poimded  ratio  — ,  we  decompound  the  latter  given  Mtfo  — ,  tine 

»8  7 

15      7      105        3 
result  will  be  (55X— =r-^=)-7=the  fbrmelr  given  ratio: 

wbeoQe  «to  subtiact  one  ratio  Cram  another,  thU  is  the  riile. 

Auxs.  I^t  the  ratios  be  represented  like  fractions.  (Art.  27.) 
Invert  the  termos  of  the  ratio  to  be  subtracted,  and  then  multi- 
*ply  the  correspondent  terms  of  both  fractions  tpgether ;  the  pro- 
duct reduced  to  its  lowest  terms  will  exhibit  the  remaining  ratio, 
or  that  which  heing  compounded  with  the  ratio  subtracted,  will 
give  the  ratio  fit)m  which  it  was  subtracted. 

ExAiifrFLBs. — 1.  S^rom  5  -.T^  let  9 :  8  be  subtracted. 

'5  9 

l%jgie  raiiot  reprsnented  like  fractifms^  are  —  and  — . 

7  " 

b4 


66  ALGEBRA.  Past  IT. 

8  5       8      40 

The  latter  inverted,  becomes  —  j  wherefore  —  ^  "5"=^*  ^ 

40 :  63,.  the  difference  required. 
9,  ¥tam  6 :  5^  decompound  7  '  10. 

Thug  —  X  — =--= — >  or  12 :  7»  <ft«  difference  required, 
o       7      oo       / 

3.  From  the  ratio  compounded  of  the  ratios  8  :  7i  3  :  4^  and 
5 :  9^  subtract  the  mtio  compounded  of  the  ratios  1:2,  8:3, 
9:7,  and  20:  21. 

Thus   _x-X-x-X-^X-^X-=.~=7:24,   the 

difference. 

4.  From  a :  b  decompound  x:y.    Ans,  asf :  bx. 

5.  From  11 :  12  dk»mpound  12  :  11.    Ans.  121 :  144. 

6.  From  3  :  4  take  3  :  4.     Ans.  1  :  1, 

7*  From  a :  x  take  3  a :  5 x,  and  from  ax :  y^  take  y  :9ax. 
S.  From  the  ratio  compounded  of  a :  b,  x :  z,  and  5 :  4,  take 
the  ratio  compounded  of  5  fr  :  x,  and  2  a  :  3  z. 

48.  If  the  terms  of  a  ratio  be  nearly  equaU  or  their  diffisrence 
when  compared  with  either  of  the  terms  very  small,  then  if  this 
difference  be  doubled,  the  result  win  express  double  the  given 
ratio ',  that  is>  the  ratio  of  the  squares  of  its  terms,  nearly. 

Let  the  given  ratio  be  a+x:a,  the  quantity  x  being  very 
sv^l  tshen  compared  with  a,  and  consequently  stiU  smaller  when 
compared  with  a+x;  then  wiU  (a+x]*,  or)  a*+2aae+x* :  a*  be 
ih^  ratio  qf  th^  squares  of  the  terms  a+x  and  a :  and  because  x  is 
small  when  compared  with  a,  xjs  (or  x^)  is  small  when  compctred 
with  ^a.x,  and  much  smaller  than  a.a;^  wher^ore  if  on  aecoumt 
of  the  ejpceeding  smallness  of  ofi,  compared  with  the  other  quantities, 
it  be  rejected^  then  {insteqd  of  a* +2  ax + a?* :  a*)  we  shall  haoe  a» 
4-2  ox  :  a^ ;  that  is,  {by  dividing  the  whole  by  a)  a+2x :  a,  for  the 
ratio  of  the  squares  of  a+x :  a,  which  was  to  be  shewn. 

.Examples. — 1.  Re(juired  the  ratio  of  the  square  of  19  to  the 
square  of  20  ? 

Here  a=s  19,  x=  1,  and  ■  ■     =g^,  ther^ore  by  the  preceding 

a  "y*  X       ^  V 

a         19 
article, — ^  ==2p5  ^^^  •*>  *^  **^**^  ^f  ^^e  square  of  19  to  the 
a  -^  2x    '2 1 


*ART  IV.  RATIOS.  6T 

/■o/^     ,..    «,  ,       T,     1^'     ,361     ,7681        .19 

square  of  20  is  19  :  21,  nearly.    For  —  =( —  =^)' ,  and  -- 

•^  *  «0«      MOO     '8400  «1 

7600  19 

^^AAivk'  ^^^^^^^^^^y  the  ratio  —  is  somewhat  too  great,  but  it 

19 

exceeds  the  truth  by  only ;  which  is  inconsiderable. 

^       ^  8400 

2.  Let  the  ratio  of  8o|* :  79l*  be  required? 

„  a-f-x     80  «4-2x 

Here  as=79,  jps=l,  ctmsequently =z:-,  ond  — r— = 

a       79  79 

81 

--,  or  81  :  79=*^  ratio  of  86l«  :  79lS  nearly. 

_     80»     ,6400    .505600         ^  81     505521      ^.  ^  ,, 

For -—=(--— -=) — -— --,   and  — 5= •  lomc/i  t/iere- 

79*  ^6241  ^493039      79  493039 

79 

fore  differs  from  the  truth  by  only 


493039' 

3.  Let  the  ratio  loS* :  ill)*  be  required  ?     jins.  ^. 

4.  Required  the  ratio  iooil « :  1000 1«  ?    Ans.  — . 

6.  What  are  the  ratios  3009)*  :  3oIo]S  and  lOOOOl*  :  100051*? 

49.  Hence  it  appears^  that  in  a  ratio  of  the  greater  inequality, 
the  above  proposed  ratio  of  the  squares  is  somewhat  too  small  ^ 
but  in  a  ratio  of  the  less  inequality,  it  is  too  great. 

50.  Hence  also^  because  the  ratio  of  the  square  root  of  a+ 
2x:al8  a-^x  :a  nearly^  it  follows  that  if  the  difierence  of  two 
quantities  be  small  with  respect  to  either  of  them,  the  ratio  of 
their  square  roots  is  obtained  very  nearly  by  halving  the  said 
di£Perence. 

Examples. — 1.  Given  the  ratio  120 :  122>  required  the  ratio  • 
1201t:122]x? 

^    120         a  a        120 

Here  a=120,  2  j:=2,  755=  -t^*  •'  —rz=7^>  ^^  130 : 

122     a+2a?        a-^-x     121 

121  ss  the  ratio  of  ISSIt  :  I22I  i,  nearly. 

2.  Given  the  ratio  10014: 10013,  to  find  the  ratio  of  their 
square  roots  ?     Ans.  20027  :  20026. 

4.  Given  9990 :  9996  and  10000 :  10000.5,  to  find  the  ratios 
of  their  square  roots  respectively  ? 


J»  AUmnUL  Paw  IV. 

51.  Bjr  fifaiiiUar  nasonb^  it  may  be  shewn^  that  the  ratio  of 
(Hie  cubes,  pf  the  fimrth  powers^  of  the  nth  powers,  is  obtained 
\jiy  taking  3^  4,  n  tiroes  the  difbrence  respectivety,  provided  S, 
4j  or  n  times  the  difference  is  afxtaSl  with  respect  to  either  of  the 
terms.  And  likewise,  that  the  ratioof  the  3rd,  4th,  or  nth  roots 
are  obtained  nearly  by  taking  ^,  -^^  i  part  of  the  difference 
respectively. 

S%.  When  the  terms  of  a  ratio  are  large  numbers,  and  prime 

to  eadi  otlier,  a  ratio  may  be  found  in  smelter  numben  nearly 

equivalent  to  the  former,  by  means  of  what  are  called  continuied 

firactions  <. 

h 
Thug,  let  ^git7€n  raiiQ  he  esftetrnd  bf  — ,  cmd  let  b  contain 

a,  c  times,  with   a  remainder 

d;  let  a  contain  d,  e  times,  with        a)  6  (c 

a  remainder/;  again,  iet  d  eon"  d)  a  (e 

tain  f,  g  times,  with  a  remainder  f)  ^  (JS 

h,  and  so  on ;  then  by  multiplying  h)  f  (k 

each  divisor  by  its  quotient,  and  I)  h  (m 

adding  the  remainder  to  the  pro*  n)JJp 

■duct,  there  arises  f,  Ac 

b=ac+d, 

a^de^f, 

d-fg^  h, 

h^s^lm-^n, 

l^np+q,  BfC. 

b      jac^d    \        d    , 
Hence  the  given  fraction  — ac(  ■     ■  =)  c-\ — ,  but  aszde-k- 
•        "^  a     ^    a  a 

fi  thU  value  substituUd  for  a  in  the  preceding  equation,  ise 
shall  have  — =(c-f-r — ■=)  c+ ji  but  since  d^szfg-^h,  by 

a 

substituting  this  value  for  d  in  the  preceding  equation,  we  shall 

■  ■■  ■'  ■  llllHllll  ||  ,1  III      IWI    »—— 1M  I        If 

,  «  Th0  acMiod  ef  4ii4iQ^  tbe  appronmate  vahit  of  a  ratio  in  small  munben, 
has  been  treated  of  bj  Dr.  Walltg,iD  his  TteaHse  tf  Jlgebra,  c  10, 11.  and 
in  a  tract  at  tbe  end  of  Horrox's  Works ;  hj  Huygens,  in  Descript,  Autom, 
Planet.  Op.  ReUq,  p,  174^  1. 1 ;  by  Mr.  Cotes  in  his  ffarmonia  JUensiSramm, 
.and  by  several  others. 


have  — =<c+  — i-y- <=)  c^-    .  ^      j  &ft<  oMte/sM+l,  4y 

«if&^£iftf  ^ifl^  /^t5  {Vo/Ke  /or  f  m  the  precedii^  equation,  we  thaU 

,       ^  1  1 

have-'=z(c+ =).c+ 5    6tt*  As|m+n> 

^^       J.  ■  ■■■  g-^  '■     '■  ■■' 

A  1 


therefore  ly  substituting  as  Iff  are,  — =(cH j ss) 

.  '      ^  l^-H — ' r^ 


gi- 


m»  i»  n 


*+     ' 


1  -      * 

cH ' r ,•>  5««  Z=«p+g,  therefore 

e+ : 


^+ 


*+-^    ■ 


n 


/ 


tc  + J =)  C4--r- 

c+    ■  j    t   ■■  ■  e-{- 

fir+ 5 «+ 


ft+ — i.^  *+- 


"^  ,,1  J ^^  Sfc,  a  continued  fraction, 

P 

Now  in  this  continued  fraction,  if  one  term  onhg  (viz.  c  or  y)6« 

h 

taken,  it  wiU  be  an  approximation  to  the  ratio  —  in  small  numr 
heri:  if  twe^fns,viz.c^^i:=^~')  be  taken,  it  wUl  be  a  ^  ^  ^ 


1 


40  ALGEBRA.  Part  IV. 

.« 

nearer  approjAmaium  than  the  farmer,  to  the  ratio  — ;  but  neces- 
sarily expressed  hy  a  greater  number  of  figures:  if  three  terms  be 

taken,  viz.   c+i      1  =(c-h -4.=c+^=)  S?!±£±£,a 

c-f—    ^     gg-H  ge+l  g«+l 

^  g 

nearer  approxitnation  to  the  ratio  —  expressed  by  stUl  more  figures; 

if  four  terms  be  taken  in,  we  shall  have  c-) 1         = 


*+T 

(c+f        1     ^c^^l     k    = 
k 

1 

CI     ^^-^'^ 

'  egk^e+k 
gk+l 

-^^egk+e+k   ' 

"^            egk-^e-^k 

• 

1 

ExAMFLSB.— 1.  Required  a  aeries  of  ratios  in  smaller  num- 
bers, continually  approximating  to  the  ratio  of  12345  to  67891  ? 

12345) 67891 (5 
61725 

6166)12345(2 
12332 

13)6166(474 
52 

"96 
91 

56 
52 

4)  13  (3 
12 

1 

Here  6=67891,  a=  12345,  c=5,  d:=:6\66,  «=2, /=rl3, 
g=:474,  ^=4,  /c=3,  /=1. 

Then  ---=-—,  an  approximation  to  the  given  ratio,  in  the  least 

whole  numbers  possible. 

^        „     cc+1      5x24-1     vll  .     ,. 

J       Secondly,  »( ^/"o"'  ^  ^^(^^^r  approximation. 

e  %  % 


Pav  IV.  BATIOS.  m 

-PL*    i;     ^«+<^+^     ,5x474x2  +  5+474     ,5919 

Thirdly,  — ~^( tzz — :: — : =) >  ^ 

^'     ge+l        ^         474x2+1  '  949 

nearer  approximation  than  the  former, 

cejr*+ce+r*+|rAf+l 


Fourthly, 


«f^+e+ip 


6x2x474x3+5x2+6x3+474x3  +  1     ^  15668 

5— ( 2s)   ,  a  still 

^  2x474x3+2+3  '  2849 

nearer  approximation  than  the  last, 

2 .  Required  approximate  values  for  the  ratio  763 1 7 1 ;  3 101000 
in  more  convenient  numbers  ? 

Operation. 

753171)  3101000<4 

3012684        • 

88316)753171(8 
706528 

46643)  88316  (I 
46643 

41673)  46643  (1 
41673 

4970)41673(8 
39760 

1913  *c. 

Here  0=753171,  5=3101000,  c=4,  d=88316,  6=8,/=; 
46643,  g=l,  A=4ie73,  kszl,  1=^4970,  ot=8,  i»s1913. 

e       4 

Therefore  — =— ,  thefint  approximation, 

ce+l      4x8+1     ,33   ^,  ,  ,     ^, 

s:( — -de)  — ,  the  $econd  <qtpfoxtmaium, 

e  ■  o  8 

cge-^C'\'g     ,4x1X8+4+1     .37  ,.    ,,.   , 

— : —  =  ( =x)  — ,  the  third  approxi-. 

ge^l        ^       1x8+1  ^9  if 

motion. 

C'6'gAf  +  (?€+ c/f-^grAf  + 1 
€gk'\-e-i-k 

4x8x1x1  +  4x8  +  4x1  +  1x1  +  1      V   70    ^.      .        . 

-s(- ■ — . =3)  •-—,  the  fourth  ap- 

^  8x1x1+8+1  '   if        ^  ^ 

proximaiion,  dsc  jjrc 

3.  The  ratio  of  the  diameter  of  a  circle  to  its  circumference 
is  nearly  as  1000000000  to  3141692653 }  required  approximating 
vjihies  of  tbw  ratio  in  smaller  numbers } 


5  ®  383 

Ah8.   TlmfirH  —,  <Ae  «ecoft<l  — ,  the  third  7-^,  **«  fourth 

1  .  7  '"^ 

355    . 

m '  *'• 

4.  Required  approximate  expr^^Oflff  in  small  numbers  for 
the  ratio  78539811635 :  10CX)O00000O,  being  tbxt  df  the  area  of 
a  circle^  to  the  square  of  its  diameter,  neady  ? 

^       1     3     4     7    11    17«  355    „    . 

^         1'  4'  6'  9'  14'«19'452' 

5.  IF  the  side  of  a  square  be  1234000,  its  diagonal  will  be 
1745139,  nearly ;  required  approximatioDs  to  this  ratio  in  smaller 
numbers  ? 

OF  PROPORTION  \ 

53.  Four  quantities  are  said  to  be  proportionals,  when  the 
first  has  to  the  second  the  same  rtttio  which  the  third  has  to  the 
fourth;  that  is,  when  the  first  is  the  same  multiple,  part,  oc 
parts  of  the  second  that  the  third  is  of  the  fourth. 

'  Ratio  is  the  comparison  of  magnitudes  or  quantities ;  proportion  is  the 
equality  of  ratios ;  hence  there  mast  be  two  ratios  to  constitute  that  equality 
which  is  called  proportion ;  that  is,  there  must  be  three  terms  at  least  to 
expresf  the  two  ratios  necessary  to  a  comparison.  Some  authors  have,  with 
the  most  unaeeounlable  nejfligeaee,  eeafonnded  and  perplexed  t)i«ir  inexpe** 
rienced  readers  with  the  definitions  they  liave  given  of  ratio  and  proportitm. 
Dr.  Hntton;  to  whose  useftil  labours  almosteriery  branch  of  the  mathematics 
is  indebted  for  elucidation  or  improvement,  in  his  system  of  Elementary 
Mathematics  for  the  use  of  the  Boyal  Military  Academy,  thus  defines  them : 
**  Ratio  is  the  proportion  which  one  magnitude  bears  to  another  magnitude  of 
the  same  kind,  with  respect  to  quantity ;"  and  immediately  after,  **  Proportion 
is  the  epuilUy  of  rattog"  Now  it  has  always  been  held  «s  a  necessary 
maxim  in  logic,  that  <*  in  every  definition  the  ideas  implied  by  the  tenna  oi 
the  definition,  should  be  more  obvious  to  the  mind  than  the  idea  of  the  thing 
defined/*  otherwise  the  definition  fails  of  its- purpose ;  it  leaves  us  just  as  wise 
as  it  found  us.  Wherefore,  supposing  the  above  definitions  of  ratio  and 
proportion  to  be  adequate  and  perspicuous,  as  they  ought  to  be,  if  we  appfy 
this  doctrine  to  them^  it  will  follow  from  the  fonb^r,  that  the  idea  of  proportion 
is  more  obvious  than  that  of  ratio ;  and  from  the  latter^  that  theiden^nf  laitip 
is  more  obvious  than  that  of  proportion ;  but  the  supposition  that  both  these 
conclusions  are  tttie,  implies  a  idanifest  absurdity,- and  consequently,  that  one 
or  both  of  these  definitioDs  must  be  fimlty.  It  iB  but  jastioe  to  suppose,  tiiat^ 
the  learned  Doctor  must  have  used  the  tenn  frvjpoHwn^  in  the  foriaier  ditff^iitiiMi^ 


64.  'ttis  prdpoftiOd>ar  equalitjof  ratkiB,  Is  taBuftfly  eiipi^ssed 
by  four  dots,  thus : :  interposed  between  the  tiro-iMios. 

Thus,  d:b::c:d,  shews  that  a  has  to  h  the  same  ratio  that 
c  has  to  d,  or  that  the  four  quantities,  a,  b,  c,  and  d,  are  propor* 
tionals,  and  are  usually  readj  a  is  to  b,  as  c  to  d. 

55.  Tht  first  and  last  terms  of  the  proportion  (viz.  a  and  d) 
are  called  the  extremes,  and  the  two  middle  terms  (6  and  c)  the 
means. 

56.  Sinte  it  has  been  shewn^  (Art.  97.)  that  any  ratio  is  truly 

taqirened  by  piwii^  its  terms  in  tlie  form  of  a  f^ntioD ;  therein 

fare,  when  four  quantities  are  propostionakir  that  is,  whto  tte 

first  has  to  the  second  the  same  ratio  whdck  the  third  has  to  tko 

fourth,  it  follows,  that  the  firaction  constituted  by  the  terms  of 

the  first  ratio,  will  be  equal  to»the  fraction  constituted  by  the 

terms  of  the  other  ratio  placed  in  the  same  order. 

a      c  b       d 

Thus,  if  ai  b::c:d,  then  will  -p-=---,  or  — = — . 

h       di         a      c 

57.  If  fow  fuaiiities  are  proportioiials>  the  priKiiiot  of  tka 

extremes  isieN|Ml  t^  the  product  of  the  means. 

a      e 
Let  a  \\  lie  id,  then  by  the  preceding  article,  -t-=s-t;  muU 

0       d 

•  ^  c 

tiply  the  terms  of  this  equation  by  bd,  and  (-r-  x  bdsz—  x  bd,  or) 

tk  d 

ad=zb€.    Euclid  16,6. 

58.  Hence,  if  three  quantities  are  proportionals,  the  product 

of  the  extremes  is  equal  to  the  square  of  the  mean. 

a      c 
Let  a:c:ic;dj  then  — s=-y,  by  what  has  been  shewns  mul" 

e       a 

a  c 

tiply  botk  sides  by  cd,  and  ( — xc<f=a— xcfll,  or)   ad:=z(^. 

Eudid  17>  6. 


■coonting  to  its  vuigar  acceptation,  (natetly,  the  oo«ywao>«f  oat  thii^  wiU» 
aaother,)  and  in  the  latter,  according  to  its  mathematical  import.  The 
leaxncv  «n§^  to  bo  eautioaed  to  ttndy  not  to  be  hnpoeed  on  by  tbe  double 
meaning  of  words,  and  especially  to  scorn  the  mean  artifice  of  araiiing  himself 
on  any  occasion  of  the  aJhbi^^ify  of  language.  A  wrangler  may  confound 
bis  opponent  by  using  the  siUne  word  in  two  or  three  different  senses ;  btit 
triitii  (whkfi  is-  tfate  gtttnd  object  of  science)  is  discovered  only  when  our 
rt^ttoniog  inidceeds  by  meafks  of  t^rms  which  are  strictly  limited  in  their 
signification. 


64  AXiGSRRA.  PaUt  IV. 

59.  Hence,  if  three  temis  of  any  proportion  be  given>  the 
fourth  may  be  found : 

Fo'^  since  I    z=hc,  if  a,  d,  and  h,  are  given,  then  ---zrzc;  if  a, 

o 

d,  and  c,  are  given,  -—  =& ;  if  a,  b,  and  c,  are  given,  — =d ^  and 

be  ■ 

if  d,  b,  and  c,  are  given,  then  -r=^  '• 

60.  if  the  product  of  two  quantities  be  equal  to  the  product 

of  two  others,  then  if  tlie  terms  of  one  product  be  made  the 

means,  and  the  terms  of  the  other  product  the  extremes,  the 

four  quantities  will  be  proportionals. 

ad     be 
Thus,  if  ad=bc,  divide  both  sides  by  bd,  and  (-rz^-n*  ^^) 
•^  •  ^  bd     bd 

a       c 

--=  --,  that  i8,a:b:;cid.     Euclid  If,  6. 
o       a 

61.  If  the  first  term  be  to  the  second,  as  the  third  to  the 
fourth,  and  the  third  to  the  fourth  as  the  fifth  to  the  sixth,  then 
will  the  first  be  to  the  second  as  the  fifth  to  the  sixth. 

Let  a:b::  c:  d,  and  c:  dii  e  if,  then  will  aibi:  e  :f;  for 
<^      c         .  c       e     ^      .    a      e      .      ,         ,  . 

b       d  d      f  of 

69.  Hence,  if  the  same  ratio  subsists  between  every  two  ad- 
jacent terms  of  any  rank  of  quantities,  that  id,  if  the  terms  are 
in  continued  proportion,  the  first  term  will  be  to  the  second  as 
the  last  but  one  to  the  last. 

Or  b 

For,  let  a,  b,  c,  d,  e,f,  g,  h,  k,  I,  Ssc.  be  such,  then  '-r-= — 

6       c 

c       d      e      f      g      h      k  a      k 

d      e      f      g       h       k       I  b       I 

b::  k:l. 

63.  If  four  quantities  are  proportionals,  they  are  also  pro- 
portionals when  taken  inversely. 


«  This  article  furnishes  a  demonstration  of  the  Rale  of  Three,  except  tint 
part  of  it  which  respects  the  reducing  of  the  terms :  but  the  latter  is  obvioas  ; 
since  in  order  to  compare  quantities,  it  is  plain  we  must  bring  them  to  a  sim- 
ple form,  and  likewise  the  quantities  compared  must  be  of  the  saqie  deaomi- 
na^ion,  otherwise  a  comparison  cannot  be  made. 


Paet  IV.  FftOFORTlON.  «} 

a      r 

Let  a:b::c:d,  then  will  b:a::d:c;  for  since  ---sr-j-,  let 

0        a 

unity  be  divided  by  each  of  these  equal  fractions,  and  the  qMotients 

(1-+— r-=)  — ,  and  (l-H-v=)  —  ^ill  be  equal,  wherefore  biaxi 
o        a  a   .      c 

d:c;  this  operation  and  property  is  usually  cited  under  the  name 

iNVBRTfiNDo.    Euclid  pr.  B.  Book  5. 

64.  If  four  quantities  be  proportionals^  they  are  also  propor- 

txDnals  when 'taken  alternately. 

^     a       c 

Let  a:  b::c:  d,  then  will  aicwbid;  for  ----r=---,  where- 

0      a 

fore  multiplying  each  of  tkbse  equals  by  — ,  we  have  (—-  x  — =» 

c       b  a      b      . 

— r X  — ,  or)  — ss-r>  Ihat  is,  a:€::b:d;  this  is  named  altbe- 
d       e  c       d 

NANDO,  or  PERMUTANDO.      Euclld    16,  5. 

$5.  If  four  quantities  be  proportionals,  the  sum  of  the  first 

and  second  is  to  the  second^  as  the  sum  of  th«  third  and  fourth 

to  the  fourth. 

a 
Let  a:b::cid,  then  will n+b:  b::c-^d:d.    Because  —  = 

b 

-—,  let  unity  be  added  to  each,  and  (--+ ls=--;-f  I,  that  is)  — r— 
a  b  d  o 

=         ,  wherefore  a-|-&:  6  ::c-f  d:  d;  this  is  named  comfo- 

NBNDO.    Euclid  1^,5. 

66.  In  like  manner,  the  first  is  to  the  sum  of  the  first  and 

second,  as  the  third  to  the  sum  of  the  third  and  fourth. 

_  a-^b    c+d  ,  ,6  d       ,     ,^  .  >.^v 

For  since  -^r^  = — -~,  mvertendo r= >  also  (Art,  62.) 

b  d  a+b     c-^a         ^ 

•  .    ,          J      ^       .    b           ,        d        ,          ^  abd        bed 
adh=zbc:  wherefore  ( ^xad^- ,xbc,  on ^-2=- — -^  m- 

a  c 

vide  these  eouaU  by  bd,  and r=: -,  or  ar  a-f  6 : :  c :  c4-  d. 

a^b  .  c-^d 

67.  If  four  quantities  be  proportionals,  (he  excess  of  the  first 
above  the  second  is  to  the  second,  as  the  excess  of  the  third 
above  the  fourth  is  to  the  fourtli. 

a 
Let  a:b::c:d,  then  v?v[l  a-^b ;  h  r :  c— d :  d.    Because  —ss 

VOL.  IX.  F 


M  ALGSfiRA.  T^n  If^ 

C  M  C 

-—,  let  unity  be  subtracted  from  each,  and  •(-r^l=-;r'~l*  ^) 

a^^b    c-^d 

— -—=—-—,  that  M,  a— 6:  6::c— d:d;    this  is  called  divi- 
b  a 

DBNDo.    Euclid  17>  5. 

68.  In  like  manner/ the  first  fe  to  its  excess  above  the  second,' 

as  the  third  to  its  excess  abo%'e  the  fourth. 

a — b    c — d  b       d 

Because  —j—  =     ,     by  the  preceding^  and  siMX  — =  — , 
b  d  a      c 

^       a^b     b      c— rf      d      a— ^     c— d  , 

therefore  — ; — -x — = — r—  X — = — • — r= --,   or  a — b  :  a  :: 

b         a         d         c         a  c 

e — d  :  e,  and  invertenda  (Art.  63.)  a:  a^^h  : :  c  :  c— d;  this  is 

CONVBRTENOO. 

69.  Hence^  because  a-^b  :  at:  «^d  :  c,  the  excess  of  the  €i«t 
above  the  second  is  to  the  first,  as  the  excess  of  the  third  above 
the  fourth  to  the  fourth. 

7X).  If  four  quantities  be  proportionals,  the  sum  of  the  first 

and  second  is  to  their  difference,  as  the  sum  of  the  third  and 

fourth  to  their  difference. 

Let  a:  b  :;  c  :  d,  then  mil  4+ 6  :  a— 6  : :  c+d  :  c— d;  for 

a-^b     c-\-d  a— 6     c — d 

since  —■ — =—--—,  (Art,  65.)  and  — r— -as         ,  {Art,  67.)  divide, 
b  d  0  d         ' 

,    ^  ,   ,      ,    •  ,    fl+ft     ^—b     c+d     c— i 

the  former  equcus  by  the  latter,  and  (— ; — i — ; — = — ; — •- — 7—1. 

•^  .  b  h  d  a 

or)  ^ — r= r,  that  is,  a-^-b  :  a — b  : :  c4-d  :  c— ^. 

a— 6     c^d 

71 .  Hence,  the  difference  Of  the  first  atid  second  is  to  thtelr 
sum,  as  the  difference  of  the  thi)*d  and  fourth  to  theh*  Sum. 

Far  since  a-j-b  :  a — b  ::  c-fd  :  c-^d,  therefore  imicrtendo 
a— 6  :  a  +  b  : ;  c—d  :  c+d. 

73'  If  several  quantities  be  pfoportlonals,  «s  any  one  of  ike 
antecedents  is  to  its  consequent,  so  is  the  sum  of  any  number  of 
the  anteoedents,  to  the  sum  of  their  respectiv«  consequents. 

Let  a  :  b  ::  c  :  d  ::  e  :f  ::  g  :  h  .:  k  i  l : :  m  ;  n,  8(c.  then 
ioiU  a:  b  ::  a+c-|-e+g^+>-f  wi :  6-f-d+/+*-f  f-hn.  Because 
a  :  b  ::  c  :  dt  therefore  ad^^bc,  and  abszba;  also,  because  a  :  b  ::■ 
e  :f,  therefore  afssbe;  in  like  manner^  ah^ssbg,  alzsibk,  and  anss 
bm:  wherefore  {ad-\-af+ah-\ral+ans=^bc^be'\-bg+bk'j^bm,or) 
flxdH-/+^+/+ii=6xc-f e+^-f/f-Hw,  wherefore  a:6::c4> 


Part  !V.  PROPORTION.  07 

H-f+*+»»  J  <i+/+^+^-f*»;  ond  the  like  may  be  proved, 
whatever  number  of  antecedent^  and  their  respective  consequents 
be  taken. 

73.  If  fonr  qaantities  be  proportitoals^  and  if  eqaimultiples 
or  «qiuil  ^arls  of  the  first  and  aecand,  and  equimultiples  or 
#qu$l  pait9  fk  the  third  and  fourth,  be  tdceD>  the  resiidting 
quaatities  will  likewise  be  proportionals. 
Thus,  if  a  :  b  i:  c  :  d. 


Tbm  will 

1. 

ma 

mb     : 

me  : 

md 

• 

2. 

ma     ; 

mb 

lie    : 

nd 

t. 

ma    : 

mb     :: 

r 
n 

r  ^ 
n 

4. 

r 

— a   : 
ft 

It    :: 

mc  : 

md 

5. 

f» 

-^tf   : 
m 

— b    :: 
m 

r 

< — c  : 
s 

s 

For  in  each  case,  (by  multiplying  extremes  and  i^eans,) 

ad=:bc,  or  -7-=--r->  or  a  :  b  ::  c:  d. 
o      a 

74'  HeMe^  if  two  quantities  be  prime  to  each  dther,  they 
Vt^  the  le«iit  in  that  proportion. 

75.  If  four  quantities  be  proportlonais^  and  the  first  aad  third 
be  multiplied  or  divided  by  any  quantity^  and  also  if  the  second 
and  fourth  be  multiplied  by  the  same  or  any  other  quantity^  the 
results  will  be  proportionals. 
Xtf t  a  :  b  ::  c  :  d. 


Then  will 

1.    ma 

• 
• 

nb    : :     mc 

nd 

• 

8.    ^ 
m 

• 
• 

b           .  c      . 

mm                   1                  y          • 

n             m 

d 
n 

ft 

3.    ma 

• 

b 

"^    : :     me    : 

n 

1            , 

4.    ma 

• 
• 

mb    :;     mc    : 

nd 

t 

5.      • 

m 

ft 
ft 

nb     ; :      —     : 
m 

nd,  8ic. 

k         1 

Bar  in  eOfdh 

case,  ^rmdtipiyingmftremesahd 

[  means,} 

ad:a^bc,0t 

a       c  ...... 

-rrsz—-',  or  a:b  : :  c  :  d. 
b      d 

76.  Hence,  if  four  quantities  be  proportionals,  their  e^ui-' 

multiplefl^  as  also  their  like  parts,  are  proportionals. 

F  2 


68  ALGBWIA.  .  Pait  IV. 

77.  Heoce  also,  if  instead  of  the  first  and  second  tenns,  or 
of  the  first  and  third,  or  of  the  second  and  fourth^  or  of  the. 
third  and  fourth,  other  quantities  proportional  to  them  be  sub- 
stituted, the  results  in  each  ca^e  will  be  proportionals. 

78.  In  several  ranks  o€  proportional  quantities,  if  the  cor- 
respcHiding  terms  be  multiplied  together,  the  product  will  be 
proportionals. 

Thus,  let  a  :  h  ::  c  :  d^ 

And  e  :f '.'.  g  ".  h\  then  tc'ill  aek  :  hfl ::  cgm  :  dhn. 
And  k  :  I  ::  m:  nj, 

aek  :  bfl  ::  cgm  :  dhn,  and  the  like  may  he  sheum  of  any  number 
of  ranks. 

79.  Hence  it  follows^  that  the  likQ  powers  of  proportional 
quantities  (viz.  their  squares,  cubes,  &c.)  are  proportionals. 

For,  let  a  :  b  ::  c  :  d 

And  a  :  h  ::  c  :  d 

Also  a:  b  ::  c  :  d,  8fC.  then  by  multiplying  two  of  these 
tanks  together,  as  m  tfie  former  article,  we  have  «*  :  6*': :  c*  :  d^, 
and  by  multiplying  all  the  three,  a^  :  6^  : :  c^ :  d' ;  and  the  like  nun^ 
he  shewn  of  all  higher  powers  whateder. 

60.  Hence  also  the  like  roots  of  proportional  quantities  are 
proportionals. 

For,  let  a:  b  ::  c  :  d,  then  will  or  :  br  n  cr:  dr^  for  -t'= 

^..       /.  tt  c  ffT     cr  III        » 

-3-,   therefore  ^-r'=-  \/--r»   ^"^^  ***  r~~3~»  ^  or  :  frr  : :  c»-  : 
a  o  a  b^-     d^ 

dr,  and  the  same  may  he  shewn  of  any  other  roots. 
The  c^ration  described  in  the  three  foreg;oing  articles.  Is 

called  COMPOUNDING  THE  PROPORTIONS. 

81.  If  there  be  any  number  of  quantities,  and  also  as  many 
others,  which  take^n  two  and  two  in  order  are  proportionals, 
namely,  the  first  to  the  second  of  the  ^t  rank,  as  the  first  to 
the  second  of  the  other  rank ;  the  secotid  to  the  third  of  the 
first  rank,  as  the  second  to  the  third  of  the  oth^r  rank,  and  so 
on  to  tlie  last  quantity  in  each  f  then  will  the  first  be  to  the  last 
of  the  first  rank,  as  the  first  to  the  last  of  the  other  rank. 


PabtIV..  PROPQRTION.  69 


»  .  ... 

d:  e  ::  k  :  I 


Then  will  a  :  e  ::f:  I;  for  if  the  above  four  proportions  hfi 
compounded^  {Art,  78.)  we  shall  have  abed  :  bcde  :  ifghk  :  ghkl, 

.abed    fghk      .   €t      f       ,      ^  ^  .       , 

^^  ^^     ghkl'  ^^  Tl'  ^*^^'**  a:e::f:l,and  the  like 

may  be  demonstrated  of  any  number  of  ranks. 

This  IB  called  sx  jeolvkli  in  fropostion£  ordinata,  or 
simply  BX  mwjo  ordinato.     Euclid  22,  5. 

82.  If  there  be  any  number  of  quantities^  and  as  many  others^ 
which  taken  two  and  two  in  cross  order  are  proportionals* 
namely^  the  first  to  the  second  of  the  first  rank,  as  the  lost  but 
one  to  the  last  of  the  other  rank ;  the  second  to  the  third  of  the 
first  rank^  as  the  last  biit  two  to  the  last  but  one  of  the  other 
rank,  and  so  on  in  cross  order ;  tben  will  the  first  be  to  the  last 
of  the  first  rank,  as  the  first  to  the  last  of  the  other  rank. 

ra  :  b  ::  k  :  I 
Let  a  :  b  :  c  :  d  :  el  ,  ^.  i,  ^i  a)  f>  -  ^  '»  h  :  k 
Andf:g  '.h'.k'.lS^*^''  ^^^\  c:d::g:h 

\d:e::f:g 

Then  wiU  a  :  e  ::f :  I;  for  compounding  the  above  four  pro* 
portions,   (Art,  78.)    there  arises  abed  :  bcde  : :  khgf :  Ikhg,  or 

(t-t-=  .,  t  j  that  is,)  — sr-^-*  wherefore  a:  e  ::f:  I,  which  was 
ifcde     Ikhg  'el  -^  '' 

to  he  shewn ;  and  the  like  may  be  proved  ef  any  numher  of  ranks. 

llib  is  called  ex  jaayALi  in  proportiokb  pbbturbata,  or 

siniply,  BX  mq,uo  pbrtukbato  ^    Euclid  23>  5* 

INVERSE,  OR  R£GIPRCX:AL  PROPORTION, 

83.  The  foregoing  artides  treat  of  the  pn^rties  of  what  Ib 
called  DiBBCT  Pbopo&tion,  where  the  first  is  to  the  second  as 
the  third  is  to  the  fourth ;  but  when  the  terms  are  so  arranged. 


^  It  must  be  undentood,  that  what  we  bate  delirered  on  proportion,  refers 
to  eommenturabU  magnUude*  only :  it  is  io  sobstaDce  tbe  tame  as  the  Slih 
book  of  Euclid's  £iemeiita,  except  that-  the  doctrine  there  deliverid  iocludes 
both  eommenmrabU  and  meommensurabie  nagnitndet ;  Eaclid  has  effectod 
this  double  object  by  means  of  his  fifth  definition,  which  although  strictly 
feneraly  has  been  justly  complained  of  for  its  ambiguity  and  clumsiness. 

F3 


'';fc 


70  .  hUSSmti^  y«w  IV. 

that  the  first  is  to  the  second,  as  the  fourth  to  the  third,  it  is 
then  oamed  Ivybbsb  PaopoKTioH»  and  the  fovri^iuMBtlties  in 
the  order  thev  stand,  are  said  to  be  rnvtasKLY  paoPonTioNAL. 
Thusy  2  :  4  : :  12  :  6^  and  9  :  5  : :  10  :  18>  *c.  are  inverseUf 
proportional. 

84.  Hence«  an  inverse  prpportion  may  be  made  direct,  by 
chaining  the  otder  of  the  terms  in  either  of  the  ratios  which 
constitute  the  proportion. 

85.  The  reciprocals  of  any  two  quantities  will  be  inversely 
proportional  to  the  quantities. 

Let  a  and  b  be  two  quojitities,  then  vfiU  a  :  (  : ;  -r* :  — ,  for 

muHipl^ing  both  terms  of  the  latter  ratio  by  aby  tee  shaH  have 

a  :  b::  (-r-  :-:-::)  a  :  b,  therefore  a:  bz:  -r-i  — ;  inlikeinanr 
o      a  o      a 

11  ... 

ner  b  :  a  ::  —  :  ^r-,  that  is,  the  direct  ratifi^ o^  tfte  qui9fi^tiB»  i^ 
a      0  V  » 

the  same  as  the  inverse  ratio  of  theit  reciprocals  ;  and  the  inverse 

ratio  of  the  quantities^  the  same  as  the  direct  of  their  reciprocals. 

Hence,  inverse  proportkn  i*  Ukt^i^  frequently  chilled  reci- 

rfiOCAL  FROPQ&TIQN. 

HAKMONICAL  PROPORTION. 

86.  Three  quantities  are  said  to  be  in  harmonical  or  mueieal 
pro[>oriion,'  tvhen  the  first  is  to  the  t^iird,  as  the  difierenee  of 
ike  fii-dtaAd  second,  toi  ihe  di^«aAe««:eiof  the  seeond  WMt  thirds 
fUid  fouii  t^nm  are  mi  to  be  in  h^H^mwical  proportipnA  i?f hen 
the  lirst  is  to  t\^  fyvLTiium  the.  dtflFwenoettf  tlw  &^  and  seoeAd 

m 

to  the  difference  of  the  third  and  fourth. 

TAds,  tf  A:  e::  a^-^b :  b^c,>  then  an  the  (htee  quantities, 

4>  bw^dsy  hafimimkaUit.  pfoppr^ipnoL 

A^d  \fia,:dr.:0r^bi^'^d,.tkm!air^'th$fQufyai,b,c,audd, 
Mrmim^flUy  proportional; 

&7.  Hence^  if  all  the  terms  of  any  harmonical  proportion  be 
either  multiplied  or  divided  by  any  quantity  whatever^  the  ropults 
.w'iU  still  be  in  hai^oiopiqal  proportion. 

88.  If.  double  the  product  of  anjf  two  quantities  be  divided 
•by  their  s«di>  the  ^otient  will  be  a  bann(»mcai  mean  betn^eeti 
the  tw'o  qtiantilies. 


9iw  IV.  VRpgmXW^  « 

duct,  and  04-1=  their  sum,  wherefore r   is  the  harmonical 

a-^-b 

mean  required,  for  (Art.  86.)  a  .  6  : :  a :  ( — xa r=: 

a^f-A       a  tt'+'O 

',  ,  =  — ; =) T^^;  that  is,  the  first  is  to  the  third,  as 

ike  d^fttence  between  the  first  and  second  to  the  difference  be- 
tween the  second  and  third. 

Examples. — 1.  To  find  a  harmonical  mean  between  9  and  6. 

«T            ^   .     «       ,  ^ab     ^  ,  .     ,    ^ 

Here  a=2,  6=6,  a»d ^-=---=3,  the  mean  required;  for 

e:©::  (3— «:6— 3  ::)  1  :3. 

%  |l«quired  a  harmonicat  mean  between  24  and  12? 
Jns.  16. 

3.  Heqttired  the  harmonieal  mean  between  5  and  20? 
Ahs.S. 

4.  Required  tbe  harmonieal  mean  between  10  and  30  ? 

« 

89.  If  the  product  ei  two  given  qaaatitiM  be  divided  bf  the 
difierence  between  double  the  greater  and  the  less^  or  double 
tfete  le$s  Mod  the  greater,  the  quotient  will  be  the  third  harnMni* 
cal  proportional  to  the  two  given  quantities. 

Let  a  ijmd  b  be  twogi^sen  quasUities,  whereof  ais  the  greater  $ 

4he»  tnU be  the  iln/rd  harmonieal  proportional  to  a  and  b : 

I  ■  ■  ■  . 

A         .  ab  ,     Mka — ^    oh—h^    b^—ah    ,        ab 

fora--: 1 1:  a— 5  :  ( — =_- r=- — ---=so^- ■  '-^j 

•^  2a-.6  a.2a^     2  a— 6     6— 2  a  6— 2a    ^ 

ab 

>  the  difference  between  ihe  secgnd  and  third  '• 


2  a— i 


I        I      I    I     III    I     PI        t       ■  .  t  ■- T        ^  **?'  — 


t  Td  wlmt  bas  been  safd  on  this  subject,  the  following  pftrtiealan  rclttinf 
to  the  comparison,  &c.  of  the  three  Ikinds  of  proportionals,  my  be  a^Asd;  viz, 

I,  The  reciprocals  of  an  arithmetical  progression  are  in  harfnooical  pro- 
gression, and  the  reciprocals  of  a  harmonif^ai  pr<^ression,  are  in  arithmetical 

pragifMioD. 

7%us,  a,  a'\'d,  a'\-Zd,  a-^-Sidt  are  arUhmeticuUy  proportional, 

^*^  T'  HM '  5+2^  ^+34'  '***^  reciprocaUy  are  hartMrncaUg 
preporHMol^  and  tibe  contforse* 

F  4 


<f 


•# 


73  ALOEXRA.  Paw  IV. 

ExAMFLBs.-*!.  To  find  a  tldrd  bsmiomcal  proportiDnal  to 
48  and  39. 

rr  .«      r      ««  J        «*  48x33  1536      ^^    • 

Here  a=:48,  6=32,  and r=- — -- — -.=s-----=54, 

2  a- 6    2x48—32       64 

the  number  required;  far  48 :  24  : :  (48—32  :  32—24  : :  )  16  :  8. 

2.  Required  a  third  hannoaical  proportional  to  2  and  d,> 

Ans»  6. 

3.  Required  the  third  harmonical  proportional  to  20  and  8 1 

Ans,  6. 

4.  Required  the  third  harmonical  proportional  to  10  and  100  > 

90.  Of  four  harmopical  proportionals  any  three  being  given^ 
the  fourth  may  be  found  as  follows. 

,  Let  a,  h,  c,  cmd  d,  he  four  quantitkn  ia  harmmical  propor* 

tion,  then  since  a  :  d::  a— 6  :  c— d,  (Art»  86.)  by  multiplying 

extremes  and  means,  ac — ad=ad — 6d;  from  this  equation  OMg 

three  of  the  qwmtities  being  given,  the  remaining  one  may  be  found. 

ac 
Thus,  a,  6,  and  c,  being  given,  we  have  ds=  - — r  one  of  the 

bd 
extremes  i  if  b,  c,  and  d,  be  given,  then  azpr-^ —  the  other  ear* 

treme;  if  a,  b,  and  d,  be  given,  then  ess——*  om  ^f  the 

a 

mean* ;  ./  a.  c.  and  d.  be  given,  then  b^^-^tl^  theother  mean. 


2.  If  there  be  taken  an  arithmetical  mean  and  a  harmonteal  mean  between 
any  two  quantities,  then  the  fonr  quantities  will  be  geMnetrically  propoctiooal* 

Thutf  between  a  and  h  the  harmonical  mean  is  — rT>  and  the  arithme' 

a'\'b  ^ab        a-^h 

Heal  mean  — - — ,  and  a  :  — r-r  : :  — - —  :  b. 
2  a-^b         2 

3.  The  following  simple  and  beautiful  comparison  of  the  three  Unda  of  pro«> 
poTtionals,  is  given  by  pappus,  in  his  third  book  pf  Mathematical  CoUeotiopt* 

Let  a,  bf  and  c,  be  thejirst,  second,  and  third  tertnt  ^  thent 

C  JrUhmetieals  a\a' 
<  Geometrieals  a 
l^Harmomcals  a 

4.  There  is  this  remarkable  difference  between  the  three  kinds  of  proportion  ^ 
namely/  from  any  given  term  there  can  be  raised 

A  continued  arithmetical  series,  increasing  but  not  decreasing,  '\  . 

A  continued  harmonical  series,  decreasing  but  not  increasing,  > 

A  continued  geometrical  series,  both  increasing  and  decreasing,  J  ^^*^/* 


In  the<  Geometrieals   a:  b>::  a-^b  :  6— tf. 


Tart  IV.  PROPORTION.  73 

Examples. — 1.  Let  there  be  given  3,  4,  and  6,  being  the 
first,  second,  and  third  terms  of  a  harmonical  proportion,  to 
find  the  fourth  ? 

Here  fl=3,  6=4,  c=6,  and  -_^=(-__=-==)  9, 

the  fourth  term  required;  far  3:9::  (4—3  :  9—6  : :  )  1  :  3. 

2.  Given  the  second,  thirds  and  fourth  terms,  viz.  4,  6,  and 
D,  to  find  the  first  ? 

Here  £>=r4,  €:^6,  thud,  vtherefore  a^s — f— -=(- -= 

36 

— =)  3,  the  first  term  required. 

3.  Given  3, 6,  and  9,  being  the  first,  third,  and  fourth  terms^ 
to  find  the  second  ? 

rr.  «  ^    J     ^        J  .     2fld— flrc      54—18     .  ^    ^ 

.rore  «=s3,  c=6,  d=:9,  and  6=—; — s=:{— — as)  4,  • 

d  9 

f/i€  second  term  required. 

4.  Given  3,  4,  and  9,  being  the  first,  second,  and  fourth,  to 
find  the  third  ? 

tr  o    1     .    J    «        J        2ad-M      54-36     .  ^ 

Acre  aa=3,  o=4,  d=9,  and  c=  — — =s( — - — =)  6, 

a  o 

the  third  term,  as  was  required. 

5.  Let  the  first,  second,  and  third' terms  in  harmonical  pro- 
portion, viz.  36,  48,  and  7^>  be  given  to  find  the  fourth  ?  .    ^ 

6.  Given  d4,  36»  and  54,  or  the  second,  third»  and  fourth 
terms,  to  find  the  first  ? 

7.  Given  97%  36,  and  81,  being  the  first,  second,  and  fourth 
tanauB,  to  find  the  third  ? 

8.  Let  48,  96,  and  144,  being  tbe  first*  third,  and  fourth,  be 
^ven,  to  find  the  second  ? 

91.  Three  quantities  are  said  to  be  in  contra-harmonical 
PROPORTION,  when  the  third  is  to  the  first,  as  the  difference  of 
the  first  and  second  to  the  difference  of  the  second  and  third. 

Thus,  let  a,  b,  and  c,l)e  three  quantities  in  contra-harmonv^ 
cal  proportion t  then  will  c  :  a  : :  acssb  :  &CV)c. 

98.  Tbe  following  is  a  syDopsis  of  the  whole  doctrine  of  pro- 
portion, as  contained  in  the  preceding  articles. 


74  AUUtSBJL  PabtIV. 

Let  fiiur  qinmtities  a,  6»  c,  aod  d,  be  pr^portionaU^  tben  are 
ttiey  also  proportionals  ia  all  the  foUowkig  fprms  -,  viz. 

1.  Directly • . .  a  :  6  : :  c  :  d. 

8.  Inversely b  :  a  ::  d:  c. 

3.  Alternately , n  :  e  : :  b  :  d, 

4.  Alternately  and  inversely  . . . .  c  :  a  : :  d  :  6. 

6.  Compoundedly a  :  a+6  : :  c  :  c+d. 

6.  Compoundedly  and  inverstcly  a-f  6  :  a  : :  c-|-d  :  c. 

7.  Compoundedly  and  alternately  a  :  c  ::  a-i-b  :  c^d. 

8.  CampouadedJy. alternately,  \^,^.,^^^,  «+j. 

and  inversely J 

9.  Dividedly a  :  a  — b  : :  c  :  c — d. 

or, a  :  b — a  : :  c  :  d— c. 

10.  IXvidecHy  and  alternately  . . . .  a  :  c  : :  a-^b  :  c— d. 

or, a  :  c  :i  6— a  :  d— c. 

11.  Mixedly a+6:  a— 6  : :  c+d  :  c— d. 

1*.  Mixedly  and  inversely a— ^ :  a-^-h  :t  c— d  :  c+d. 

13.  Mixedly  and  alternately a-f  6  :  c-f  d  : :  a*-&  :  c— d. 

14.  By  multiplication ra    :    r6    : :    «c    :    sd, 

15.  By  division : —    :    —    : :    —    :    — , 

r  r  $  s 

IS.  By  invidution a*    :    *■    : :     c*     :    d". 

17.  By  evolution av   :    ^r  : :     c*^    :    dy. 

18.  They  are  inversely  proportional  when  a  :  b  ::  d:  c. 

19.  They  are  in  harmonical  proportion  when  a  :  d  :  t  tf  wo  6  : 
€^d. 

Sa  Three  numbers  are  in  contra^hsnaon]^  proportion 
when  c  :  a  : :  a  c/)  6  :  c  c/)  d. 

The  14th>  15th,  leih,  and  17tb  partieidaiB  admk  of  inver- 
sion, alternation,  composition,  division,  &c.  in  the  same  mnncr 
with  the  foregoing  ones,  m  is  evident  from  the  niBtare  of 
proportion. 

The  comparison  of  VARIABLE  and 
DEPENDANT  QUANTITIES  \ 

93.  A  quantity  is  said  to  be  variable,  when  from  its  nature 

and  coDstitution  it  admits  of  increase  or  decrease. 

_ —  '  ■ 

^  TM  doetrine  of  Tariable  aofl  depeadBiit  qinntitieB,  «»  laid  doMm  in  the 
fuUowiu{;  articles,  sbo«M  bo  v«ll  iui4«nt90cl  hyaU  tki^te  vho  intwA  i^ntd 


PaktIV»  variable  ahp  DJ^SNPAMT  QUANTinES.  n 

94.  A  (juaatity  is  sajki  to  be  hmatitMe  or  eMittoiil»iidien  its 
Ofiture  is  such  that  it  do«s  not  cbaoge  its  value. 

95.  Two  variaUe  q^Hiatitisft  are  aaid  to  be  depend€mi,  whett 
ent  of  tbi^iii  being  increased  or  decreased,  the  other  k  Increased 
(BT  decureafi^d  reepectiveljF,  in  the  same  ratio. 

Thus,  let  A  and  B  be  two  variabU  qumiiUM,  mtch,  tM 
when  A  i«  changed  into  any  other  value  u,  B  u  necessarUf  ch^niged 
mtQ  a  ^corresponding  value  b,  (in  which  oast  A  :  a  ::  B  :  b,)  ihm 
A  and  B  are  said  to  be  mutually  d^^itndant.r 

d6r.  To  every  proportion  four  terms  are  necessary,  but  in 
af^lyijlg  the  dnrfiiiie  td  pvaetice,  although  four  quantitks  are 
always  understood,  two  only  are  emplc^ed.  This  concMe  mode 
of  expression  is  found  to  possess  some  advantages  above  the 
common  method,  as  it  saves  trouble,  and  likewise  assists  the 
inind,  by  enabling  it  to  conceive  more  readily  the  relations 
which  the  variable  and  depeadaol  quantities  under  coinsideratioA 
bear  to  each  other. 

97.  Of  two  variable  and  dependant  quantities,  each  is  aaid  to 
vary  directly  as  the  other,  or  to  vary  as  the  other,  or  simply  to 
be  as  the  other,  when  one  being  increased,  the  other  is  neces- 
sarily increased  in  the  same  ratio,  or  when  one  is  decreased,  the 
other  also  is  decreased  in  the  same  ratio. 

Thus,  if  r  be  any  number  whatever,  and  if  when  A  is  in^' 
creased  to  rA,  B  is  Tiecessarily  increased  to  rB,  (that  is,  when 

A 
A\r4'.vB\  rB,)  <?r  p^hm 4 is  docreoMd  to—,  B  is  necessarii^ 

r 

B  A  B 

decreased  to  -r-,  (iluit  is,  when  A  :  ■*-::  B  :  — ,)  then  A  %s  said 
r  r  r 

to  vary  directly  as  B:  or  we  say  simply,  A  is  directly  as  B. 

Example.  A  labpor^r  agrees  tp  work  a  week  for  a  certain 
sum  ;  now  if  he  work  2  weeks,  he  receives  twice  that  sum,  if 
ke  work'  trtit'half  a  week,  he  receives  but  half  that  sum,  and 
*o  on ;  in  tWs  cstse,  the  sum  he  receives  is  directly  as  the  time 
he  works. 


tUti  Isaac  I^ewton's  Principla,  or  any  other  scientific  treatise  00  Natoral 
Philotopby  or  AstroDomy.  See  on  this  subject,  JUtdlamfs  Rudiments,  hth 
M'lt,  p.  S3.*>— 250.  and  If^ocMTs  Algebra^  3d  Edit.  p«  103 — 109* 


1 


76  ALGEBRA.  Part  IV. 

98.  Ohe  (piantity  is  said  to  vary  inversely  as  another,  when 
the  former  cannot  be  increa8ed>  but  the  other  is  decreased  in 
the  same  ratio ',  or  the  former  cannot  be  decreased,  but  the  other 
must  nccessprily  be  increased  in  the  same  ratio ;  that  is,  the 
former  cannot  be  changed,  but  the  reciprocal  of  the  latter  is 
changed  in  the  same  ratio. 

Example.  A  man  wallu  a  certain  distance  in  an  hour;  now 
if  he  walk  twice  as  hst,  he  will  go  the  given  distance  in  half 
an  hour  -,  but  if  h&  walk  only  half  as  fast,  he  will  evidently 
require  two  hours  to  complete  his  journey  i  in  this  case  his  rate 
of  walking  is  inversely  as  the  time  he  takes  to  pei*fiirm  it. 

99.  The  sign  ec  placed  between  two  quantities,  signifies  that 
they  vary  as  each  other. 

Thus  A  K  B  implies  that  A  varies  as  B,  or  that  A  is  as  B; 

ulso  A  K  -^  skews  tlmt  A  varies  as  the  redprocal  of  B,  or  that 

■ 

A  is  inversely  as  B, 

100.  One  quantity  is  said  to  \'ary  as  two  others  jointly,  when 
the  former  being  changed,  the  product  of  the  two  latter  must 
necessarily  be  changed  in  the  same  i*atio. 

Thva  A  varies  as  B  and  C  jointly,  that  is,  A  9^  BC,  when 
A  cannot  be  changed  mto  a,  hut  the  product  BC  must  be  changed 
into  be,  or  that  A  :  a  ::  BC  :  be. 

101.  In  like  manner  one  quantity  varies  as  three  others 
jointly,  when  the  former  being  changed,  the  product  of  the 
three  latter  is  changed  in  the  same  ratio. 

Thus  Ak  BCD,  and  the  like,  when  more  quantities  are 
concerned. 

Example.  The  interest  of  money  varies  as  the  product  of 
the  principal,  rate  per  cent,  and  time,  or  I  ic  PRT. 

« 

loss.  One  quantity  is  said  to  vary  directly  as  a  second,  and 
inversely  as  a  third,  when  the  first  cannot  be  changed,  but  the 
second  multiplied  by  the  reciprocal  of  the  third,  (that  is,  the 
second  divided  by  the  third,)  is  changed  in  the  same  ratio. 

B 

Thus  A  varies  directly  as  B,  and  inversely  as  C,  or,  A  tc  -t7# 

B      h 

when  A  :  a::  -^  :  — . 

C      c  •  ' 


Part  IV.  VARIABLE  and  DEPENDANT  QUANTITIES.  77 

Example.  A  fermcr  must  einploy  as  many  reapers,  as  are 
Erectly  as  the  number  of  acres  to  be  reaped,  and  inversely  as  the 

number  of  days  he  alV;>ts  for  the  work,  or  B  jc  — . 

103.  U  JtQ  B,  and  ^  oc  C,  then  wiU  ^  *  BC 

For  smce  B:b::A:  -^=ra, and  C  :  c  :: -r^- i  -57.=v<»=  <*<? 

i*  jj      BC 

final  value  of  A  arising  from  iU  successive  changes  in  the  ratios  of 

v4hr 

Bil^andC:  c;  wherefore  smce'^;r;:sza,  or  Abc^aBC,  A  :a:: 
BC :  be,  or  A  fKi  BC. 

104.  in  like  manner  it  may  be  shewn,  that  if  ^  oc  B,  A  u:  C, 

s 

and  A9i  D,  then  A  oe  BCD  -,  also  if  ^  «c  B,  and  ^  ec  ~,  then 

B  1 

-^  *  "^i  and  likewise  li  A  tt  B,A  ec  C,and-4  cc  -yr.  then  A  « 

BC 

-gj-,  the  proof  of  all  which  is  the  same  as  in  the  former  article. 

104.  B.  If  ^  cc  BC  and  B  be  constant,  then  ^  oc  C5  if  Cbe 

B 
constant,  then  A  k  B-,  if  -rf  «c  -tt  and  C  be  constant,  then  A  « 


B^  if  B  be  eonstant,  then^^  tc  -r;. 


For  since  the  product  BC  varies  by  the  increase  or  decrease  of 
C  only,  when  B  is  constant,  and  A  varies  aJs  that  product,  there* 
fore  when  B  is  invariable,  A  must  evidently  vary  as  C,-  and  when 
B  alone  is  variable,  and  C  constant,  A  {varying  as  the  product 
AB)  must  in  like  manner  vary  as  B:  after  the  same  manner 
it  may-  he  shewn,  that  when  A  ee  BCD,  if  B€  be  constant,  then 
A  ^  D  i  if  D  be  constant,  then  A  k  BC;  if  C  he  constant,  then 
A  ee  BD ;  and  if  B  be  constant,  then  A  cc  CD  ;  and  in  general^ 
if  A  be  as  any  product  or  quotient^  and  if  any  of  the  factors  be 
given,  A  will  be  as  the  product  or  quotient  (as  the  case  tfiay  be) 
of  all  tfie  rest, 

105.  If  the  first  quantity  vary  as  the  second,  the  second  as 
the  third,  the  third  as  the  fourth,  and  so  on,  then  will  the  first 
vary  as  the  last. 

Let  A,  B|  C,  a«d  D,  he  any  number  of  variable  quantities. 


m  AtOSBRA.  PaktIT. 

and  a,b,t<md  d,  torfespondit^  mlues  of  them ;  and  let  A  ^  B, 
Bit  C,andCtt  D;  then  teiU  Ate  D. 
Because  A:a::  B :b. 
And  Bib::  C:c. 

And  C:c::  D-:  d,  therefore  ex (cquo  (Art,  81.)  A:  an 
D :  d,  that  is,  A  k  D  ;  and  the  same  may  he  shewn  to  be  true  of 
any  nufiAer  of  variable  quantUies, 

106.  If  the  first  be  as  the  second^  and  the  second  inversely  as 
the  thirds  then  is  the  first  inversely  as  the  third. 

1  I  • 

l4et  A  n  By  and  B  ti  -—,  then  is  A  ^t  -^> 

For  since  A:a::B:b, 

And  B  :  6 : :  —  :  — ,  therefore  ex  aquo  A:  a::  —  :  — , 
o     c  \^     c 

1       • 
that  is,  A  96  -j;, 

167.  If  eadi  of  two  quantifies  Vary  as  a  thiiti,  then  will  both 
their  sum  and  difference^  and  also  the  square  root  of  their  pro* 
dnct,  vary  as  the  third. 

Let  A  9c  C,  and  B  9^  C,  then  will  A;j^B  K  C,  and^AB 
n  C. 

Because  A  :a:t€;e,^  i    ,       .,    . 
AndC:c::B:bJ^^y^^^''' 

Therefore  ex  aqucUi  Aia::fiib^qnd  aUemmdH  AnBi: 
a:b,  wherefore  componendo  et  dividendo  A±B :  B  i:a-^b:b, 
whence  altemando  A±B  :  a±b  ::B:b;  but  B:bi:C:c,  where- 
fore ex  aquali  A±B :  a±b  i:C:c,  that  is,  A^  9fi  C,  or  the 
sum  and  the  difference  of  A  and  B  will  each  be  as  C. 


Again,  because  A 
And    B 
Therefore  (Art.  78.)    AB 
Whence  (Art,  80:)  ^AB 


a  ::   C  :  c, 
b  ::   C  :  c, 
aJb:\  O  :'c^, 
^ab  : :  C  :  c,  that  is,  ^AB  cc  C. 
108.  If  one  quantity  vary  as  another,  it  will  likewise  vaiy  aa 
any  multiple  or  part  of  the  other. 

Let  m  be  any  constant  quantity,  and  let  A  9^  B,  then,  wUl 

A  ee  taS,  and  A  ec  — . 

m 

'  Because  A  :  a  ::   B  :   b,  by  hypothesis,  and 

B  :  b  ::  mB  :   mb.  Art.  73. 

Ther^ote  A  :  a  :;  mB  :  mb,   that u,  A  tn  mB, 


PabtIV.  variable  ani>  DBPSKBANT  OUANTITIES.  19^ 

And  B  :  b  ::  —  :  — . 

m      m 

Therefore  A  \  a  :\  —  :  — . 

mm 

Thai'ts,A9^  ~. 
m 

Since  A^  B,AiM  tquml  to  B  imdHfi^ed  €t  ^vkM  tf  $ofm' 

R       h 
constant  quantity  j  for  A  :  a  ::  mB  :mb  ::  —  :  — ,  whence  alter-* 

m     tn 

nando  A  :  mB  : :  a  :  mb  :i 

B  b 

And  A  '.  —  : :  a  :  — ^  if  m  b%  uummd,  ao  thai  Av^mB,  ar 
m  m 

.     B  b 

A= — ,  then  will  a^smb,  4tr  a= —  reipectively, 
m  ffi  ^ 

110.  If  the  corresponding  values  of  A  and  B  be  known^  then 
will  the  value  of  the  constant  quantity  m  be  likewise  known. 

For  if  a  and  b  be  the  known  corresponding  values  of  A  and 

B,  then  since  A^mB,  or  A=^ — j  by  substUuting  a  and  b  for  A 

m  ■ 

cmd  Bi  we  shall  hate  a^s^mbf  or  a=; — ;  whence  m=-;-^  or  «!« 

m  b  ' 

b  H  /I 

—  .•  wherefore  dUo  (since  As^mB,  ot  Aa-^)  -rfat-r  M  t,  ^)r« 
a  '  ^  m  6 

a 

111.  If  the  product  of  two  quantities  be  coBttaot/  iNn  will 
the  fiietOTs  be  inversely  as  each  other. 

1  1 

Let  AB  be  a  constant  quantity,  then  is  A  t^  ~  and  B  m  -^ 

/or  AB  being  coMPant,  it  mm/  be  OMsider^  ae  1 5  iha$  is,  AB  « 

1,  whence  A  «  -^,  and  JB  oc  ~ . 

B  A 

119.  ileiiQs,  ia  the  cefMtant  product  ABC,  A  m  -^^^  B  « 

1  1  I  1  1 

AC  ^  *  'jW  S€  t^  -*j,  AC  9c  -^,  4md  AB  n  -^i  9tw^  U>e  Uk« 

may  be  shewn  wh^n  the  product  consists  of  any  number  of 
fectors.  ^       ^ 


8a  ALQBBKA.  PAsrlt. 

113.  If  the  quotient  c^  two  quantitks  be  oooatMt^  tbeo  %xe 
those  quantities  directly  as  each  other. 

Let—  ec  1«  then,  (multiplying  both  sides  by  B,)  wiUA  ce  B, 

and  B  K  Af  and  the  like  may  be  shewn  wJien  the  quottent  is  com^ 
posed  of  any  number  of  quantities, 

1 14.  If  two  quaotities  vary  as  each  other^  their  like  multiples 
and  also  their  like  parts  will  vary  ^$  each  other  respectively. 

Let  A  K  B,  and  let  m  be  any  quantity  constant  or  variable, 

A       B  * 

then  will  mA  ec  niB,  and  —  aq  —  . 

m       m 

.    For  since  by  hypothesis  A  :  a  ::  B  :  b,  therefore  mA  :  ma:: 
mB  :  mb  {Art,  73.)  that  is,  mA  «  mB, 

Also  —  :  —  : :  —  :  — ,  therefore  —  «  — . 
m      m      m      m  mm 

1 15.  If  two  quantities  vary  as  each  other^  their  like  powers 
and  like  roots  will  vary  as  each  other  respectively. 

Let  A%B,  then  since  A:a::  B:  b  {Art,  95.)  A^  :  a"" : :  \B"  : 
b\  and  A^  :a^  ::B~-:  6v,  {Art.  79.)  that  is.  A'  k  B\ 


Iff  v«  >n 


ec  B^ 


116.  If  one  quantity  vary  as  two  others  jointly^  then  will  each 

of  the  latter  vary  as  the  first  directly,  and  as  the  other  inversely, 

A  A 

Let  A  fic  BC,  then  £  «  77,  and  C  «c  —  . 

For  since  BC  oe  A,  divide  both  by  C,  and  B  «e  77  ;  divide 

both  by  B^  and  C  cc  -^ . 

B 

117*  If  the  iirst  of  four  quantities  vary  as  the  second*  pind 
the  third  as  the  fourth^  then  will  the  product  of  the  first  an^ 
third  vary  as  the  product  of  the  third  and  fourth. 
Let  A  ti  B.andCK  D,  then  is  AC  k  BD. 
For  A:  a::  B:b. 
And  C  :  c  ::  p  :  d, 
"    Therefore  {Art  79.)  AC:  ac::  BD:  bd:  or  AC  ce  BD,  - 
118*  If  four  quantities  be  proportionals^  and  one  or  two  of 
them  be  constant,  to  determine  how  the  others  vary. 

Let  A  i  B  ::  C  :  D,  then  will  AD==  BC,  and  therefore  AD 
ce  BC,    Let  A  be  constant^  then  D  ce  BC,  {Art.  104.)  let  D 


^ART IV.  VARIABLE  AND  DEPENDANT  QUANTITIES.  81 

^  coiiBtani,  then  A  oe  BCx  lei  B  be  constant,  then  C  ee  AD;  let 
C  be  constant,  then  B  k  AD,  Next,  let  A  and  B  be  both  constant, 
then  D  k  C;  let  A  and  C  be  constant,  then  D  oc  B;  let  D  and 
B  heconstant,  then  A  «e  C;  lei  D  and  C  be  constant,  then  A  %  B> 

let  A  and  D  be  constant^  then  B  and  C  will  be  both  constant,  or 

%. 

vary  inversely  as  each  other,  that  is,  B  k  -^»  and  C  te  -^  ; 

(Art.  111.)  in  like  manner,  if  B  and  C  be  constant,  then  A  and  D 
vUl  both  be  constant,  or  vary  inversely  as  each  other,  nam 


A  «  ~,  and  D  «e  -j.    lastly,  if  three  of  the  quantities  be  con- 
stant, the  fourth  will  evidently  be  constant. 

119.  To  shew  the  use  and  great  convenience  of  the  conclu- 
sions deiived  in  the  preeediog  artides,  the  following  examples 
are  subjoined. 

Examples. — 1.  Let  Pssany  principal  or  sum  of  money  lent 
out  at  interest^  i{=the  ratio  of  the  rate  per  cent.  T=the  time 
it  has  been  lent  at  interest^  and  J=the  interest;  to  determine 
the  relative  value  of  each. 

First,  supposing  all  the  quantities  variable. 

Then  Ice  PRT {Art.  22.)  whence  Pss-—-,  R  m  :—^  and 
T  «6  — ,  (Art.  114.)  Let  I  be  given,  then  P  «c  ^,  R  «c  p^,  and 

T  te  •^^,  (Art.  104.)  let  P  be  given,  then  I  9^  RT,  R  k  -=,   «id 

I  I 

T  ic  -=-,  (Art.  111.)  let  R  be  given,  then  I  tc  PT,  P  «c  -^^  and 

\ 
I  I 

r  Bc  -5",  (^rt.  111.)  let  T  be  given,  then  I  ec  PR,  P  «  -5-,  ond 
P  jK 

R  «e  -^;  let  I  and  P  be  given,  then  R  «c  -=r>^ui(£  T  ce  s";  let  I 
P  T  H 

and  R  be  given,  then  P  cc  -=;,  and  3*  oc  -5-;  let  I  and  Tbe  given, 

then  P  flc  ~>  and  R  tn  —;  let  P  and  R  be  given,  then  I  k  T; 
R  V 

let  P  and  T  be  given,  then  1 9t  R»    Lastly,  let  R  and  T  be  given, 

then  I  9c  Pi  and  if  any  three  of  the  quantities  be  given,  tbe 

fmurth  wiU  be  given. 

VOL.  II.  O 


%  SuppcMf  the  qiiuidtie$  of  inotioii  in  taro  monipg  htOm  tfft 
be  in  the  ratio  comppunded  of  the  qqantitie^  of  0i9Uer«  «nd  tin 
veloekiesj  to  determine  the  other  dicgnvtances. 

Brst,  let  Msithe  qumtUy  Qfmotitm,  Q^zfuamtUiy  ^  mUter, 

Vzsvelociiy;  then  M  9^  QV  by  hypothesis,  wherefore  Qm-prs 

and  if  Mbe  given,  Q  «  j^  ;,  also  ^  «  -g-*  ««d  M  being  given, 

yK^;ifQbe  given,  then  M  9c  V;  and  if  Vbe  given,  M  k  Q* 

Secondly,  suppose  the  quanUty  of  matter  Q  to  be  in  the  com* 

pound  ratio  of  the  magnitude  m,  and  density  D,  or  Q  %  mD; 

by  substituting  mD  for  Q  in  the  abov^  expre$sions  where  Q  is 

M  1 

found,  we  shall  have  M  ce  mDV,  mD  «  j^,  mD  st  rp-,  M  bang- 

M  1 

given:  Fee  ^--r^rfWVm  —^^  M  bnng  giutsifrom  ibete  ii  is 
mil  mU 

plain  that  a  great  variety  of  other  expressions  may  be  obtained,  qni 

still  more,  by  considering  one  or  more  of  the  quantities  invariable* 

Lastly,  since  the  magnitudfss  qf  bodies  are  as  the  cubes  of 

their  homologous  lines,  {or  d^,)  that  is,  (P  k  m;  if  d^  be  substi" 

iut^dfor  m,  by  proceeding  as  before,  toe  $bull  obtain  at  length  aU 

the  possible  relations  of  the  above  quantities :  but  the  prosecution 

of  this  is  left  as  an  exercise  for  the  learper. 

GEOMETRICAL  PROGRESSION. 

120.  To  investigate  the  rules  and  theorems  of  Geometrical 
Progression, 

Let  aszthe  least  term,      1     u  j    i    *i    ^  - 

z^ihe  greateH  term,  T "^  '^  ***  «**'«^- 

n=^the  number  of  terms, 

r=the  common  ratio, 

s=zthe  sum  of  all  the  terms, 

*  Then  will  a4-ar+or*H-ar*>  ^c.  to  ar'^'^^^ be  m  increasing  geo* 

metrical  progression. 


*  A  progression,  consisting^  of  three  or  four  terms  only,  is  nsually  Galle4 
geometrical  proportion,  or  %im^\f  proportion.  One  important  property  of  s 
gttomftrieal  progression  is  tbis,  namely,  the  product  oC  the  tw«  extreme  tern* 
is  equal  to  that  of  any  two  terms  equally  distant  6om  tlw  cadrHDea :  hmos^  ia 


U»  IV.      GSOMETRrCAL  PROGRESSION.  8S 


'  K  Z  Z  Z 

And  z-\ 1 — 5-I--JJ  *c.  to-—-^  will  be  a  decreasing  geO' 

From  the  farmer  of  these  we  have  ar^'-'^szthe  last  term  of 
the  series,  hut  z^  the  last  term  by  the  notation,  wher^^e  ar"—  *=c2  ; 

from  this  equation  we  obtain  a=-j~-j,  (theor.   1.)   zs:iaf'-^ 


(theok.  2.)  r=~ 

a 


r 


(theob.  3.)   and  since  l:  riia+ar-^' 
«r* :  ar-^-ar^+ar^,  (Art  72.)  that  i*,  1 :  r :  r  i—x  :  s-^a,  therefore 

9-'-aszr,s~'Z.whencer= (theoe.4.)  a:=s—r^'~z  (theor.  5.) 

_  ^ — z^ 

^ — l.#+a-  -  rz— a  .    , 

xs (THEOR.  6.)  and  s= (theor.  7.)  out  smce 

r         ^  '  r — 1 

«=rar"— >  by  th.  2.  substitute  this  value  for  z  in  th.  7,  and  szs 

7-  (theor.  8.)  whence  a= (theor.  9.)  and  since  rr= 

^i3* ...    ^ .        ,        rz — a 


(th,  3.)  and  sss-. (th,  7.)  if  for  r  in  the  latter ^  its 


1^^   he      


«.±|--i-a 


value  —1"—'   be  substituted,  we  shall  have  *= 
a 


a 


(theor.  10.)  and  because  (th.  4.)  s—az^sr-^zr,  and  (th,  1.) 

z  z  .  . 

«= r,  therefore  (s — a=)  » --^isr-^zr.  or  sr-~s=i  (zr^ 


z        zr» — z     .  T*—  1.Z      ,                 r* — l.z    ^ 
"r-r=s    „    .  =)  — r-T-  *  whMmce  s=z (theor.  11.)  con* 

r^  1  !*"-»'  J 
sequenHy  ztsz ^^         (thbor.  12.) 

The  dhove  theorems  are  all  that  can  be  deduced  in  a  general 
manner^  without  the  aid  of  logarithms  in  some  cases^  and  of 
equatioDs  of  several  dimensions  in  others.    The  theorems  want- 
ing are  four  for  finding  n,  two  for  r,  one  for  a,  and  one  for  z  t 
the  fout  theorems  for  finding  the  value  of  n,  may  be  expressed 

four  proportioDals,  ihe  product  of  the  two  extremes  is  equal  to  the  product  of 
the'  two  means';,  and  in  three  proportionate^  the  product  of  the  extremes  if 
etpttl  to  the  ^tputt  ef  tile  liteall. 

6  9i 


^ 


84  ALG£BRA.  Pakt  IV. 

logarithmically;  the  remaiidiig  four  cannot  be  g^ven  in  a 
general  manner,  but  their  relation  to  the  other  quantities  maj 
be  expressed  in  an  equation,  by  means  of  which  any  particular 
value  will  be  readily  known. 

121.  We  proceed  then,  first,  to  deduce  the  equations  from 
whence  the  remaining  values  of  r,  a,  and  z,  may  be  found  in 
any  paiticular  case ;  next,  we  shew  how  the  theorems  found  are 
to  be  turned  into  their  equivalent  logarithmic  expressions; 
and  lastly^  we  shall  deduce  logarithmic  theorems  for  the  four 
expressions  of  the  value  of  n. 

Firsts   because  2=ar»— *  (th.  2.)  and  z= {th.  6.) 

sr^8'{-a 
therefore  ar^'-^^i ,  whence  ar"=fr— t +a,  or  ar»— sr=s 

rs     a—s  ,  ,^ .      ,.  ,   . 

a— «,  w  r* = (theor.  13.)  which  u  as  near  as  we  can  * 

a        a 

get  to  the  value  of  r,  and  which  (supposing  a,  s,  and  n,  given 

in  numbers)  if  n  be  greater  than  2,  will  require  the  solution  of  a 

high  equation  to  find  its  value, 

Secof{dly,  because  «— a=«r — zr^  {th,  4.)  and  (fh.  1.)  a=5 

z                                                z 
7,  therefore  (<— a=)  s -=ssr— zr,  and  zf^x=:sr*  — 

fP^vaal  V  \  f  |-TT  1 


z 


sr^-^^,  or  2— <.r»— «r*"-'s=— «r  to^ccr*— r"— *=— , 

z^s  z—s 

(theor.  14.)  this  equation  being  solved,  the  value  of  r  wiU  be 

known,  ^^ 

TUrtUy,  since  s—a=tr—xr,  (th.  4.)  and  r=— |*~S  («A.3.) 

a 

71.  _  . 


z  \  z 

therefore  s-^a^s — ■■— >— «. — 

a  1  a 


»— 1 


(theor.  15.)  by  the  solution  of  which  equation  («,  fi,  and  z,  beisig 

given)  a  will  be  found.  

Fourthly,  by  the  same  equation,  viz,  a,s — il"— '=2.4 — 2''— ', 
(theor.  16.)  s,  n,  and  a,  being  given,  2  will  likewise  be  known. 

1^^.  It  remains  now  to  put  the  above  theorems  into  a  loga- 
rilhmical  form>  to  place  the  whole  in  one  point  of  view,  and  to 
deduce  the  four  theorems  for  finding  the  value  of  n  :  observing 
that  to  multiply  two  factors  together,  we  add  their  logarithms 
together  3  to  divide,  we  subtract  the  logarithm  of  the  divisor 
from  that  of  the  dividend ;  to  involve  or  evolve^  we  multiply 


Pa«t  IV.      GEOMETRICAL  PROGRESSION.  85 

or  divide  respectively  the  logarithm  of  the  root  or  power  by  its 
index^  as  directed  in  Vol.  I.  Fart  2. 
Let  A- 


represent  the  logarithm  of 

And  L  the  logarithm  of  the  (juantity  to  which  it  is  prefixed; 
then  will  the  following  synopsis  exhibit  the  whole  doctrine  of 
geometrical  progression^  as  investigated  in  the  preceding  arti- 
cles i^. 


k  Some  of  the  foUowidg  logarithmic  expresftkmt  are  extremely  inconTenieiity 
particularly  theor.  10.  Th«  batt  method  of  computing  the  ?aloe  of  t  in  that 
theorem,  will  be,  first  to  find  the  log.  of  z,  subtract  the  log.  of  a  from  it,  add 
this  remainder  to  the  log.  of  z,  and  divide  the  sum  by  Hf—  ] ;  find  the  natural 
number  corresponding  to  the  quotient,  from  which  subtract  a,  and  find  the  log, 
of  the  remainder.  Secondly,  from  the  log.  of  2,  subtract  the  log.  of  a,  divide 
the  remainder  by  n-^  ],  find  the  natural  number  corresponding  to  the  quotient, 
subtract  I  from  it,  aad  subtract  the  log,  of  this  ^emahider  from  that  of  tho 
former;  and  thellM  ill  other  oases. 


QS 


■    V 


86 


Theor. 


II. 
VIII. 


VU. 
XVII. 

VI. 
XIX. 


w. 


xvm. 


Given. 


a,r,n 


a,  r,  z 


a,8^r 


Req. 


s 


n 


ALGEBB4. 


Solution  by  Numbers. 


Pav 


a,z,s 


XIIL 
XVI. 


III. 


X. 


I. 


XI. 


a,n,  8 


a,n,z 


r,  n,  z 


IX. 


XII. 


V. 
XX- 


XV. 
XIV.  I 


n 


n 


8 


z^zar^-^ 

r— 1 

rz — a 

'~r-l 

r 

«■ 

*— a 

• 

5—2 

.     r8     a'-^8 

a         a 

2.«-2l"-»=ai- 

ra|"-i 

I 


i^^"-»i>»<^^i^«^P^»«^^^pW 


Solution  by  Logarithms. 


Z=^A+R.n^l 


S^sA-k-  i.^—  I— X.n—  l.ttA«ra  JBi 


iSs^s  L.TZ  ^a'^Ls'^  1 


+  1 


Z=sl».r— 1^+a— fi 


nil  I  «— ^«». 


n=? 


JL-f— i.<»+a— -4 


As:  L.f'-^a  -^X.«*— « 


n=:- 


Z-A 


L.S — i— £.5— z 


+  1 


■I—* 


R= 


g.—l'-'-a 


<=' 


1^- 


a=- 


r»— 1 


r,w,  * 


r,  2,  « 


»,«,< 


n 


«•— ; 


r"— 1.Z 


«: 


r-l.r»-» 


Z^A 

n— 1 


5=:I..2.i]a-»-.a-.L.3*-*-.l 


-4=Z-.fi.n— 1 


r— 1.« 


a= 


.B.^1 


z= 


a=« — r.»— 2 


.1    I 


a.,— a^"-i=2iZ^"-. 


*— 2  «--2 


5=  t.?*— 1  +  Z— I..r— I  +  JB.n-] 


^=I..r— 1+5— R.r"— 1 


Zrsli.r-.  1+ jR.«— 1  +S-.X.r"-i 


-4=£.«— r.#— .« 


RUt  IV.      GEOMETBJiGAL  FBOGKBSSION.  m 


L  To  sliew  how  tie  17^,  IBth,  lOfhi  snd  9(Hh  th^rana 
are  derived. 

Z— -el  . 

«=— ^  +  1    (TfliOK.  17;)  and  because  R=zLa^a^Lj^t 

(th.   4.)  suhsiituie  this  value  for  R  in  theor.  I7.  and  ns 

2 ^ 

7 =  +  1  (thbok,  18.)  again,  for  Zin  theor.  17.  sub' 


»■  « 


ifillifi^  ii«  raZtte  /row  theor,  6.  aHct  »!=:  (— ^^^^ '         « ^  1 

*=) ^ (thkor.  19.)    Lastly,  for  A  xa  theor.  17. 

2 A 

substitute  its  value  from  theor.  5.  and   n=(--^ — 1-1=) 

— ^t h  1.  (theob.  20.) 

£tAMPLE8.-^l.  Given  the  ratio  %  the  number  of  terms  6, 
and  the  last  term  96>  of  a  geometrical  progression^  to  find  the 
first  term>  and  the  sum  of  the  term^  ? 

Bete  rss^,  ttss^,  zae96>  whence  (theor.  1.)  as:— ^=: 

By  IiOgarithnH). 

Z:=... 1 .9822712 

g.n— lasO^SOlOSOOx  6atl.6051500 

«-*-f*-*=aa3 0.4771212 

pfience  as=3« 

I.1*— lssX.2«-l=£.6S=1.799S406 

+2ai;N9gg 1.9822712 

X.f-l+Zs:.,: 3.7816117 

^  f  Lrr-lsLass 0.0000000 

1  +RM^lsiL.^  X  5=  ... .  1.5051500 

<« 2.2764617 

fshence  ssslB9t 
04 


88  .  ALQEBRA.  Part  IV. 

2.  Given  the  ratio  2>  the  number  of  terms  6,  and  the  sum  of 
the  terms  189>  to  find  the  first  and  last  terms  ? 

Here  rsz%    nss6,  J33^89^  and    (theor.  9.)   «=-; — r  = 

1x189    V    1Q9    ^  •      r— l.f--'^  ,  . 

(-—r— —=)   -^=3  J    alsQ  «= — - — : —  (Umr.    12.)= 

^2«— 1      '    63  r"— 1        ^  ' 

1 X  2*  X  189     ,32  X  189       ^ 

^      2^-1       =)nS3-=^^' 

By  Logarithms. 


L.r— 1= 0.0000000 

4-^= 2.2764617 

—  L.r'— 1= 1.7993405 

-^= 0.4771212 

whence  a=3. 


X.r— 1 +  5.11—1= ...  1.5051500 
4-iS= 2.2764617 

I..r-1  +  fi.n— 1  +  5=  3.78161 17 

-Lr*— 1= 1.79984(^ 

Z= 1.9822712 

whence  zss96. 

3.  Given  the  first  term  3^  the  ratio  2>  and  the  last  term  96» 
to  find  the  number,  and  sum  of  the  terms  ? 

vr           ^        ^          ^        ,     ,           V  r«— «     ,2x96—3 
Here  a=3,  r=2,  2=96,  and  (theor.  7.)  — -r  =  { ; 

=)  189=». 

By  Logarithms. 


Z= 1.9822712 

—-4= 0.4771212 

-♦-iJ=0.3010300)  1.5O515Cl0(5 


L.rz— a=I..189=  2.2764617 
— L.r— IssJL.ls      Q.000000O 

5=      2.2764617 
whence  «=189. 


therefore  n= 5  + 1 = 6,  theor  17. 

4.  Given  the  first  term  4,  the  ratio  3,  and  the  sum  of  the 
terms  484,  to  find  the  last  term,  and  number  of  terms  ? 

Here  a=4,  r=3,  «=484,  and  {theor.  6.)  2=^""    '     ■  = 


^        3  'a 


Logarithi 


L.r— l.«+a=L.972=2.9876663 
— ie= 0.4771212 

Z=  . . .  fl.5 105461 

whence  z=324. 


L.r— 1.5+a=L.972=2.9876663 
--4= 06020600 

-i-i2=. . . .  0.4771212)2.3856063(5 

whence  n=5,  ^Aeor.  19. 


PiBT  IV.      GEOMETRICAL  PROGRESSION.  89 

5.  Given  the  first  term  %  last  term  2048,  and  sum  of  the 
terms  2730,  to  find  the  ratio,  and  number  of  terms  ? 

vHere  a=:2,  2=2048,  «=2730,  and  Uheor,  4.)  r=-^^— ss 

2730-2    _   2728  _ 

^730-2048""^  682  ""  ' 

By  Logarithms. 


2=3.3113300 

—-4=3.3010300 

Z— -4=3.0103000 


I.«-a=L.2728=3,4358444 
-L.*-.z=  1.682=2.8337844 

B= 0.6020600 

whence  r=4. 

L.Jira=L.2728=3.4358444 
-.L.»^=  X.682=2.8337844 

X,  .THi— i,.«-.z=a602oeoo 

therefore  .6020600)3.0103000(5 
whence  n=5  +  l=6,  theor.  18. 

6.  Given  r=4,  n=:6,  and  ^=2730,  to  find  a  and  z.  iliu.  a= 
2,  z=2048. 

7.  Given  rsx2,  n=6,  and  z=96,  to  find  a  and  «.    .4n«.  a=3, 

«=189. 

8.  Given  the  ratio  5,  last  term  12500,  and  sum  of  the  terms 
15624,  to  find  the  first  term,  and  number  of  terms.  Ans.  a =4, 
n::ze. 

9.  Given  a=:4,  n:=:6,  and  z=:  12500,  to  find  r  and  t.  Answer 
r=5,«=  15624.  • 

10.  Given  r=3,  n=4,  and  z=81,  to  find  a  and  *. 
ll.^Given  r=i6,  w=5,  and  «=1555,  to  find  a  and  z. 
12.  Given  a=3,  r=10,  and  n=20,  to  find  «  andz. 

124.  PROBLEMS  IN  GEOMETRICAL  PROGRESSION. 
1.  Of  three  numbers  in  geometrical  progression,  the  difference 
of  the  first  and  second  is  4,  and  of  the  second  and  third  12  j 
required  the  numbers  ? 

Let  X,  y,  and  z,  be  the  numbers. 

Then  y— «=4,  or  xz=zy—4',  z—y=  12,  or  z=y4-12. 

Wherefore  since  by  the  problem  x  :y::y:z,by  substituting 
the  values  of  xaadz  in  this  analogy,  we  shall  have  y — 4  :  y  :  :y: 
y+ 12  5  wherefore,  (by  multiplying  extremes  and  means,)  y— 4 
.y+12=)  y*+8y— 48=y*,  or  8y=48;  wherefore  y=6,  ar=2, 
«=18. 


M  ALGEBBA.  Part  IV. 

%.  The  product  of  three  numbers  in  geptnetricfd  ^rogfesftion 
is  1000^  and  the  sum  of  the  first  and  last  25  5  required  iht 
numbers  ?  « 

Let  X,  y,  and  x,  be  the  numbers ;  then  since  xiyiiyiZjwe 
have  xz=iy^,  {Art,  120.  Note,)  and .  {xyzzsixz.yss)  ^^ssiooa, 
whence  ^=10;  also  xzTz(y^=)  100^  and  by  the  problem  X'\-z^ 
25 :  from  the  sqwire  of  this  equation  subtract  four  times  the  pre" 
ceding,  and  x*— 2x2+2*— 225:  extract  the  square  root  of  this, 
and  X — 2=15  5  add  this  to,  and  subtract  it^from,  the  equaHixm 
x-f  2=25,  and  2x=40,  or  x=20,  also  2 z=10,  or  2=5  j  whence 
5^  10^  and  20,  are  the  numbers. 

3.  To  find  any  number  of  mean  proportionals  between  two 
given  numbers  a  and  b. 

Let  n— 2=i/ie  number  of  mean  proportionals,  then  will  n= 
the  number  of  terms  in  the  progression :  also  let  r=  the  ratio,  then 

(theor.  3.  Geom,  Prog.)  r= — 


B-.1 


5  and  by  logarithms,  log.  b — hg.  a 


H-n— 1=20^*.  r ;  whence  r  being  found,  if  the  less  extreme  he  coff- 
tbaudiy  muUvpUed,  or  ike  greater  divided^  6y  r>  ifte  retsUU  miU 
he  the  mean  proportionals  required, 

BxAMPLKs.— »1.  To  find  two  mean  proportionate  betw^n  12 
and  4116. 

4116)7         ^ V 

Here  ac:12,  6=c4U6,  ♦»=4,  and  r=r{J±J  =3431t=)7  ; 

12  ' 

whence  12x7=84>  the  first  nXan,  and  84x7=5S8>  the  secofid 

mean, 

2.  To  find  four  mean  proportionals  between  2  and  48^.   An». 
6,  18,  54,  and  162. 

3.  To  fibd  five  mean  proportionals  between  1  and  G4* 

4»  There  are  four  numbers  in  geometrical  progresBion'^  the 
^uiB  of  the  extremes  is  9,  and  the  suqqei  of  the  cubes  of 'th» 
means  72 }  what  are  the  numbers  ? 
Let  X,  y,  u,  and  z,  be  the,  numhers. 
Then  by  thepix>blem, 

arH-2=9,  or  x.=9*— 2. 
X  :  y  : :  m.:  2,  op  xz^uy,  whence  xz=  (9— 2.2=). 9 2—^2*. 

x'.ywyiu^or xu^y^ (J?«y=)  xH^y^, 

y:  u::u:z,or  zy=zu'* ...  {zyuss)  X2*=«i 

%tt  {xz.x+zsz)  92— *«.9=X*2+XZ«. 


>lnf  the  problem. 


l^AXTlF.  COMPOUND  INTEREST.  91 

^n4  (y^+fi'ss)  T^sssfla^+xz^,  and  things  that  are  egico/  to 
the  same  are  equal;  therefore  9«— «*.9=72,  or  9z— z*=8,  or 
2>— .92=^8;  iphence  bff  oowtpleting  the  square,  4rc.  zss:S,  xss 

(9-2=)  1,  y=(V^*«)  «*  tt=(V^«*=)  4. 

5.  Of  foiff  numbers  in  geometrical  progRtsion,  tbe  product 
of  tlie  two  least  k  8,  and  of  the  two  greatest  1S8  j  what  are 
the  numbers  ? 

Let  X,  y,  «,  and  x,  be  the  numbers. 

^  8 

Then  xy^B,  or  xss^r- 

y 

198 

VflsiVie,or  z:s. 

u 

8  198 

ORCsttif^  or  — . sstttf 

y    tt 

2%6rc/are  (8  x  128=)  1024=ttV,  or  uy=3%  and  «=—. 

f 
8  39 

J?a^  (x :  y  : :  y :  ttj  that  is,)  —  :  y : :  y  :  — ,  where  miuUipUfing 

956 
extremes  and  means,  y^s— j>,  or  y^=956i   whence  y=4^  a:= 

8  39  198 

( — =)9,  tfss  ( — =)  8,  z=( — =s)  16,  (i^ntun^tf  required^, 

6.  The  sum  of  3  numbers  in  geometrical  progression  is  14, 
and  the  greater  extreme  exceeds  the  less  hj6;  what  are  the 
numbers  ?    Ans.  %  4>  and  8. 

195.  Def.  Compound  Interest  is  that  which  is  paid  for  the 
«se»  not  only  of  the  principal  or  sum  lent,  but  for  both  princi-^ 
pal  and  interest,  as  the  latter  becomes  due  at  the  end  of  the 
year,  half-year,  quarter,  or  other  stated  time. 

To  investigate  the  rules  of  Compound  Interest, 

Let  p:=sthe principal,  r^r:the  rate  per  cent,  t^the  time,  i2= 
(14-r=)  the  amountoflLfor  a  year,  called  the  ratio  of  the  rate 
ppr  cent,  a^the  amount. 

Then  since  1  pound :  is  to  its  amount  for  any  given  time  and. 

rate : :  so  are  any  number  of  pounds :  to  their  amount  for  the  sam^ 

time  and  rate^  therefore  as 

p     ipRssthefost, 
pR  I  pR^:=  second,    | 

>  year's  amou^t,. 


p     ipB^the  first 
ipR  I  pB^:=  second, 
I:  R::2pE^:  pR^szthird, 
\  pB? :  pR^^fourth, 


92  ALGEBBA.  Part  IV. 

Whence  we  have  theorem   1.  pR^=a,  theor.  2.  ~=p, 

theor.  3.  V^=^.  THEOR.  4.  ^f^^^^^S'P^^     ^j^  ^y^^  ^^^^ 
P  log.  R  * 

of  which  follow  immediately  from  the  first;  the  fourth  cannot  be 
conveniently  €xhU}ited  in  nutnbers  without  the  aid  of  logarithms. 

By  means  of  these  four  theorems,  all  questions  of  compound 
interest  may  be  solved. 

Examples.— 1.  What  is  the  amount  of  1250i.  lOu.  6d,  for  5 
years,  at  4  per  cent,  per  annum,  compound  interest  ? 

Here  p:sz(UBOl.  lOs.  6d.=)  1250.525,  ^=5,  J«=±1.04. 
Thentheor,  1. (p/J*=)  1250.525 x  foS^s:  1250.525 x  1.2166 
.  =1521.388715=1521/.  7«.9^.=a. 

2.  What  principal  will  amount  to  200Z.  in  3  years,  at  4  per 
cent,  per  anniun  ? 

Here  ar=200,  JR=1.04,  teS,  emd  theor.  2.  (^=)  ?^  = 

1.124864  =^7y«y^92=17y/.  155.  U^d.^ip. 

3.  At  what  rate  per  cent,  per  annum  will  500i.  amount  to 
578/.  I6s.  3d.  in  3  years  ? 

Here  p=500,  fl=(578/.  16*.  3d=)  578.8125,  ^=3;  and, 

^r        «   /♦      fl     V  •     578.8125        1 

theor.  3.  (V-=)V      gQQ  "=(y  V144.7031.  5ee  FoZ.  J. 

P.  3.  ^r*.63.=)yx5.25=1.05=12..  te^^orc,  (*ince  jR-l 
=r,)  we  Aare  fi— l=.05=r,  «w.  5  per  cent,  per  annum. 

4.  In  how  many  years  will  225Z.  require  to  remain  at  interest, 
at  5  per  cent,  per  annum,  to  amount  to  260/.  9s.  3^d.  ? 

Here  p=225,  -R=1.05,  a=(260/.  9s.  34d.=)  260.465625; 

whence,  theor.  4.  (^t^-^^P^^S-  260.465625- fo^.  225 

^g'  R  log.  1.03  ■"' 

2.4157506-^2.3521825     0.0635681 

0.021 1893  ""0.0211893  "^^  ^^"'"'^  ^• 

5.  What  sum  will  500/.  amount  to  in  3  years,  at  5  per  cent. 
per  annum  ?    Ans.  578/.  16«.  3d. 

6.  What  principal  wiU  amount  to  1521/..7*.  9id.  in  Syeare, 
at  4  per  cent,  ptr  annum  ?    Ans.  1250/.  lOs.  6d. 


Part  IV.       PROPERTIES  OF  NUMBERS.  93 

7.  At  what  rate  per  cent,  will  7912.  amount  to  16421.  I99.9id. 
in  21  years  ?    Jm.  4  per  cent 

8.  In  how  many  years  will  7^11.  be  at  interest  at  4  per  cent, 
to  amount  to  1642/.  I9s,  9^d.    Ans.  21  yean. 

If  the  interest  be  payable  half-yearly,  make  ^ssthe  number  of 
half-^years,  that  isstwice  the  numbir  of  years,  and  r=:half  the 
rate  per  cent,  but  if  the  interest  be  payable  quarto*]?,  let  lasthe 
number  of  quarter-years^  viz.  4  times  the  number  of  years,  and 
r=one-fourth  of  the  rate  per  cent,  and  let  JRsr-f- 1  in  both 
cases,  as  before  ^ 

126.  To  determine  some  of  the  most  useful  properties  of 

numbers. 

Def.  1.  One  number  is  said  to  be  a  multiple  of  another^  when 
the  former  contains  the  latter  some  number  of  times  exactly, 
without  remainder. 

Thus  12  t«  a  multiple  of  I,  2,  3,  4,  and  6. 

CoR.  Hence  every  whole  number  is  either  unity,  or  a  multiple 
of  unity. 

2.  One  number  is  said  to  be  an  aliquot  part  of  another,  when 
the  former  is  contained  some  number  of  times  exactly  in  the 
latter. 

Thus  1,  2,  3,  4,  and  6,  are  aliquot  parts  of  12,  for  1  is 
tV,  2  is  ^,  3  w  ^,  4  M  4^,  and  6  is  ^  of  12. 

Cor.  Hence  no  number  which  is  greater  than  half  of  another 
number^  can  be  an  aliquot  part  of  the  latter. 

3.  One  number  i»  said  to  measure  another  number,  when  it 
will  divide  the  latter  without  remainder. 

Thus  each  of  the  numbers  1 ,  2,  4,  5, 10,  and  20,  measures  20. 

4.  One  number  is  said  to  be  measured  by  another,  when  the 
latter  will  divide  the  former  without  remainder. 

Thus  20  is  measured  by  1,  2>  4,  5,  10,  fsnd  20. 
Cor.  Hence  every  aliquot  part  of  a  number  measures  that 
number,  and  every  number  is  measured  by  each  of  its  aliquot 
parts,  and  by  itself. 

^  It  was  at  first  intended  to  investigate  and  apply  every  rule  in  aritbmeticy 
but  want  of  room  obliges  us  to  omit  Equation  of  Payments,  Loss  and  Gain, 
Barter,  Fellowship,  and  Exchange;  these  will  be  easily  understood  from 
the  doctrine  of  proportion,  of  which  we  have  amply  treated. 


^ 


M  ALGEBRA.  Past  IV. 


6.  Any  nttmbtr  which  lAesiftttret  two  or  mor^  numbers^  is 
called  their  common  measure;  aM  the  greatest  nuttiber  tbftt 
will  raeasttre  theoi^  is  cslM  ih^it  greatest  conmion  measure. 

Thus  1,  2,  3,  and  6,  are  ihe  common  measures  cf  12  and  18  i 
mtd  6  tf  thevr  greatest  common  measure. 

Cot.  Heoce  the  greater  common  m«asmre  of  several  num^ 
bers  cannot  be  greater  than  the  least  of  those  numbers  \  and 
when  the  least  number  is  not  a  common  measure,  the  greatest 
cdomoQ  measure  caiinot  be  greater  than  half  the  least.  Def.  ^. 
cor. 

6.  An  even  number  is  that  which  can  be  divided  into  two 
equal  whole  numbers. 

ThMs  6  is  an  even  number,  being  divisible  into  two  equal 
whole  numbers,  3  and  3,  8se. 

7.  An  odd  number  is  that  which  cannot  be  divided  into  tw6 
equal  whole  numben }  or^  which  differs  from  an  even  number 
by  unity.     Thus,  1»  3,  5,  7,  &c.  are  odd  numbers. 

Cor.  Hence  any  even  number  may  be  represented  by  2  a^  i^nd 
any  odd  number  by  2  a+ 1,  or  2  a— 1. 

S.  A  prime  number  is  that  which  can  b6  measured  by  itself 
and  unity  only  \ 

Thus,  I,  2,  3,  5,  7,  1 1,  13,  17,  19,  23,  &c.  are  prime  num- 
bers. 


1  Hence  it  appears,  that  no  even  nniiiber  except  3  can  be  a  prime,  or  thai 
all  primes  except  3  are  odd  ttumben ;  Imt  it  doea  not  fbttow  that  all  the  odd 
numbers  are  primes :  every  power  of  an  odd  nniibcr  ia  odd,  odaseqiieBtly  the 
powers  of  all  odd  kwmbers  greater  than  1,  after  the  first  power,  will  be 
composite  numbers. 

Several  eminent  mathematicians,  of  both  ancient  and  modem  times,  have 
made  fruitless  attempts  to  discover  some  general  expression  for  finding  the 
prime  numbers :  if  n  be  made  to  represent  any  of  tbe  nambers  1,  2,  3,  4,  &c. 
then  will  all  the  taDtes  of  6  n  + 1  •»!  6  n-^  I  constitute  a  series,  including  all 
the  primes  above  S;  but  this  series  will  have  some  of  its  terms  composite 
numbers:  thus,  let  ns=I,  then  6ii+l»7  and  6ft— l«B5y both  primes;  if 
n=2,  then  6n 4- 1  =  13,  and  6 n—  1 » 1 1 ,  both  primes  ;  if  iib3,  then  6n+l 
=  1.9,  and  69t— 1 » 17,  both  primes,  Sec.  Let  »s6,  then  6ft-|- 1  ssST  a  prime, 
but  6  }i— 1  s35  (::35  X  7)  a  composite  number;  also  if  irsg,  then  6ii-{->  1 »" 
49  a  composite  number,  and  6  n — 1  se47  a  prime,  Stc.  For  a  talble  of  wB  tbei 
prime  numbers,  and  all  the  odd  composite  numbers,  undcfT  10,000,  see  j^. 
HuttmCs  MathemaHcal  Dtctionafy,  1795.  Vol.  H.  p.  276,  378. 


Sair  1%         FROPfiRTlSS  Q^  NUMBERS.  9h 

9.  Namben  are  said  to  be  prime  to  each  #dier,  when  unity 
IS  their  gi-eatest  common  roeasture  ». 

Thus,  11  and  26  are  prime  to  each  other,  fm'  no  uwmber 
greater  than  1  will  divide  both  without  remainder, 

la  A  composite  number  is  ^atwhkh  is  measured  bf  any 
ownber  greater  than  unity. 

Thus,C  i9  a  composite  mmber,for  %  and  3  wiU  each  meeh 
mreit. 

Cob.  Hence  every  composite  number  will  be  measured  by 
two  numbers :  if  one  oi  these  numb^B  be  known^  the  oflMf 
wiU  be.  the  quotient  arising  from  the  division  of  the  eottiposite 
Dumber^  by  the  known  measure. 

Thus,  6=3  X  2,  and-^-z^^y  also  -^=2. 

2-  3 

11.  The  component  parts  of  any  number,  are  the  numben 
(eacb  greater  than  unity)  which  multiplied  toget^er^  produce 
that  number  exactly. 

Thus,  2  and  3  are  the  component  parts  of  69  for  2x3cb6; 
3,  4,  and  5  are  the  component  parts  of  60,  for  3  x  4  x  53=60,  &c. 

12.  A  perfect  number'*  is  that  M^iiefa  is  equal  to  the  sum  of 
all  its  aliquot  parts. 

■  Nombcn  which  are  priaie  to  erne  another,  mre  not  aeceMarily  pritme$  in 
the  sense  of  def.  8.  thus  4  and  15  are  composite  nnmbers  according  to  def.  10. 
bnt  they  are  prime  to  each  ethers  since  unity  only  will  divide  both.  Hence  two 
even  nujjabers  cannot  be  prime  to  each  other. 

In  the  Scholai's  Guide  to  Arithmetic,  7th  Ed.  p,  104.  9.  it  is  asserted, 
tiat "  If  a  number  cannot  be  divided  by  some  nnmber  less  than  the  square  root 
thereof,  that  nnmber  is  a  pnmc."  Now  tbia  cannot  be  troe ;  for  neitber  of  the 
sqaavs  nnmbers  &»  3&9  49>  4fe.  fte.  can  be.  neaturcd  by  any  number  Icaa  than 
its  square  root,  and  yet  these  numbers  are  not  primes :  a  slight  alteration  in 
tbe  wording  will  however  make  it  perfectly  correct ;  thus,  *<  If  a  number  which 
is  fM  a  Sfuair09  cannot  be  divided  by  some  number  less  than  the  square  root 
thereof,  that  nnmber  is  a  prime.**  This  interpretation  was  undoubtedly  in^ 
tended  by  the  learned  author,  akhongh  his  words  do  not  seem  to  warrant  it. 

■  The  IbUowing  table  is  said  to  coatain  all  the  pex&ct  namben  at  present 

6  8589869056 

88  IS7438691328 

406  2305843008 1399^1^ 

8128  S4178516398381.58837784576 

33550336  9903530314283971830448816128 

These  nnmbers  were  extracted  from  the  Ada  of  the  Petersburg  Academy,  in 

several  of  the  Tolnmes  of  which^  Tracts  on  the  subject  may  be  feond* 


96  ALGEBRA.  Past  It. 

\ 

Tims,  6  is  a  perfect  number,  for  its  aliquot  parts  ute  !(= — 

6 

of  6)  2  (=—  of  6)  andS  (=--  of  6)  and  1+2  +  3=6. 

13.  An  imperfect  number  is  that  which  is  greater  or  less 
than  the  sum  of  its  aliquot  parts ;  in  the  former  case  it  is  caUed 
jan  abundant  number,  in  the  latter,  a  defectine  nunU^er. 

Thus,  8  and  12  are  imperfect  numbers;  the  former  (viz.  8) 
is  an  abundant  number,  its  aliquot  parts  being  1,  2  and  4,  the 
9um  of  which  l-h2+4=:7>  is  less  than  the  given  number  8.  7%e 
loiter  (viz,  12)  is  a  defective  number,  its  aliquot  parts  beia^  I,  % 
3,  4,  and  6,  the  sum  of  which,  vix,  16,  is  greater  than  the  given 
number  12. 

14.  A  pronic  number  b  that  which  is  equal  to  the  sum  of  a 
square  number  and  its  root 

Thus^  6,  12,  20,  30^  8sc,  are  pronic  numbers;  for  6=s(4+ 
^4=)  4+2;  12=(9+^9=)  9+3 5  20=:(16+Vl6=)  16 
+  4i  30s=(25+  V26=x)  25  +  5,  *c. 

Property  1.  The  sum^  difiference^  or  .product  of  any  two 
whole  numbers^  is  a  whole  number.  This  evidently  follows  from 
the  nature  of  whole  numbers,  for  it  is  plam  that  fractions  cannot 
enter  in  either  case, 

'    CoK.  Hence  the  product  of  any  two  proper  fractions  is  a 
fraction. 

2.  The  sum  of  any  number  of  even  numbers  is  an  even  number. 
Thia,  let  2  a,  2  b,  2  c,  8fc,  be  even  numbers,  (See  def,  7*  cor.) 
Then  2a+2&+2c+,  ^c,z:^their  sum;  but  this  sum  is  eoi- 

dently  diioisihle  by  2,  it  is  therefore  an  even  number;  def,  6, 

CoR.  H^[ice  if  an  even  number  be  multiplied  by  any  number 
whatever,  the  product  will  be  even. 

3.  The  sum  of  any  even  number  of  odd  numbers  is  an  even 
number. 

Thus,  (def  7.  cor.)  Iet2a+h  2  6+ 1,  2  c+ 1,  and  2  d+ 1, 
be  an  even  number  of  odd  numbers. 

Then  will  their  sum  2  a+2  6+2  c+2  d+ 1  + 1  + 1  + 1,  6e  m 
even  number;  for  the  former  part  2a+26+2c+2d  is  even,  by 
def  6.  and  the  latter  consisting  of  an  even  number  of  units  is  like* 
wise  even ;  wherefore  the  mm  of  both  will  be  even,  by  property  2. 
Con,  Hence  if  an  odd  number  be  added  to  an  eveo>  the  sum 
will  be  odd. 


fhRT  ly.         PROPERTIES  OF  NUBfBERS.  9f 

4.  The  sum  of  any  odd  number  of  odd  nuinben»  is  an  odd 

number. 

For  let  ^a-^l,  2  6-4-1,  Sc+1,  be  an  odd  number  of  odd 
numbersy  then  2a+2  6H-2c+l+l  +  l==<A«ir  9um,  the  former 
part  of  which  2a+26+2c,  being  divisible  by  2,  {def  6.)  a  an  even 
number,  and  the  latter  part  1  +  1  +  1,  comisting  of  an  odd  number 
of  units,  is  odd  :  now  the  sum  of  both,  being  that  of  an  eten  num- 
ber added  to  an  odd,  wiU,  by  the  preceding  corollary,  be  an  odd 
number. 

5.  The  di&rence  of  two  eren  numbers,  will  be  an  even 
number. 

For  let  2  a  and  2  6  6e  two  even  numbers,  then  since  2  a->2  b 
and  2  6+2  a  will  each  be  divisible  by  2,  it  is  plain  that  the  difftt- 
rence  of  ^  a  and  2  6  wUl  be  even,  whichever  of  them  be  the- 
greater, 

6f  The  di£Eerence  of  two  odd  numbers  is  even. 
jFbr  let  2a+l  and  2  6+1  be  two  odd  numbers,  whereof  the 
former  is  the  greater;  then  stftc«2a+l— 2 6+  ls2a— 2 bis  the 
proposed  difference,  which  is  divisible  by  2,  it  is  therefore  an  even 
number. 

7.  The  difference  of  an  even  number  and  an  odd  one  will  be 
odd,  whichever  be  the  greater. 

Let  2  a  be  an  even  number,  2  6+ 1  an  odd  number  greater 
than  2  a,  and  2  c+1  on  odd  number  less  than  2a;  wherefore  (2  6 
+  1—2  a=)  2  6^2  a+ 1  ss  efte  difference,  supposing  the  odd  num- 
bet  to  be  the  greater ;  and  (2— 2c+l=)  2  a— 2  c-^l=sthe  diffe- 
rence, supposing  the  even  number  the  greater.  Now  each  of  these 
differences  differs  from  the  even  numbers  26— 2  a,  or  2a— 2c  6y 
unity :  the  difference  therefore  in  both  cases  is  an  odd  number. 

9.  The  product  of  two  odd  numbers  is  an  odd  number. 

For  fel  2a+ 1  and  2  S+ 1  6e  any  two  odd  numbers,  then  wiU 
(2a+1.26+l  =  )  4ab-^2b+2a-{-l=:iiheir product ;  butthesum 
of  the  three  first  terms  is  evidently  even,  being  divisible  by  2,  cmd 
the  tohole  product  exceeds  this  sum  by  unity,  the  product  is  there' 
fore  an  odd  number,  (def.  7 .) 

0.  If  an  odd  number  measure  an  odd  number,  the  quotient 
will  be  odd. 

For  let  a  + 1  be  measured  6y  6+ 1,  and  let  the  quotient  be  q ; 

(J  J.  \  

*fcw,  7 — -=9  5  then  will  bssl.qssa+i  ',    and  since  6=1,  apd 

O  "^  JL 

VOL.  II.  H 


9S  ALGEBRA.  Fait  it. 

d-f  1  are  odd,  it  is  plain  that  q  must  he  odd,  othervnse  an  odd 
number  multiplied  by  an  even  number,  would  produce  an  odd  num- 
ber, which  is  impossible,  (proper.  2.  cori) 

10.  If  an  odd  number  measure  an  even  number^  the  quotient 
will  be  even  •. 


2a 


fibrtet— — =g,  then2b+l.q=i2a;  and  since  2fc+l  is 


^  Mr.  Boanycastle,  in  treating  on  this  subject,  (Scbolar's  Gaidey  5th  Eifit.  p> 

S03.)  has  committed  a  tiifting  oYeniglit.    Plop.  10.  in  hit  book  is  as  ioUnirc  i 

"  If  an  odd  or  even  number  measures  an  even  one,  the  quotirat  will  be  even." 

The  fermeir  p<Miit«on  is  here  shewn  to  be  true,  but  the  latter  is  evidently  £ilse« 

namely,  "  if  an  even  number  measure  an  even  number,  the  quotient  is  even.** 

2a 
In  proof  of  his  assertion  be  says,  « let  r-r*  q ;  then  2  (.9«t8  a  ;  and  siaoe  ftm 

and  2  b  are  even  numbers,,  q  must  likewise  be  an  even  number."    This  oenso^ 

qnence  however  does  not  necessarily  follow ;  q  may  be  either  even  or  odtf,  for 

any  even  number  (2  b)  multiplying  any  odd  number  (q),  will  evidently  pro- 

duce  an  even  number.    (See  proper.  3.)    Henoe  the  quotient  of  an  «vea  nwa- 

8 
ber  by  an  even  number,  may  be  either  even  or  odd  ,•  thus,  ~=*4  an  even  num^ 

dfr;  but  -rr^S  anoddnnmber.    Mr.  Keith  has  fidlen  into  the  fame  error, 

or  (whicfi  is  more  probable)  has  copied  it  from  the  above  work.    See  his  Cbm* 
plete  Practical  jirithmeticiany  3d  Edition,  p.  283.  Cor.  to  Art.  S2. 

The  first  named  Author  is  likewise  mistaken  when  he  says,  (Prop.  II.)  ''  If 
nn  odd  or  an  even  number  meaaiires  an  even  one,  it  will  al«o  measure  the  half 
of  it."    Now  the  half  of  any  number  will  evidently  measure  the  whole,  and  the 
half  measures  itself,  that  is,  it  is  contained  once  in  itself;  wherefore  it  follows, 
according  to  the  tenor  of  the  reasoning  there  employed,  that  if  one  quantity  be 
contained  once  in  another,  the  former  quantity  measures  the  latter,  but  the 
whole  is  contained  once  in  the  whole,  and  therefore  measures  it :  but  what- 
ever measures  the  whole  meastures  its  half,  says  Mr.  B.  whereiore  the  whoU 
must  necessarily  measure  the  half!   Thi<  nftiittfce  seems  to  have  arisen  from  »- 
circumstance  which  might  easily  have  happened—that  of  confounding  the  idea 
of  a  measure  with  that  of  an  aliqtcot  part :  bad  it  been  said  that  every  aliquot 
part  of  the  whole  measures  the  half,  ^^^  assertion  would  have  been  perfectly 
accurate.    Should  the  freedom  of  the  above  remarks  require  an  apology,  I  feel 
it  necessary  to  testify  my  unreserved  admiration  of  the  eminent  talents  of  the 
teamed  and  respectable  authors  in  qaestion,  and  to  assure  them  tibat  nothing 
invidious  can  possibly  be  intended :  but  truth  is  the  grand  object  of  the  sciences, 
^nd  he  who  is  engaged  in  the  arduous  and  important  office  of  instruction,  forfeiti 
alt  claim  to  fidelity  and  confidence,  if  he  does  not  point  out  error  wherever  he 
may  happen  to  find  it ;  and  he  is  scarcely  less  blameable  who  omits  to  do  it  with 
becoming  caqdour,  and  under  a  sense  of  his  own  fallibility. 


hmT  IV.         PROPERTIES.  Of;  NUMBERS.  .  » 

4ui  odd  ftufnto-,  wkd^mm  eem  m$,  iifbU9w$  thai  q  rniUl  b$ 
m>en ;  f^henoise  the  product  of  two  odd  mumb^n  mould  6t  km*^ 
tDhich  u  impossible,  (proper.  8.) 

11.  An  even  aumber  caoxiot  measure  aa  odd  oamber. 

2a+l 
Jf  possible,  let      ^  ^  ■■=?;  wherefore  V{a-|-I=S6.g.*  hut 

since  2  b  is  an  even  numbeff  2  b.q  is  also  even,  (proper.  2.  cor.) 

that  is,  an  odd  number  (9  a+ 1)  »  eguoZ  to  an  even  one,  (8  b,q,) 

which  is  absurd ':  wherefore  an  even  number,  8fC^ 

I'd.  If  one  nurmber  measure  another^  it  will  measure  everj 

multiple  of  the  latter. 

*  fl  na 

Let  nssas^  idude  number,  and  -r^qf  ^^^  ^^  T^*^' 

But  since  ^  is  by  hypothesis  a  whole  number^  nq  must  be  a  whofe 
number,  (proper.  1 .)  thai  is,  b  measures  n  times  a. 

13.  That  number  which  measures  the  whole,  and  also  a  part 

of  another  number,  will  likewise  measure  the  remainder. 

a-^-b  a 

Fbr  let asid  --*  be  each  a  mhelU  number. 

c  c 

Then  wiU  (- — =)  —  be  a  whale  number,  (pnfper.l.} 

^     C  C  .0  \r     r  ^ 

14.  If  one  number' measure  two  other  numbers^  it  will  like* 
wise  measure  their  siun  and  diffiifenCe. 

Let  e  measure  bo^  a  mtd  k^  tibea  wiU  —  and  —  be  both 

c  c 

a       b         ci-f-5  a       b      . 

whole  numbers  z  wherefore  ( — | — =) ,  and  ( =) 

•^         ^  c       c    '        c  ^cc 

~I^,  will  also  be  whole  numbers,  (proper.  I.) 
c 

CoR.  Hence  the  commoti  measure  of  two  numbers  will  like* 

wise  be  a  common  measure  of  the  sum  and  di&rence  oi  SBf 

multiple  of  the  one,  and  the  other. 

Thus,  if ,  and <-,  he  whole  numhers,  then  w%U 

c  c  c 

and  "^ be  whole  numbere. 

c 

15.  If  the  greater  of  two  numbers  be  divided  by  the  leas, 
and  if  the  divisor  be  divided  by  the  remtdnder,  nhd  the  last  di« 
▼isor  by  the  last  remainder  continually,  until  nothing  remain^ 


100  AiLGBBRA.  PaetIV, 

the  last  diVisor  of  aH  will  be  the  greatest  common  measure  of 
the  two  given  numbers. 

Let  a  and  b.  be  two  numbers,  and  let  a  be  contained  in  b,f 
times  with  c  remainder ;  let  c  be  contained    a)  b  (p 
ina,q  times  with  d  remainder ;  and  let  d  be         c)  a  (q 
contained  in  c,  r  tiines  exactly ;  then  will  d)  c  (r 

d  be  the  greatest  common  measure  of  a  and  b.  .       o 

For  since  (6=:ap-\-c,  or)  b—ap^ss^c,  and  a-^qcszd,  it  follows 
{from  proper*  12.)  that  every  quantity  which  measures  a  askd  b, 
will  likewise  measure  ap,  and  also  b-^ap  or  c,  (proper,  13.)  in  like 
manner,  whatever  quantity  measures  a  and  c  wiU  also  measure  a 
und  qc,  and  likewise  (a^qc,  or)  d;  wherefore  any  quantitff  which 
measures  <2,  must  likewise  meeuure  c  and  a  and  b,  but  d  measures 
d,  therefore  it  is  a  common  measure  of  a  and  b.  It  Kkewve 
appears,  that  d  is  the  greatest  comnum  measure  of  a  and  b; 
for  since  rd=sc  and  (c^-|-(J=)  rdq+dz=:a,  and  (ap+c^)  rdqp-i- 

dp+rd^tby  that  is,  rq+l,d=ia,  and  r9p-fP+''-^=^>  it  follows 
that  d  is  the  greatest  common  measure  of  these  two  vtUues  of  a 
and  b,  or  that  it  is  a  multiple  of  all  the  common  measures,  except 
the  gres^est,  of  a  and  b. 

Otherwise,  since  it  appears  that  every  common  measure  of  a 
and  b  measures  d,  and  d  itself  measures  a  and  b,  it  follows  that  d  is 
the  greatest  common  measure  of  a  and  6'. 

16.  The  sum  and  the  diffiavnoe  of  two  numbers  will  each 
measure  the  difiference  of  the  squares  of  those  numbers. 

For  smce  a+6.a— 6=a»— 6»,  it  follows  that 7— =a— 6> 

find -— =a+6. 

a— 6 

17.  The  suni  of  any  two  numbers  measures  the  sum  of  their 
cubesi  and  the  difiference  of  any  two  numbers  measures  the  dif- 
fepence  of  their  cubes. 

_,    a*4-6«  ,  a* — 6» 

For ----=:a*— a6+ft*;  ond  — — r-.=aH«6+fcS  asap- 

pears  by  actual  d^ision. 


9  See  Wobd'r  Ahrebrd^  tWrd  Edition,  p. «.    The  «boTe  is  a  demonstratFon 
of  the  Tole  ia  page  I48r  of  the  ^t  volume. 


Paxt  IV.         PROPBRTIBS  OP  NUMBERS.  101 

^  ^CoR.  Hence  if  the  (MPoduet  of  any  two  tuimben  be  tubtracted 
from  the  sum  of  their  squares,  the  remainder  mcafores  the  sam 
of  their  cubes ;  and  if  the  said  product  bte  added  to  the  stmi  of 
the  8quares>  the  sum  measures  the  difference  of  their  cubes.    . 

1 8.  If  any  power  of  one  number,  measure  the  same  power  of 
another,  the  former  number  measures  the  latter. 

JFor  let  —  be  a  whole  number  produced  by  -r^T-'T^'  *^«  ^^  ^ 
tr  bob 

o  ^ 

term$;  then  will  *t-  ^  a  whole  number  ;  for  if  not,  let  it  if  pom' 

ble  he  a  fraction,  then  thu  fraction  being  multiplied  continually 

a* 
into  iteelf,  wiU  at  length  produce  {-tA  a  whole  number,  which  i$ 

tr 
C 

abewrd:  wherefore  ~is  a  whole  number,  or  b  meaeuree  a. 

b 

Cob.  Henoe  if  one  number  measure  another,  any  root  or 
power  of  the  former  will  measure  the  like  root  or  power  of  the 
latter  respectively. 

19.  If  the  similar  powers  of  two  numbers  be  multiplied  toge- 
ther, the  product  will  be  a  power  of  the  same  kind  with  that  of 
the  &ctors. 

For  if  a^  be  multiplied  by  6',  the  product  a*"  b^  is.  likewise  an,  ^ 
n^  power,  the  root  of  which  is  ab. 

Cor,  Hence  e?ery  power  of  a  square  number  is  a  square, 
every  power  of  a  cube  number  a  cube,  and  in  geneial  eveiy. 
power  of  an  »*^  power  is  an  n^  power  \ 

20.  If  any  power  of  one  number  be  divided  by  the  sama 
power  of  another  number,  the  quotient  will  be  a  power  of  the 
same  kind  with  that  of  the  said  numbans. 


0^ 

Let  (f  and  b*  be  the  n^  powers  of  a  and  b  ;  then  is  -r^  also 


a 


an  n*  power,  for  its  root  is—. 

Cor.  Hence  the  quotient  of  one  square-  by  another,  is  a 
square ;  the  quotient  of  one  6ube  by  another  is  a  cube,  &c. 


«  And  it  it  obvieottluit  all  Hm  powtrt  •£  a  piinc  number  (eacMpt  the  fin( 
power)  will  be  eompottte. 

h3 


103 


AL6EBRA.  P^^t  IV. 


«l.  If  two  Bvmbeis  dMfer  by  unity,  their lum  if  fpal  to  the 
difference  of  their  Mjuares. 

Ltt  a  and  a+l  be  any  im  numbm  J^ffkrmg  fty  unity :  thm 
toiU  ««+!  be  tkeir  mm,  also  (a+  lj*-.^«<^-f  «a-|-l— 0^=) 
5Ja-M«f*«  <fcy«f«iceo/<Acir  t^iMrtf « MipA  u  the  $ame  as  ikeir 
sum. 

C0R..I.  Henee  the  differences  of  0*,  !•,  9^,  S«»  4*,  &c/ 

»  •  . 

(ssO,  1,  4,  9,  1.6,  &c.)  are  the  odd  numbers  1,  3,  5,  7,  &c. 

Cor.  2.  Hence  the  squares  of  all  whole  numbers  may  be  found 
from  the  series  of  odd  numbere  1,  3,  5,^,  9,  &c.  by  addition 
only. 

Thus,  1=1«;  l+3=(4=)?«j  1+3+5=^(9=)  3*;  1+3 
+  5+7=(16=:)  4«5  l+3  +  5+7+9=(25=:)  5»i  and  so  on  at 
pleasure, 

92.  An  odd  number  which  is  prime  to  another  number,  is 
nicewise  prime  to  double  the  latter. 

For  let  a  be  an  odd  number,  and  b  any  other  number  ;  then 
since  a,  being  odd,  cannot  be  measured  by  any  even  number^  (proper. 
11.)  it  must  be  measured  by  an  odd  one:  wherefore  if  a  and  9  b 
have  a  common  measure^  it  must  be  an  odd  nunther ;  but  9  bis  eri- 
dently  even,  (def  6.)  and  if  an  even  number  be  measured  by  an 
odd  one,  the  quotient  toiU  be  even^  (proper.  10.)  and  since  this  even 
quotient  can  be  halved,  it  is  plain  that  the  foremeniloned  odd  num» 
her,  which  meaeures  9  b,  mill  be  cteltriwati  hi^f  ess  many  tinms  in  h 
amitis4$^9b,,  that  at,  it  -  meaeures'  b^  whence  a  and  b' have  a  com* 
men  measure;  but  they, are  pwimq  to  audi  Mier^uiherefoTe  a am^ 
%h  have  no  cdmtiKMi  meaiUre. 

'  Cob.  Hence'  if  an  odd  nuniber  be  prime  to  any  other  num* 
ber>  it  is  prime  to  twb^  ftnat,  eight/ sltteen^  &c.  tunes  the 
latter. 

23.  If  each  of  two  numbers  be  prime  to  a  third  number^  their 
product  is  prime  to  it. 

Let  a  and  b  be  each  prime  to  c,  then  will  ah  be  prime  to  c. 

Then,  since  neither  a  and  c,  nor  b  and  c,  have  any  common 


'  I|i  the  Scholar's  Qai4e,  p.  204.  prx>|».  19.  cpr,  lite  0*  UJbj  RUitsbi 
but  with<{at  it,  the  eondasion  doee  not  follow.  , .    , 


V4BT IV.         PROPERTIES  OF  KUMBERS.  103 

measure,  it  is  pUdn  that  ah  and  c  can  haoe  no  eomuum  measure; 
wherefore  ab  is  prime  to  c. 

34.  If  one  number  be  prime  to  another,  every  power  of  the 
Ibnner  will  be  prune  to  the  latter. 

Let  a  be  prime  to  b,  then  wHl  a"  be  prime  to  6,  For  since  a 
and  hhaoe  no  common  measure^  a.a.a.a»  SfC,  and  b  Cjonnot  hove  a 
eommon  measure;  wherefore  {a,a>a.aj  SfC.z^)  a"  is  prime  to  b. 

fid.  The.8mn  of  two  numbers  wl^cli  are  prime  to  each  other^ 
18  prime  to  each  of  the  numbers. 

Let  a  be  prime  to  b,  then  wUl  a^^b  be  prime  to  a  and  b.  For 
if  not,  let  e  be  their  common  measure;  wherefi^re,  since  c  measures 

a-^b  a 

both  a+b  and  a,  that  is, and  —  are  whole  numbers,  by  jub* 

c      •      c 

•      b 
tracting  the  latter  from  the  former,  the  remainder —  is  a  whole 

c. 

nunUfcr,  (proper.  1.)  In  like  manner,  because and  —  are  whole 

CO 

a 

numbers^  by  subtracting  the  latter  from  the  former,  —  will  be  also 

V 

a       '  b 

a  whole  number;  wherefore  —  and  —  are  both  whole  numbers, 

c  c 

that  is,  thenismbersa  and  b,  which  by  hypothesis  are  prime  to  each 

other,  haoe  a  common  measure  c,  which  is  absurd. 

,    CoR.  Hence  if  a  part  of  any  number  be  prime  to  the  whole^ 

the  remaining-  part  is  prime  to  the  whole. 

£6.  In  a  series  of  continued  geometrical  proportionals  begin- 
ning at  linity,  all  the  odd  terms  will  be  squares  j  the  first,  fourth, 
•eventh,  tenth,  &o..  terms  will  be.  cubes  s.  and  the  seventh  term 
will  be  both  a  square  and  a  cube. 

Thus,  letl,r,  r^,  r^,  r*,  r^,  r®,  r',  r*,  r^,  8fC.  be  an  increasisig 
geometrical  series,  beginning  at  1.  Then  wiU  I,  f^,  r*,  r^,  r*,  4kc. 
{that  is,  all  the  odd  terms)  be  squares ;  I,  r^,  r^,  r^,  (or  the  1st, 
4tK  7th,  and  lOth,)  wUl  be  cubes ;  also  r^,  (or  the  Jth  term,)  is 
both  a  square  and  a  cube:  and  the  like  may  be  shewn  in  a  decreas* 
ing  series, 

Sr.  Every  square  number  o^st  end  in  either  1>  4>  5^  6,  9, 

orO. 

The  truth  of  this  will  appear  by  Sj^wisrii^  the  first  ten  numr 
bers\,^,^,^iuto\D.    * 

h4 


104  ALGEBRA.  Part  IV. 

Cob.  Hence  no  square  can  end  in  9, 3^  7>  or  8. 

28.  A  cube  number  may  end  in  either  of  the  ten  digits. 
This  voiU  likewise  appear  by  cubing  those  numbers, 

Coa.  Hence  2>  3,  5, 6,  7,  B,  10^  &c.  can  have  no  exact  sqaars 
root,  nor  can  3>  3,  4,  b,  6, 7, 9, 10,  &c.  have  an  exact  cube  root. 

29.  All  the  powers  of  numbers  ending  ih  0>  I,  S,  and  6,  vnXi 
end  in  the  same  figures  respectively  5  and  all  powers  ending 
in  the  above  figures,  will  have  their  roots  ending  in  the  same 
figures  respectively. 

Thus  iol*=100,  10l'=1000,  l9|«ssl<KXX)«  SfC.  ending  in  0. 

ll]«=:121,  in»=1331,  m^= 14631,  Sfc.  ending  in  I. 

5l*=  25,    5l*=125,      5]*=:625,      SfC.  ending  in  S. 

6)*=  36,    6)«=216,     6?*=  1296,    8(C.  ending  in  6. 

and  the  like  for  the  roots  of  powers  ending  as  abovCf  as  is  plain. 

SO.  All  numbers  ending  in  4  or  9,  will  have  their  even  powers 
end  in  6  and  1  respectively ;  and  their  odd  powers  the  same  ss 
their  roots,  viz.  4  and  9,  respectively. 

7%M»  il«^=;16,4?»=64,    4l*=266,    *c. 
9l«=81,  §?'=729,  9l*=6561,  «rc. 

31.  The  powers  of  numbers  ending  in  2  will  end  in  4,  8,  6, 
and  2,  alternately ;  numbers  ending  in  3  will  have  their  powers 
ending  in  9,  7,  1>  and  3,  alternately;  numbers  ending  in  7  will 
have  their  powers  ending  in  9,  3,  1,  and  7^  alternately  5  and 
numbers  ending  in  8  will  have  their  powers  ending  in  4,  2,  6, 
and  8,  alternately. 

77^19  will  appear  by  involving  such  numbers. 
Cor.  Hence  numbers  ending  in  1  and  9  will  have  their  even 
powers  end  in  the  same  figure,  viz.  1  -,  numbers  ending  in  3 
and  7  will  end  their  like  even  powers  with  the  same  figure,  vis. 
their  squares  with  9»  their  4th  powers. with  1,  &c.;  numbers  end- 
ing in  2  and  3  will  end  their  even  powers  alike,  viz.  their  squares 
with  4>  their  4th  powers  with  6 ;  numbers  ending  in  4  and  6 
will  have  their  even  powers  end  alike,  viz.  with  6 ',  and  in  gene^ 
ral,  the  like  even  powers  of  any  two  numbers  equally  distant 
from  5,  will  end  in  the  same  figure. 

32.  The  right  hand  places  of  any  number  being  ciphers,  if 
the  right  hand  significant  figure  be  odd,  the  number  will  be  divi*- 
sible  by  unity,  with  as  many  ciphers  subjoined  as  there  are  d- 
j>hers  on  the  right  of  the  saifj  number  -,  if  the  right  hand  signi* 


Pabt  IV.  PROPERTIES  OF  NUMBERS.  lOS 

ficant  figure  be  even,  it  wiU  be  divkiUe  b^  2,  with  as  many  ci- 

phera -subjoined. 

Thvs  12S0  is  dwiiible  hy  10, 3100  hf  100, 7000  by  1000,  «c: 
Also  1240  is  divisible  by  30,  S£00  by  900,  8000  by  2000,  4c. 

Off  d  </ie  ZiAre  is  true  in  all  simUar  cases. 

33.  Every  number  ending  in  5,  is  divisible  by  6  without 
i^mainder. 

This  is  plain,  since  all  such  numbers  are  either  5,  &r  multiples 
of  6. 

Cor.  Hence,  numbers  ending  in  O  or  5  are  divisible  by  5, 

34.  If  the  two  right  hand  figures  •£  any  number  be  measured 
by  4,  the  whole  is  measured  by  4  j  and  if  the  three  right  hand 
figures  be  measured  by  8,  the  whole  is  measured  by  8. 

Thus  the  two  right  hand  figures  of  each  of  the  numbers  184, 
2148,  37128,  13716,  71104,  *c.  being  divinble  by  4,  each  of 
these  numbers  is  measured  by  4. 

jilso  the  three  right  hand  figures  of  each  of  the  numbers 
13398,  97464,  9916,  100800,  9040,  4c.  being  measured  by  6, 
each  of  the  numbers  is  measured  by  8  j  and  the  same  is  true  in  all 
similar  cases. 

35.  In  any  even  number,  if  the  sum  of  its  figures  be  measured 
by  6,  the  number  itself  is  measured  by  6. 

Thus  the  sum  of  the  figures  in  the  eten  number  738  t«  I85 
which  b&ng  measured  by  6,  the  number  738  itself  is  likewise  mea^ 
sured  by  63  and  the  like  of  all  other  similar  numbers, 

36.  If  the  sum  of  the  figures  in  the  first,  third,  fifth,  &c. 
places  in  any  number,  be  equal  to  the  sum  of  those  in  the 
second,  fourth,  sixth,  &c.  places,  the  number  itself  is  divisible 
by  11. 

Thus  the  number  4759  is  divisible  by  11,  because  44*5  {the 
sum  of  the  first  and  t^ird)s=7+9,  {the  sum  of  the  second  and 
fourth ;)  in  like  manner  1934563  is  divisible  by  II,  for  1  +3-f  5-f 
3=9+4+6 ;  and  the  same  is  true  of  all  similar  numbers, 

37*  Any  part  of  the  sum  or  difierence  of  numbers  is  found 
by  dividing  each  of  the  given  numbers  separately  by  the  num^ 
ber  denoting  that  part  3  and  any  part  of  their  product  is  found 
by  dividing  one  only  of  the  numbers  by  the  number  denoting 

the  part  *. 

■  '■  ill'  I ■— III       II...  ■     «i  «      1 1 1.» 

•  The  properties  32  to  37  iaclasivt,  with  some  others^  are  iotrodaced  in  « 


109  ALGSS&A.  PAKTlir. 

TkH9  half  the  sum  ^  ea-i-Ab^Scii  Sa+S6^4e. 
Jnd  half  the  product  of  6ax4bxSc  u  Sax4hx8c,or 
6ax26x8c,  or6ax46x4Cj  ^ach  be'mgsa^S  abe. 

38.  Every  even  square  number  is  measured  by  4,  and  erery 
odd  square  divided  by  4  leaves  1  remainder. 

For  nnce  the  root  of  an  even  square  must  be  even,  (proper.  8.) 
let  2n  be  its  root;  then  ^^s4n^  the  square,  which  is  evidently 

divisible  by  4. 

Again,  since  the  root  of  an  odd  square  must  be  odd,  (proper. 
Il.)let2n  +  lbe  such  root,  *ik€n^»+ll*=4n«-i-4n+ 1  thesqwxre; 
Ujhich  being  divided  by  4,  wj^l  evidently  leave  1  remaining. 

39.  If  any  number,  and  also  the  sum  of  its  figures,  be  each 
divided  by  9^  the  remainders  will  be  equal. 

Met  n  he  any  number  composed  of  the  digits  a,  h,  c,  and  d; 
then,  according  to  the  establisfied  principles  of  notation,  1000  a + 
1006+  10c+d=:n;  but  1000a=(9994-La=)  999a-i-fl;  1006= 
(994-1.6=;)  99  6+6;  10c=(9+l.c=)  9c+c:  therrfore  n=: 
(1000  a+ 1006+ 10  c+ds=)999a+996  +  9c+a+6+c+diCOtt- 

sequently — =111  a+11  6+c-| — — i—,  or  the  number  n 

being  divided  by  9  leaves 1- —  remainder,  which  is  the  same 

as  tlie  remainder  of  the  sum  of  its  digits  divided  by  9',  as  was  t0 
be  shewn, 

CoR.  Hence  the  operations  of  addition^  either  of  whole  num- 
bers or  decimals,  may  be  proved  by  casting  out  the  nines;  for  it 
is  plain  that  if  the  excess  of  nines  in  two  or  more  numbers  be 
taken,  and  likewise  the  excess  of  nines  in  these  excesses,  the 
last  excess  will  equal  the  excess  of  nines  in  the  sum  of  the  given 
numbers  j  since  the  sum  of  the  excesses  of  the  parts  (taken 
feparately)  is  evidently  equal  to  the  excess  of  the  whole  t. 

t  • 

note  on  p.  155,  156.  Vol.  I.  as  usefal  for  readily  finding  the  measures  of  nam* 
bcrs,  and  fpr  redueiog  fcactions  to  their  lowest  terms. 

>  To  shew  the  method  of  proving  addition  by  casting  oot  the  nines,  the 
following  examples  are  subjoined. 

£x.  1.  Ex.  2. 

357S  ..  •  *  8  68.496    ....  6 

6832  ..  ..I  I  Excenes  ^^^fj    •  •  -  •  «    |  Ercestes 


7654  ....4  V^      ^^  4.7121 6 

8323 

563«T 


••••       >       of  4.7121 6    V    «;. 

••  -I  \mnes,  ^^«  ^^^    •  •  •  •  JL  \niL. 

....2^  8S7.S091  .,..8-^ 


Pabt  IV.         PROiȣRTIBS  OF  NUMBERS.  lOf 

40.  If  each  of  two  nnmbeTB  be  ^fivMed  by  9>  and  the  product 
of , the  semainden  also  divided  by  9*  this  remaiiidei'  shall  equal 
the  remainder  ariaiiig  from  the  product  of  thie  two  given  num-* 
hers  divided  iof  9. 

F&r  ifit  0^-).  a  and  &B+b  be  tlm  ttpo  numben,  whkk  being  dU 

ab 

vid^d  by  0,  toill  evidently  leave'a  and  bfor  remainders^  and — ^ 

the  product  of  the$e  remMnden  diaided  by  9. 

^+MX9B^b_»\jiB^9aB'{'9Jh^ab 

9  ""^         '  9  ""^ 

aB+Ah-i ;  wherefore  —  is  the  remainder  of  the  product  of  the 

two  given  numbers  divided  by  9«  and  it  equals  the  product  of  the 
remainders  of  the  two  given  wam^beri.diindfd  by  9^0$  found  ahomf 
which  was  to  be  shewn  *^. 


'  WII'M 


In  Ex.  I.,  the  nines  b«ing  cast  ovt  of  the  top  tine»  the  8  placed  opposite 

remains  in  excess;  in  like  manner  1,  4,  and  7>  are  respectively  the  excesses  «l 

thel  second,  third ,  and  jfourtb,  lines:  now  these  foar  excisies  heing  added 

together,  and  the  nines  cast  ont  of  the  sum,  the  excess  will  be  9,  and  if  the 

nines  be  cast  oat  of  the  sum  of  the  numbers  proposed,  (263^1«)  the  excess  ia 

Ulewise  2,  which  two  excesses  agreeing,  the  work  is  presumed  to  be  right  for 

the  reasons  gireti  in  property  39.  and  its  corollary.   But  there  are  two  cases  in 

iM^ich  Hiis  mode  of  proof  does  not  succeed ;  the  first  is  idien  a  mistake  of  9y 

or  any  nultij^  of  ^s  lias  been  made-  in  tlie  addiag  ;  and  tbe  second  Is  whea 

all,  or  any  of  the  figures  haine  beei^  transposed:  in  each  of  tbese  cases,  al» 

thongfa  the  work  is  mantfiestly  wrong,  the  proof  will  make  it  appear  right. 

Subtraction  may  likewise  be  proved  by  the  same  method,  but  this  will  be  con- 

aidered  rather  as  a  natter  <d  coyiosity  than  use :  in  subtradtog  tfae  ezcesees^ 

if  the  Viwes  one  be  the  greater^  9  mqst  be  borrowed,  as  in  Ex.  2.  below. 

E«.  1.  Ex.  3. 

From  237165  ....  6")  37.4&     ....  11 

Take  123428  ...  ,2  >JBreiiwse.  3.12^4  • . .  .^  > 

J!?e»t.  1  I37a7  . .  . .  4  J  .  34.326^.  ...63 

In  Sx.  1.  basving  ea«t  tbe  nines  ont  of  tbe  t^o  given  numbers,  the  lower 
czeess  2  is  subtracted  from  tbe  upper  excess  6 ;  then  the  difference  4  being 
c^nal  to  the  excess  of  nines  in  (1 13737)  the  remainder,  shews  the  work  to  be 
jpgbt)  «Db)e^  b^wever  to  tbeiexeeptions- stated  above. 

In  £x.  2.  the  4  cannot  be  taken  from  1,  therefore  9  is  borrowed ;  the  rest 
•V  Mk  tbe  preeeAnif  example. 

-  i  «  Tbe  pTttctieal  plication  of  tbts  property  of  the  number  9,  is  fully  exem< 
pHfiini  in  tlM.pM>o£i  suls^iaed  to  tfae  operations  of  nuUipKcation  and  divisioQ 
ef  both  whole  numbers  and  decimals.  See  Vol.  I.  p.  34—38.  47—49^^15. 319. 


10$  ALOERRA.  Part  IV. 

41.  Any  ariUimetical  pragretekm  cui  be  increased  m  tfj^i- 
turn,  bat  not  decreased;  a  barmonical  prqgreauon  can  be  de* 
creased  in  infimiium,  but  not  increased;  bat  a  geometrical  pro- 
gression  can  be  both  increased  and  decreased  in  it^bniMm  *. 

First;  let  a-|-a+r+a+2r-f-,^.  be  an  arithoietical  progres- 
sion ;  this  series  can  evidently  be  increased  at  pleasure  by  the 
constant  addition  of  r :  but  if  you  take  the  series  backwards, 
and  decrease  its  terms  suooessively  by  r,  it  will  become  <i4'r+ 
a-f  a~r-f  a-*-8r-h,  8fC.  now  when  ei/ft«r of  the  quantities  r,  2r» 
3  r,  becomes  equal  to  a,  that  term  is  equal  to  O,  and  (he  series 
evidently  can  proceed  no  further. 

Secondly,  let — | 1 -f,  4rc.be  a  barmonical  series,  in 

•'  a     a-fr     a+^ 

which  the  last  term  is  the  least ;  this  can  evidently  be  decreased 

at  pleasure  by  the  constant  addition  of  r  to  the  denominator.  Now 

taking  this  series  backwards,  and  continually  subtracting  r  from 

the  denominator,  it  becomes H h h  — tt'^*  *^-  ^^ 

Q'^r      a     a — r     a**%r 

when  r,  9  r,  S  r,  or  some  multiple  of  r,  becomes  equal  to  a,  it  is 
plain  tbe  next  term  of  the  series  will  be  negative,  or  the  series 
terminates,  without  the  possibility  of  further  increase. 

Thirdly,  let  a+ar-^-ar^'^ar^,  be  a  geometrical  series;  thi» 
series  may  be  increased  by  constantly  multiplying  by  r,  or  de- 
creased by  constantly  dividing  by  r,  as  is  evident,  without  the 
possibility  of  its  terms  becoming  negative. 


The  nuiuber  3  poticMes  tbe  tame  property,  bat  9  is  mwiUr  prdemd,  at  being 
tbe  moBt  convenient  for  practice :  we  may  add,  that  tbe  tame  incoDTenience 
attends  the  proving  of  multiplication  and  division  by  this  method,  as  that  men- 
tioned in  the  precediiig  note. 

Tbe  rate  for  proving  addition  by  casting  out  the  nines  was,  according  to  Mr. 
Bonnycastle,  first  pablishcd  by  Dr.  Wallis  in  1657  ;  but  the  property  of  the 
number  d»  on  which  tbe  rule  is  founded,  was  most  piohably  known  to  tbe 
Arabians  long  before  that  time :  Lucas  de  Bmgo,  who  wrote  in  1494,  was 
well  acquainted  with  this  property,  and  shewed  the  method  of  proving  the 
primary  operations  of  arithmetic  by  it,  as  is  witnessed  by  Dr.  Uutton. 
Matf^.  Diet.  Vol.  I.  p.  66. 

X  This  property  of  the  three  kinds  of  progressioas  was  first  noticed  by 
Pappus,  a  Greek  Mathematician  of  tbe  Alexandrian  School,  who  flourished  in 
the  latter  part  of  the  fourth  century,  in  the  third  book  of  his 
Collections. 


Past  IV.         PROPERTIES  OF  NUMBERS.  109 

49.  If  a  harmonicai  mean  and  an  ariUunetical  mean  be 
taken  between  any  two  numljln^  the  four  terms  will  be  pro- 
portionals. 

Let  a  and  h  he  any  two  nutAers,  then  will  — —-  ^  ^a  arith^ 
metical  mean,  and  — tt  <>  harmankal  mean  between  a  and  b:  then 

wiw  a :  ■■*      : :  :  by  for  the  product  of  the  meam  (ab)  it  equal 

to  the  prodmct  of  the  extremee  {ab),  which  is  the  criterion  of  pro- 
porOonality.    (Art.  56.) 

43.  The  square  root  of  a  rational  quantity  cannot  be  partly 
rational,  and  partly  a  quadratic  suri^ 

For  if  possible,  let  ^xssa+  jy/%  of  which  jjb  is  an  irredu- 

cible  surd  ;  square  both  sides,  and  x^m^  +9  a ^^6+ 6^  or>  9  a^6 

X— a*  — & 
=*— a*  — 6>  V  j^b^ — ~ ,  that  is,  an  irreducible  surd  equal 

to  a  rational  quantity,  which  is  absurd;   wherefore  ^x  cannot 
equal  any  quantity  of  the  form  ofa-^  ^b, 

44.  If  each  side  of  an  equation  contain  rational  quantities ,  and 
irreducible  surds^  then  will  the  rational  parts  be  equal  to  the  ra- 
tional^  and  the  surd  parts  to  the  surd. 

Lei  4?+  ^«=a+  ^b,  then  will  x=a,  and  V'=*  V^- 

For  if  x  be  not  =a,  let  x^a-^m,  then  a+iii+  ^z:sa+  j^b, 

^  ^  +  iv/2=  ^^b,  that  is,  j^b  is  partly  rational,  and  partly  surd, 

which  is  proved  to  be  impossible  in  proper.  43. 

45.  From  the  forgoing  property  we  derive  an  easy  method 
for  extracting  the  square  root  of  a  binomial  surd^  as  follows. 

Example.  To  find  the  square  root  of  m+  ^n. 

First  assume  ^x+  V*^  V^wH"  V*  **^  squaring  both  sides 

x4-2^«B-f  «=:m+ v^n;  wherefore  {proper.  44.)  x-^-z^m,  and 

9  ^xzx  ^n;  these  equations  squared  gioe  x'  +  9  xz^z*  =sm*,  and 

4xzszn;  subtract  the  latter  from  the  former,  and  x* -^2  xz-^-z* 

ssm*— «,  V  by  wofoilioa*— xss^ia*— «;  but  X'\-z^m,  v  t= 
^ ,andz^ ^ /.•  vm+  ^n^{^x+  ^z=) 


^ Z-- ^  V 21- ,  the  root  required. 


PART  V. 

ALGEBRA. 


OF  EQUATIONS  OF  SEVERAL  lilMENSIONS. 


A  GENERAL  view  of  the  nature^  fonnaticm,  mnd  roots  of 
•qaations. 

1 .  A  simple  equation  is  that  which  contaiiii  the  unknown 
quantity  in  its  first  power  ohly. 

Tku9  cur+ftssc. 

2.  A  quadratic  equation  is  that  whick  contains  the  second 
power  of  the  unknown  quantity^  and  no  power  of  it  higher  than 
the  second. 

Thus  ta^-^bx^c, 

3.  A  cubic  equation  is  that  which  contains  the  thirds  and  no 
higher  power  of  the  unknown  quantity. 

Thus  a3fi^bx*'\-cx=::d,  or  ax^  +  bx^=::c,  or  wfi-^bx=sc. 

4.  A  biquadratic  equation  is  that  which  contains  the  fourth^ 
and  no  higher  power  of  the  unknown  quantity. 

Thus  ac^-h&a?*— cr®+(ir — c=o,  8fc. 

5.  In  like  manner^  an  equation  of  the  fifth  degree  is  that 
which  cooftains  the  fifth,  and  no  higher  power  of  the  unknown 
quantity j  an  eqtiation  of  the  sixth  degree  contains  the  iixth 
power  J  one  of  the  seventh  degree  the  seventh  power  of  the 
unknown  quantity^  &c.  &c.  i 

6.  All  equations  above  simple^  which  contain  only  one  power 
of  the  unknown  quantity^  are  called  pure. 

Thus  ax^=b  is  a  pure  quadratic,  a3?i=:h  is  a  pure  cuhie, 
ua^zsih  a  pure  biquadratic,  S(c. 

7*  All  equations  containing  two  or  more  different  powers  of 
the  unknown  quantity^  are  called  affected  or  adfected  equations. 
Thus  aot^-^hx^s^e  is  an  adfected  quadratic;  ckc*— iBr*s3C,  amd 
aa:'  +  &r=c  are  adfected  cubics ;  a^'^sf^-i-ax^sb,  and  a**-^to*aac;, 
and  ax^-^bx^  +  cx*^dx-^esso,  are  adfected  biquadtFodct, 


112  ALGEBRA.  Part  V. 

8.  An  equation  is  said  to  be  of  as  many  dimensions,  as  there 
are  units  in  the  index  of  the  highest  power  of  the  unknown 
quantity  contained  in  it.  - 

Thus  a  quadratic  is  said  to  be  an  equcUion  of  two  dimensions ; 
a  cubic  of  three ;  a  biquadratic  of  four,  <rc. 

9.  A  complete  equation  id  that  which  contains  all.  the  powers 
of  the  unknown  quantity »  from  the  highest  (by  which  it  is 
named)  downwards. 

Thus  ax^—bx+cszo,  is  a  complete  quadratic ;  ax^—hs^-bcx 
— dsso,  is  a  complete  cubic  ;  a?*— Jf*— ac^+a?— a5=o,  a  complete  ii- 
quadraiiCy  ^c. 

10.  A  deficient  equation  is  that  in  which  some  of  the  inferior 
powen  of  the  anknown  quantity  are  wanting. 

As  aa?*— 6a:*+c=so,  a  deficient  cubic;  aa:*— 6a;*-hca?— d=o, 
a  deficient  biquadratic,  S;c, 

11.  An  equation  is  said  to  be  arrsMEiged  according  to  its  di- 
mensions, when  the  term  containing  the  highest  powet  of  the 
unknown  quantity  stQSids  first  (on  the  left) ;  that  which  contains 
the  next  highest,  second  ;  that  which  contains  the  next  high^, 
third  ;  and  so  on. 

Thus  the  equation  x*— ar♦4■6a^'— ca7®-fd|3P— ^«=o>  m  arranged 
according  to  its  dimensions, 

Cos.  Hence  every  complete  equation  of  n  dimensions  will 
contain  n-i-l  terms. 

12.  The  last  term  of  any  equation  being  always  a  known 
quantity,  is  usually  called  the  absolute  term :  and  note,  this  last 
or  absolute  term  may  be  either  simple,  or  compound,  consisting  of 
leveral  known  quantities  connected  by  the  sign  +  or  —  5  ^which 
t€>gether  are  considered  as  but  one  term. 

13.  The  roots  of  an  equation  are  the  values  of  the  unknown 
quantity  (expressed  in  known  terms)  contained  in  that  equa- 
tion  ',  hence,  to  find  the  roots  is  the  same  thing  as  to  resolve 
the  equation. 

14.  The  roots  of  equations  are  either  possible,  or  imaginary. 
Possible  roots  are  such  as  can  be  accurately  determined,  or  their 
values  approximated  to,  by  the  known  principles  of  Algebra. 

Thus  y^a,  ^^a-^b,  *^c,  ^c.  are  possible  roots. 

15.  Imaginary  or  impossible  roots  ar^  such  as  come  under 
the  form  of  an  e»en  root  of  a  negative  quantity,  which  cannot  be 
determined  by  any  known  method,  of  analysis. 

Thus  V**"**  *  V***^*  *  V"~^/  *^*  ^^  impossible  roots* 


Paut  V.  NATURE  OF  EQUATIONS.  IW 

16.  The  limits  of  the  roots  of  an  equation  are  two  quantities, 
one  of  which  is  greater  than  the  greatest  root  3  and  the  other, 
less  than  the  least.  The  greater  of  these  quantities  is  called  the 
iuperior  limits  and  the  less,  the  inferior  limit.  Also  the  limits  of 
each  particular  root,  are  qutotities  which  &11  between  it  and  the 
preceding  and  following  roots. 

17*  The  depression  of  an  equation  is  the  reducing  it  to 
another  equation,  of  fewer  dimensions  than  the  given  one 
possesses. 

18.  The  transformation  of  an  equation  is  the  changing  it  into 
another^  differing  in  the  form  or  magnitude  of  its  roots  from 
the  given  equation. 

OF  THE  GENERATION  OF  EQUATIONS  OF 
SEVERAL  DIMENSIONS. 

19.  If  several  simple  equations  involving  the  same  unknown 
quantity  be  multiplied  continually  together,  the  product  will 
form  an  equation  of  as  many  dimensions  as  there  are  simple 
equations  employed  '. 

Thtis,  the  product  of  tmo  simple  equation»  is  a  quadratic ; 
the  continued  product  of  three  simple  equations  is  a  cubic;  that  of 
four,  a  biquadratic;  and  so  on  to  any  number  of  dtmensUms, 

For^  let  X  be  any  variable  unknown  quantity,  and  let  the 
given  quantities  a,  b,  c,  d,  Ssc  be  its  several  values,  so  that  xs^a,, 
x^b,  xssic,  x^d,  SfC.  these  by  transposition  become  x-^as^o, 
x^b^o,  X— csso,  x-^d^o,  8(C.  if  the  continued  product  of  these 
simple  equations  be  taken,  (viz.  x^ajr— 6.x— cor— d.  Ssc.)  it  will 


m^f 


f  This  metikod  of  gemsntmg  roperiot  tqiiations  by  the  eontimul  maltipli- 
catioo  of  inferior  oaei ,  was  the  invention  of  Mr.  Thomas  Harriot^  a  oelc« 
brated  Xnglish  mathematician  and  philoeopher,  and  was  first  pnbUsbed  at 
JjondoQ  in  the  year  163 1*  beinf  ten  years  after  the  antbor^s  decease,  by  his 
friend,  Walter  Warner,  in  a  folio  woik,  entiUed,  Artis  Jnafyiice  Praxis^  ad 
/B^uatumes  AlgebraiettM  nova,  expeHtay  et  generdU  metkodo^  t^emh^emdas^ 
By  this  excellent  contrivance  the  relations  of  the  roots  and  coeiBcients,  and 
the  whole  mptery  of  equations,  are  completely  developed,  and  their  rarions 
relations  and  properties  discovered  at  a  single  glance.  See  on  this  subject 
iSitr  Isaac  Newton's  Ariihmetica  UmversaUt,  p.  256,  257.  Madaurin** 
jRgebra,  p.  139.  ^»  Huiton't  Mathematical  Dictionary^  Vol.  I.  p.  90. 
;^mpaon*9  Algebra,  p.  131.  &c.  Dr.  WaSHtU  Algebra  ;  Pr^essor  yilantU 
Elememis  qf  Matkematieal  Ana^sit,  p.  48.  and  various  other  writers. 

VOL.  II.  1 


114  ALGEBBA.  Part  V. 

m 

constitute  an  equation  (=zo)  of  qs  many  ^mennons  as  there  are 
factors,  or  simple  equations,  employed  in  it^  composition:  for 
example. 

Let  X — a=o 
Be  multip,  info  x—b^^o 

The  product  U  ^'-«|,+«t^^,  „  quadratic. 

Multiplied  into  x—c=io 

The  product  is  a?'— a"|      +a6^ 

—6  >3i^+ac  >x—abc=o,  a  cubic, 
— cj      +bcj  

Multiplied  into  x — dsso 


The  product  is  x*'~a'^     +a^T      ^abc\ 

,-i-flc  J     "Obd  \x+abcd=zo,  a 
-f-fld  I -pft— acd  f  biquadratic. 
+  6r  {     — 6cdJ 
+  bd\ 
-t-cdJ 

*c,  S(C. 

From  the  inspection  of  these  equations  it  appears^  that 

SO.  The  product  of  two  simple  equations  b  a  quadratic. 

91.  The  continual  product  of  three  simple  equations^  or  of 
one  quadratic  and  one  simple  equation,  is  a  cubic. 

22.  The  continual  product  of  four  simple  equations^  or  of  two 
quadratics^  or  of  one  cubic  and  one  simple  equation^  b  a  biqua- 
dratic 5  and  so  on  for  higher  equations  '. 

^.  The  coefficient  of  the  first  term  or  higher  power  in  each 
equation  b  unity. 

84.  The  coefficient  of  the  second  term  in  each,  b  the  sum  of 
the  roots  with  their  signs  changed  \ 

Thus,  in  th4(  quadratic,  whqse  roots  are-^-amnd'^b,  the  coefi" 
eientis.'^a'^b^in  the  cubic,  whose  roots  aTe'\-a,  +  b,  and-i-c,  it 


■  It  M  in  like  manner  eTideot,  that  the  roots  of  the  componnded  equatioot 
will  have  not  only  the  same  roots  with  its  component  simple  e^ationsy  but 
that  its  roots  will  hare  the  same  signs  as  those  of  the  latter. 

■  Hence,  if  the  sum  of  the  affirmative  roots  be  equal  to  the  sum  of  the  ne- 
fattve  roots,'  tlie  coefficient  of  the  second  term  will  be  0  ;  that  li,  the  icoQiid 
tenn  will  vanish :  and  conversely,  if  in  an  equatioa  the  second  term  be  wantr 
ing,  the  sum  of  the  jaffirmative  roots  and  the  sum  of  tl^e  negatiYe  loota  ate 
equal. 


/ 


Paet  V.  NATURE  OP  EQUATIONS.  lis 

is — fl— fc— c;  in   the  biquadratic,  whose  roots  are+af  +  bt-^-Cj 
and+d,  it  is  — a— fc— o — d,  8(C.  ' 

25.  The  coefficient  of  the  third  term  in  each^  is  the  sum  of 
all  the  products  that  can  possibly  arise  by  combining  the  roots, 
with  their  prober  signs,  two  and  two. 

Thus,  in  the  cubic,  the  coefficient  of  the  third  term  M+a6-f 
ac-^be;  in  the  biquadratic,  it  iS'{'ab+ac+ad+bc-{'bd-{-cd,  SfC. 

26.  The  coefficient  of  the  fourth  term  in  each,  is  the  sum  of 
all  the  products  that  can  possibly  arise  by  combining  the  roots, 
with  their  signs  changed,  three  by  three. 

Thus,  in  the  biquadratic,  the  coefficient  of  the  fourth  term 
18  — abc^ahd^acd-^bcd. 

In  like  manner,  in  higher  equations,  the  coefficient  of  the 
fifth  term  will  be  the  sum  of  all  the  products  of  the  roots, 
having  their  proper  signs,  combined  four  by  four  \  that  of  the 
sixth  term,  the  roots,  with  their  signs  changed,  five  by  five,  &c. 

27.  The  last,  or  absolute  term,  is  always  the  continued  pro- 
duct of  all  the  roots,  4^aving  their  signs  changed. 

Thus,  in  the  quadratic,  whose  roots  are -^^  a  and-^-b,  the  last 
term  is-^ab  (or—ax  —b) ;  in  the  cubic,  the  absolute  term  is  —abc 
(=: — ax— fcx— c);  in  the  biquadratic,  ^e  absolute  term  is-\- 
abed  (=— a  X  — 5 x  — c x  — d),  ^c. 

28*  The  first  term  is  always  positive,  and  some  pure  power 
of  X. 

2S.B.  The  second  term  is  some  power  of  x  multiplied  into 
^a, — b,—c,  ifc.  and  since  x  is  affirmative^  and  each  of  these 
quantities  negative,  it  follows  that  the  second  term  itself  is 
negative,  since  4- X  —  produces — . 

29*  The  third  term  wUl  be  positive,  for  its  coefficient  being 
the  sum  of  the  products  of  every  two  of  the  negative  quantities- 
— a,— 6,— c,  4rc.  and  (since-*- X— produces +)  therefore  these 
sums,  multiplied  by  any  power  of  x,  (which  is  always  positive,) 
will  always  give  a  positive  result. 

SO.  For  like  reasons  the  fourth  term  will  be  negative,  the 
fifth  positive,  the  sixth  negative,  and  so  on  i  that  is,  when  ,tbe 
roots  are  all  positive,  the  signs  of  the  terms  of  the-  equation 
will  be  alternately  positive  and  negative :  and  convei'sely,  when 
the  signs  of  the  terms  of  the  equation  are  alternately + and  — , 
all  the  roots  will  be  positive. 

12 


lie  ALGEBRA.  PaktV. 

Cor.  Hence,  if  the  signs  of  the  even  terms  be  changed,  the 
signs  of  all  the  roots  of  the  equation  will  be  changed. 

31.  Let  now  the  roots  of  the  equations,  above  referred  to*  be' 
supposed  negative  5  that  is,  x= — a,  a?= — b,  a?=  -r  c,  x=: — d,  4rc. 
then  by  transposition,  x-)-a=:o,  j:+&=:o,  x4-c=ao,  x+d^o,  4rc. 

^i^^^tm^mm   ^m^t^-^n^^     fl^H^^^p*  «^i^H^^^ 

the  product  of  these,  or  x+a.x+b.x+cjB+d,  Sfc,  wiU  bean 
equation,  having  all  its  terms  affirmative;  for  since  all  the 
quantities  composing  the  &ctors  are  +,  it  is  plain  that  the  pro^ 
ducts  will  all  be  -h . 

Cor.  Hence,  when  the  signs  of  all  the  roots  (in  the  above 
simple  equations,  having  both  terms  on  one  side)  aj^e  -<• ,  the  signs 
of  all  the  terms  of  the  equation  compounded  of  them  will  be-f  ^ 
and  conversely,  when  the  signs  of  all  the  terms  of  an  equation 
^*e  4^,  the  signs  Of  all  its  roots  will  be  — . 

32.  If  equations  similar  to  the  foregoing  be  generated, 
having  sotne  of  the  toots +,  others  ^,  it  will  appear,  th^  there 
will  be  as  many  changes  in  the  signs  of  the  terms,  (from + to  —  y 
or  from  —  to+,)9s  the  equation  has  positive  roots  3  and  as  inlany 
continuations  of  the  same  sign,  (-hand+/or  —  and  — ,)  as  the 
^quatiom  has  negative  roots :  and  conversely,  the  equation  will 
have  as  lAanjr  affirmative  roots  as  it  has  changes  of  signs,  and  as 
many  negative  roots  as  it  has  continiiations  of  the  same  sign  \ 

Cor.  It  follows  from  what  has  been  said,  that  every  equation 
has  as  many  toots  as  its  unknown  quantity  has  dimensions. 
To  be  particular  j  a  quadratic  has  two  roots,  which  are  either 
both  affirmative,  both  negative>  or  one  affinnatite  and  one 


i^Hb 


^  ThU  supposes  the  roots  to  be  all  possible.  Ererj  equation  w3(  have 
either  an  even  number  of  impossible  roots,  or  node :  hence  a  quadratic  wSl 
bare  both  its  roots  possible,  or  both  impossible ;  a  etibfc  one  ot  thYee  possible 
roots^  and  twof  or  none  impossible ;  a  biqnadratie  will  have  eHhet  fdar^  two, 
or  none  of  its  roots  possible,  and  none>  two,  or  fouSr,  impoisib^  *^  and  the 
like  of  hig^her  equations.  An  impossible  root  may  be  considered,  either  as 
affirmative  or  ne^tire.  The  di  Acuities  attending  the  doctrine  of  impoa^le 
or  imaginary  roots,  have  hitherto  bid  defiance  to  the  skill  and  address  of  the 
^rned :  a  great  number  of  theories  atid  invesfigations  have  appeared,  it  is 
tfne ;  bat  our  knowledge  of  the  origin,  nature,  properties,  &c.  of  imaginaiy 
roots  i»  sUU  very  imperfect.  The  following  Authors,  among  others,  have  treated 
on  the  sttl^ect,  via.  Cardan,  Bembelli,  Albert  Oirard,  Wallis,  Newton,  Mao- 
laurin,  James  Bernoulli,  Emerson,  Euler^  D'Alembert,  Waring,  Hnttoo, 
Sterling,  Playiair,  &c. 


PahtV.     depression  of  equations.  lir 

i^egative.  A  cubic  has  three  roots,  which  are  either  all  afErma- 
tive,  all  negative;  two  affirmative,  and  one  negative;  ot  one 
affirmative,  and  two  negative :  and  the  like  of  higher  equations. 
33.  If  one  root  of  an  equation  be  given,  the  equation  may  h^ 
depressed  one  dimension  lower ;  if  two  roots  be  given,  it  may  be 
depressed  two  dimensions  lower ^  and  so  on,  by  the  following  rule  *. 

RuLB.  When  one  root  is  given,  transpose  all  the  terms  to 
one  side>  whereby  the  whole  will=o;  transpose  in  like  manner 
.^e  value  of  the  root>  then  divide  the  former  expression  by  the 
letter,  and  a  new  equation  will  arise=o^  of  one  dimension  lower 
than  the  given  equation. 

Examples. — 1.  Let  of*— 9x*-|-36x— 24=o  be  an  equation, 
whereof  one  of  the  roots  is  known;  namely,  x=33. 

By  transposition  x— 3=o,  divide  the  given  equation  by  this 
quantity. 

Thus,  jr— 3)a:'— 9  a:«+26a?— 24(x*— 6x+8=o,  the  resulting 

a:*— 3  a:*  equation,   which   being   re* 

-r.6x*4-26  «  solved  by  the  known  rule  for 

c-6x*+18j;  quadratics j  lis  two  remain^ 

"  8^—24   ing  roots  will  be  found,  viz, 

8  07^24   x^4,  and  xsS. 

5.  Letap*-h4«'+19a«— 160«=140p,  whereof  one  root=  — 
B,  be  ^ven,  to  depress  the  equation. 

Here   by   transposition,  a?*+4x'  +  19a:*— 160  a?— i400=o, 
/md  ar+5=o;  then,  dividing  the  former  by  the  latter,  we  have 

— I 1 rsif'— a?*-f  24  x— 380=0,  the  re- 

a?+5  '    . 

sultvf^  equation, 

3.  Given  x=3  in  tb^  equation  x^-**5x 4-6=0,  to  depress  it. 

4.  If  jr— 4=sob«  ft  divisor  of  the  equation  a:*— 4  a:*— x-|-4=d, 
to  de^ness  the  equation,  and  determine  its  two  remaining 
roots.    Ans,  the  resulting  equation  is  jr*^l=:o,  and  its  roots  -^l 

\ 

'  When  the_  absolute  term  of  an  equation  so,  it  is  plain  that  one  of  the 
roots  is  0,  and  consequently  the  equation  m^y  be  divided  by  the  unknown 
quantity,  and  reduced  one  dimenslpn  lower.  In  lika  loanner,  if  the  two  last 
lerms  be  wanting,  the  equation  may  be  reduced  two  dimensions  lowe?)  if 
^hrec;,  three  dimensions,  &c. 

»3 


118  ALG£BRA.  Part  V. 

5.  To  depress  the  equations  a?*— 5a?®+2x+83=^,  aod  j:*— 
sis  oi^+ 18  j?4-40=o,  on^  root  of  the  former  beiog  +4,  and  one 
of  the  latter  —5. 

34.  If  two  of  the  roots  be  given,  x-f  ^=o,  and  xHh<=Oy  the 
given  equation  being  divided  by  the  product  of  these^  x+rjc+*, 
will  be  depressed  thereby  two  dimensions  lower ;  thus, 

6.  To  depress  the  equation  x'--5j?*+2x+8=o,  two  of  its 
roots,  —1  and  -f-2,  being  given. 

Thus,  x-f  1=0,  and  j:— 2=0|  then  x+lj:— 2=a^— a?— 2, 

o:^— 5a?*-|-2j:+8 
the  divigor ;  wherefore •- =rx— 4,  whence  x— 4=o 

is  the  resulting  equation, 

7.  Given  jt'—S  a:*— 46x— 72=o,  having  likewise  two  values 
of  X,  viz.  —2  and  —4,  given,  to  depress  the  equation.  An" 
swer,  X — 9=0. 

B.  Given  a:*— 4x'  — 19  jr» +46  x  4-120=0,  two  roots  of  which 
are  +4  and  — 3,  to  depress  the  equation. 

35.  To  transform  an  equation  into  another,  the  roots  of  which 
u  ill  be  greater,  by  some  given  quantity,  than  the  roots  of  the  prO' 
posed  equation* 

Rule  I.  Connect  the  given  quantity  with  any  letter,  different 
from  that  denoting  the  unknown  quantity  in  the  proposed  equa* 
tion,  by  the  sign  — ,  and  it  wiU  form  a  residual. 

II.  Substitute  this  residual  and  its  powers,  for  the  unknown 
quantity  and  its  powers  in  the  proposed  equation,  and  the  result 
will  be  a  new  equation,  having  its  robts  greater,  by  the  given 
qiiantity,  than  those  of  the  equation  given'. 


'  The  truth  of  this  rale  is  clear  from  the  fivst  example,  where  since  y — 3  »!', 
it  is  plain  that  y^x-\-  3,  or  that  the  equation  arising  from  the  substitution  of 
y— 3  for  X  will  have  its  roots  (or  the  Talue's  of  y)  greater  by  3,  than  the  values 
of  X  in  the  proposed  equation :  this  will  be  still  more  evident,  if  both  the  fiven 
and  the  resulting  equation  be  solved ;  the  roots  of  the  former  will  be  found  to  be 
—7  and  +  3,  those  of  the  latter  —4  and  -|>  6.  Let  it  not  be  thought  strange  that 
the  negative  quantity  ~7,  by  being  increased  by  3,  becomes  —4,  or  a  less  quan- 
tity than  it  was  before ;  for  a  negative  quantity  is  said  to  be  increased,  in  pro- 
portion as  it  approaches  towards  an  affirmative  value ;  thus,  — 3  is  ssud  to  be 
greater  than  —4,  —2  than  ^3,  —1  than  —3,  and  0  than  —  1  :  in  the  pre- 
sent instance,  it  is  plain  that  »7  added  to  +  3  will  give  —4  for  the  sum. 
Hencef,  if  the  roots  of  an  equation  be  increased  by  a  quantity  greater  than  tb)^ 


Part  V,     TRANSFORMATION  OF  EQUATIONS.         119 

ExAifPLBS. — 1.  Given  a:*+4a?— 21=o,  to  ti'ansform  it  into 
Another  equation,  the  roots  of  which  are  greater  by  3  than  those 
of  the  given  equation. 

Operation.  Explanation, 

Let  y-3=x,  then  Having  substituted  y-3  for  x^  I 

-r«—  (iZIil  »  — ^i/«— 6 «/  -I-  O  substitute  y-3)9  for  jfi,  y-3.4  for 

*  —  ^y     ^'    ""^a'  — ^y-f-y  4^^  and -.21  for  itself;  I  then  add 

4-  4l^=  (y — 3.4^)     +  4  y — 12  ^1  the  quantities  arising  from  these 

J«2J  -— ^  ^  ^ 2]^  substitutions  together,  and  make  the 

— * result  y*  — 2y — 24«»o,  which  equa* 

J?*4-4j?--21=sy^— 2y— 24=  tion  wiU  have  its  roots  greater  by  3 

„„        .          »     ^  than  the  roots  of  the  equation  given 

Wherefore  y*^^y-^U=zo,  in  the  quegtion. 

is  the  equation  required. 

2.  Given  the  equation  a^'+a?*— JLOa?+4=o,  to  transform  it 
into  another,  the  roots  of  which  ai*e  greater,  by  4  than  the  va- 
lues of  X. 

Let  y— 4=x,  then 

X^z=(y^4\S=:)        y3  — I2y«4.48y— 64 

■f x^=(f^^=^)  ...      4.^2-  8 y4- 16 

— lOx  =:(y— 4.— 10=)  ....  — lOy+40 

+  8= +  8 


This  transformed  equation  is  evidently  divisible  byy(ory  +  o, 
ory^o)'y  therefore  0  is  one  of  its  roots:  by  this  division  U  be" 
comes  y*— 11  y+30=o,  the  two  roots  of  which  are  +6  and  +5  j 
hence  the  three  roots  of  the  equation  y'— 11  y*+30y=o,  being  o 
.  +  6,  and  +  5,  those  of  the  proposed  equation  x* + x**-  10a?+  8=0 
are  known;  for  {stwe  xsBy<^4)  its  roots  ioill  be  0-^4^  6—4^  and 

5— 4;  or —4, +2,  <MMi  4-1. 
CoR.  Hence,  when  the  roots  of  an  equation  are  increased  by 

a  quantity  equal  to  one  of  the  negative  roots,  that  root  is  taken 

away,  or  becomes  0  in  the  transformed  equation  ^   and  in  this 

case,  the  transformed  equation  may  be  depressed  one  dimension 

lower. 

3.  To  increase  the  roots  of  the  equation  x?-^6a?*4-12x-^8 
s=o,  by  1. 

■  '■■■'■'■'  I   ■■    ■     111  II  I  <       I     1 1  1  I  III       III     I  ■    I'l 

greatetjfc  negative  root,  the  negative  roots  will  be'  changed  into- affirmative 
ones. 

It  may  be  likewise  useful  to  remark,  that  a  de&cicnt.  equation  may  be  made 
complete  by  this  rule. 

14 


120  ALGEBRA.  Pakt  V. 

4.  To  increase  the  roots  ofa:*— 4  j:' -1-6  j:*— 13=0,  by  5. 

36.  To  transform  aw  equation  into  another ^  the  roots  of  which 
will  be  less  than  tlwse  of  the  proposed  equation,  by  some  gioen 
/quantity, 

KuLB.  Connect  the  given  quantity  with  some  new  letter  bf 
the  sign  +,  and  proceed  as  directed  in  the  preceding  rule  *. 

Examples. — 1.  Transform  the  equation  x*--2  j?— 24=:o  into 
another,  the  roots  of  which  will  be  less  by  3  than  those  of  the 
given  equation. 

Operation. 

Let  y-^S^X,  then  EspUmatum. 

— 2j:=(y-|-3.— 2=)— 2y—   6     and— 24foriUclf,thefumoftbe8c 
■^24= -*24     i«  y*  +  4y— 21=0,  the  equation 

ia-.2x-24=r  y.+4y-21     ^^^^^«*- 

I  I.I  ■■■■-—■» 

Wherefore  y*+4y^21=:o,  is  the  equation  required. 

This  equation  being  solved,  the  roots  wUl  be  found  to  he  -^S 

and  —7;  wherefore  those  of  the  given  equation  are  +3+3  and 

—7+3,  or  +6  and  —4. 

2.  To  transform  the  equation^^— a«p^+&r— c:^o  to  another^ 
the  roots  of  which  shall  be  less  by  e. 

Let  y+eszx,  then 

*Srr(^+;|s=:)y9  +  Sy«C+3yc*+e'    -j 

+fca:  =a(y+c.6=) by+be  I  quired. 

— c= — c   -^ 

3.  Duninish  the  roots  of  a^— 6«^+9ap— 12ssa,  by  6. 

4.  IMminish  the  roots  of  a7^+5a^— 6x*+7x— Sso,  by  10* 

37.  To  exterminate  the  second  term  of  an  equation. 

RuLB  I.  Divide  the  coefficient  of  the  second  term,  by  the  in* 
dex  of  the  highest  power  of  the  unknown  quantity  in  the  given 
equation. 

II.  Change  the  sign  of.  the  quotient,  and  then  eonnect  it  with 
some  new  letter ;  tins  will  form  a  binomial. 

•  The  trath  of  this  mle  will  be  plain  from  ex.  I .  for  y + 3  being  made  equal 
to  JT,  or  ifssjr— 8,  that  is,  y  less  than  or,  by  8 ;  the  roots  or  values  of  y  in  tba 
transformed  equation,  will  be  less  by  3  than  the  corresponding  values  of  x  in 
|hf  proposed  f  tj^uation,  ^s  is  eridentf 


Pa«t  V.     TRANSFOBJilATlON  OF  EQUATIONS.         l«l 

m.  Substitute  this  binomial  and  its  powers^  for  the  unknown 
quantity  and  its  powers  in  the  given  equation^  and  there  will 
arise  a  new  equation  wanting  its  second  term  '• 

Examples. — 1.  To  transform  the  equation  a^-|-12  *•— 8x— 9 
=o>  into  an  equation  wanting  its  second  term. 

Operation. 

13 
First  — =  +4.    Let y— 4=:ir. 

Then,  a^^(y^4)p^)  y^— 12y«  +  48y—  64* 

—  Sx  =(y— 4.— 8=)  —  8y+  32. 

—  9     = —     9. 


jr»  +  12a«— 8x— 9=  ....  y  *— 66y  +  151=o. 

Explanation* 
I  first  divide  tbe  coefficient  12  of  the  second  tenn  by  the  index  3 ;  the  qno* 
tient  4  I  annex  to  a  new  letter  y,  first  changing  its  sign  from  +  to  —-a 
making  1^—4 ;  this  quantity  and  its  powers  are  next  substituted  for  x  and  ita 
powersy  as  in  the  two  foregoing  rules ;  then  adding  the  like  quantities  together, 
the  sum  b  the  equation  y^  *— 56y  -|- 15 1  no»  wasting  its  second  term,  as  was 
proposed. 

2.  To  destroy  the  second  term  from  the  equation  a:*— 0x^4- 

fcc»— ca?+d=o. 

a 
First,  — —  is  the  coefficient  of  the  second  term  dimded  by  the 

index  of  the  first. 

Let  y  be  the  new  letter,  then  by  the  rule,  y-{'—zsix,  whence 

■      3y«o«    3ya»      a* 
^4  16        64 

ca 

"~CX    SBr        ••«.•••      ••"     Cy    "~  ■-—;* 

4 
4"d=     ...,. -f-rf 

^.     3y«fl«    3y»a»  ya»  ^  6ya         "S^       a*  , 

^*--i — r+'^*-T+T-^+^~64+ 


'  Thb  rule  is  necessary  to  the  solution  of  cubic  and  biquadratic  equations  ; 
and  the  truth  of  it  will  appear  from  an  attentive  examination  of  the  process  in 
ex,  1.    Tbe  third>  fourth,  and  fifth>  &c.  terms  may  be  exterminated  from  auf 


1»  ALGSBAA.  Part  v. 


[•  dsso,  which  J  properly  contr  acted  »,  becomes  jf*  -f  ^—  -^ 

•y^ — S'+'S — ^-y ^?7S ™^>  '^  c^tta^ion  regKtrea. 

o       "  256 

3.  Given  ar* — 4af+8=o,  to  exterminate  the  second  term. 

—4 

Thus, =  —2  ',  then  let  y  +  2=x,  and  proceed  cw  before. 

4.  Given  a:*+ 10  x— 100=0,  to  destroy  the  second  term. 

10  '  , 

Tfitis,  -jrss-l-Sj  te^y--5=x,  and  proceed. 
2 

5.  To  exterminate  the  second  term  from  x'—S  x'-j*4x— 5=o. 

3 

Thus,  — =*:— 1,  let  y  +  l:=x,  and  proceed.    . 

6.  Let  the  second  term  be  taken  away  from  the  equation 
x*4-24x5— 12a?*+4x— 30=0*. 

7.  To  take  away  the  second  term  from  the  equation  x^— 
50x*+40x^— 30x«+20x— 10=0; 

38.  To  multiply  the  roots  of  an.  equation  by  any  given  quantity, 
that  is,  to  transform  it  into  another,  the  roots  of  which  will  be  any 
proposed  multiple  of  those  of  the  given  equation. 

Rule  I.  Take  some  new  letter  as  before^  and  divide  it  by 
the  given  multiplier. 

II.  Substitute  the  quotient  and  its  powers^  for  the  unknown 
quantity  and  its  powers^  in  the  given  equation^  and  an  equation 

equation,  but  these  transformations  being  less  nseful  and  more  difBcult  than 
the  above,  we  have  in  the  text  omitted  the  rales :  in  general,  to  take  away 
the  second  term  reqnires  the  solution  of  a  sioif  le  equation ;  to  take  away  the 
third  term,  a  quadratic ;  the  fourth  term,  a  cubic  ;  and  the  n^  term  requires 
the  solution  of  an  equation  of  n —  1  dimensions.    See  the  note  behw, 

f  This  contraction  consists  in  the  reducing  of  the  fractional  coefficients  of  the 
same  powers  of  y  to  a  common  denominator,  and  then  adding  or  subtracting, 
according  to  the  signs;  putting  the  coefficients  of  the  same  power  ofy  under 
the  vinculum,  &c.  &c. 

i>  In  like  manner,  to  take  away  the  third  term  from  the  equation  x^ — ax' 

-^hx^c=o,  we  assume  y4~^=^>  where  e  must  be  taken  such  that  (suppos- 

»— I  7 

ing  ai=the  index  of  the  highest  power  of  x)  n,  -3—  « •  .^»—  I,  ae+b=:o»    In 

which  case  a  quadratic  is  to  be  solved ;  and  in  general,  to  take  out  the  m*^ 
term,  by  this  method,  an  equation  of  m—  1  dimensions  must  be  solved,  as  was 
observed  in  a  preceding  note.    See  Wbod^s  Algebra,  p.  141. 


Past  V.     TRANSFORMATION  OF  £QUATIONS.        1S3 

win  thence  arise^  whose  roots  are  the  proposed  nnJtsple  of  those 
of  the  given  equatioa.  .  . 

Rule  I.  Assume  some  new  letter  as  before,  and  place  the 
given  quantity  under  it,  for  a  denominator. 

II.  Substitute  this  fraction  and  its  powers,  for  the  unknown 
quantity  and  its  powers  respectively,  in  the  given  equation,  and 
a  new  equation  will  arise,  having  its  roots  respectively  equal  to 
the  given  equation  multiplied  by  the  given  quantity*. 

Examples. — 1.  To  transform  the  equation  x*+5j:— 3=o,. 
into  another,  the  roots  of  which  are  10  times  as  great  as  those 
of  the  given  equation. 


o 

—2      = -2 


Lei  r-=ap. 

10 

Then  j?«      = 

100 

-f5x  = + 


Whence  a«+5x  ^2=i!-+-^^ — 2=o,  that  is,  y«+50y-200 

100     2 

=0,  the  equation  required  \ 

2.  liCt  the  roots  of  3  0^—12  a?* +  15  X— 21=0,  be  multiplied 
by  3. 

9 
Thus,  -|-=*' 


t 


Then3a^^C'p=^y^ 


4y< 


+  15x  = +5y 

-21      =: -gl 

3%ere/are  (^-^+5  y-21,  or)  y«-12y*+45y 
— 189:=o,  the  equation  requured* 


<  This  nile  reqairct  neither  pro«f  Dor  explaiMtion  ;  it  it  fometimet  ufeful  for 
freeing  an  equation  from  fractions  and  radical  qnantities. 

k  Hence  it  appearf,  that  to  mnltiply  tfae  rooU  of  an  equation  hy  any  quan- 
tity, we  have  only  to  n^ultiply  its  terms  respectively  by  those  of  a  geometrical 
progression,  the  first  term  of  which  is  1,  and  the  ratio  the  mikltiplying  qoMi- 


124  ALGEBRA.  Part  V^ 

4.  Let  the  rooU  of  x'  -*3  x+4=so^  be  doubled. 

5.  Let  the  roots  of  ar'-flSa:*— 20x-f  50=o,  be  multiplied  by 
100. 

39.  To  transform  any  given  equation  into  another j  the  roots  of 
which  are  any  parts  of  those  of  the  given  equation. 

Rule  I.  Assume  a  new  letter  as  before^  and  let  it  be  multi-r 
plied  by  the  nimiber  denoting  the  proposed  part. 

II.  Substitute  this  quantity  and  its  powers^  for  the  unknown 
quantity  and  its  powers>  in  the  given  equation  ',  the  result  will 
be  an  equation,  the  roots  of  which  are  respectively  the  parts  pro- 
posed of  those  of  the  given  equation  ^ 

ExAMPLBS. — 1.  Let  the  roots  of  »•— x— 6=o,  be  divided 

toys. 

Assume  3  y=x ;  then  wiU 

x«:s  9y* 
—X  =  .  .  —3  y 

fFA«ice  (9y*— 3y— 5=0,  or)  y* — ^ =o,  is  the  equa^ 

•  3       9' 

tion  required, 

2.  Let  the  roots  of  x»+7x*— 29x+2=:o,  be  divided  by  5. 

3.  Given  x*— 2x^—3  x+4=o,  to  divide  its  roots  by  8. 

40.  To  transform  an  equation  into  another,  the  roots  of  which 
are  the  reciprocals  of  those  of  the  given  equation. 

Rule  I.  Assume  a  new  letter,  and  make  it  equal  to  the  reci- 
procal of  the  unknown  quantity  in  the  given  equation. 

tlty .  thus,  in  ex.  1 .  the  roots  of  the  equation  are  to  be  multiplied  by  10 ; 
wherefore  mnltiplying  the  given  equation  x*  +    5  :r—    Saso 

by  the  geometrical  progreision  1        10    ,    100 

The  product  is  x'  +  sOr— 200so,  as  above,  where 
y  in  the  above  example  answers  to  x  in  this  ;  and  the  like  in  other  cases. 

1  This  rule  is  equally  evident  with  the  foregoing ;  and  in  like  manner,  the 

roots  of  an  equation  are  divided  by  any  quantity,  by  dividing  its  terms  by  those 

of  a  geometrical  progression,  whose  £rst  term  is  1,  and  ratio,  the  said  quantity : 

Thus,  ex,  1.  to  divide  the  roots  of  x'  —  ar  —  5  bo  by  3, 

pivide  its  terms  respectively  by  I        3       9 

X        5 
The  qnotientf  are  x  • — "5"— "T""*  <>>  w  above  ; 

where  y  in  that,  answers  to  x  in  this.    It  is  sometimes  necessary  to  have  rci 
codrse  to  this  rule,  to  exterminate  surds  from  an  equation. 


Pabt  v.    transformation  of  equations,      iss 

II,  Substitute  the  reciprocal  of  this  letter  and  its  powers^  for 
the  unknown  quantity  and  its  powers^  in  the  given  equation  j 
the  result  will  be  an  equation,  having  its  roots  the  reciprocals 
of  those  of  the  g^ven  equation. 

Examples.— 1.  Let  the  roots  of  «*— 2j;a-h3«— 4r=p,  be 
transformed  into  their  reciprocals. 

Assume  y=:— ,  that  is  »=— ,  then  will 

X  y 

y 
y 

+3«=  —  +1 

y 

-4     =  .  ; -4 

^'^^  (77— r7+— — 4a=o,  or  muUiplying  by  f,  ehang-^ 
9      tf       if 

kig  the  signs,  and  dimding  by  4,)  yS-.i-y«+i.  y^L  -<,,  the 
equation  required, 

2.  Let  the  roots  of  a^+lOa?— 25=o,  be  changed  into  their 
reciprocals. 

3.  Change  the  roots  of  a?— ac«+fcxr-c=so,  into  their  reci- 
procals. 

4.  Change  the  roots  of  «*-f  at»-»— fca;"r-«+caf-«— d=ao,  into 
their  reciprocals. 

41.  To  transform  an  equation  into  another,  the  roots  of  which 

are  the  squares  of  those  of  the  gioen  equation* 
RuLjs.  Assume  a  new  letter  equal  to  the  square  of  the  un- 
known quantity  in  the  given  equation  5  then  by  substituting  as 
in  the  preceding  rules  an  equation  will  arise^  the  roots  of  which 
are  the  squares  of  those  of  the  ^ven  equation. 

Examples.— 1.  Let  the  roots  of  the  equation  x^+9:r— 17so^ 
be  squared. 

Assume  yj=^x^ 
Then  x«=sy 

—17= —17 

Whence  y-^O^y'^lT^^o,  the  equation  required  ". 


II.  1 1 1 


*  The  roots  of  the  propoied  equation  fro  1.6  «d4  ^lOSi  those  of  th« 


126  ALGEBRA.  Pakt  V. 

2.  Let  the  roots  of  ar*— a?*+r— 7=o*  be  squared. 
Assume  yssj^ 
Then  a:^s=yi 

-7  =  ....-7 


Whence  y^^y-^  sjy^T^o^  the  equation  required. 

3.  Square  the  roots  of  x^+Sx*— 3a?-.12=o. 

4.  Square  the  roots  of  x*— (mp*4-^— cx+d=:o. 

5.  Square  the  roots  of  xr — 7xt— 8=o. 

OF  THE  LIMITS  OF  THE  ROOTS  OF 

EQUATIONS. 


42.  Let  x— a.x--6j7— c^-hd=o,  be  an  equation^  having  the 
root  a  greater  than  h,  b  than  c,  and  c  than  d*;  *'hk  wfaich^  if 
a  quantity  greater  than  a  be  substituted  for  x,  (as  every  factor 
i^^  on  thb  supposition,  positive,)  the  rescdt  will  be  positive;  if  a 
quantity  less  than  a,  but  greater  than  b,  be  substituted,  the  re- 
sult will  be  negative,  because  the  first  factor  will  be  negative, 
and  the  rest  positive.  If  a  quantity  between  b  and  c  be  sub- 
stituted, the  result  will  again  be  positive,  because  the  two  first 
fsuctora  are  negative,  and  the  rest  positive ;  and  so  on  ^.    Thus, 


transformed  equatiuD  are  2.56,  and  113.36,'  which  are  the  squares  of  the  for- 
mer respectively. 

*  *'  In  this  series  the  greater  is  <f,  the  less  is  —  </ ;  and  whenever  a,  b,  c,  —  <f, 
&c.  are  said  to  be  t^e  roots  of  an  equation,  taken  in  order,  a  is  supposed  to  be 
the  greatest*  Aiso  in  speaking  of  the  limits  of  the  roots  of  an  equation,  we 
understand  the  limits  of  the  possible  roots."  This  note,  and  the  article  to 
which  it  refers,  were  taken  .from  Mr.  Wood's  Algebra ;  see  likewise,  on  this 
subject,  Maclaurin* 8  Algebra y  part  %  cb.  5.  Pf^olfius's  Algebra,  part  1.  sect.  2. 
ch.  5.  Sir  Isaac  Newton* sArithmeiica  Universalis,  p.  258.  &c.  JCh\  J9^arwg*s 
AMUcUuma  AlgebraictB,  8cc 

•  To  illustrate  this,  let  the  roots  of  the  equation  x*  — /»x*  +  ?* '  — rx-^-s^o 
be  a,  b,  e,  and  if,*  then  x— aso,  x—b^o,  x — cso,  and  x^^dsso  ;  and  let 
g,  which  we  will  suppose  less  than  a,  but  greater  than  6,  be  substituted  for  x 
in  the  latter  equations  ;  then  will  ^— a  be  negative,  and  the  rest,  viz.  g—h, 
g'^c,  and  g — d,  positive,  and  consequently  their  product  will  be  positive ;  and 
g'^Oy  (a  negative  quantity,)  multiplied  into  this  positive  result,  will-  therefore 
give  a  negative  product:  if  h,  which  is  less  than  6,  but  greater  than  c,  be  sub*, 
stitttted  for  Xj  we  have  A— a  and  h^-^b  both  negative,  and  their  product  posi- 
tive} but  A"-»c  and  A»- (fare  both. negative,  therefore  their  product  isitosi- 


Part  V.  LIMITS  OF  THE  ROOTS.  IftT 

quantities  which  are  limits  to  the  roots  of  an  equation^  (or 
between  which  the  roots  lie^)  if  substituted  for  the  unknown 
quantity^  give  results  alternately  positive  and  negative.** 

43.  *'  Conversely,  if  two  magnitudes,  when  substituted  for 
the  unknown  quantity,  give  results  one  positive  and  the  other 
negative,  an  odd  number  of  roots  must  lie  between  these  mag- 
nitudes :  and  if  as  maoy  quantities  be  found  as  the  equation 
has  dimensions,  which  give  results  alternately  poiitive  and  ne- 
gative, an  odd  number  of  roots  will  lie  between  each  two  suc- 
ceeding quantities  5  and  it  is  plain  that  this  odd  numb^  can- 
not exceed  unity,  since  there  are  no  more  limiting  terms  than 
the  equation  has  dimensions.** 

44.  If  when  two  magnitudes  are  severally  substituted  for  the 
unknown  quantity,  both  results  have  the  same  sign,  either  an 
even  number  of  roots,  or  no  root,  lies  between  the  assumed 
magnitudes. 

Cor.  Hence,  any  magnitude  is  greater  than  the  greatest  root 
of  the  equation,  which,  being  substituted  for  the  unknown  quan- 
tity, gives  a  positive  result. 

45.  To  find  a  limU  greater  than  the  greatest  root  of  an  equation. 
Rule.  Diminish  the  roots  of  this  equation  by  the  quantity 
6,  (Art.  36.)  and  if  such  a  value  of  e  can  be  found,  as  shall 
make  every  term  of  the  transformed  equation  positive,  all  its 
roots  will  be  negative,  (Art.  31.  Cor.)  consequently  e  will  be 
greater  than  the  greatest  root  of  the  eqtuition. 

ExAMi>LE8. — 1.  To  find  a  limit  greater  than  the  greatest  root 
rfa*— 5ar+6=o. 

Let  a?=:y-fe 
TZien  iriM  jt«=:y«+2  ye+€* 
— 5a:=    — 5y— 5e 
+6  = +6 

Whence  (y*4-2ye— 5y  +  ^— 5e+6=a,  or)  ys+ge— 5^ 


+e.e— 5+6=0,  is  the  transformed  equation  ^  now  it  appears  by 
trudsj  that  4  being  substituted  for  e  in  this  equation,  it  will  be* 

five ;  and  theie  two  products  mnUipiied,  give  likewise  a  poMtirc  product.  In 
like  manner  it  may  be  shewn,  by  substituting^  k,  which  is  less  than  c,  and  great- 
er UiaB  1/,  the  result  will  be  negative ;  and  substituting  m,  less  than  the  least 
root,  the  result  will  be  positive. 


138  ALGEBRA.  Paet  V. 

come  y^+3y+3=d,  of  which  all  the  rooU  are  negative;  where- 
fore 4  Is  greater  than  the  greatest  root  of  the  equation  a^— 5x4- 
.6=:o,  '  - 

2.  To  find  a  limit  greater  than  the  greatest  root  of  x'— l^x^ 
-f  41x«-43sse;o. 

Let  xssjr-f  6,  a$  before. 
Then  ioii/x»=sy*-f  3y*e+Sy€«+«» 

-|-41x= 4-41y  +41e 

—43    = -43 

JfTAerc/bre  (y«+3y«e— I2y*+3ye*  — 24ye+41  y+c»— I2c«+41c 

—43=0,  or)y'+3.c— 12.y«+3e»— 24c+41.y+e.c'  — 12e-f41 
•^43=^0,  is  the  transformed  equation j  where  (by  trials)  it  isfoundp 
that  if  S  be  substituted  for  e,  the  terms  will  be  allposUive;  viz. 
^  +  12y'+41y+29=o;  whence  S  is  greater  than  the  greatest 
root  of  the  given  equation, 

3.  Required  a  limit  greater  than  the  greatest  root  of  x^— 6  x* 
— 25  X— 12=0.     Ans.  9. 

4.  find  a  limit  greater  than  the  greatest  root  of  x*— 5x'+ 
6x*— 7x+8=o. 

5.  To  find  a  limit  greater  than  the  greatest  root  of  x^+3  x'— 
5x«+8x— 20=0. 

46.  To  find  a  limit  less  than  the  least  root  of  an  equation. 
KuLB.  Change  the  signs  of  the  even  terms,  (the  second, 
fbuirth*  sixth,  &c.)  and  proceed  as  before  ^  then  will  the  limit 
greater  than  the  greatest  root  of  the  transformed  equation,  with, 
its  sign  changed,  be  less  than  the  least  root  of  the  given  equa- 
tion.   See  Cor.  to  Art.  30.  and  Art.  45. 

ExAMPLBs.— 1.  Let  X*— 7x+8=o,  be  given  to  find  a  limit 
less  than  the  least  of  its  roots. 

This  equation,  by  changing  the  sign  of  its  second  term,  becomes 
x'-f7x-f8=o. 

Let  x=y-f  c. 
Then  x*=y»+2ye-|-c« 
-|-7x=     +7y+7« 
+  8  =...........+8 

^yhence  {y  •  4-2  ye4-r  y+c'  4-7  e4-8=o,  or)  y '  +2e4-7.y 

+«+ 7»e4-8=o,  is  the  transformed  equation;  and  i/"— 1  be  substi- 


/ 


P^ET  V.  LIMITS  OF  THE  ROOTS.  189 

bUed  for  e,  aU  Us  terms  will  be  posiiive^  for  the  equatum  he- 
€Oi»e«y^4-5y«f  ftsco;  whetefwre'^l  ualimiU  less ikan the leasi 
root  of  the  equation  s'  — 7  J'-f  8=5«. 

9.  To  find  a  limit  less  than  the  least  root  of  x«-f-x'«»lOjr4^ 
€=so. 

Changing  the  signs  of  the  second  and  fourth  terms,  the 
e^aHon  becomes  x' ^3f* ^lQx^6:=iQ» 
Lei  x=cy  4-  e,  then  voill 

— «*=s    —   y*  —  2ye  —  €» 
—  lOopss — lOy  — lOe 

-6= -6 

**  «  — .  I         ■       I    III      ■  ■■  ■  — 

ff^hetue  y'-fSe— l.y»-f 3€'— »e— 10.y+«'— e— IQ.c— 6 
=0^  is  the  transformed 'equation,  in  which  4  being  substituted  for 
e,  U  becomes  y«-f  11  y'-^S0y'\'^:s;o$  wherefore  —4  is  less  than 
the  least  root  of  the  equatiofi  x^-^x' ^  10  x -^6=^0, 

3.  To  find  a  limit  less  than  the  least  root  of  x'-f- 12«— 90 
=0.     Ans.  —14. 

4.  To  find  a  limit  less  than  the  least  root  ci  x'-^Ax'-^Sx-^ 

6=0. 

5.  To  find  a  Hmit  less  than  the  least  root  of  a?* —5  a?'  —3=0. 

6.  To  find  the  limits  of  the  roots  ot  jr'+«»— 10«  +  9=o. 
Ans.-^Z  and'^6. 

7.  Baqnirfdthttttfliitoiof  a»«-^4«»4*8a?'-14«+^=o? 

a  What  are  the  limtta  of  the  fools  c^jp* ---2a;' — 5  x+ 7^0  ? 
9.  What  are  the  limits  of  the  roots  of  a?«+ 3  a;'— 5x4- 10«o? 

RESOLUTION  OF  EQUATIONS  OF 
SEVERAL  DIMENSIONS. 

47*  When  the  po^ibte  roots  of  an  equation  are  integers,  either 
positive  or  negatiuoe,  they  may  be  discoffered  as  follows, 

RuLB  I.  Find  all  the  dt^^sors  of  the  last  term,  and  suhsdtnte 
them  soceessively  fyr  the  imkaown  ^^uaatitj^  In  the  proposed 
equation. 

II.  When  by  the  substitution  of  either  of  these  divisoiB  for 
the  rooty  the  rewilf  ing  equation  becomes = o,  that  divisor  is  a  root 
ei  &e  giifM  eq^aadoA',  otherwise  it  is  not. 

HI.  U  none  <tf  the  (Nfison^  suooeed,  the  rools  are  either 
fractional,  irrational,  or  impossible. 

VOL.  I.  K 


ISO 


ALGEBKA. 


Pabt  V. 


IV.  When  the  last  term  admits  of  a  great  aumber  of  diilaors,! 
It  will  be  convenient  to  transform  the  given  eqiMttion  into  ano- 
ther, (Art.  35,  36.)  the  last  term  of  whid»  will  haye  femx 
divisoTB. 

Examples. — 1.  Let  x'— ^a:'--5x+6=sa,  be  given,  to  find 
its  integral  roots  by  this  method. 

First,  the  divisors  of  the  last  term  6,  ore-f  1,-1,  +2,—^, 
-f3,— 3,  +  6,  and—6-y  now  +  l  being  substituted  for  x  in  the 
given  equation,  it  becomes  +  1  —2—5  +  6=o ;  wherefore  -^1  is  a 
root. 

Next  J  let  —  l  be  substituted,  and  the  equation  becomes  —  1— S 
-J-  5  +  6s=8 ',  wherefore  — I  is  not  a  root, 

Thirdly  y  let -^-^  be  substituted,  and  the  equation  becomes 
$ — 8— lO-^er:— 4;  wherefore +  ^  is  not  a  root. 

Fourthly,  let  —2  be  iubstituted,  and  the  ^nation  becomes 
—8 — 8  +  10  -f  6=0;  wherefore  —2  w  a  root* 

Fifthly,  let+S  be  substituted,  and  the  equation  wiU  then  be- 
come+27— 18—  H^  +  6=^0  *,  wherefore  +  3  i«  likewise  a  root. 

Thus,  the  three  roots  of  the  given  equation  are'\- 1,-2,  and 
+  3  3  and  it  is  plain  there  can  be  no  more  than  three  roots,  since 
the  equation  arises  no  higher  than  the  third  degree  f  consequently 
there  is  no  necessity  to  try  the  remaining  divisors, 

2.  Givenx*— 6  0?'  — 16  a? +  21=0,  to  find  the  roots. 

The  divisors  of  the  last  term  21,  are+ 1,— 1,  +3,— 3.  +  7> 
—  7>  +  2 1^  and  —21 ;  these  beif^  successively  substituted  for  x,  we 
shall  have 


SubstitmioDs. 


i^^to 


+  1 


—  I 


+  3 


Results. 


+  1—       6—   16+21=0 


+  1—       6+16+21=32 


+81—     54—  48+21=0 


—3 


+  7 


-7 


+^1 


—21 


4-81-     .54+  48+21=96 


+2401—  294—112+21=2016 


+  2401—  294+li2+2!=2240 


+  194481-2646—336+21=191520 


+  194481—2646+336+21  =  19219^1 


M'fterefore  + 1  and  +3  are  the  only  roots  which  pan  befdmnd 
by  this  method;  the  ttoo  remaining  roots  are  therefore  impo^silde^ 

*ein^— 2+^^—3.  .. 


Pakt  V.  RtoOLUTlON  OF  EQUATIONS.  ISl 

3.  Given  x*— 4  j!9*-19 x«+ 106  J?—  l«Oajo,  to  find  the  roots; 
S'mce  the  last  term  1^0  has  a  great  number  of  divisors,  it 

wiU  be  proper  to  transform  the  equatim  into  another,  whose  abso* 
lute  term  will  have  fewer  divisors ;  in  order  to  which,  let  xsz^-f  2> 
then  (Art.  36.) 

j^=:j(*+8y»+^4y«+  3Sy+   16 

—  4x3=5:     _4yS_^y«_  48y^  3«' 

— 19x«= — 19y*—  76y—  76 

+  106j;=: +I06y+212       ' 

—  1%   =? ..—120 

y*+4y'— 19y«+  14y=o 
Here  ^Ae  last  term  vanishing,  the  number  assumed,  viz, +2,  is 
mi€  of  the  roots  of  the  oiigiwU  equation,  (Art,  33.  note,)  and  the 
transformed  equation  being  divisible  by  y,  will  thereby  be  reduced 
one  dimension  lower :  thus,  y^  +  4  y^— 19  y  +  14=o ;  the  divisors 
of  <Acto^/crml4,arc+l,— 1,+2,-*2,+7,— 7.+  14,— 14j  each 
of  these  being  substituted  for  y  in  the  last  equation,  +1,4-2,  and 
^7  are  found  to  succeed,  they  are  therefore  the  roots  of  the  transi* 
formed  equation  ^^4-4^*— 19y-hl4=o;  wherefore,  since  x=y4- 
2,  three  of  the  roots  of  the  original  equation  will  be  (l-^-^sz)  S, 
(2+2=)  4,  and  (—7+2=)— 5,  which  with  the  number  2  <w- 
sumed  above,  gioe  +  2>+3^  +  4>  and  — 5>  for  the  four  roots  re- 
quired. 

4.  Given  x'— 3ax^— 4a^x+12a'sso,  to  find  the  roots. 

The  numeral  Visors  of  the  last  term  are  + 1,-1, 4-2,-2, 
+3,— 3,+4,—4, 4-6,-6,4- 19>  antfi —IS ;  and  of  ^toe, 4-2,-2, 
ojid— '3  are  found  to  succeed  ji  wherefore  the  roots  are  4- 2  a,— 2  a, 
and — 3  a* 

5.  Required  the  roots  of  x^4-a?— 12=a?     Ans,  3,  and  —4. 

6.  What  are  the  roots  ^f  a:»4-4x«4-a?— 6=o?     Ans.  1,-2, 
and  —3. 

7.  What  are  the  roots  of  a!5  4-2jr*— 19x-20=o?     Ans.-^l, 
—4,  aitd+5. 

8.  Required  the  roots  of  a?>— 14 «« +51x4-126=0?     Ans. 

—2^+7,  and+9. 

9.  Whataretherdotsofx*— 15x^+10x+24=o?    Ans,--!, 
+2»+3,  and  —4. 

10.  Required  the  roots  of  x'+4x'— 7x— 10=o? 

K  2 


la 


ALOBBIU. 


Pa»t  V* 


4g.  SIR  ISAAC  NBWTONS  M£THQI> OF  DISCOVI^mG 
THE  ROOTS  0£  EQUATIONS  BT  MEANS  OF 

DlViBOKS; 

Rule  I.  For  the  unknown  qiiMitity  In  the  given  equatkni^ 
substitute  three  or  more  terms  of  the  arithmetical  progresiioil 
2>  1>  0^— 1>— 2^  &c.  and  let  these  t«nni  lie  placed  in  a  column 
one  under  the  other. 

il.  Substitute  each  number  in  this  column  successively  fo^ 
the  unknown  quantity  In  the  proposed  equation ;  collect  all  the 
terms  of  the  equation  arising  from  each  substitution  into  one 
sum^  and  let  this  sum  stand  opposite  the  number  substituted 
from  whence  it  arises :  these  sums  wiH  form  a  second  cc^iinm* 

III.  Find  ail  the  divisors  of  the  8ums>  and  place  th^ai  ill 
lines  opposite  their  respective  sums :  these  will  form  a  third  co* 
lumn. 

IV.  From  among  the  divisors  collect  one  or  more  aritlimeti- 
cal  progressions^  the  terms  of  which  difieir  either  by  unity>  or 
by  some  divisor  of  the  coefficieifit  of  the  highest  power  of  the 
unknown  quantity,  observing  to  take  one  term  only  (of  each 
progressioh)  out  df  each  line  of  the  divisors :  eaeh  of  these  pro* 
gressions  will  form  an  additional  column. 

V.  Divide  that  term  of  the  progression  thus  found>  (or  of 
each  progression,  if  there  be  more  than  one,)  which  stanifii 
against  O  in  the  assumed  pitogreadon,  by  the  conmion  dift^rvnoe 
of  the  terms  of  the  fortner }  and  if  the  ]progres6ion1te  increas- 
ing, prefix  the  sign  -|-  to  the  quotient ;  but  if  it  be  decreasifig;^ 
prefix  the  sign  — :  this  quotienf  will  be  a  i^oot  of  the  equatnM». 

Hence  there  will  be  as  many  roots  found  by  this  ma^iod)  ii 
there  are  progressions  obtained  fl*om  the  divisMi. 
EixAMVLES^ — 1.  Givenx'— 24:— 24s«^  to  &d  tte  tdiies  of  x. 

Operation. 

I,  2,  3,  4,  6,  8,  12,  24 

l>5,2g. 

1,  2,  3,.  4,  6,  8,  12,  24 

1.  3,  7,  21 

1,  2,  4,  8,  16 

Whence,  the  roots  are  +6  and  —4. 

JExpUufotion, 

The  left  haod  column  is  the  assumed  progreition»  the  tevms  of 
rabilltnted.  successively  for  x  in  the  given  equation:  firsts  by  subslitatiiii^  2 


Substitutions, 

Results, 

2 

—24 

I 

—25 

0 

-24 

—  1 

—21 

—2 

-1(5 

Prog'i  deritmd.\ 

4 

a 

'      & 

& 

6 

4r 

'     7 

3 

8 

2 

sli 


hich 


txw  V.    N£WTON*S  METHOD  OF  DIVISORS. 


133 


iir  j%  tiKB  aqikftti0a  atamiBlbt  to  —04,  wbkh  h  tli«  'nmtit  io  tliit  ossc ; 
this  I  put  in  the  seoood  colnmn,  and  itg  divisors  1  >  S,  3,  4,  H^  &c.  in  the  third. 
Secondly,  I  substitute  1  for  s,  and  the  whole  equation  amounts  to  —35,  viitdi 
is  the  second  retuit^  and  it«  divisors  are  J  ^  5,  and  25.  ThirdJl)r»  bjr  svbstitut- 
iug  0  fpr  Xy  the  result,  is  —34,  and  its  divisors  1 ,  3,  3,  4,  6,  &«.  as  in  the  first 
case.  Fourthly,  by  sabstitnting  —  1  for  x,  the  result  is  «-'21 ,  and  its  dansors 
aue  I,  3,  ?•  and  SI.  Ftflhly,  by  svbstiUAior  '-^^  ^  'V**'^  is  "-I6»  the  d^ 
vu#rf  «f  which  are  1,  S,  4,  8,  and  16.  Sixthly,  I  try  t9.obtM  a  progreswm, 
hj  taking  one  number  out  of  each  line  of  the  (divisors :  and  first  I  tvy  for  an  in- 
creasing one  ;  the  only  one  that  can  be  found  is  4,  5,  6,  7t  and  8,  Tis.  4  out  af 
Che  first  line,  5  out  of  the  second,  6*  out  of  the  third,  7  out  of  the  fourth,  and  % 
iMit  of  the  fifth  $  these  numbera  eonstitute  the  fourth  column.  SeTenthly,  I  tiy 
fisr  a  decreasing  progression,  and  (proceeding  as  belbre)  find  that  6,  5,  4,  Sf 
mild  2,  which  constitute  the  fifth  column,  is  the  only  one  that  can  be  obtmined. 
Eighthly,  the  number  6  and  4,  standing  opposite  the  0  in  the  assumed  progresik. 
sion,  divided  by  the  common  difference  I ,  gives  6  and  4  for  the  roots  of  the  equa- 
tion. The  former  being  a  tenn  of  the  incrcasmg  progression,'  must  have  4>  pre- 
fixed to  it ;  the  latter  being  a  term  of  the  decreasing  progression,  must  have  — 
prefixed  ;  wherefore  the  roots  are  +  6  and  —4. 

2.  Givea  of^— 64?'  -7  jp+60*=#,  to  find  the  roots. 

Opbration. 

Dwisions, 
I,  2,  3,  5,  6,  10,  15,  30 
1,2,3,^,6,8,  12,  16,  &c. 
1,2,3,4,5,6,  lO,  15,  &c 
1#  2,  3^  4j  5,  6,  10,  15,  &c. 
1,  2,  3,  6,  7>  14,  21,  42 


Substitutions, 

ResuUs. 

2 

30 

1 

48 

0 

60 

—  I 

60 

-2 

42 

Prog:d(rived.\ 

2 

3 

5 

3 

4 

4 

4 

5 

3 

5 

6 

2 

6 

7 

I 

Roots  4,  5,  and 

Expktnaiion, 
Proceeding  as  before,  I  obtain  three  progressions,  two  increasing,  and  one 
decreasing,  and  the  numbers  4, 5,  and  3,  standing  opposite  the  0,  bein^  dirided 
Vj  1  the  common  difference,  the  quotients  are  the  soots,  nz.  4-  4  and  -f  5  in 
ihe  increasing  progressions,  and  —3  in  the  decreasing  one. 

3.  Given  «*— x*  — 10a:+6=<>,  to  find  the  root$. 


Substitutions. 

Results, 

2 

-10 

1 

—  4 

^       0 

+  e 

—1 

+  14 

-« 

+  14 

Divisors, 
1,  2,  5,  10. 
1,2,4 
1,  2,  3,  6 
1,2,7,14 


Progressions. 
5 
4 

I  > 

I 


1,  2,  7,  14 

Here  vfe  can  derive  only  one  progression,  and  ikat  a 

one;  wherefore  t/ie  only  root  discovered  l^  this  s^ethod  ja  i^jS : 

but  by  means  of  this  root  the  given  equation  may  be  depressed  to  a 

quadratic,  (Art,  33.)  and  the  two  remaining  roots  found  by  the 

Jmown  rule  far  quuiraties;  thus,  Mee  x-f^seo,  d^idmg  the  pror 

^H-ir'  •A^lOxcf^ 
posed  equation  by  this,  we  obtain  {^ ZTZ ss);^*  — 4«+ 


s^^ 


k3 


134 


ALGEBRA. 


Paet  V, 


5=:o,  the  two  roots  of  which  are  (2+  v^.=t)  3.4142135624  onrf 
.6857864376. 

4.  Required  the  roots  of  6  x*— 20  x» — 12  x*  —  1 1  x— 20=o  > 

Dioison.  \PTOg, 

1,2,7,  11,  14,22,77,  154  2 

I,  3,  19,  57  3 

1,  2,  4,  5,  10,  20  4 

1,5  5 

1,  2,  3,  5,  6,  7,  10,  14,  15,  21,  30,  &c.  6 

Here  we  obtain  only  one  progression,  consequently  -^^  A  is  the  only 
Toot  found, 

5.  Given  j?*+a?'-- 29 x»— 9x4-180=0,  to  find  the  roots. 


Subsiit. 

Results. 

2 

—  154 

1 

-57 

0 

-20 

-1 

+  5 

-2 

+210 

Subst 

Results. 

2 

70 

1 

144 

0 

180 

-1 

160 

-2 

90 

Divisors, 


Progressions, 


\i 


2 
3 
4 
5 
6 


5 
4 
3 

2 
1 


7 
6 
5 
4 
3 


1,  2,  5,  7,  10,  14,  &c.    1 
1,  2,  3,  4,  6,  8,  &c.        2 
1,  2,  3,  4,  5,  6,  &c.        3 
1,  2,  4,  5,  8,  10,  &c. 
1,  2,  3,  5,  6,  9,  &c. 

Here  are  four  progressions,  two  increasing  and  two  deereasi$ig, 
and  the  roots  are  3,  4,-3,  and  —  5. 

6.  Required  the  roots  of  x« — x — 12a:o  ?     Ans,  +4  and  —3, 

7.  Required  the  roots  of  x* +2  x»  —23 x— 60=0  ?    Ans,  +5, 
—4,  and  -^3. 

8.  What  are  the  roots  of  2x5— 5x*-|-4x— 10=o  ?     An^ 
swer,  one  root  +  24-. 

9.  Required  the  roots  of  x' — 3  x»  — 46  x— 72 = o  ?    Ans,  +  9, 
—2,  and  -B-4. 

10.  Tofindtheroots  ofx*— 6x*  +  10x— 8=0? 


RECURRING  EQUATIONS. 

49.  A  recurring  equation  is  one  having  the  sign  and  coeffi- 
cient of  any  term,  rec]coning  from  the  banning  of  the  equa- 
tioil,  the  same  with  those  of  the  term  equally  distant  from  the 

end  5  and  its  roots  are  of  the  form  a,  — ,  b,  -r-,  or  the  recipro- 

a         b 

.cals  of  one  another. 

»        •  • 

..  50.  If  the  recurring  equation  be  of  an  odd  number  of  dimen^ 
sions,  + 1  or  —  J  is  a  root  y  and  the  equation  may  be  depx^esed 
to  one  of  an  even  number  of  dimensions.  (Art.  33.) 


Tamt  ▼.  RECURRING  SQUATIONS.  135 

Thus,  let  x^^^x'-^-lcno',    +i   »   evidenibf   one   rool; 
ihatefwre,  (Art  32.) 

—    «'  +  ! 


TAtf  equation  x'— x^l=o,  6et»^  resolved  hy  the  rule  for 
f iia<2rafuSf,  it$  roots  wiU  be  found  to  be    -^  "^    . 

Cor.  Hence,  a  cubic  equation  of  the  form  Ji^±px'±px'^i 
may  always  be  reduced  to  a  quadratic*  and  its  roots  found. 

51.  If  the  given  equation  be  of  even  dimensions  above  a 
quadratic,  its  roots  may  be  found  by  means  of  an  equation  of 
half  the  number  of  dimensions. 

Thus,  by  supposing  the  equation  to  be  the  product  of  thefae* 

i    i 

tors  X— flj? ,  X— 6jr— T-,  4c.  by  actual  multiplication,  and 

a  0 

putting  m=:a-l ,  n=r&-| — r-,  4c.  we  obtain  x*  — mx-h  1,  x*  — nx 

a  0 

+  1,  4c.  wherefore  by  multiplying  these  quadratic  factors  toge^ 

ther,  and  eqiictting  the  coefficients  of  each  term  of  the  product, 

with  that  of  the  corresponding  term  of  the  given  equation,  the  t)a- 

lues  of  m  and  n  will  be  readily  found :  and  since  for  every  single 

value  ofm  there  will  be  two  values  of  x,  it  follows  that  the  equc^ 

tion  for  finding  m  will  be  of  but  half  the  number  of  dimensions  ne- 

cessary  for  finding  the  value  of  xby  other  methods. 

Examples.— 1.  Let  x*— 3x'+3x»— 3x  +l=obe  the  pro- 

posed  equation. 

Assume  the  product  (x*— nix+  Ijf*— nx-f  1=)  x*  ^m-j-n,3i^-{' 


pm^^^jc'-^m'^n.X'^'l^the  proposed  equation:  then  making  the 
coefficients  of  like  powers  of  x  in  this  product  and  the  given  equa^ 
tion  equal,  we  shall  have  m-|-n=s3,  and  mit4-2=2^  or  nussto-, 
wherefore,  if  n^:zo,  then  m:=z3,  and  the  two  equations  x'— -mx 
•fl=o^  and  X*— nx-flssoj  become    respectively    x'— 3x+I 

3+ a/5 
=0,  and  x'  +  l^oj  from  the  former  of  these  x=g(  -^^  -as) 

k4 


IS6  ALdBBAA.  FaktT. 

^SieasaiSSSt,  imd  ^rt^MOllS  -,  whieh  tw6  taluei  of  «  mre  the 
reciprocals  of  each  other.  From  the  latter,  ifiz,  a^  +  la^o,  we 
obtain  *=  +  v^ — 1,  or  +  ^  —  1,  bud  — ^-*1,  /oi"  the  two  re- 
maining  values  of  x, 

2.  Let  a:'— 1=0  be  given^  to  find  the  values  v£x. 

Here  it  is  plain  that  -f  1 1#  a  root,  or  x-^-lsto,  wherefore  di* 

viding  the  given  equation  by  this,  we  have  (- si)3fl-{'X+lsso, 

the  two  roofs  of  which  are     ■   ■     ■  ~"  .  cM  ^   ""'      «  . 

1-4-  ^•^ft 

3.  Given  a:*  -f- 1  =:o,  to  find  the  values  of*.  Arts.  —1, ^ , 

4.  Let  the  equations  or^—lsso,  «•+ lwo>»*-— lr*o,  andV-h 
Iso,  be  proposed^  to  find  the  values  of  « in  eaieli. 

Literal  equations^  wherein  the  given  quantitj^  and  the  lan- 
known  one  are  alike  afiected^  may  be  reduced  to  others  of  fewer 
dimensions^  by  the  following  rules. 

52.  H^hen  the  given  equation  is  bfevin  ^mensUms, 
Rule  L  Divide  the  equation  by  the  equal  powers  of  its  two 
quantlti^  in  the  middle  tenh. 

II.  Assinne  a  new  equation,  by  putting  some  letter  equal  to 
the  sum  of  the  quotients  arising  fh)m  the  division  of  the  given 
and  unknown  quantity,  alternately,  by  each  other. 

III.  Substitute  in  the  former  equation  the  values  of  its  terms 
^ound  by  the  latter,  and  an  equation  will  arise  of  half  the  di- 
mensions of  the  given  one^  from  the  solution  of  which  the  roots 
of  the  given  equation  may  be  detennined. 

Examples. — 1.  Required  the  roots  of  ar*— 4al?^-5a«x•— 
irf'J;4■a*=o? 

Fir^j  dividing  the  whole  equation  by  the  equal  powers  in  the 

*P*     4j?  4fl     fl' 

UtiddU  term,  it  becomes  ( |-.^-f  5— — ^--ssio:  or^  which  is  the 

-a»      a  XX' 

9mte,)  -r^ 4. — f- 5s=o.  Let — | b=«> thenhysqutfr- 

a      X*         a      X  ax 


x^     a' 


ing,  -^.f  ~«f  ftiaap',  and  by  sulistitutmg  z'  and^for  th^r  va&tes 


a?'     a* 


in  the  equation —--{ —4. — | h5=o,  it  becomes  «•— 4f+3 

a'     X'         ax 


KwtT.  HECUBttlNQ  EOUATIONS.  137 

=0,  whence  z=3,  or  1  y  but  since  — + — =x,  if  the  former  valae 

d      X 

be  taken,  then  — | — Ss3  ;  whence  «*-^3  axss  — 1^»  fMch  eokei, 

a      X 


a 


gives  xss{-^3±^bB^)  3.618034a,  or  S61966  eu    But  if  the 

X      a  \   _D 

latter  value  ofz,  namely  1,  be  taken,  then  ( — | — =1,  or)  ar— 
•^  ax 

ax=:^a\  whence  j=s    —    "^ are  1^  ^too  remaining  roots, 

.  «.  Gii«B  7*?*^— ^««*— 8fi«'*JP+7«^=o,  to  fiod  the  faliM 
of  X.  

This  divided  by  aV  becomes  7—+ 26. h- t==<>'  -^* 

^  a*     a?»  a«     x' 

*s       a  x^     k*  X      a 

z'sz 1 >  then  2*— 2= — h^ — ,  which  multiplied  by  z^--^^ — , 

ax  a^     X'  ^         "^        a      X 

<p3  a  «p  £|»  3p3  |i|3 

a*      X      a     x^      a^  x^ 

X*     a* 
3z=s— +— . 
a^     X* 

}%e»e  wttecj  ^fe^ifwfed  a*  before,  we  obitnn  72'— 262*— 21  z 

+52=0,  one  root  of  whWh  {by  Art.  47)  e<  4,  and  by  means  of 

this,  the  equation  may  bedepressed  to  the  quadratic  If-^^z-^  IS 

=0,  {Art.  32.)  the  two  roots  of  which  are  +1.2273804,  and  — 

1.5130947.     Wherefore,  since «=— H — ,  or  jc*— aarss— «%  by 


a       X 


4^  M2t£tio»  of  this  we  obtain  xs=:      —"^ ,  i«  which,  if  the 

three  values  ofzbe  successively  substituted,  the  six  roots  of  the 
given  equation  will  be  obtained, 

S.  To  find  the  roots  of  «*+6aa?*-20a*«*+6a*a?-fa*=sp. 

4.  To  find  the  roots  of  a;*-204ia:»  +  1««^x*-20«»«+«*sb:o. 

5.  Hcqtnrcdtheroot8  6fa^-aa?*-fl*x+rf^=^ 

63.  When  the  given  equation  is  of  odd  dinunsixms. 

Rule.  Divide  the  equation  by  the  sum  of  the  known  ^nd  un- 
koown  quantities,  and  proceed  as  before. 

aLAMPi.Bs.— 1.  Given  «?*-3  ax^+e.a's^-^e  f^x'  -3  a^x+a*, 
to  find  the  roots. 


1S»  AliGfiBBA.  Paut  V. 

First,  dividing  by  X'\'a,ihe  quotient  is  x^  — 4x^a+10£'a'  <* 
4jra'+(r*=o;  wherefore  dividing  this  by  x*a*,  according  to  the 

fretting  ruUt  the  quotient  is  — | 4* — I 4-  10=o  ;  let  z 

a^     X*         a      X 

X      a       .  X'     a*  - 

«= — I ,   then  z*= — I \'2,  and   substituting   these   va- 

ax  a*     a?* 

lues  as  before^  2*— 4z-|-6=o;  whence  2=5:2+ v'— 5;  but  si 


since 


ar       a  aZ'\'a>/z' — ia 

z  = — ,  we navex'  '-azx^-^a*  ;  whence x= — = — ^  ■» 

ax  2 

and  substituting  for  z  its  values  found  above^  we  obtain  four  of  the 

roots,  which  together  with  —a,  (since  x+asso,)  make  up  the  five 

roots  of  the  equation, 

2.  Given*'— or*— a'-'jj-Ha'sso,  to  find  the  roots.  Ans,  a, 
a,  and  —a. 

3.  Required  the  roots  of  Jc*H-4  OB*-- 12  a'r*  — 12  «•*» +  4 a*« 
-^a^sio} 

4.  To  find  the  roots  of  x^— or*— o^x+a'sso. 

CARDAN'S  RULE  FOR  CUBIC  EQUA- 

TIONS. 

54.  Let  X*  -}-cLP=s6  be  any  cubic  equation*  wanting  its  seeond 
term  >  it  is  requii*ed  to  find  one  of  jfs  roots,  according  to  Car- 
dan's method  ^ 

P  This  rule  bean  Cardan's  name  from  the  circumstance  of  bis  baring  been 
-the  first  who  published  it,  namely  at  Milan  in  1545»  in  aivork  entitled.  An 
Magna :  but  it  was  invented  first,  in  or  about  the  year  1505,  by  Scipio  Ferreas, 
Professor  of  Mathematics  at  Bononia;  and  afterwards,  v'is.  in  1535,  by  Nicholas 
Tartalea,  a  respectable  mathematician  of  Brescia;  from  the  latter  Cardan  con- 
trived to  extract  the  secret,  which  he  afterwards  published  in  violation  of  the  most 
solemn  protestations.  The  rules  which  Cardan  thus  obtained  were  for  the  three 
cases  j^  +  hx»€,  X^^hx-^e,  and  afi -f  c»»hx ;  and  it  must  be  acknowledged  m 
justice  to  him,  that  he  greatly  improved  them,  extending  them  to  all  forms  and 
variatiM  of  cubic  equations,  in  a  manner  highly  creditable  to  his  abilitiei  as  a 
mathematician.  See  'nurtalea's  QumHti  H  JmfeiUiam  diverse,  ch.  9«  Boesut's 
Hist,  of  the  Math.  p.  907.  Montucla's  Hut,  desMath,  t,  1.  p.  591.  Pr.  Hut- 
ton's  Math,  JDiet.  vol.  1.  p.  68—77. 

The  root  obtained  by  this  method  is  always  real,  although  not  always  the 
greatest  root  of  the  equation :  and  it  is  remarkable,  that  this  rule  always  exbi* 
bits  the  root  under  an  imaginary  form,  when  all  the  ro«>ts  of  the  equation  are 
real ;  and  under  a  real  form,  when  two  of  the  roots  are  imaginary.  See  Dr. 
Button's  Paper  on  Cubic  Equations,  in  the  Philotoph,  Trans,  for  17^. 


Part  V.  CUBIC8,    CARDAN'S  RULE.  139 

Assume  y-^-zszx,  and  3  yz=  —a;  suUthute  these  values  for 

X  and  a  in  the  proposed  equation,  it  becomes  (y^+^y'x+3pz* 

Hhf^+a.y+«=y»  +2*  +3  yz.y+z-|-a.y+z=y»  +2»— a.^47+a 

.jf-f-z=r)  ys-fz'=6;  from  the  square  of  this  tidce  four  times 

a  4  a' 

the  cube  of  yz=  — ~,  and  the  result  is  y«— 2  y  V + 1"=5»  +— - , 

4  d* 

the  square  root  of  which  is  y'— z»s=:^6«-j j  buty'+z*=:6; 

*7 

wherefore  the  sum  and  difference  of  these  two  equations  being 

taken^  the  former  is  2tf*=i+  A/^^+-7ziry  and  the  lattw^z'se 

x7 


4  a'  111 

*—  V^+i^^^^  Whence  is  found  y=»  V— 6+  ^/-r^*+sr«'i  and 
^7 »  4         3/ 

«=^  Vy^— ^-4-^'+^«'>  whence  j?=(y+z=:) 

Vy^^  VT^'+^^'  + '  'v/-^^-  -^^T^' +^«'»  ^^^^  ^  ^*'- 

dan*s  theorem  :  but  the  rule  may  be  exhibited  in  a  form  rather 

more  convenient  for  practice ;  thus,  because  z=r — — -,  we  have  x 

3y 

4« 


W'rrb-\'  J — 6*  +r-:rt' ;  whcucc  the  rule  is  as  follows. 
^2        ^4         27 

55.  Rule  I.  If  the  given  equation  have  all  its  terms,  let  the 
second  term  be  taken  away  by  Art.  37. 

II.  Instead  of  a  and  h  in  either  of  the  above  general  theorems^ 
substitute  the  coefficients  of  the  corresponding  terms^  with  their 
proper  signs»  in  the  transformed  equation;  then,  proceeding 
according  to  the  theorem,  the  root  will  be  obtained. 

If  a  be  negative,  and  —a'  greater  than  —h*»  the  root 

37  4 

cannot  be  found  by  this  rule^. 


4  This  is  called  the  Irreducible  Case ;  it  exhibits  the  root,  although  real, 
under  an  impossible  form  :  thus  the  root  of  the  equation  xs— 1 5x^:4  ii  4,  but 
by  Cardan's  rule  it  is'  >v/2+  ^  —  121  +  »  -/2— V*  121,  an  impossible  form. 


140  ALGEBRA.  Paut  V. 


ExAVFLSS.— 1.  CSfwen  x'-^-S  rssd8>  lo  fifid  the  volue  of  x. 
Here  the  second  term  is  wanting,  wherefore  a=6»  &^88>  and 


^a 


•^^•"m^m  aa^HiVawr^i^a 


.^W~b+V-^'+^'-W^+V^'+^'=- 


88         88)'     Si'  88         Sii'     6)' 


Let  the  cube  root  of  each  of  these  imaginary  expressions  be  extracted,  thej  be- 
come 8-f  v^— 1  +  2—  V^—  1 9  which  being  added  together,  the  impostiUepftrls 
destroy  eadi  other,  and  the  Mm  is  4,  agreeably  to  what  has  been  obsenred.  It 
is  remarkable,  that  this  case  never  occurs  except  when  the  equation  has  three 
real  roots,  as  we  bsre  before  obsenad. 

The  irreducible  case  has  exercised  the  abilities  of  the  greatest  algebraists 
for  these  three  hundred  years  past,  but  its  solution  still  remains  among  the  de< 
^iderata  in  science.  Dr.  Wallis  thought  he  had  discovered  a  general  rule,  but 
it  was  afterwards  found  to  apply  only  to  particular  cases.  Baron  Maseres  gare 
a  series,  which  he  deduced  by  a  laborious  train  of  algebraic  reasoning  from 
Newton's  BinomialTheorem»  whereby  this  case  is  resolved  without  theintervea- 
tion  of  either  negative  or  impossible  quantities.  Dr.  Button  has  likewise  disco- 
Tered  several  series  applicable  to  the  solution :  (see  Philoi.  Traru,  vol.  68.  and 
70.)  other  series  for  tlHs  purpose  may  be  seen  in  Ctmrmilfs  Afy^bra,  p.  S. 
Art.  19.  Soma's  jflgebra.  Art.  178-9.  Landen's  lAicuiratitms,  Zm  CaUU*t 
Le^ontde  Math,  Art.  399.  &c.  ,  Lorgna's  Memoirs  qfthe  HaKan  ^ctsdewy, 
t.  i.  p.  707.  &c. 

The  irreducible  case  may  be  easily  solved  by  irigonometry ;  as  «arly  as 
1579)  BombeUi  shewed  that  angles  are  trisected  by  the  resolution  of  a  cubic 
equation.  Vieta,  in  161 5,  shewed  how  to  resolve  cubics  and  higher  equations 
by  angular  sections.  In  1639,  Albert  Girard  solved  the  irreducible  case  by 
a  table  of  sines,  giving  a  geometrical  con&tniction  of  the  problem,  and 
exhibiting  the  roots  by  means  of  the  hyperbola  and  circle,  Halley,  De 
ttotvre,  Emerson,  Siikipson,  CrakeK,  Cagnoli,  Wales,  Madielyne,  Tbacker, 
:Sic.  hawte  employed  the  eaue  method  of  sines :  and  lastly,  Mr.  Bonaycastif , 
Professor  of  the  Makhanatkies  at  the  Royal  Ifilitary  Academy,  «Woolwicii,  has 
communicated  additional  observations  on  the  irreducible  case,  and  an  improved 
solution  by  a  taUe  of  natural  sines.  See  HuttorCs  Math.  DicU  vol.  2.  p.  743^. 

When  one  root  is  obtained  by  Cardan's  rule,  the  two  other  roots  may  be  de- 
rived not  only  by  depressing  the  equation,  as  in  ex.  1 .  but  likewise  as  follows  : 
let  r=>  Cardan's  root,  and  v  and  tr^stbe  two  other  roots,  then  will  v  4-  w=  -^r, 

N  r        1        r3--46  r  _    \        r"*— 4ft 

ami, vwrssiy whesoe  »«  — -^r H- -t-V  ■"  ' '  •  ,  and w*  — -^  +  o   v  '  '  '""". 


PAn  y.  CUB1C&    CASDANS  RULE.  itt 

3 


*V'^^M^^i4^^l6=s4<449--.449s4s^^  root  required. 

If  the  two  remaining'  roots  be  required^  deprei^  the  given  e^iia^ 

fion,  (-rfr^  33.)  thus  {--^ 7—=)  ^•+4ar4.22=o,  of  which 

|A«  roote  (/ottiul  by  the  rule  fitr  quadratia.  Vol.  I.  P.  3.  Art, 
97.)  are  —2+3^—2. 
3.  Given  y' — 6  j^*'^-  3  y  **  4=0'>  to  find-  the  value  of  y. 

First,  to  take  away  the  second  term,  {Art.  37')  let  y^{x^ 

Then.  y5=:^+da?*+12«+  8 

—6  y*=  .  —6  a?«— 24  a? —24 

-|-3y  =....+  3jc-h  6 

-4     = -  4 

Whence  x^  *     —  9  a:— 14= o,  or  j?*— 9  a:=s  14. 

-3 


Here  a=:^-*9,i»14,  aiMix=V74-  V49--27 
-3 


-^  :Vu.«90415--~^;^=2,269- 


3^+4.690415       ^  V^l^^^l^ 

—3 

=2.269+ 1.322»3.591^  the  root  ear  vahie  of  x;  mber^re 


2.269 

y^(x+2=)  5.591  =  ^^  root  of  the  proposed  equation, 

3.  Let  y'+3y«+95f=13begiven,  tofindy. 

Here,  putting  y^x^  I,  the  equation  is  transformed  (Art.  SJ.Y 

into  a;'H-6j:=20j  whence  asset,  6=20,  and  xss'^lO+^/WS 

^T ;;='  V20.3923— r-:Tj--^=:2.732-.732=2 1 

wherefore  y=(jr— 1=)2— 1=1,*^  'root  required. 

4.  Given  x*— 12a;=16^  to  find  x.     Ans.  a:=4. 

5.  Given  j:^— 6j?=— 9,  to  find  x.     Ans.  x=— 3. 
6*  Giveq  y5+30y=117.  to  find  y.    Ans.  y=3. 

7.  Given  ^54.^^—350,  to  find  y.    Ans.  yrs&OS. 

8.  Given  y^^  15  y«+81 9=s243,  to  find  y.     Ans.  y=9. 

9.  Given  y»-.6y«+10y— 8=0,  to  find  y,    AnK^y^^. 
10.  Given  y« + 20  y  ^  100,  to  find  y .  " 


14C  ALGEBRA.  Pau  V. 

COliJPLETING  THE  CUBE. 

"  55.  B.  In  eveiy  complete  cubic  equation,  haying  its  signs 
cither  all  -f  >  or  alternately  +  and  -— ,  if  the  coefl|cient  of  the 
third  term  be  equal  to  three  times  the  square  of  one  third  of  the 
coefficient  of  the  second  term>  the  cube  may  be  completed  by 
adding  the  cube  of  one  third  the  coefficient  of  the  second  term, 
with  its  proper  sign,  to  both  sides  of  the  equation  j  and  then,  by 
extracting  the  cube  root  from  both  sides,  the  root  of  the  equation 
will  be'found '. 

ExAMPLBs.— 1.  Given  j^  +  6j^  +  12xs=56,  to  find  the  value 

of  J7. 

Here  i  of  6sz2,  and  12=3x2*;  wherefore  adding  2?^  io 
both  sides,  the  given  equation  becomes  jr'4-6i:*+12x+ 8=  (56 + 
S=)  64.     The  cube  root  of  this  is  :r4-2=4;  wherefore  J?s2. 

2.  Given  a^— 12a:«+48jr=:61,  to  findx. 

Here  i  o/— 12=— 4,  and  3.— 4l«=483  wherefore  ^V^^ 
—64  is  to  he  added,  and  the  equation  becomes  x' — 12x*-f  48l:— 
64=(189— 64=)  125.  The  cube  root  of  which  is  j— 4=5; 
whence  ai;=9. 

3.  Given  6x»  —  90 jr» +450 ar= 729.75,  to  find  x. 

First,  dividing  by  6,  we  have  x*  — 15x*-f76x=121..625. 
Also  4.  0^—15=^5,  3.-5|«=+75,  ond-.5l«=— 125,  to  he 
added;  wherefore  x»  — 15x«-|-75x— 125=(121.625— 125=)- 
3.375;  andx— 5=(v'— 3.375=)  — l.5,t(7^cre/'or«x=(5— 1.5=) 
3.5. 

4.  Given  r»  -f.3x'+3x=26,  to  find  x.    Ans.  x=3, 

5.  Given  x»  — 18  x"  -f  108  x=  189,  to  find  x.     Ans,  x=  —3. 

6.  Given  x»  +21  x*  + 147^=400,  to  find  x. 

7.  Given  x^  — 2 1  x«  + 147  x= —64,  to  find  x. 

2x      1  • 

8.  Given  2  x*—x» +--=—-,  to  find  x. 

27     2 

»  This  rale  is  evident ;  for  let  (r +«!*=*)  x* +3aa:» +  3««x^tfa  be  a 
complete  cube,  it  is  plain  that  +  a  is  4.  the  coefficient  of  the  second  term,  3 
.+aS»the  coeiScient  of  the  third  tenn,  and  the  cube  of+a,  w^a^  the  third 
term  ;  wherefore  if  jr4  +3  ojt*  +  Za*x^h  be  given,  it  is  plain  that  the  cube 
is  completed  by  adding  the  cnbe  of  one  third  the  coefficient  of  the  second  term 
to  both  sides,  making  x*  +*'*  +  rt^»x+<i*  =6+rt«;then  extracting  the  cube 
root  xHtfl=^  ^fc-f-a*,  and  x«  +a  +  3  ^ft^i|3,  which  is  the  rule. 

The  root  of  aj^mplete  cnbe  is  found  by  taking  the  root  of  the  first  term  and 
the  root  of  the  latt^  and  ooaiMsctinf  them  by  the  s\gn  of  the  last. 


PaitV.     BIQUADRATIOS.    DSS  CARTES' ftULE.      145 

56.  DES  CARTES'  RULE  FOR  BIQUA- 
DRATIC EQUATIONS '. 

RuLV  1.  Take  away  the  second  term  from  the  given  equa* 
tion,  (Art.  37.)  and  it  will  be  reduced  to  this  form,  x*-\-ax^  rbx 
-^-c^szo;  wherein  the  coefficients  a,  b,  and  c,  may  represent  any 
quantities  whatever,  either  positive  or  negative. 

II.  Assume  the  prodact  x'-J-fxr+9.a:'+rx+«  equal  to  the 
transformed  equation  j?*4-flu?*+6r+c=o,  and  let  the  two  fac- 
tors be  actually  multiplied  together ;  then  will  the  product 


•  Lewis  Ferrari,  the  friend  and  papil  of  the  celebrated  Cardan,  was  the 
firat  who  discovered  a  mle  for  the  solutioD  of  biquadratics ;  nsmelyi  aboat  tib« 
year  1540.  His  rule,  which  is  called  the  liaHan  method^  was  first  published 
bj  Cardan  with  a  demonstration,  and  likewise  its  application  to  a  great  va- 
riety of  suitable  examples :  it  proceeds  on  a  very  general  principle,  completing 
•oe  side  of  the  equation  up  to  a  square  by  the  help  of  multiples,  or  parts  of  its 
own  terms,  and  an  assumed  unknown  quantity  ;  the  other  side  is  then  made 
a  square,  by  assuming  the  product  of  its  first  and  third  terms,  equal  to  the 
square  of  half  the  second :  then  by  means  of  a  cubic  equation,  and  other  cir- 
cumstances, tlie  management  ot  which  greatly  depends  on  the  skill  and  judg- 
ment of  the  operator,  the  root  is  found. 

The  mle  we  have  given  above  was  invented  by  that  eminent  French  philo- 
sopher and  mathematician,  Ren^  Des  Cartes,  whose  name  it  bears ;  and  was 
first  published  in  his  Geometry,  lib.  3^in  1631 ,  but  without  any  investigation : 
like  Ferrari's  method.  It  requires  the  intervention  of  a  cubic  and  two  qnadraticr; 
both  methods  are  sufficiently  Uborions,  but  that  of  Des  Cartes  has  in  some 
respects  the  preference. 

The  reason  ol  the  rule  is  extremely  obvious  ;  for  it  is  plain  that  any  biqua- 
dratic may  be  eonsidered  as  the  product  of  two  quadratics ;  and  if  the  coeflB- 
cients  of  tte  terms  of  these  latter  can  be  found  in  terms  of  «,  fr,  c,  &c.  the 
coefBcienti  of  the  transformed  biquadratic,  (as  we  have  shewn  they  can  by 
maaiM  «#  a  cubic,  &c.)  then  those  quadratics  being  solved,  their  roots  wi|l 
evidently  be  those  of  the  transfoxmed  biquadratic,  from  whence  the  roots  of  the 
givett  equation  will  be  known. 

All  the  roots  of  a  complete  biquadratic  equation  will  be  real  and  unequal. 
'  l^t,  when  4  of  the  square  of  the  coefficient  of  the  second  term  is  greater 
than  the  product  of  the  coefficients  of  the  first  and  third  terms.  Secondly, 
wlien  ^  the  square  of  the  coefficient  of  the  fourth  term  is  greater  than  the 
product  of  the  coefficients  df  the  third  and  fifth  terms.  Thirdly,  when  4  the 
si|uar«  of  the  coefficient  of  the  third  term  is  greater  than  the  product  of  the 
coelBcients  of  the  second  and  fourth  tertns  r  in  all  other  cases  besides  these 
three,  the  complete  biquadratic  equation  will  have  imaginary  roots^. 


144  ALQEBRA.  PaktT. 


**:?}.' +^|.-+j}x+^= 


X*      *       +aar»     +  &r   +    c. 

III.  Make  the  coefficients  of  the  9«ne  power  of  x  on  each 

ode  this  equation  equal  to  each  other,  in  order  to  find  the 

vafaies  of  the  aflsamed  coeffidents  p,  q,  r,  and  $;  then  will 

p4-rs=o,  f  ^.g^-jM-ssOy  jm4-^=^  and  qs^^c;  from  the  first  of 

these  we  get  rs — p,  from  the  second  s+q=(a — ^pr =since  r= 

b 
— p)  a+p',  and  from  the  third  «— 9s;— -. 

IV.  From  the  square  of  the  last  hut  one,  subtract  the  square 

b* 
of  the  bflty  and  4f«33a'  +2 «|>'  +p^— — ,  or  (since  ^azzc)  4tf 

6» 
^a*  +S  ap'  +p* ^,  which  equation  reduced,  is  p*  +2  ap*  + 

a' '-4  c.p*  =z&',  from  the  solution  of  which  (by  Cardan*s  rule  er 
otherwise)  the  vahie  of  p  will  be  found. 

V.  Having  diseovered  p,  the  value  of  *='X"+^+5~»  *^ 
that  of  ^=5-^+^ — ^-,  will  likewise  be  thence  determined;  that 

%       3      2p 
&,  (since  r=:^p,)  sdl  the  quantities  in  tiie  two  assumed  Catctoi? 

j?»  +pa?+9ur' +rx+*,  excq>t  the  value  of  x,  are  known. 

VI.  Next,  liiid  the  roots  of  the  two  assumed  quadratics  x*  + 
pX'\-q=o,  and  x»  4-rx+#=o,  and  we  shall  have,  from  the  for- 

9  P'  T 

mer,  0?=—- ~+ ^"2 — 9*   *"^  ^o™  ^^®  latter,  x=  ( jh 

^.^^Mt  or  since  rs;:— ps5j)^4- V^*^*-  Wheroftwe  the  fiwf 

4  3  4 

roots  of  the  transformed  biquadratic  equation  x*+ax'  +  bX'^c 

p         p*  P         P*  P        p' 

and  — ^ —  v'^""  9  **  *^®  roots  of  the  proposed  equation. 

»  4 ' 

SKAimns.~l.  To  find  the  fonr  roots  of  the  biquadrstie 


FaktV.    biquadratics.    SBS  CARTES' RULE.      146 

rmi^  io  iake  uHBOf  the  setmi  term,  {AH.  37»)  Mt  z^x^ 

z*x3a:*H-4x*-f  6«»-f  4«:+l 
^4z*±s     -'4««'^I2x'^13jr— 4 

•— *8z  »  . —  8 J: — 8 

+  3S  = +32 

«*— 6^'  — 16j:  4-21    =o 
Here,  putting  a=—6,  b^^l6,  and  c=+21j  the  assumed 
cubic  (p°+2ap*+a*— 4c.p»=&')  becomes  by  substitution  p*  — 
1^|»* — 48 p' =256.    ^om  tAtf^  /e^  the  second  term  be  taken 
away,  by  putting  p'=:y-f-4  3  then  will 

p«=y'-|-12y«+48y+e4 
—  12p*=     —  12y'— 96y— 192 

— 48|>»=s — 48y— 192 

—256    s=  .  . —256 

""*  y*— 96y=576 

To  find  the  root  of  tMs  equation  by  'Cardah*s  rule,  {Jfrt, 

h^  55^  here  «=— 96,  6=576,  and  ^.Z— 6+^—6*+— a'  — 

2  4  27 

1 

■  i.iainpi.    ".!■   Ill     'iPl     lilt 
I  .  1.1 


' '^"i**^  'v^T^'  +2r**^'  Ar««+  v^2944-S2768 


27 
-32 


^^+V^9ii=32^=^'^=^'   "^^>^  P=(^y+4:±:) 

^      t  ,«.P*.^      -^-6     16     —16    .  /«.P' 

^2^2^2p      2^2^    8       ^'^^2^2 

Wherefore  the  two  quadratics  to  be  solved  s  viz.  x*  -^px-^-q 
zso,  and  x'  +rX'^.87so,  (&y  sub^ituting  the  abope  values  of  p,  q, 
r,  and  s,)  become  a?»  +4  j:=:  —7*  ond  x*  —  4  x=  —3  -,  the  two  roots 
qftheformerofthesearex:=i'-'^^^'^3',  andoftheiatter,x^S; 


•«-a>.*Mt<M*ai<MM«iifc>«««-«rfB.*«.^M«>MaM«a« 


*  We  have  the  solution  of  both  these  quadratics  (or  rather  th^  ttttstrers) 

iajgMMul  teniB»  in  the  »ol« f  tie. -^±  v''4^«>  "^  ^T4  '^^ — *'  ^* 

which  the  valaes  of  p,  q,  and  i,  being  subttitotedy  the  roots  of  the  transformed 
eqaation  will  come  out  as  before* 

vol..  JU  L 


146  ALGEBRA.  Fast  V. 

and  1.    Wherefore  the  four  roots  of  iht  tT€ai»fofrmed  eqfauOiwn 

X*— 6a:*— 16j:+21=o,arc  —2+  v^— 3 3—  ^—3 . .  .Sand 

1 5  hut  iwce  z=x+ 1^  by  aiding  unity,  to  each  of  these  roots,  we 
shall  have  the  four  roots  of  the  gwen  equation  z«— 4  z* — 8  z-^SZ 
=0,  as  follows;  2=  —  1-f-  V^^t  2=— 1— V^*  «=4,  and  zss 
2,  (M  tea*  required  ". 

2.  Given  z*— 42'— 3z*— 4z+l=o,  to  find  the  values  of  z- 

i^n«,  z=-=~ —  ana =^^2: . 

2  2 

3.  To  find   the  roots  of  x*— 3  a?*— 4«— 3=o.    Jns.  «= 
2  « 

57.  EULER'S  RULE  FOR  BIQUADRATIC 

EQUATIONS '. 

Rule  I.  Let  x*-^ax'  -^bx-^-cszo,  be  a  general  biquadratic 

^     a         a'      c 
equation  wanting  its  second  term,  and  let  J^-^j  ^==75+  T* 

and  h^-. 

II.  With  these  values  of/,  g,  and  A,  let  the  cubic  equation 
z*  — /z'  -f-gz— A:=o  be  formed,  and  let  its  three  roots  (found  hj 
any  of  the  preceding  methods)  be  p,  q,  and  r. 

III.  Then  will  the  four  roots  of  the  proposed  biquadratic  be 
as  fblloWSj  viz. 


When  -^^  is  positive 

l-st  root,  ar=s  ^p-k-  ^q-k-  ^r 
2nd  root,  x=  ^p-f  ^9—  ^r 
Srd  root,  a:=VP-"  a/9'+  V 
4th  root,  x»  v'p*-^  V9~  v''" 


Wlien  —h  is  nqgative^ 

a:=     ^/V—A/q-V^r 

x^^  ^/P'^r  ^q-¥  V^ 
x^^^p^^q^^r 


ii   I 


>   "  This  rale  applies  to  that  casa  only  in  which  two- of  the  roots-are  potsible, 
and  two  impossible. 

▼  The  learned  and  renerable  Leonard  Euler,  joint  Professor  of  Mathematics 
alt  the  University  of  Petersbarg,  was  the  inventor' of  this  method;  which  he 
first  published  in  the  6th  volume  of  the  Petersburg  Commentaries  for  the 
year  1738 ;  and  afterwards  in  bis  Algebra,  translated  fifom  the  German  iot» 
£reneh,  in=  1774»<and  lately  into  English^ 


Paet  V.       BIQUADRATICS.    SIMPSONS  RULE.        147 

Examples.— 1.  Given  x«— 95;r'+60x— 36ss:o^  to  find  the 
four  roots. 

a        25 
Here  a=i^,  6= — 60,  and  c=36 ;  wherefore  f:=z(—s:z)  —, 

a'      c        769  225 

g= (—+--=)  ----,  and  A=  —  3  consequently  by  substituting 
16      4  16  4 

<Aese  values  in  the  cubic  equation  z'-^fz*  -j-gz — hzso,  it  becomes 

25   ,  .  769      225 

a* z*  -\ z szo, 

2  16  4 

The  three  roots  of  this  equation  being  fotmd,  foiU  be  z^ 

9  25  ] 

■-r=p*  Jf=4=9,  a»dz=---=r;  and  since -- b  is  negative,  the 

four  roots  will  be 

9  25 

9  ^ 

9  25 

9  25 

«=     ^^^g-.^r=-^— -^4-v'-4-=— 6 

2.  Given  a?*— 6a?"4-4=o,  to  find  the  roots.  Ans,  x=  +  l, 
+2,-1,  and  —2. 

3.  Given  a?«— 3 x'— 36 a?' +68x^*240=0,  to  find  the  roots, 
jfni.  xss— 2,— 5,+4,  and-^e. 

4.  Findtheroot8of««+x««-29x'—9x-f  180=0.  Ans.x^ 
3,4^—3,011(2—5. 

6,  Findtherootsofy«—4sr'— 19^^+46^+120=0. 

58.  SIMPSON'S  RULE  FOR  BIQUADRA- 
TIC EQUATIONS  \ 

This  method  supposes  the  given  biquadratic  to  be  equal  to 
the  difference  of  two  assumed  squares  >  thus, 

-  ^^— ^— — ^— -— ^^^— — ^— — ^— ^— ^— — — ^— ^— —  —    —  I 

X  This  rule  was  first  giveii  by  Mr.  Thomas  Simpson,  Professor  of  the 
MaftheiDatics  at  the  Royal  Military  Academy,  Woolwich ;  and  published  in  the 
second  edition  of  his  Algebra,  about  the  year  1747 :  it  is  in  some  instances  pre- 
ferable to  either  of  the  preceding  methods,  and  some  trouble  is  saved  by  it,  as 
here  we  are  net  under  tha  necessity  of  exterminating  the  second  term  from  the 
complete  biquadratic  equatiooi  which  in  the  preceding  rules  is  indispensable. 

L2 


148  AWVnUL  Faet  V. 

BuLE  I.  Let   X* +031' -hfar* '♦riar-f  4ag»^  >e  tte  proposed 
equation,  and  equal  to  the  difEerence  x*  +—  <v+<^ ' — Bx-i-  C| '. 


II.  Square  the  two  latter  quantities,  making  the 
the  squares  equal  to  the  imposed  equation,  and  jou  will  have 
j:*+aa*+2-4r' ^ 

—  B*jr»— 2BCr— C»  J 
in.  Blake  the  confident  cix  in  each  terpi  on  one  side  of  the 
equation,  equal  to  the  coeflkient  of  the  same  power  of  x  on  the 
other  j  then  will 

1  1 

First,  2.*+— o«— J5«=fc,  or  2^+— a«-6=:B«. 
4  4 

Secondly,  aJ— 2  BC=c,  or  aA—c:si9  BC. 

Thirdly,  A^»^0=zd,  or  i<*— d=C». 

IV.  Multiply  the  first  and  last  of  these  equations  together, 

and  the  pioduct  (B^O)  will  evidently  h«  «q^  to  (—AB^O) 

4 

one  fourth  the  square  pf  the  «Moad  \  that  f9»  2iC>h— 4^— 6^ 

4 


1  1  ' ' — '-'— ^ «^ 

4  4 


•••ti 


V.  Let  ifc=s— oc— d,  i=— c/-f"A-T-4*^^i  and  hy  this  sub- 
stitution, the  preceding  equation  will  become  A^^^'^kA*  '^kA 

'1=0. 


2 

VI.  Find  the  root  or  value  of  ^  in  Hm^  cubic  equation,  by 
any  of  the  foregoing  methods ;  which  being  done,  B  and  C  will 

■  I   L  -  ■      ■  I.  -  > 


2B 


likewbe  be  known,  since  ^=s  v84+'T'<»'-^A»  I»4  C=5* 

VII.  And  sinoe  the  proposed  qunitityjp«-Hur*  +  to' -fcar+d 

is  equal  to  nothing,  its  equ?a  a;*+4'ax+-rfl*-ftc4-Cl'  irtH 

2 

likewise  be  equal   to  nothing;    wherefore    it   follows^  that 
1 


x'-^'—ax-^A 


«=&*♦- cj*. 


Tkn  V.      BIQUADRATICS.    SIMPSONS  RULE.        14» 

VIIL  Ettfact  the  square  root  fi^m  both  sides  of  this  equation, 

and  or' +~-ax+^=r  + Br +C,  whence  a?»H a+Bjp=4-C— 

i#;    tvhich    equation   solved,  gives   xss'\ — B a-{- 

""2         4     — 

VT^a'-f — aB-f--B«-|-C— -rf;  wherein  all  the  four  roots  of 
*o  4  4        — 

the  given  equation  are  exhibited*  according  to  the  variations  of 
the  signs  ^. 

ExAMFLM.-^!.  Giv^   ar<—Gi*—-58««—H4«— 11=50,  to 
find  the  values  of  x. 

Hire  asu-^e,  &98-<-Sd,  e3«-114,  and  d=s-ll,  whence  k 


1  11 

==(^  ac-d=)  182,  Z=(--c»+d.— a»— 6=)  2512  j  whence  by 

iubstituting  these  values  in  the  cubic  equation  A' bA'  +  kA-^ 

-^1=0,  it  6flcome« -rf* +29-4* +  182^^—1256=0,   the  root  of 


y  Dr.  HttttoD  remarkti  that  Mr.  Simpson  has  subjoined  aa  observation  to 
this  rule,  which  has  since  been  proved  to  be  erroneous  ;  namely,  that  **  the 
▼alue  of  A,  in  this  equation,  will  be  commensurate  and  rationai,  (and'  there- 
fore the  easier  to  be  disoovered,)  not  only  when  all  the  roots  of  the  fiv«9 
ofoattoo  are  eemmetmnriiief  but  when  they  are  trrsrfMNO^and  even  impossible  ; 
aa  wiU  appear  from  the  ^camples  wah^lptdm"  This,  oootinues  the  I>octor,  is 
a  strange  rcaeon  for  Simpson  to  give  in  proof  of  a  proposition :  and  it  is 
wooderftd  that  he  |sU  on  no  examples  that  di^rove  it,  as  the  instances  in 
which  hia  assertion  holds  true,  are  veiy  few  indeed  in  comparison  with  thosa 
ia  which  it  feib.    MttK  JDki.yoh  h  f,  m. 

When  dthcr  jS^e,  jBa>a,  or  €>■•,  the  roots  of  the  proposed  biq^adratio 
win  be  obtaiUBd  by  the  resohitiao  of  a  quadratic  only.  Simfton'$  Alg. 
«l*  tekt.  p.  16$. 

Besides  the  rules  by  Ferrari,  Des  Cartes,  Eulcr,  and  Simpson,  two 
other  rules  ibr  the  solution  of  biquadratics  have  been  discovered  x  one  by 
La  Foati^ne,  of.  the  Royal  Academy  of  Sciences  at  Paris,  and  inseipted  in  the 
Ifcamin  of  that  leanud  society  for  1747  ;  ud.  the  other  by  Dr.  Edward 
Waring,.  LuCMlftn  Professor  of  Blathematies  at  Cambridge,  iq  a  profowi4 
waak,  cntilled^AIMiiMiMiM  jagehmemf  published  in  the  year  1770.  AXt 
tempt* 'havu  not-  been  wanting  to  diseover  methods  of  resolving  equations  of 
the  h%her  orders^  but  they  have  hitherto  been  unsuooessful;  no  general  rule 
Urn  the  solution  of  adfeeted  eqnolions  above  tho  fsorth  ponret^  has  y«^  heiy 
discovered. 

1.3 


150  ALGEBRA.  Past  V. 

which  (found  hy  Cubics)  is  A^A;  whence  B^(,^/2A'\"2<^*'^h 

=)    5^3,    C=(?^=)     3^3,     and    t  :=z  ±-LB^l.a± 
1      Zn  i  5  3  21 

=  11.761947,  or  3.101693,  or  +2.830127+  ^—1.1865334798, 
for  the  four  roots;  the  two  latter,  expressed  by  the  doMe  sign,  are 
impossible, 

2.  Let  the  roots  of  j;«— 6x'+5f'+2x— 10=0,  be  fband. 
Jns.  x=5,  —  1,  1  +  i/ — 1,  and  1— ^—l. 

3.  Givenj;'*— 12  JT— 17=0,  to  find  tbe  values  of  X.    Ans.T=i 
2.0567,  or  .6425,  or  .7071+^—4  7426406. 

4.  Given  x«— 25x'  +60x=  -36,  to  find  the  roots.    Ansmer 
x=3,  2,  1,  and  —6. 

5.  Given  x*— x»+2x«— 3x+20=o,  to  find  the  roots. 

RESOLUTION  OF  EQUATIONS  BY 
APPROXIMATION  •. 

59.  The  foregoing  rules  require  for  the  most  part  great 
labour  and  circumspection,  and  after  all^  they  are  applicable 

»        ■»  ■■       ■  —  -  I.  ■   I  ■       ,       .    ■  

■  Methods  of  apprmamatiiu^  to  the  roots  of  nunhen,  were  enplojed  ss 
early  as  the  time  of  Lacas  de  Bnrgo,  who  flourished  in  the  ISth  eenUiry;  bat 
the  first  who  are  known  to  hare  applied  the  doctrine  to  the  resolotion  of  eqas* 
ttons,  were  Sterinns  of  Bruges,  and  Vieta,  a  cclehrated  mathematician  of 
Lower  FiAtoa  ;  the  former  in  bis  Arithmetic,  printed  at  Leyden,  in  1585,  and 
in  his  Algebra,  pablished  a  little  later  ;  and  the  latter  in  his  Opera  Math^ 
tnatiea,  written  about  the  year  1000,  and  pablished  by  Van  Schooien,  in  1646. 
.  Their  methods,  although  in  some  respectaimprored  by  Ooghtred  in  his  Key  t» 
the  Mathematies,  1648,  were  still  very  tedious  and  imperfect:  to  remedy 
these  defects.  Sir  Isaac  Newton  turned  his  attention  to  the  subject,  and  it  is  to 
his  successful  application  to  this  branch,  that  we  are  principally  iadebted  for  a 
general,  easy,  and  escpeditious  method  of  approximating  to  the  roots  of  all  sorts 
of  adfected  equations,  as  may  be  sten  in  his'  tract  De  Anafyn  per  EquaHenui 
^umere  terminerum  infSMitat,  1711,  and  elsewhere.  Dr.  Halley  inrented 
two  roles  for  the  same  purpose,  one  called  his  rmHemal  ikeerem,  and  the 
other,  his  irrational  theorem,  both  of  whkh  are  still  justly  esteemed  for 
their  utility.  This  necessary  part  of  Algebra  is  likewise  indebted  to  the  labours 
of  WaUis,  Raphson,  De  Lagni,  Thomas  Simpson,. and  others ;  whose  methods 
have  been  given  by  various  writers  on  the  subject. 


Fait  V.  APFROXniiOION.  151 

onfy  to  eqoations  of  particular  Idods^  all  of  which  taken  toge- 
ther^ form  but  a  small  part  of  the  numerous  kinds  and  endless 
variety  of  algebraic  problems^  which  may  be  proposed.  But  as 
we  have  no  general  rules  whereby  the  roots  of  high  equations 
can  be  founds  we  must  be  content  to  approximate  as  near  to  the 
required  root  as  possible^  when  it  cannot  be  found  exactly. 

60.  The  methods  of  approximation  are  general,  including 
equations  of  every  kind  and  description,  applying  equally  to  the 
foregoing  equations,  and  to  all  others  which  do  not  come  under 
the  preceding  rules :  hence  approximation  is  the  most  general, 
easy,  and  useful  method  of  discovering  the  possible  roots  of 
numeral  equations,  that  can  be  proposed. 

61.  It  must  be  observed,  that  one  root  only  is  found  b^  these 
methods,  and  that  not  exactly,  but  nearly.  We  begin  by  making 
trials  of  several  numbers,  which  we  judge  the  most  likely  to 
answer  the  conditions  of  the  proposed  equation;  then,  (by  a 
process  to  be  described  hereafter,)  we  find  a  number  nearer  than 
that  obtained  by  trial ;  we  repeat  the  process,  and  thereby  ob* 
tain  a  number  nearer  than  the  last  5  again  we  repeat  the  pro* 
cess,  and  obtain  a  number  still  nearer,  and  so  on,  to  any  assign^ 
able  degree  of  exactness* 

62.   The  simplest  method  of  approximation, 

KuLB  I.  Find  by  trials  a  number  nearly  equal  to  the  root  of 
the  proposed  equaticm. 

If.  Let  r=the  number  thus  found,  and  let  zsthe  diffierence 
between  r  and  the  root  x  of  the  equation :  so  that  if  r  be  less 
than  X,  then  r'{-zssx;  but  if  r  be  greater  than  x,  then  r^zzsx, 

III.  Instead  of  x  in  the  given  equation,  substitute  its  equal 
r+x,  or  r^z,  (according  as  r  is  less  or  greater  than  x,)  and  a 
new  equation  will  arise,  including  only  z  and  known  quantities. 

IV.  Reject  every  term  in  this  equation  which  ccmtains  any 
power  of  z  higher  than  the  firsts  and  the  value  of  z  will  be  found 
by  a  simple  equation.    > 

V.  If  the  sign  of  the  value  of  2  he  -f,  this  value  must  be 
added  to  the  value  of  r;  but  if—,  it  must  be  subtracted,  and 
the  result  will  be  nearly  equal  to  the  root  required. 

VI.  If  this  root  be  not  sufficiently  near  the  truth,  let  the 
operation  be  repeated ;  thus,  instead  of  r  in  the  equation  jus^ 
paw  resolved,  substitute  the  corrected  root,  apd  the  secon4 

l4 


15S  hJ/RSKBA  fjtn*  T 

mine  of  z  being  added  or  rabtiaettd  accordtog  toi^agft,  a 
nearer  af^roxnnatioa  to  the  root  wifl  be  haA,  and  if  a  still 
nearer  appeoxiniation  be  required,  the  operation  may  be  re- 
peated at  pleasure^  observii^  alwa^  to  sufaetltiite  Ite  last  cor- 
reeted  root  for  the  new  iraloe  of  r. 

Examples. — 1.    Given  x*+x=:14,  to  find  x  by  approxi- 
mation. 

By  trials  it  soon  appears  that  x  must  he  nearly  equal  to  3.; 
let  therefore  r=3^  oad  r+2=x;  wherefore  substituting  this  value 
of  X  in  the  giten  eqiuition,  it  becomes  r + rl*+  r + z=  14,  that  is, 
r'+2r2:  +  r*+r+z=14}  whence  by  transposition,  and  rejecting 

,     .  ,       14— r»—r     14—9—3 

«*,  we  ODinfii  2  rz -^xs:  14 — r»  -?- r,  oaa  xs=  — ,  ■  sb-    ^  .■  ^ — 

?.r-fl  6  +  1 

2 
SS-— 3S.28,  and  a:=:(r4.z=3+.28=)  S.28>  aeariy. 

/• 

For  a  nearer  value  of  or^  let  the  operatimi  be  repeated. 
Thus,  let  r=3.28  *3  and  substituting  this  valufi  for  rintht 
14-r»— r     .^    ^                      14— 10.7584— S.28: 
equatum  ^=-^;rfr'  **  **^^^^  "=( e^eTI ■== 

—  .0384  .  ^  ^_  .  ,       , 

=s)— .00508,    nearly;     wherefore   jr=(r+r=s3.28— 


7.56 
.00508=)  S.27492»  extremely  near. 

2*  J»et  «'^-*-2:x*  +3ss:5  be  giseo,  t»  find  dr. 

/^  appears  by  trials,  that  x^S  nearly,  wherefbre  lei  f  b=5, 
nad  r+z3=dr  as:before;  then  wiltf 


jp»s5:r«-h3r'z-f3faf»+  z?^ 
— 2jr?=     — 2r*  — 4rz  —2a*  >=>£n 
4-30?  = ,3r    +32  J 


From  which,,  rejecting  tUl  the  terms  which  contain  z*  or  9^,  we 
obtain  (r»+3r'z— 2r»— 4rz+3r+3z=5»  or)  3r»z— 4rz+ 


■  Sometimes  it  happens  that  the  correction  consists  of  several  figures ;  in 
that  case,  if  a  second  operation  be  necessary,  it  will  be  convenient  not  to  snb- 
alitute aU  the  %nrea  for  r, but ooIjl oneflgore, or  two^  spdi  as  will  nearly 
express  the  valneof  the  wbo)^ :  thus,  if  x  alter  the  first  opeiatioa  be  3.5^ 
for  a  second  operalioal  will,  piit  r»(not  3.68,  bttt).3.^  if;  «t  the  ooiiQliisioQ 
of  this  second  process  ;r^  3.648917,  and  a  third  be  deemed  neeesswy,  I  will 
not  employ  all  these  agnres,  but  instead,  of  them  put  rs  3.65,  and  proceed. 
This  method  is  to  be  attended  to  in  all  cases,  as  it  saves  miich  trouble,  and 
prodtices  searcely  any  effect  on  the  approximation. 


PXnT.  APFROXIMAtlC»f.  15S 

3z=5— r»+2r'— Sr  :    whence    z= — -— ^ — : r— ^ 

t   g7-.lg^3    "^     jQ=)-.7;  »A«icea?={3— .y-)2^ne(irfy. 

For  a  nearer  approximation. 
Let  fs2.3,  iAt«  vff/ue  mbetiiuted  for  r  in  the  preceding 

5— W.167+10.58— 6.9     -3.437       ' 
ecmatum,  we  heme  z=:( as— ^ =s) — 

^  ^         15.87—9.2+3  9.67       ^ 

S6,  iotoicex=3(^3— .36aB)  1;.94,  f<tJ<  neater  Ann  hefion;  and 
i/*  1.94  6e  substituted  for  r  in  i^  eguaiion  above  aUuded  to,  a  third 
approximatkm  wiU  be  had,  wkerebf  a  nearer  value  of  a  wiUhe  o6- 
tained, 

3.  Given  x'  — 5  xssSl,  to  find  x.    Jus,  X78i603S77S» 

4.  Given  x«  +  2  a7--40:? 0>  to. find  x.    Am.  xis5.403135. 

5.  Given  x*  +  j:'  +x=:90,  to  find  x.    Ans.  x=74.10283> 

6.  Given  2x'  4: 4 x'  —245  x-*-70^o>  tafind x.  Jfi».  x=s  10.265. 

7.  Given  x*—  12x+7=o>  to  find  x.    ^)m.  x=:2.0567- 

8.  G^een  x'  -4>10x^20a:9>  to  find  ^le  value  of  x. 

63.  The  following  method  affords  a  motfter  approximatum  to  - 
the  unknown  quantity  than  the  former  rule  \ 

Rule  I.  Let  a  number  be  found  by  tmls  nearly  equfd  to  the 
required  root,  and  let  z=tbe  diSerence  of  the  assumed  number 
and  the  true  root,  as  before. 


<>  This  method  is  given  by  Miu  Simpcon  in  p.  162.  of  his  Algebra,  where  be 

has  extended  the  dpctrine  beyond  what  our  limits  wiU  admit :  the  above  rule 

is  in  its  simplest,  form,  imd  triples  the  number  of  figares  tme  in  the  root,  at 

p 
every  operation ;  he  calls  it  an  approximation  of  the  teeond  degree^  (s«  -^ 

p 
being  the^rj*,-)  and  since  g«^^^^^_^,  ^    j^,  if  the  first  value  of  z  (vi<, 

-^]  be  substituted  in  the  second  term  of  the  denomii^^tor,  and  the  following. 

op 
terms  be  rejected,  it  will  become  z»  -^ — ^,  an  approximation  of  the  second  de- 

grtty  the  same  as  the  above  rule.    If  for  z  its  second  value  ^— —  be  substi- 

p 

toted,  then  gg-  ,  an  approximation  of  the  third  degree^  which 

h* 


154  ALGKRRA.  Pait  V. 

II.  Sidistibite  the  ttBumed  quantity  -jhz,  in  the  given  equa* 
tion,  as  directed  in  the  preceding  mle;  and  the  given  equation 
will  be  reduced  to  this  fimn,  iiz+6z'-|-cz'  +,  &c.=sp. 

o      bz*     cz* 

III.  By  transposition  and  division  we  have  z=<^- , 

a       u       Q 

&c  where,  if  aU  the  terms  after  the  first  be  rejected,  we  shall 

P  P 

have  z= —  ;  and  if  9  be  put  for  -=--,  and  its  square  substituted 

bq' 
for  z'  in  the  seobnd  term,  we  shall  have  zso— •-^. 

a 

EzAMPLBS. — 1.  Given  x'-*2jr«  +3  xs5,  to  find  ar. 

Hare  x=:3  nearly;  let  3+2=jr,  then, 

«*=s     27+27  z+9z«+z»-| 
^5x'  =  -18— 12z— 2z*  .  .    V=5,  that'u, 
+3j:  =       9-f  3z J 

18+18z+72'+«*=5,  or  18z4-7«*+2'  =  -13. 
Here  a=18/t=7,  c=l,  p=-13,  9=(^='ZH-:)-.72. 

9 — ^'^^"'•'^^ 18 =)— -^SlCsz;  wherefore  x=s 

(3+z=3— .9216=)  2.0784. 

For  a  second  approximation, 
Let2-^z=six;  then 
a?»=     8+12z+6z»+«'^ 
-2  x^=  --8^8  z-2  z«  .  .    V  =5,  that  is, 

+3x=     6+  3z J 

6-f7«-h4z«+z»=s5,  Of  7«+4z«+z«=  —  l. 


0        c 
fcy  making  «»-^^rr^,  araltiplying  both  tenns  of  the  Craciion  hj  l  +  tq, 

and  rejecting  ht'q»  (as  very  small)  from  the  product,  becomes  — fll^r-* 

a*  +b+as.f 

By  similar  methods*  and  by  putting  «-—"+ r; ,  the  approximating  mlt 


of  thc/owfAdegreeis  ap.a  +  wp_  ^^^^^  quintuples  the  nunH 

her  of  figures  true  at  every  operation. 


Bkit  V.  APPROXIMATION.  155 

Here  fl=7,  6=4,  c=l,  p=— l,  and  q:=(—ss^^ss)^ 

a       Tf 

.14285;   wherefore  a— !!il=(— .14285— -~X— .I428a•=)  — 
a       ^  7 

.15451064=2. 

^dj;=(2.0784-. 15451064=)  1.92388996,  very  nearly. 

^.  Given  a7»4-20ar=100,  to  find  the  talue  of  x.    Am.  a?= 
4.1421356. 

3.  Given  a:*— 2  r=5,  to  find  r.    ^rw.  x=2.094551. 

4.  Given  a?'— 48  x«+200=o,  to  find    x.    Ans.  i= 
47.91287847478. 

5.  Given  «♦— 38  af'+SlO a:'  +  538  xH-289=o,  to  find  x.    An- 
swer,  07=30.5356537528527. 

6.  Given  j?*+6a?*—10a?s-112ar«-207a?-110=o,  to  find  x. 
Ans.  a?=4.4641016151. 

7.  Given  2  a?"  +3  x+4=50,  to  find  the  value  of  a:. 

64.  BERNOULLrS  RULE 

Has  been  sometimes  preferred  on  account  of  its  great  shnpli* 
city  and  general  application :  it  is  as  follows. 

Rule  I.  Find  by  trials,  two  numbers  as  near  the  true  root 
as  possible  ^ 


*  This  is  perhaps  the  most  easy  and  general  metbod  of  re9olYixig  equations  of 
ererj  kind,  that  has  ever  yet  been  proposed ^  it  was  invented  by  John  Bernooliiy 
and  published  in  the  Leipsic  Acts,  1697.  The  most  intricate  and  difficult  forms 
of  equations,  however  embarrassed  and  entangled  with  radical,  compound,  and 
mixed  quantities,  readily  submit  to  this  rule  without  any  previous  reduction 
or  preparation  whatever ;  and  it  may  be  (Conveniently  employed  for  finding  the 
roots  of  exponential  equations. 

The  rale  is  founded  on  this  supposition,  that  the  first  error  is  to  the  second, 
as  the  difference  between  the  true  and  first  assumed  number  is  to  the  diffe- 
rence between  the  true  and  second  assumed  number :  and  that  it  is  true  accord* 
ing  to  this  supposition,  may  be  thus  demonstrated. 

liet  a  and  6  be  the  two  suppositions ;  A  and  B  their  results  produced  by  si« 
wilar  operations ;  it  is  required  to  find  the  number  from  which  N  is  produced 
by  a  like  operation :  in  order  to  which. 

Let  N—A^  r,  N^  B^s^  and  x » the  number  required ;  then  by  hypothesis, 

r :  *  : :  ar— «  ;  x^h,  whence  dividendo  r—  * :  «  : :  i— a  :  jr— ft,  that  is, --• 

^x^hf  which  is  the  rule  when  both  the  assumed  quantities,  a  and  6,  are  (ett 
than  the  tme  root  ^. 


159  AUEOmk.  Vkm  r. 

II.  Substitute  these  assumed  numbers  for  the  unknown  quan« 
tttjr  m  the  ^ven*  equatidn^  and  mark  the  errdr  which  arises 
from  each  with  the  sign  +>  if  it  be  loo  greats  and  — ^  if  too 
Itttle. 

III.  Multiply  the  difference  of  the  assumed  numberf  bjr  fife 
least  error,  and  divide  ihe  product  bj  the  difimnee  df  the  er- 
rors when  they  have  like  signs,  but  by  their  sum  when  they 
have  unlike. 

IV.  Add  the  <|uotient.  to  the  assumed  number  beloii^n^  to 
the  least  error,  when?  that  number  is  too  littld*^  but  subtiact 

'when  it  is  too  great  5  the  result  will  be  the  root^  nearly. 

V.  The  operation  may  be  repeated,  if  necessary,  as  in  Ihe  Ibr- 
mer  rules>  either  by  taking  two  new  assumed  numbers^  or  using 
one  of  die  fiormer  numbers^  and  assumiog  a  new  one. 

Examples. — 1.  [Given  10jr*+9a:'  +  8  j:«+7Jf=1234,  to  find iT. 
Here  hy  triah  k  appears  to  be  greater  than  3  ;  wkerefof&let 
3  and  4  be  the  two  aswmedr  numbers*. 


Next,  let  ji  and  B  be  eaeb  greater  Hiaa  JIT,  then  wifl  N'^A^  — r,  Ari 
N^B^  —Si  but  — r :— *  : :  +r ;  +*,  wherefore  r— *  :  #  :  :  a— ^ :  h — x^  nt 

a — 6'j 

< ss  &•— Xy  which  is  the  nde  when  the  assumed  quantities^  a  and  Vy  aie  each 

greater  than  jr. 

Lastiyy  M  oviief  result  ^  be  too  little,  and  the  other  B  too  great ;  then  will 

rbe  positfre  and^  negative.    Wherefore  r-f-«':  (— r,  oi*,  which  is  the  samej 

a — .h9 
41  •: :  a'^h :  h^as  <>iM  «*> <**  i>^«V  wfakh  it  the  iQk^-  wHefl«0ii«  of  tlM 

assumed  quantities  is  too  great,  and  the  other  toe  small.    Q.  £.  D.    All  qpes- 
tions  in  double  position  are  resolved  by  this  method. 

^  The  convenience  of  substituting  two  numbers  which  differ  by  unity  is  this, 
it  saves  the  trouble  of  multiplying  the  least  error  by* that  difference.  If  the 
numbers  substituted  have  decimal  ^aces,  the  same  method  is  to  be  observed : 
thus,  suppose  they  are  1 .34  and  1.35,  and  the  least  error  12<5794,  in  this  case 
the  diffbrence  of  the  supposed  numbert  ia  .01,  and  the  multiplication  is  per- 
formed by  simply  removing  the  decimal  mark  two  places  to  the  left,  makiag 
the  product .  1 25794  ;  and  the  like  in  other  instances. 


Famt  V.  APPROXIMATION.  157 

first  8mfp9ei^4m,        Eque^ion.        Second  Supposition, 
or  9^$.  Wff=4. 

810 =10a:«= £560 

243 =  9x^= 576 

72 ,  .  SK  8jp«s= 138 

21 =  7a?  = ^8 


1146 :ssrmtltss 3292 

—  88  '  '  ' =  error=  ......  4-^58         , 

Difference  of  the  assumed  numbers  4+3=1. 

Least  error  88.     Sum  of  the  errors  {they  Mug  unlike)  88+ 

1  X  88       88 
2058=21463  wherefore  "^7:^^=2777^=  .041,  the  correction  to  be 

%i4o      %14o 

added  to  3  the  number  from  whence  the  least  error  crises,  3  being 
too  little;  wherefore  3.041  is  the  root  or  value  ofx,  nearly. 

2.  Given  ^l+a?+ V2+«'+ V3+^=16,  to  find  x. 

Firom  a  few  trials  it  appears  that  x  is  somewhat  greater  than  8, 
rnhfirefm-e  assuming  S  and  9  for  the  values  ofxy  the  work  uMl 
stand  thus  *. 

Erst  Supp.  Equation.  Second  Supp. 

<w  *»8.  or  irasO. 

3 =  v^l+j?  = 3.16228 

4.041^4 =»v^+j»= 4.36207 

4T6378 s=;V3+y'= 5.20149 

11.80502 =zresult^  .  .  .  .12.72584 


—4.19498 z::error=sz —3.27416 


•  The  logarithms  are  of  excellent  service  in  all  cases  of  this  rule,  where  rooU 
and  powers  are  required  to  be  foond,  op  where  the  terms  are  mixed  and  com- 
plicated: thus  in  the  pf<«seat  instaBce,  supposing  ar«8,  then  1  +x=9,  the 
square  root  of  which  (vl«.  8)  imm^iately  ooeivrs  ;  but  let  ar-9,  then  I  +»- 
10,  to  find  the  square  root  of  which,  by  the  conimon  method,  xeqoires  rather  a 
long  process.  I  therefore  take  the  logarithm  of  10,  divide  it  by  3,  (the  index  of 
the  square,)  and  the  quotient  is  a  logarithm,  the  natural  number  cprrespond- 
ing  to  wfai^b  (s  3.16228,  as  above.    Next,  supposing  ar«8,  then  »  V'SHhrT^ 

V^66.  I  find  the  logarithm  of  $6,  divide  it  by  3,  and  the  natural  number 
%reqiDg  with  the  quotient  is  4J04m,  «5  above.  Let  ^«9>  Mi«P  *  ^/U^  » 
'i/83,  which  by  a  similar  process  is  found  to  be  4,36207,  as  aboye.  Lastlp 
if  jr*8,  «han  *V«  +  a:«-^V'5l«5  if  ar«9,  then  ♦^5T*9«*>/7«2#  «» 
roots  of  both  which  are  found  by  a  similar  operation,  and  ar^  as  above,  viz. 
4.76378  and  5.20149.    8ee  VoL  I.  Part  2.  Art.  38. 


158  ALGEBRA.  Pakt  V. 

Diff.  of  assumed  numherszsil,  least  error  3.^7416,  diff.  of 

the   errors    {having    like    signs)    4.19498— 3.274 16=  .9^062; 

3.^7416 
wherefore  =3.5309^  the  correction  to  be  added;  com- 

quenthf  12.5309  is  the  value  of  x  nearly. 

For  a  second  approximation^ 
Let  the  numbers  11  and  12  be  assumed,  then 

First  Supp.  Equation.  Second  Supp, 

or  0?=  11.  orx=12. 

3.38525 =  ^1+jp  = S.60555 

4.9732 =V2+j;«= 5.26563 

6.0435. =♦^3+^:'=: 6.4502 

14.40195 =rc»ttZf= 15.32138 

—  1.59805 :zzerTor= —.67862 

Least  error  .67862,    diff.   of  errors   (1.59805— .67862=) 

67862 
491943;  w^nce^--— -=.73809,  the  correction  to  be  added  l» 

» 

12  5  to^cforc  a?=  12.73809,  rery  nearZy. 

3..  Given  ai — -xr+a^^j^.^^x^'j^^ ^"^ =45.  to 

^  x^x—l 

find  the  value  of  x. 

Here  x  will  be  found  by  trials  to  be  nearly  equal  to  10» 
wherefore  let  10  and  II  be  two  assumed  numbers;  then^ 

First  Supp.  Equation.  Second  Supp. 

or  x=zlO.  or  ar=  II. 

7.74264 =:A=: 8.42718 

4    g 

—4-14358 as — -^Ts= —4.43549 

5 


67.6616    3=  +x  3 ^x'  -H2 ar v'*'  H*a?  ss79.S363 

x+l 

^.seeee =5 — = —.34497 


70.894 ^result=i 82.88302 


4-25.894 ^errorsz  ....  +37*88302 

Least  error  25.894,  dij^.  of  errors  (37.88302—25.894=) 

25  894 
11.08902;  «*'^^/^''«  iY-^gQ^=2.1598,  to  be  substracted  fron 

10 :   consequently  »= (10— 2.1598=)   7.8402  nearly  ;   and  if 


Part  V.  APPROXIMATION.  15^ 

greatet  exactitess  be  required,  the  operation  may  be  repeated  at 
pleasure,  08  in  the  second  example. 

4.  Given  d^+3a;s20^   to  find    the  value  of  x.    Am.  x^ 
3.13939. 

5.  If  a:'+a;«+a?=20,   what  is  the  value  of  4:?    An$.  xsa 
2.3^174. 

6.  Let  2a:»+3x«+4a?=100  be  given,  to  find  Jl.    Am.  x^ 
3.0696. 

7«    Given  -—a?*— 12  a?*— 50=0,    to    find   x.     Answer,  X3» 
4 

11.9782196186948. 

x^ 
8.  Given  — +3x*— 5a^— 56a!«— 10S4.a:=55,  to  find  x.  Ans. 

ar=2.2320508075. 


9.  Given  >v^l+a?'  + v'2+a;*+^3+a:*=l0,  to  findx.    Ans. 

a?= 3. 0209475.  

IOOjp  «  /5-4*.ir" 

EXPONENTIAL  EQUATIONS. 

By  the  foregoing  rule,  the  roots  of  Exponential  IJquationB 
may  be  approximated  to,  with  the  assistaiice  of  logarithms* 

65.  An  exponential  equation  is  that  in  which  the  indices,  as 
well  as  some  of  the  quantities  themselves^  are  unknown  qu8Q"> 
titles  to  be  determined. 
Examples. — 1.  Given  x*=sl000,  to  find  the  value  of  x. 
li  appears  by  trials  that  x  is  greater  than  4,  but  less  than  5. 
Let  4.4  and  4.5  be  the  numbers  proposed. 

Then  since  x  x  log.  ofx^log.  of  1000,  that  is, 
Rrst,  (4.4xlog.  o/4.4s)  4.4 X0.6434527» 2.83 119188 

But  the  log.  of  iq0Oaa3.O000000O 
Error  --^0.16880612 

Secondly,  (4.5  X  tog.  of  4.5=)  4.5x0.6532125=2.93945625 

Log,  of  1000=3.00000000 

Error  —0.06054375 

StAtract  this  error  from  thefonntr,  and  the  dJiff.  is  0.10826437 

Then  4.5— 4.4s.l= di/f.  of  numbers  found  by  trial,  and 

1  X  06054375 
.06064375,  kast  error ^  therefore  '-  .naa^A^^'  =.055922,  the 


160  AL6EBBA.  PartV* 

correcUon;  wherefore  js:xz (4,5 +,0^^922:=)  4.559^2^  i^  cmswer, 
very  nearly i  for  4.SbO^^^'^^^^=^(Jby  logarithms)  1009.315, 
which  reeuU  exceeds  the  truth  by  9.315. 

To  repeat  the  operation. 
Let  4.55  and  4.56  be  the  assumed  numbers. 
Then  (4.55  x%.  4.55) =4.55x0.65801 14 =2.99395 187 

Log.  of  1000  K  3^0000000 

Error— 000604813 

Also  (4,56  X  log.  4.56=)  4.56  X  0.6589648=3.00487948 

Log.  of  1000=3.00000000 
Error  (least)  ^0X)04S7945 
Then  0.00604S13  +0.00487945= .01093758,  sum  of  the 
errors. 

Tkerefore  :25iii^i^!?:5?=:2^;g^l=. 00234.  cor- 
•^  .00487945  .00487945 

rectum. 

Wherefore  ^=4.56-^.00224=4.55776,  nearltf. 

For  4.5.5776l*-5*776=  1005.6,  which  is  too  great  by  5.6;  and 
for  a  still  nearer  approximation,  the  operation  may  again  be  re- 
peated; thus,  let  4.556  and  4.557  be  proposedy  and  proceed  as 
brfore. 

2.  Given  x^^lQO,  tx>  find  x.    Aks.  ^7=3.597285. 

3.  Given  a»=7837577897,  to  find  z.    AM.  «=  11.295859. 

4.  Given  x*as  123456789,  to  find  x.    Ans.  d^c±8.6400268. 

5.  Given  y'=3000,  and  a?y=5000,  to  find  x  and  y.  Ans.  «*» 
4.691445,  and  y=5*510132. 

a.  Given  a?*s=400,  to  find  x,    Ans.  d?=2.32443i8. 

66.^  Two  or  more  equation^  imvohing  ««  many  unknown  quantities, 
may  he  resolved  by  a^itpreximaUen,  as  follows^ 

RvLM,  I.  Reduce  M  the  equatiods  to  one,  (by  either  of  the 
methods  for  redncing  equations  containing  two  or  more  un- 
known quantities.  Vol.  I.  Part  3.  Art.  90 — ^95.)  this  equatien 
viU  contain  only  one  unknown  quantity. 

II.  Find  the  value  of  this  unknown  quantity  by  one  of  the 
preceding  rules  ^  from  whence  that  of  the  others  may  be 
obtained. 

Examples. — 1.  Given  x-fy+z=^2,  2t— 3y+5z=40,  and 
3«4-4y**2«^afc— IQD,  to  find  x,  y^  and  t. 


ftAMT  T.  APPROXIBIATION.  let 

Erom  eq»  I.  :p»n— y-^2;  iukitUuie  ikii  value  of  s  in  the 
second  and  third,  and  (44-~2y— 9s--df +5c=:40,  or}  6y-*3z 
=45  aba  (M— 3y*-3z+4y-S<*s--100j  «r)^j:»4-3«— fs= 

166 ',  let  now  the  value  of  y  (= — - — )  in  the  last  but  one  he  sub* 

d  r4-4 
stituted  in  the  last,  and  it  becontes  {2z»  +3z =166,  or) 

10z'H-12z=834.  • 

Now  it  appears  from  trial,  that  z  is  greater  than  4,  but  less 
tJian  5 ;  fee  tkes9  two  numbers  therefore  be  substituted  for  t,  then 
by  the  last  rule, 

\st  Supp,  Equation.  Znd  Supp, 

or  2=4.  or  2=5. 

640. =10«»= 1260 

48 s»  +  J2«ss 60 

688 r=  result  = 1310 


— 146 =  error  z:^ +476 

For  a  nearer  approximation.   Let  4.2  and  4.3  be  put  for  z,  and 
1st  Supp,  Equation.  ^nd  Supp, 

740.88 =102^= 795.07 

50.4    s=-f  122= 51.6 

^91.28 =zresult=sz 846.67 


—42.72 sserror:^ +12.67 

^     ^    12.67 x.l       1.567     v^ooo-r^  ,. 

'T^  (  •  ■  1  rL« =)X)22874, 1^  oorreciuis. 

^42.72+12.67     65.39     ' 

Wherefore  a= (4.3— .022874=)  4.277126,  «ciy  nearly. 

Whence    y=(-^^ac)  3.366275,   cwd    x=(22— i^— 2=) 

6 

14366599^  Msr^. 

2.  Given  «— x=10>  x^+x2=900,  and  xyzvtzSOOO,  to  find  x, 

y,  and  2. 

Erom  eq.  1.  t=10+a?;  ^«H<ttfe  this  value  for  2  ill  ^A« 

900— IOjc— x« 
second,  and  it  becomes  xu + 10  x + x* =900,  and  y = ; 

X 

write  this  value  for  y,  and  10 +x  for  z  in  the  third,  and  it  will 
become  (9000+800*— 20 «*—x'= 3000,  or)  a^+20««— «00«= 
6000. 

VOL.  II.  M 


I 

IGZ  ALGEBRA.  Pakt  T. 

&re  hf  trials  x  isfimnd  to  he  greater  than  93,  bnt  leu  than 
24  3  then  Mtmg  these  two  numbers  as  snpposUians,  and  proceed- 
ing as  before,  x =23.923443456^  9s3.69655893S,  and  zsl 
33.923443456,  nearly. 

3.  Given  jc^+y=157>  and  y'— 2r:s6,  to  find  x  and  y.    Jnt, 
j:=  12.34,  y =4^21. 

4.  Given  x+xy=&0,  and  jr^— y*=495,  to  find  x  and  y. 
Ans.  xs=8>  y=:9. 

5.  Given  i^+3r'=12,  and  i'+y'sS,  to  find  xmnd  y. 

6.  Given  ar+yzs20,  y-|-2z=22,  and  x+xy=:28,  to  find  x,f, 
andz. 

67.  Dr.  BUTTON'S  RULE  for  extracting  the  rooU  of 

numbers  by  approximation. 

Rule  f .  Let  N=the  number  of  which  any  root  is  required 

to  be  extracted,  — =the  index  of  the  proposed  root,  r=the 

number  found  by  trials,  which  is  nearly  equal  to  the  root, 
namely,  r^=:N  nearly,  and  let  x=the  root,  or  i^zs^N  exactly. 

11.  Then  will  x='*"^^'^^"""— V r,  neariy '. 

n+l.r*+fi— l.A^ 


'  The  rale  is  thnB  demonstrated;  let  iVathe  given  ntimber,  the  root  of 

I 

which  it  is  proposed  to  evolve;  — sthe  index  of  the  root,  r  as  the  nearest  it- 

tional  root,  v= the  difference  hetween  rand  the  exact  root,  x^r + v^the  enct 
root;  then  since  i^^a^r+v,  we  shall  have  i\r=r+t;J»=r"+iir*— »v+» 

•--^~r«  -  ' V'  +  ,&c.  (Vol.  I.  P.  3.  Art*  54.)  and  hy  transposition  and  diTision, 

TV"— I*  Ji— 1 «»  fi^ltf> 

■■ggp+-*'-— « — ^y&c.  in  which,  rejectingr  —r-' —  on  acconnfc  of  its 
Mr"  -*2r  ''        ^     2      r 

saiallness,  v  may  be  considered  as  «       ^  .    Bat  from  the  first  eqnatioB, 

ff-l  n— 1 

JV— r«=itr«  —  » t;  +  ».-g-»«- •»»  +  ,&«.  =  (iir"-»  +11.-2"'* ""**')  X^,* 

which,  if  the  former  value  of  v  (vie.   r r)  be  substituted,  we  shall  have 

«— I  N—r^^           2nr^  +  n—  l.iV— »r«  +  r" 
iVr-r»=.j:»r«»-»+-^ —)Xv^ ^ Xf= 

— y  y;  consequently  t;^ — :- — ,  and  are(r+v*; 

^»*  «+l.f*  +  »— l.-Y 


l^ART  V.  APPfiOXIMATJON.  163 

III.  To  find  a  nearer  value,  let  this  value  of  a?  be  subetituted 
for  r  in  the  above  theorem^  and  the  result  will  approach  nearer 
the  root  than  the  former. 

IV.  In  like  manner,  by  continually  substituting  the  last  value 
of  X  for  r,  the  root  may  be  found  to  any  degree  of  exactness. 

Examples. — 1.  Let  j:*=19  be  given,  to  find  the  value  of  x. 
Here  iV=rl9,  ns=4,  and  the  nearest  whole  number  to  the 
fourth  root  of  19  is  ^',  let  therefore  r =2,  then  iciW  r"=  16,  and  xzs 

n+l.iV+^^l.r-  5  X  19  +  3x16*     ^      286     ^ 

— ^-- xr=(- — -__^>_.-_x2t=)-— =2.08,  nearly. 

;r4rr.r»-n-l.2\r  '5X16+3X19  ^37  ^ 

To  repeat  the  process  for  a  nearer  approximation. 

Let    rst^,OS,     then    r"  =5  (2^08/*=)    18.71773696  j    these 
numbers   being  substituted  in   the  theorem,  we  shall   fiave  xz=^ 
5x19+3x18.71773696     ^    ^     ,151.15321088     ^ 
^6x18.71773696+3x19  ^150.5886848 

2.0677975,  extremely  near ;  and  if  a  nearer  value  of  x  be  require 
'ed,  this  number  must  be  substituted  for  r,  and  repeat  the  operation. 

2.  Given  rc'ssSlO,  to  find  x,    Ans.  a?=7.999,  ^c. 

3.  Given  x*=790O,  to  find  x,    Ans.  j:=6.019014897. 

4.  Extract  the  sixth  root  of  262140.     Ans.  j:=3.9999,  ^c. 

5.  Required  the  sixth  root  of  21035.8  ?    Ans.  a?=5.254037. 

6.  Extract  the  sixth  root  of  272. 

es.  PHOBLEMS  PRODUCING  EQUATIONS  OF  THREE 

OR  MORE  DIMENSIONS. 

1.  What  number  is  that,  which  being  subtracted  from  twice 
its  cube,  the  remaipder  is  679  ?    Ans,  7. 

2«  What  number  is  that,  which  if  its  square  be  subtracted 
from  its  cube,  the  remainder  will  exceed  ten  times  the  given 
number  by  1100  ?     Ans.  1 1. 


r+==. = —   =: == — .»',  which  is  the  rule.    This  is  the 

«+!.»*  +  «— I.A^    w+ l.r^+H— l.AT 

inTcatigation  of  the  rule  io  Vol.  I.  page  260 :  the  theorem  was  first  i^iven  b^ 
Dr.  Hntton,  in  the  first  Volatne  of  his  Mathematical  Tracts  j  it  includes  all  the 
rational  formulae  of  Halley  and  De  Lagni,  and  is  perhaps  more  convenient  foi^ 
nemery  and  operation  than  any  other  rule  that  has  been  discovered. 

M  2 


164  ALGEBRA.  Part  V. 

5.  What  number  is  that^  whieh  being  added  to  its  8<]uare^  the 
sum  will  be  less  by  56  than  —  its  cube  ?    Am.  8. 

4.  There  is  a  number,  thrice  the  square  of  which  exceeds 

9 
twice  the  cube  by  .972  j  required  the  number  ?    Am.  —. 

5.  If  to  a  number  its  square  and  cube  be  added,  four  times 

43 
the  sum  will  equal  —-  of  the  fourth  power ',  required  the  num- 

54 

bet  ?     Ans,  6. 

6.  If  the  sum  of  the  cube  and  square  of  a  number  be  mt^i- 
plied  by  ten  times  that  number,  the  product  shall  exceed  twice 
the  sum  of  the  first,  second,  third,  and  fourth  powers  by  180; 
what  is  the  number  ?     Ans,  2. 

7.  Required  two  numbers,  of  which  the  product  multiplied 
by  the  greater  produces  18,  and  their  diffierence  multiplied  by 
the  less,  2  ?     Ans,  3  and  S. 

8.  The  di^s  being  16  bouts  long,  a  persM  ntfao  was  asked 
the  time  of  day,  replied,  *'  If  to  the  cube  <tf  the  hours  passed 
since  sun-rise  you  add  40,  and  from  the  square  oi  the  hours  to 
come  before  sun-set  you  subtract  40,  the  results  wrill  be  equal  *' 
required  the  hour  of  the  day  ?     Ans.  Sin  the  Tiwming. 

9.  To  find  two  mean  proportionals  between  I  and  2.  Ans^ 
r. 25992,  and  1.5874. 

10.  The  ages  of  a  man  and  his  wife  are  such,  that  the  sum  ef 
theur  square  roots  is  11,  and  the  difference  of  their  cubes  31031  f 
what  are  theif  ages  ?     Ans,  36  and  25. 

1^1.  If  the  cube  root  of  a  lather's  age  be  added  to  the  square 
root  of  his  son's,  the  sum  will  be  8  $  and  if  twi6e  the  cube  root 
of  half  the  son*s  age  be  added  to  the  square  root  of  the  fiitha^'s, 
the  sum  will  &e  IS  3  what  is  the  age  of  each  i  Ans.  thefaihefs 
e^,  the  son's  16. 

13.  There  are  in  a  statuary's  shop  three  cubical  blocks  of 
marble,  the  side  of  the  second  exceeds  that  of  the  first  by  3 
inches ;  and  the  side  of  the  third  exceeds  that  of  the  second 
by  2  inches  5  moreover,  the  solid  content  of  all  the  three  to- 
gether is  1136  cubic  inches  3  required  the  side  of  each  ?  Afi^> 
4,  7,  and  9  inches. 


PART  VI. 

ALGEBRA. 


THE  INDETERMINATE  ANALYSIS 


!•  A  PROBLEM  18  said  to  be  indeterminate,  or  unlimited, 
when  the  number  of  unknown  quantities  to  be  found  is  greater 
than  the  number  of  conditions,  or  equations  proposed  ^ 


■  For  some  accouat  of  the  subject,  see  the  note  on  Diopbanttne  problems. 

^  If  the  namber  of  putsita  exoec4  the  nvmber  of  datm,  the  problem  is  nn- 
limited.  If  the  qtutrita  be  equal  in  number  to  the  data,  the  pioblitm  is 
limited.  If  the  data  exceed  the  quauita,  the  excess  is  either  deducible  from 
the  other  conditions,  or  inconsistent  with  them ;  in  the  former  case  the  excess 
is  redaadant,  and  thnreibre  unnecMsary  ;  in  the  latter  it  renders  the  problem 
absurd,  and  its  solution  impossible.  To  give  an  example  of  each. 
-    1,  Lei  x-i-y^S  hegivtHtto/indtke  wUmes^  X  andy. 

Here  we  haye  but  one  condition  proposed,  and  two  quantities  required  to  ba 
fonndy  the  problem  is  therefDre  unlimited;  for  (admitting  whole  numbers  only) 
X  may  si,  then  ys5 ;  if  xs»9,  then  jf»4 ;  if  x^a,  then  y^S  $  if  xa>4» 
then  jr»9 ;  if  jr^S,  then  jr^s  1. 

3,  Lei  x+yssS,  arndx^y^A,  he  given. 

Here  we  have  iwa  conditions  proposed,  and  #100  quantities  to  be  found, 
whence  the  problem  1ft  UmUed;  (see  Vol.  I.  P.  3.  Art.  89.)  for  «r»5,  jf«l : 
and  no  other  numbers  can  poasibly  be  found,  that  will  lulil  the  eonditions. 

3.  Lei  *+y«6,  «—y«4,  iwrf«y —5,4*  ^w«i. 

Here  is  a  redundancy,  three  conditions  are  laid  down,  and  but  two  quantitiey 
to  be  found.  By  the  preceding  example  x—h,y^\  \  wherefore  Ay —5  X  1  — 
5,  or  the  latter  condition  {xy^h)  is  deducible  from  the  two  former. 

4.  Let  x+y=6,x--y^4,  and xyisli,  he  given. 

Here  is  not  only  a  redundancy,  but  an  inconsisteney ;  for  the  grntest  pro- 
duct that  can  possibly  be  made  of  any  two  parts  of  6,  is  9,  that  is,  Ay  »9 ;  it 
cannot  then  be  divided  into  two  parts,  x  and  y,  so  that  «y— 18;  wherefore 
the  latter  condition  is  inconsistent  with  the  two  former,  and  renders  the  pro- 
blem impossible.  There  is  a  mistake  in  the  appendix  to  Ladlam'e  Rudimentif 
5th  edit.  p.  338.  Art.  107'  by  which  the  subject  is  altogether  perverted. 

M  3 


\ 


166  ALGEBRA.  Pabt  VT. 

2.  An  indeterminate  problem  will  frequently  admit  of  innu« 
merable  answers^  if  fractions,  negative  quantities^  and  surds  be 
admitted  3  but  if  the  answers  be  restricted  to  positive  whole 
numbers  J  the  number  of  answers  will  in  many  cases  be  limited. 

3.  The  indeterminate  analysis  is  the  method  of  resolving 
indeterminate  problems  3  it  depends  on  the  following  self-evident 
principles^  viz. 

'^  The  sum,  differences  and  product  of  two  whole  numbers^ 

are  likewise  whole  numbers." 

'^  If  a  number  measure  the  whole^  and  likewise  a  part  of 
another  number^  it  will  measure  the  remaining  part." 

4.  In  the  given  equation  ax^^by'\-c,  to  find  the  values  of  x  and 
y  in  positive  whole  numbers. 

Rule  I.  Let  W  stand  for  the  words  whole  number,  then 
(since  x  and  y  are  by  hypothesis  whole  numbers)  the  above 

equation  aj=s6y  -|-c  reduced^  will  be  a?s= =fr, 

II.  If  JZf  be  an  improper  fraction,  reduce  it  to  its  equi^'a- 

a 

lent  mixed  quantity;  (see  Vol.  I.  p.  880.  ex.  9,  10.)  that  is,  let 

Jj^^ifn-^-^^ :  from  which  rejecting  m,  we  have =  ^ 

a  a  ^  « 

by  Art.  3. 

III.  Take  the  difference  of  -^^  or  any  of  its  multiples,  and 

a 

y  or  any  of  its  multiples,  viz.  — ,  -^,  — ,  &c.  in  order  to  re- 

•^  a      a       a    ^ 

duce  the  coefficient  of  y  to  unity,  or  as  near  unity  as  possible, 
and  the  remainder  will  he^W. 

IV.  Take  the  difference  of  this  remainder  and  any  of  the 
foregoing  fractions,  or  any  other  whole  number  nearly  equal  to 
it,  then  will  the  remainder  ;=  W, 

V.  Proceed  in  this  manner^  till  the  coefficient  of  y  becomes 
unity,  or  ?^  =  fr. 


VI.  Let— ^=»,  then  will  yszap—g;  and  if  any  whole  num- 
'a 


Pabt  VI.        IND£TERA1INATE  ANALYSIS.  167 

ber  whatever  be  substituted  for  p,  the  value  of  y  wUl  be  known  ^ 

whence  x  (= )  will  likewise  be  known. 

a   ^ 

Examples. — 1.  Given  4x=5y— 10,  to  find  the  values  of  x 
and  y  in  whole  numbers. 

^    ,         5y  — 10     „,    ,      5tf— 10  ^    y— 2 

First,  x=-^ =^i  6tt<-^ =sy— 2+     '■  9  fohenc9 

«— 2 

(rejecting  y— 2)  ^ =sW^=p,  therefore  y— 2=54p,  oiid  y=4p 

4 

.«     ,.              ^            ^      ,                ,5y— 10     10-^10    ^ 
+  25  te*  /)=o,  then  y  =2>  whence  x=  (— ^ = — - — =)  O- 

Secondly,  letp  he  taken=:ly  then  ^=(4^+2=;)  6,  and  x= 
5y--aO_20_ 

5v— 10 
T^irdty,  letp^%  then  y=(4y^2=)  10^  and  j=(  ?    i 

=^=)  10. 

4      ^ 

Fottr*%,  fe^  p=S,  ^/i£»y=(4p+2=t)  14,  and  x=(-=— — 

4 

=)  15- 

Fifthly,  let  ptB4,  then  y=i}8,  and  j;=20. 

Sixthly,  let  p=5,  then  y=22,  and  07=25.   8sc.  4e. 

Hence  it  appears,  that  the  values  of  x  (viz.  0,  5,  10,  15,  2Q, 
25,  jS^.)  di£fer  by  the  coefficient  (5)  of  y  ;  and  the  values  of  y 
(viz.  2,  6,  10,  14,  18,  22,  &c.)  by  the  coefficient  (4)  of  x; 
and  it  is  plain,  that  this  will  be  the  case  universally  in  every 
equation  of  the  form  axzs,hy — c,  viz.  the  successive  values  of 
X  will  di£fer  by  h,  and  those  of  y  by  a. 

2.  Given  17^=13^—14,  to  ^n4  ^  ^nd  y  in  positive  whole 
numbers. 

13t/— "14  17  n 

First,  x=  — ^- —  s=Fr,  afap  ^^W;  wherefore  (Art,  3.) 

17J|_13y-14^4jH:14^  ^  4jH^  lJy+56 

17  17  17  '•17  ^      vT^ 

^W,  that  i,,i£?ii+3==»r,»fce«ceH^ti=Fr,.  and  (i^^ 

17  17  17 

M  4 


we 


led  AJLGEBRA.  PaetVI. 

^ 

.-. — ?[Z— rs)  lUssWszp,  whence  y=17p+55  lei  p^o,  then 

13tf-14    65-14     ,„ 
y=5,  and  j?=( — j^  =  --^^  =)  3. 

And  by  continually  adding  13  to  the  value  of  x,  and  17  to  the 
▼alue  of  y,  we  obtain  the  following  values^  viz. 

x=3,  16,  29,  42,  55,  68,  81,  94,  107,  &c. 

y=5,  22,  39,  56,  73,  90,  107,  124,  141,  &c. 

3.  Let  4x+7y=s23,  be  given,  to  find  x  and  y. 

^  23— 7y    X  ^  3— 3tf      ,  .   ..     K 

First,  x=( ^=)  5— yH --^,  whence  rejecting  5— y, 

4  4 

have  '-=^=ir,  »A*re/Te  (ll?+«-Ziii=)  y±?=ir=p; 
4  4  4  4 

consequently  y-\- 3=4 p,  and  y=Ap—Z;  let  p^l,  then  y=:(4jj— 

3=)  1,  andx=z{ ?= — =)  4-,  which  are  the  only  affirvM' 

4  4 

^ii7#  answers  the  question  admits  of, 

4.  Given  19a?+14^=1000,  to  iind  jrandj^. 

First,  x=:.{ j= — ??=)  52+— ^^-^5  r6;«c«i?^  52,  ^ 

12— 14y     „^  ^,     19y     12r-14y    ^5y+12 

hxive  —^^^W,  cojisequently  (^+       ^^    ■  =)  -^5—= 

TMr     »      .5v+13  20y-f48    .20y-fl0    ^     „,      , 

JT,  a^o   (    ^^      x4=      ^J      =).^-ZL^+g=:y,   „fc«we 

— j_.  =  jrr;  wherefore  {—^ j^=)  ^^=  ^=P'  ^'^ 

y=19p  — 10^  ief  p=l,  theny=z9,  and  x=z{ "^ — ^=)  46. 

Let  p=:2,  /^«n  ^=28,  and  ^s32. 

Ze^  p=3,  t^en  y=:47f  andxsz  18. 

X,e<  p=s4,  then  y=^G6,  and  x=:4. 

These  are  all  the  cffirmative  values  of  x  and  y ;  for  if  pbe 
' taken  :=:^,  then  u;tZ/y=85,  and  ^  =r  — 10,  a  negative  quantity, 

Th£  above  values  will  be  obtained  by  adding  the  coefficient 
(1 9)  of  X,  to  the  preceding  value  of  y ;  and  subtracting  the  coeffi- 
cient (14)  of  y,  from  the  corresponding  value  of  x;  and  the  same 
is  universally  true  of  every  equation  of  the  form  of  <fcr+fey=<?. 
6.  Given  13  a?=21  y — 3,  to  find  the  least  values  of  x  and  y  in 
whole  numbers.    Ans.  a?=3,  y=2. 


Past  VI.       IND£T£BMINAT£  ANALYSIS.  109 

6.  Given  41jrs43y— 53>  to  find  x  and  y.    Jm.  xalO, 

7.  Given  8a;+9y=25^  to  find  x  and  y.    ^w.  xs=2>  ysl. 

8.  How  many  positive  values  of  x  and  y  in  whole  numbers 
can  be  found  from  the  equation  9x=2000— 13y?  Ans.  17 
values  of  each, 

9.  Given  13jr=14y+36^  to  find  J?  and  y. 

10.  Given  101  x=s4331-.177y,  to  find  j?  and  y  ^ 

5.  To  find  a  whole  number,  which  being  divided  by  given  numbers, 

shaU  leave  given  remainders. 

Rule  I.  Let  x=the  number  required;  a,  b,  c,  ^rcsrthe 
given  divisors;  f,  g,  K  ^c.=the  given  remainders;   then  will 

a  b  c 

11.  Make  the  first  fraction =p,  find  the  value  of  x  from  it^ 
and  substitute  this  value  for  x  in  the  second  fnictioo. 

III.  Find  the  least  value  of  p  in  the  second  fraction,  (Art.  4.) 
in  terms  of  r,  and  thence  x  in  terms  of  r. 

IV.  Substitute  this  last  value  for  x  in  the  third  fraction, 
whence  find  the  least  value  of  r  in  terms  of  s^  and  thence  the 
value  of  X  in  terms  of  s, 

V.  Substitute  this  ^-alue  in  the  fourth  fraction,  &c.  and  pro- 
ceed in  this  manner  to  the  last  fraction,  from  whence  the  value 
of  X  wiU  be  known. 

£xAMPL£s.-^l.  What  number  is  that,  which  being  divided 
by  3^  will  leave  9  remainder,  and  being  divided  by  2,  will  leave 

1  remainder  ? 

J— 2  X—  1      _^ 

Let  xz=the  number,  then  will  — -— =IF,  and  — ---=rF| 

3  2 

let =p,  then  wiU  x=3p+2  ;  substitute  this  value  forxin 

the  frac^n  ^^,  and  it  becomes      ^       ^W:  but  -^^^t 

wherefore  (^-^^'■-'^=)^^^  tr^cep=2r-l;  let 

%         %  % 


c  By  similar  metbods  indeterminate  equations,  involving  three  or  mor« 
unknown  quantities,  may  be  resolred. 


170  ALGEBRA.  Paht  VI. 

r  be  takenszl,  then  p=:(3r— ls=2— 1=)  1,  and  x=(3f)4-2=) 
5,  the  number  required, 

2.  What  is  the  least  number  which  can  be  divided  by  2,  3, 
5,  7,  and  11,  and  leave  1,  2,  3,  "4,  and  5,  for  the  respective 
remainders  ? 

Let  x:=zthe  number,  then  fZLL=:  fT, -— =  ^,  fZ?  =  ^, 

3  '3  *     5  ' 

«C"'""4  X     5  T     1 

—— =  /f;;  and  __s=:^^  fcy  //,g  problem.    Let  -— -=:p,  M01 

«c— 2 
a;=2pH-l  5  substitute  this  value  for  x  in  the  fraction ,  and 

o 

it  becomes^-l^:=zfV;  but  ^=W,  wherefore  (gP^^P-^^.,) 
-3  3  33 

0+1 

4— -=fF=r,  and  p=3r— 1,  wherefore  a?=(2p4-ls=)  6r— 13 

$tt6£<i/tt^e  this  value  for  x  in  the  third  fraction ,  and  it  he* 

5 

f;omes  ^Irl^fv    but  ^=:fV,  wherefore  (?Iri-.^=)  !JZi 
S  5  "^        ^     5  5      ^     6 

=  ^='>  a«d  r5=:5»4-4,  consequently  a7=(6r— 1=:)  30<+23| 

/^i*  value  being  substituted  for  x  in  the  fourth  fraction  ^^,  it 

-  30«+]9  2*4-5 

becomes  — =4<+2-f  =:W,  whence  (rejecting  4»+2) 

-     — #r;  a«o  ( — -; — X3= — - — =) — ' 1-2,  wherefore 

(rejecting  the  2)  -Jl^zzztV;  but  y=^,  consequently  (— - 

—_ ss)  -^— =fr=^,  wAcncc  5=7^+1,  and  Jr=(30«-f23=) 

210  *  +  53  5  /Aw  value  substituted  for  x  in  the  fifth  fraction  ^^, 

.,  .  210/-h48  t4-4  ^ 

it  becomes =  19 1  +  4+--Y-,/rom  whence  rejecting  19 1 

t+4 
+  4,  we  have  —-z=zfV:=zu,  whence  /=n  u— 45  let  «=!,  iAcii  ^ 

=(11  tt— 4=)  7i  and  a:=(210  ^+63=)  1523. 
3.  Required  the  least  whole  number^  which  being  divided  by 


Part  VL  INDETERMINATE  ANALYSIS.  17I 

Sf  will  leave  2  remainder  3  but  if  divided  by  4,  will  leave  3  re- 
mainder ?    jins,  11. 

4.  Eequired  the  least  whole  number^  which  being  divided  by 
6,  5j  and  4,  will  leave  5,  %  and  1^  for  the  respective  remainders  ? 
Am.  17. 

5.  To  find  the  least  whole  number^  which  being  divided  by  3, 
5,  7>  and  %  there  shall  remain  2^  4^  6,  and  O,  respectively. 
Am,  104. 

6.  Required  the  least  whole  number^  which  being  divided  by 
16^  \7,  IS,  19^  and  20^  will  leave  the  remainders  6,  7>  8>  9^  and 
10,  respectively  ? 

6.  Any  equation  involving  two  difierent  powers  only  of  the 
unknown  quantity^  may  be  reduced  by  substitution  to  the  form 
of  an  indeterminate  equation,  involving  two  variable  quan- 
tities. Hence,  all  commensurate  quadratic  equations,  commen- 
surate cubics  wanting  one  term,  commensurate  biquadratics 
wanting  two  terms,  &c.  may  be  resolved  by  this  method.  It  will 
be  proper  for  the  convenience  of  reference,  to  premise  the  fol- 
loviring  table  of  roots  and  powers  *'. 

Roots  1,2,3,  4,  6,  6,  7,  8,  9,  10,  11,  12. 
Squares  1,4,  9,  16,25,  Se,  49,  64,  81,  100,  121,  144. 
Cubes    1,  8,  27,  64,  125,  216,  343,  512,  729, 1000, 1331, 1728. 

Examples. — 1.  Let  aj*+4x=32  be  given,  to  find  x. 

32 j;«    '  4 

Ftrst,  by  tramposition  and  divmouj —T"'   Secondly, 

X  1 

it  %$  plain,  that  whatever  equimultiples  of  4  and  1  be  taken,  the 

fractiom  whose  terms  are  constituted  of  these  equimultiples  re- 

4  4       8 

spectively  will  be  equal  to  --  and  to  one  another,  that  is,  -T"— "o" 

12     16    20   „        „„      .       ^, .    .,     ..  ,,  ...    32— J* 

= — = — = — ,  ^c.     Wherefore,  thirdly^  if  the  quantity 

3-45  ^ 

he  made  equal  to  aether  of  these  fractiom,  which  (after  transposing 
the  known  quantity  32)  will  give  the  resulting  numerator  equal  to 
the  square  of  the  denominator,  that  denominator  will  be  the  value 


*  See  on  this  subject,  Dodson's  Mathematical  Bepontory,  Vol.  I.  Emerson's 
Algebroy  Simpson's  Algebra  and  Select  Exercises,  Vilaut's  Elements  ef  Ma^ 
tkematieal  Analysis,  &c. 


IW  ALG£BIIA.  Pabt  VI. 

of  X  in  the  proposed  equation ;  that  is,  — Z£.  a=  JL— Z.— 1?=:!5 

X  13      3     4 

5=-T->  *c.  here  it  is  plain,  that  if  the  fraction  —  be  taken,  we 

shall  ^i?cS9-«*=sl6,  or  jr«=(S3— 16=)  16,  whence  xs=4. 
%.  Given  s^^Sxss40,  to  find  ar. 

By  transposition  and  division,  as  before,  we  have = 

6      19     18    34  . 

r_„-r  40-24=16, 

'^''•'  I  and    xss4,  the  answer. 

3.  Given  «*+S  arsSS,  to  find  «. 
„^^8S>~j'_3_g_9      13     15     18     gl      24 

*  1      S      3""4""5""6""7""8* 

And  /^-«*  =^*> 

\  whence  x=s8>  ^Ae  answer. 

4,  Given  a:^— 5^ q?El44  to  find  :p. 

Here  fIllil=-l=i?=i5=:??-?5-£2-.?5 

HfA     ^        r  »*r=(36+14=)49. 
Wherefore  {      j       J  ^i. 

^        I  oita  ;r=7,  ^A«  answer. 

6.  Given  «• — -Hr=118^,  to  find  x. 

4 

Here  il=i^=±=±=s±=l=l*=li-ii_l_!: 
«  12       3       4      5       6~7""8~9 

10    ir  "^  -^        i  and xzsll,  the  imswer. 

6.  Given  4s^— 5  «--6ao,  to  find  x. 

Her  ^ilf =-i=— -15 

fVkerefare  (  ^7^'^'"^''  ->r  I  ^*7(^^7^=)^> 
•^        I  ana  x=2,  I  and  ar=3. 

Consequently  a:=  +2,  or  +3. 

7.  Given  y«4-4y'=96,  to  find  y. 

96'"C)'      4      8 

ie<  r=y^  tA«n  will  tj'+4r=96,  and = — =— =s 

»  1       2 

12     16    20    24    2&    32       ,  r  96— »«=(96— 32=)644 

345678  \  and  9=8,  ^^  oiMw^. 


Pabt  Vf.        INDETERMINATE  PROBlJaiS.  17S 

But  vzsf,  whence  jf s=»  ^v^Q  v^=)  «. 

Orihui, 

«•        4      16    36    64 


Because  t^+4oss96,  therefore  _     _     _     _     . 

•^       24-17     1       4      9     16 

I  24— VBS16,  or  rsaS  j  wAence  y=2,  cw  i^e. 

8.  Given  jf*— 7sf=36,  to  find  y. 
^^^  y'-36_7_^14^gl    gg 

y  12       3""4* 

I  ofid  y=s4,  tAe  a$uwer. 

3 

9.  Gi?cn  z?— l^zs— -,  to  find  2. 

4 

— -SS-—    «ik€ncez3al. 

X         1 


Here  ^i5^if!=s  £.—??— ?1     ^^    ^^ 


JO.  Given  9z*— z'slOO,  to  find  t. 

4 

f*25- 
r  .^=*(«25-100=)125, 
^"^^  I  and  zsx  (»  ^25=)  5. 

1 1.  Given  «>+2  «sS^  to  find  x.    Am.  xs2. 

12.  Given  s^ — 5  xai6>  to  find  x.    Aw.  xs6. 

18.  Given  «*+30=9  x,  to  find  x.    Ans,  x=iB,  or  4. 
14.  Given  y'+70s39y>  to  find  y.    Ans.  y=:5. 
16.  Given  2^—21  z+20sko,  to  find  z,    Ans.  zss4. 
16.  Given  60—^=11  x,  to  find  x.    Ans.  x=3. 

7.  INDETERMINATE  PROBLEMS*. 


1.  How  must  tea,  at  7  shillingn  per  pounds  be  mixed  with  tea 
at  4  shillings  per  pound*  so  that  the  mixture  may  be  worth  6 
shillings  per  pound  ? 

Let  xisthe  mmber  of  pounds  at  7  slullings,  then  7xsztheir 
vahie;  yssthe  number  at  4  shMings,  then  4y^their  value. 

Whence  by  the  problem  7 x+4yss{6*x+y:=2)6x+6y,  or  xsz 
2y»or  l:xx=2xy  v  4? :  y  : :  2  :  1  */  there  must  be  twice  as  much 
in  the  nuxture  at  7  shUiiags,  as  there  is  at  4  shillings. 


•  These  problems  «tc  of  the  kiad  which  belong  to  the  rale  of  Alligation. 


/ 


174  ALGEBRA.  Part  VI 

^.  Twenty  poor  persons  received  among  them  20  pence ;  the 
men  had  4d.  each^  the  women  i^d.  each^  and  the  children  -^cf. 
each ;  what  number  of  men>  women,  and  children,  were  re- 
lieved ? 

Let  x=the  number  ofmeuy  y=zthe  number  of  women,  z=zthe 
number  of  children;  then  by  the  problem  x-Hy-h2=20,  and  {4x-\- 
4.y-|-4.z=20,  or)  16x+2y  +  z=80:   subtract  the  first  equation 

V 

from  this,  and  15a:+y=60,  or  y=(60— 15a:=)4— ar.l5,  or  --2- 

15     SO     45 
=-—=—=—-,  3fc.  but  by  the  problem  y  -^  20  */  y=15  j  and  since 

Ji  M  *J 

4— j:=1  \'  x=zS,  a»dj;=:(20— x— y=)20— 18r=2. 

3.  How  many  ways  can  1002.  be  paid  in  guineas  and  crown- 
pieces  ? 

Let  x=:the  number  of  guineas,  y^the  number  of  crowns. 

Then  by  the  problem  21x+5^=:(100x  20=)2000. 

2000— 6 tf     ,   ^      5-5y  5— 5y     „, 

Whence  a7=( ^=)96  + ^,  v -  =  W,  v 

^21  '      ^     21  21 

.5-5y     ^     .    20-20y     „       »      21y     „_  20-20y , 

21  tf      20+y 

-5—=)— -2-=^r=p,  vy=21p— 20;  letpznl,  then y^i crown, 

andxss( ?=)  95  guineas:  and  if  {^\)  the  coefficient  of 

21 

x,  be  continually  added  to  the  value  of  y,  and  (5)  the  coefficient  of 
y,  continually  subtracted  from  that  of  x,  the  corresponding  values 
ofx  and  y  will  be  as  follows,  viz. 

ir=95,  90,  85,  80,  75,  70,  65,  60,  55,  50,  45,  40, 35, 30,  25, 
20,  15,  10,  6,  0.  ^ 

y=l,  22, 43,  64,  &y,  106,  127,  148,  169,  190,  211, 232,  253, 
274,  295,  316,  337,  358,  379,  400.     Jns,  19  ways. 

4.  To  divide  the  number  19  into  three  parts,  such  that  seveo 
times  the  first  part,  four  times  the  second,  and  twice  the  third, 
being  added  together,  the  sum  wiB  be  90. 

Let  the  parts  be  x,  y,  and  z  ;  then  by  the  problem  x-hy-f  z=: 
19,  a»d7ar-f-4y+2«=90;/rom*^/«*»=19 — y— «,  thisvalue 
being  substituted  for  x  in  the  second^  it  becomes  (133— 7y— 7a:-h 

4y+2«=)133-3y-5z=90j  or  (3y=r43-5z,  or)y=^— 


Pabt  VI.       INDETERMINATE  PROBLEMS.  175 

1— Sz        1— Sz  Sx  1— Sz     Sz 

=  14-z+-^,  ...  -f^^W;  also  ^=ir,  •/  (^-4-y 

1+z 
=)"-^— ==^=P  '•'  x=3p— 1 J  if /)  66  to/fc«=l,  then  z=2,  yss 

43—5* 

( — - — =)11,  and  J?=(l9-y — 2=)  6;  ifp^2,  then  will  2=5, 

o 

y=6,  and  ^=85  i/'p=3, ^/»«n  2=8,  y=l,  and  a:=10:  <Ae«e  ar« 
a/2  the  possible  values  in  whole  numbers. 

5.  How  many  ways  is  it  possible  to  pay  100/.  in  guineas  at 
21  shillings  each,  and  pistoles  at  17  shiUings  each  ?     Jns.  6. 

6.  If  27  times  A/s  age  be  added  to  16  times  B.*s,  the  sum  will 
be  1600  5  what  is  the  age  of  each  ?     Jns.  J/s  48,  B:s  19. 

7.  A  Higler*s  boy,  sent  on  a  market  day 

With  eggs,  fell  down  and  smash*d  them  by  the  way } 

The  news  reached  home,  and  Master,  in  a  rage, 

Vow*d  him  a  whipping,  bridewell,  or  the  cage  : 

*'  'Tis  through  your  negligence  the  eggs  are  lost, 

''  So  pay  me  if  you  please  the  sum  they  cost." 

The  boy,  since  nought  avail  his  tears  and  prayers. 

Fetches  his  leathern  bag  of  cash  down  stairs ; 

The  cash  a  year's  hard  earnings  had  put  in. 

But  much  he  wisb*d  to  sleep  in  a  whole  skin. 

*'  How  mai^  were  there.  Master  ?*'  In  a  doubt. 

The  Higler  calls  his  wife  to  help  him  out  $ 

Says  she,  **  I  counted  them  by  twos,  threes,  fours, 

''  fives,  sixes,  sev*ns,  befoi'e  he  left  these  doors ; 

*'  And  one,  two,  three,  four,  five,  and  nought,  remained 

*'  Respectively,  nor  more  can  be  explain*d." 

At  nine  a  groat,  ingenious  Tpros,  say. 

What  sum  will  for  the  sad  disaster  pay  ? 

Ans,  4<.  4d^. 

8.  Is  it  possible  to  pay  lOOZ.  with  guineas  and  moidores  only  > 
jins.  It  is  impossible. 

9.  A,  who  owes  B  a  shilling,  has  nothing  but  guineas  about  him, 
and  B  has  nothing  but  louis  d'ors  at  17  shillings  each  -,  how» 
under  these  circumstances,  is  the  shilling  to  be  }»aid  ?  Ans.  4 
must  give  B  13  guineas,  and  receive  16  lonis  d'ors  change. 

10.  With  guineas  and  moidores  the  fewest,  which  way 
Three  hundred  and  fifty-one  pounds  can  I  pay  ? 


176  ALGEBRA-  Part  VI. 

And  when  puid  ev'ry  way  *twi]l  admit,  the  amount 
Of  the  whole  is  required  ?— Take  paper  and  count 


8.  DIOPHANTINE  PROBLEMS. 

Unlimited  problems  relating  to  square  and  cube  numbers, 
right  angled  triangles,  &c.  were  first  and  chiefly  treated  of  by 
Diophantus  of  Alexandria,  and  from  that  circumstance  they 
are  usually  named  Diophantine  Problems  '. 

These  problems,  if  not  duly  ordered^  will  firequently  bring  out 
answers  in  irrational  quantities  5  but  with  proper  management 
this  inconvenience  may  in  many  cases  be  avoided,  and  the  an- 
swers obtained  in  commensurable  numbers. 

The  intricate  nature  and  almost  endless  variety  of  problems 
of  this  kind,  render  it  impossible  to  lay  down  a  general  rule  for 
their  solution*  or  to  give  rules  for  an  innum^able  variety  of 
particular  cases  which  may  occur.  The  following  rules  will,  per- 
haps, be  found  among  the  best  and  most  generally  applicable  of 
any  that  have  been  proposed. 

RuLB  I.  Substitute  one  or  more  letters  fix*  the  req[aired  root 
of  the  given  square,  cube,  &c.  so  that,  when  involved,  either  the 
given  number*  or  the  highest  power  of  the  imknown  quantity^ 
may  be  exterminated  from  the  given  equation. 


'  Diophaotnt  lias  been  considered  hf  aoue  writers  m  the  mruAoK  of  Alge- 
bra; others  have  ascribed  to  him  the  inventioa  of  unUmited  problenM :  bat 
the  difficult  nature  of  the  latter,  and  the  masterly  and  elegant  solutions  he 
has  given  to  most  of  them,  plainly  indicate  that  both  opinions  are  erroneous. 

Diophantus  flourished,  according  to  some,  before  the  Christian  sra ;  some 
place  him  in  the  seooul  eantury  after  Christ,  others  in  the  fourth,  and  others 
in  the  eighth  or  ninth.  His  Arl^hmeticsp  (out  of  which  ba^e  been  extracted 
most  of  the  curious  problems  of  the  kind  at  present  extant,)  consisted  origi- 
nally of  thirteen  books,  six  of  which,  with  the  imperfect  seventh,  were  pub- 
fished  at  Basil  in  1575,  by  Xylander ;  this  fifagneot  is  the  only  work  00  Alge- 
btn,  which  hat  detoended  to  us  firomthv  aneieiitst  the  TCBuuMog  books  luive 
■ever  been  discovered.    See  f^ol,  I.  p.' 337. 

Of  those  who  have  written  on,  and  MoocMl^y  e«ltivated,  the  Diophantiae 
Algebra,  the  chief  are»  Bachet  de  Meseriacy  Bxaacker,  Bernoulli,  BoonyGastle, 
De  Billy»  Euler,  Fermat,  Kersey,  Ozanam,  Frettet^  Saundenon,  Vleta,  and 
Wolfius. 


V^t  VI.  DIOPHANTINE  PROBLEMS.  177 

II.  V,  after  this  open^ooy  the  unknown  quantity  be  of  but 
one  dimension^  reduce  the  equation^  and  the  answer  will  be 
found. 

III.  But  if  the  unknown  quantity  be  still  a  square*  cube^  ftc. 
substitute  some  new  letter  or  letters  for  the  root^  and  proceed 
as  before  directed. 

IV.  Repeat  the  operation  until  the  unknown  quantity  is  re* 
duced  to  one  dimension ;  its  value  will  then  readily  be  found, 
from  whenoe  the  values  of  all  the  other  quantities  wiU  likewise 
be  known. 

1.  To  divide  a  given  square  number  into  two  parts^  so  that 
each  may  be  a  square  number. 

Analysis.    Let  a'^szthe  ginen  $quare  number,  «*3soiie  oflhe 

parts  f  then  wiU  iifi'^:fiszthe  other  part,  which,  by  the  problem^ 

mutt  likewise  he  a  square*    Let  rx — assnthe  side  of  the  latter 

square,  then  wiU  (rx— 0]*=:)  r*a5*— 2  ara?+a*=tf*— x*,  whence xa 

^ar  "3  at* 

-^ — -s:^^  side  of  the  first  square,  and  ra:— a=(-^— j— a=) 

as^'^a  ^  dor ) 

■       ^'szthe  side  of  the  second  square;  wherefore  «  and 

r«+l  ^        f  J        r*4-ll 

GIMP'S  ^^^_  ^V I 

'—j^'^  mre  tiie  parts  re^ed ;  where  a  and  rmmf  be  any  numbers 

taken  at  pleasure,  provided  rbe  greater  or  less  tthon  unity  '.  Q,£,L 

~  4  a«i* 


Synthesis.    First,  -- — 1'*4 


r*+V       r*-hl 


=( 


r*-f2»*+i 


rr.  ■  ..,1  ]  _i_fl*_  lOAiCA  if 

the  first  condition. 

Secondly,  -5 — -.|«  and  -7— rf  0*"^  evidently  both  squares^  which 

is  the  second  c^dtHon.    Q,  E,  D, 

£xAMPLE8.— Let  the  square  number  100  be  proposed  to  be 
cMeM  into  two  parts^  whkh  will  be  squares. 


ii»4- 


f  Mr.  Bonnycastle,  in  his  solution  of  the  problem^  (Algebra,  third  Edit, 
p.  143.)  has  omitted  this  restriction,  which  is  evidently  necesMury  ;  for  if  r  be 

9i|fpoMd^ltiMici^«ttltlM;itni«falwr«ftiwfraat|Mi  ::r-~r  vaiusli,  and  tha 

sttetioo  become  nugatory. 

VOU  II.  ■$( 


178  ALGJ^BRA.  Past  VI. 

Here  a^sslOO,  and  aszio.    First,  a$tume  rs2>  then  wiU 

^ar       40 
xsz-^ — !=( — sz)S=the sideofthejirttiquare,  and ra?— a=6s=: 

the  iide  of  the  second  ^jitare;  for  8)*+d'=:(64r|-S6=)100,  as 
was  required. 

eo 

Secondly,  assume  r:s:3,  then  wiU  «s(— c=)6,  and  rr— a=8> 

as  htfore. 

80  380  . 

Thirdly,  assume  r=:4,  then  a?= — ,  aifrfra?— a=(-— — 10=) 

\i55|»      6400+22500_2890O_ 

^^      ^        289  «89       ^ 


150     „    80 


17 
Divide  36  into  two  square  numbers. 

Bere  a*=36,  a:^^'^  assume  r=2,  then  xss—,  and  rx^a^ 

To  divide  25  into  two  square  numbers.    Ans.  16  and  9. 
To  divide  81  into  two  square  numbers^ 

2.  To  find  two  square  numbers  having  a  given  diffieienee. 
Let  dvsthe  given  difference,  axbssd,  whereof  a  y.b,  and  let 
x=iihe  side  of  the  less  square,  and  x-^bs=the  side  of  t?te  greater  ; 
thenwiU  jr+Al*— 3?«=(a;«+2&F+6*— a?«=)  2  &r4-6«=a6;  dwide 

this  by  b,  and  2x+bsza,  v  xss ^i  the  side  tf  the  less  square; 

2 

a — If  a-^b 

and  a7-j-fcxi(— - — 1-&=)---— =*/ie  side  of  the  greater  square: 

2  2 

,      .      i+I]'     a'+2a54-&'     ^.  ,  .    .' 

wiierefore         J  = =riAe  greater  square  required. 

Synthesis.  l*5r*<,  — — I  a«d         i  ore  €tHdei>%  MA  ^guares^ 

Secondly, — I r=(— --=sa6=s)d  ;  itiicc 

4,  4'  '      4 

fy  hypothesis  abzs^d.     Q,  E.  D. 

Example^.— To  find   two  3qi|«re   numbeis^  whereof  the 
greater  exceeds  the  less  by  11. 


PartVJ;  DIOPHANTINE  PBOBLEMS.  W 

Here  dsll(sll  x  1)^  Ut  asU>  ^^l- 

Then  — ^ — =( — —  sA)6ssMe  of  ih^  greater  square. 

jiPid  -^I^s( — ^^=s)5=«td6  of  frtHen  square* 

Whence  6]  *  =36,  and  5l  *  =25,  are  the  squares  required. 
To  find  two  square  numbers  difiering  by  6. 
Here  d=6  (=3x2),  a=3,  6=2. 

Then  -i— =-—=«<ie  of  <Ae  greater. 

Jlnd  ^-^^= — =Mde  of  *^  less;  •.•  —  and  -—  are  the  squares 

required. 

To  find  two  squares,  whose   difference  is    15.     Ans.  64 
and  4^. 

To  find  two  squares  differing  by  24. 

3.  To  find  two  numbers,  whose  sum  and  difference  will  be 
both  squares. 

Let  xzsaneofthe  numbers,  s'-^xssthe  other ;  then  wiU  their 
sum  (x+jr*— x=)  x',  eoidentlff  be  a  square  number. 

And  since  (*/  — «— »jr»  )  »*  ^2  xs^their  d^fisrence,  must  U^e- 
wise  be  a  square;  let  itg side  be  aeswnedssx'^r,  then  wiU  (x— r)  »' 

=)x«— 2aT+r*=x»— 2x,  or2xr— 2x=r»,  v  xss  ,  and 


r*}«         r*  .         r*  r» 


2r-2     ^4r«  — 


**    *"2r— 2|        2r-2      ^4r«— 8r+4.    2r-2 

o»  T^-.*      .«       _^j — +r=: — thenumbersrequiredfWhere 


4r»— 12r»  +  12r— 4    /4      f:ri)s.4 

r  ffM^  be  any  number  greater  than  2  K 

36  X  45 

Examples.— Let  r=3,  <fe«i  wi/Z  ir=--,  and  x«— x=-^  ^^c 

ID  iO 

46-1-36    81      J  9    -  ,. 
niifwfcers  sottg^e ;  for      ^    =—  ^'^^  le'         '^w^'^^*- 


*  If  3  be  snlntitated  in  this  example  for  r,  both  numbers  will  come  out»3  i 
that  is,  their  sum  will  be  4,  and  difference  0 ;  wherefore  r  must  not  ooFy  be 
greater  thi&a  1,  <a»  is  asserted  in  Bonnycastle*s  A%tfbra,  p.  146.)  hut  greatet : 
thanS. 

N  2 


ISO  ALGI&m.  Fait  VI. 

Let  f  3s5,  to  find  th*  nvmbcn. 

4.  To  divide  a  givcB  nuooberj.  which  is  the  sum  of  two  known 
squares,  into  two  other  squares. 

Let  a' +b'ss the  number  given,  rx-^aszihe  tide  of  ihe  first 
required  square,  sx—b^the  side  of  the  second,  where  r  ^s. 

Then  will  rj;— al  *  4-  «i— 6^  *  =  (f*J?*— 2  arj:+ a*-f-  *•  x*  —2  bsx 

— 2ar+*2  6#^=o,  or  r»+*».a?*=:2ar+2  6«.x;  •••  dividing  ijf  x. 


2.ar+6< 


tt7C  ^at>c  r*  -|-«*  j:=2  ar+2  6*. '/  x=— ^ :  consequently  r«— 

a= — riri a^stde  of  the  first  square,  and  sx-^bsz — j— 

^b=zside  of  the  second. 

42 
Examples. — ^Let  a=6>  6=:4>  rzsS,  <=:3;  (A^  loiU  ^»7^> 

108  «      ^     58 

fx— a=--— ,  and  w:— o=— -. 
17  1/ 

Let  asz4,  b=sS,  r=:2,  and  sszl,  be  given. 

6.  To  find  two  aumbeiB,  of  whieh  the  sum  is  equal  td  the 

square  of  the  least.    Ans.  6  and  S. 

6.  To  divide  the  nnmber  Sa  into  two  partst^iudi  that  their 
product  IwiU  be  a  square  Munfaer.    Ans.  27  smd  3. 

7.  To  (fivide  the  number  129  into  two  parts,  the  difference  of 
which  will  be  a  square  number.    Ans,  105  and  24« 

8.  What  two  numbers  are  those,  whose  product  added  to  the 
sum  of  their  squares,  will  make  a  square  ?    Ans.  5  and  3. 

9«  To  find  two  squares,  such  that  their  sum  added  to  their 

\S  1 

product  may  likewise  make  a  square.    Ans,  —  and  ---. 

If  8f 

10«  To  find  two  mimbeis,  one  of  which  being  taken  from 
their  product,  the  remainder  will  be  a  cube.    Ans,  3  and  108. 

11.  To  find  two  numbers^  such  that  either  of  them  being  ad- 
ded to  the  square  of  the  other,  the  sum  will  be  a  square.    An-' 

16      .43 

^er-and^. 

.  12.  To  find  three  numbens,  such  that  their  su^xif  an4  likewise 
the  aim  of  every  two  of  them,  mil  eaeh  be  a  J<piare  numbinr. 
Ans,  42,  684,  and  22. 


PART  VII. 

ALGEBRA. 


INFINITE  SERIES  •. 


1.  A  SERIfiS  is  a  ntak  of  quantities,  which  usually  proceed 
according  to  some  given  law,  increasing  or  decreasing  sucoea- 
sively;  the  sin|de  quantities  winch  constitute  the  sories  are 
caOad  its  terms. 

9.  An  increasing  or  diverging  series  is  that  in  which  tha 
tanna  suiicesaiTBly  incraaae*  €t$  I,  8,  S,  4,  isc  a-f-3  a-f  7  a^  3re. 

S.  A  decieasiiig  or  conveigii^'  aeries  is  that  in  which  tba 
ttnoa  sttceeasiveljF  decrease,  as  d>  3^  1, 4c.  lOa^^Ja^^  a,  Use. 


*  The  doctrine  and  application  of  infinite  series,  justly  considered  as 
the  greatest  improvements  in  analysis  which  modern  times  can  boast,  were 
mtrodneed  about  the  year  166»8,  by  Nicholas  Mercator,  who  is  supposed  to 
have  taken  the  first  bint  of  such  a  method  from  Dr.  Wallis^s  Arithmetie  of 
Inteitw;  bat  it  waa  tfce  genius  oi  Ktntan  that  first  gave  it  a  body  and  fofm. 

The  principal  use  of  infinite  serie%  is  to  approximate  to  the  valoet  and 
sums  of  such  fractional  and  radical  quantities,  as  cannot  be  determined  by  any 
finite  ezpreuions ;  to  find  the  fluents  of  fluxions,  and  thence  the  length  and 
quadrature  of  curves,  &c.  Its  application  to  astronomy  and  physics  is  very  ex- 
tensive, and  has  supplied  the  means  whereby  the  modem  improvements  in 
those  sciences  have  been  made.  The  intricacy  of  this  branch  of  science  has 
exercised  the  abilities  of  some  of  the  most  learned  mathematicians  of  Europe, 
and  its  usefulness  has  induced  many  to  direet  their  chief  attenlioB  to  iti  te- 
provement :  among  those  authors  who  have  written  on  the  sulyject,  the  follow- 
ing  are  the  principal ;  D'Alembert,  Barrow,  Briggs,  the  BemonUis,  Lord 
Bronncker,  Bonnycastle,  Des  Cartes,  Clairant,  Colson,  Cotes,  Gfaaier,  Cob* 
dorcct,  Dodson,  Euler,  Emerson,  Fermat,  Fagnanus,  Goldbacb,  Oiavesande, 
Gregory,  Haltey,  De  lUdpital,  Harriot,  Huddens,  Huygens,  Horsley,  Hotton, 
Jones,  Kepler,  Keill,  Kirkby,.  Lan#ai,  De  Lsfns,  Leibdita,  Lorgna,  ManfiredV^ 
Monmort,  De  Moivre,  Maclaurin,  Montano,  Nichole,  Newton,  Oughtred,  Ric- 
catl,  RegnaM,  ftranderson,  Stusius,  Sterling,  Stuart,  Simpson,  Taylor,  Varig- 
nbn^  VioUy  WaUis,  Waring,  fto.  &«• 

N  3 


183  ALGEBKA.  PaxtVII. 

4.  A  neatnd  serin  is  tliat  in  whidi  the  terms  neither  increase 
nor  decrease^  as  I,  1, 1,  1^  Sgc.  a+a4»a+a«  4rc. 

.  5.  An  arithmetical  series  is  that  in  which  the  terms^incveaae 
or  decrease  hy  an  equal  difference,  a$  I,  S^  5>  7»  4rc.  9,  6, 3, 0, 
8(C,  11+2  a+3  a,  lire. 

6.  A  geometrical  series  is  that  in  which  the  terms  increase  hy 

constant  multiplication,  or  decrease  by  constant  division,  oi  h 

3 
3,  9,  27,  3fc.  12,  e,  3,~,  *c.  a+3tf+4«+8tf,  *c. 

7.  An  infinite  series  b  that  in  which  the  terms  are  supposed 
to  be  continued  without  end ;  or  such  a  series,  as  from  the  nature 
of  the  law  of  increase  or  decrease  of  its  terms  requires  an  infi- 
nite number  of  terms  to  e^qiress  it. 

8.  On  the  contrary,  a  series  which  can  i>e  completely  ex« 
pressed  by  a  finite  number  of  terms,  is  called  a  finite  or  termi« 
nate  series. 

9.  Infinite  series  usuaUy  arise  fitim  the  division  of  the  name- 
rator  by  the  denominator  of  such  inctions  as  do  not  give  a 
terminate  quotient,  or  by  extracting  the  rootof  a  surd  quantity. 

10.  To  reduce  fractions  to  inJinUe  series. 

Rule  I.  Divide  the  numerator  by  the  'denominator,  until  a 
sufiicient  number  of  terms  in  the  quotient  be  obtained  to  shew 
the  law  of  the  series. 

II.  Having  discovered  the  law  of  continuation,  the  series  may 
be  carried  on  to  any  length,  without  the  necessity  of  forther 
division. 

1,  Reduce  -—— -  to  an  infinite  series  \ 
l+« 


^  If  »  be  aa  integer,  theo  wiU 

1.  — j7-=sa»-- »  +  a»-- «*+«■-- S6«  +  ,&c. to ^. *■--•»,  which  aerie*  e¥i- 
dentiy  termiiiatet. 

2.  "^^^  ^tf"-  «—*■- 86+ «"—sft2-,&c which termiBttes  in-4"-  », 
when  n  is  an  even  number,  bat  goes  on  inMnitelf  when  n  is  odd. 

3. r  ~«*  —  *  "-"«■  --  »ft+  a"  7- **»  — ,  Sbc,  which  series  terminates  tn 

+b'^  ^i,  when  n  is  an  odd  namber,  bnt  goes  on  indefinitely  when  n  is  n«i». 


PAtT  Vn.  INFINITE  SERIES.  183 

Opbbatiom'. 

I + x)  1     *  (1  —  *+ «*— ap* + ,  4c.  t}^  series  required. 
^•4"^  Expkttudion* 

.^x— j:*  .  This  operation  18  similar  to  those  in  Art.  50. 

.^X—'X*  ^'^  ^*  ^^*  ^    '^  ^*  unnecessary  to  proceed 

.  ■•  farther  in  the  work,  since  we  can  readily 

X  discover  the  law  by  which  the  terms  of  the 

x'+Jc'  quotient  proceed,  vis.  by  constantly  mnlti- 

^__  5  plying  by  x,  and  making  the  terms  alter* 

nately  +  and  — ;  knowing  this,  we  may  oon- 
— 3r  —  J*        tinue  the  quotient  to  any  length  we  please, 
X*  ^c.^^^^^  troubling  ourselves  with  the  work 

2.  Reduce to  an  infinite  series. 

X— « 

Operation. 

d        CZ      HZ*  ■     QZ* 

ar— «)  a      *  ( h— .  + h— —+*  *c.  the  <eri€|  required. 

az 
a 

X 


~  ExplmuaUm. 

,  Here  the  law  of  continuation  is  mani- 

^_f!f^  fest,  the  signs  being  all  +,  and  each 

X  '    X*  term  arises  by  multiplying  the  nume- 
•   ,                        •  rator  of  the  term  immediately  preceding 

^  it  by  z^  and  its  denoipinator  by  «;. 
X' 
az*     az* 


az* 

Id*" 

fu*     az* 

X*    '     X* 


az* 
X*  8fc* 


4.  The  difference  a*  —6*  is  not  measured  by  the  sum  ai-b, 

Hencey  first,  the  difference  of  th§  nth  powers  of  any  two  numbers  is  mea- 
sured by  the  difference  of  the  numbers,  whether  f»  be  even  or  odd. 

Secondly,  it  is  measured  by  the  tmn  of  the  numbers,  when  n  is  even,  bu^ 
not  when  n  is  odd. 

Thirdly,  the  ntm  of  the  nth  powers  is  measured  by  ^he  «mm  of  the  numbers 
when  n  is  odd,  but  not  when  n  is  even.  In  each  of  the  quotients  which  <er- 
mmniCf  the  number  of  terms  is  equal  to  the  index  ji.  See  an  ingenious  appli- 
cation of  these  condnsioiu  in  the  Rar.  Mr.  Bridga's  Loetunt  on  Alg^a^ 
p.  248. 

n4 


11.  When  any  qaantity  is  common  tommftmm,  the  seriM 
may  be  simplified  by  dividiiig;eYery  term  by  that  ijuantity^  putting 
the  quotients  under  the  vinculum,  and  placing  that  qoanti^ 
^      before  the  vinculum,  with  the  sign  x  between. 

Thus,  in  the  above  series  —  is  canmum  toaU  the  temu,  mid 


dividing  hif —,  ihe  qwtiemt  tf  1+— +^-f— +,*c.«Aicfcmioti. 


I 


emi  put  under  the vincuium and  amnected  mlh  thedioiew—  ha  the 

a    ^ 


z      z*     z' 


sign  X,  the  series  becomes  —  x  l-f— H — + — ^,  Sfc.  wHxch  is  a 

X  X       X*      X* 

simpler  form  than  that  in  the  example. 

3.  Reduce- to  an  infinite  series.  A*.l+x+««+*»-f  ,*c. 

X  "*"  X 

4.  Reduce to  an  infinite  series.    Jns.  zH 1- 1 — 

fl— z  n     a*     a* 

5.  Let  -—  be  converted  into  an  infinite  series.    Jns.—-^ 

*+« X 

az    az'     az'       ^  a  z      z»     z» 

p+-:^-i;r+'*«-o^-xl--+---+,*c.  See  ex.9. 

-        a'  ijt 

"Jf+6        ^«roed  into  an  infinite  series.    Ans.  —  x 


:r 


6      6*     6» 


7.  Reduce  — ,  and  likewise  its  equal ,  to  infinite  series, 

3  »+l 


3  10'  IpO^  1000^  10000 


1111 

10    iol*    idp    lot* 


1111  1  111 

Ana  II       i»       -I-    I  t'.Ac.  ac— -v  1  m<     4...,——^ 

■  I     I      IIU         II        ■■  11     »  I.  I 

12.  To  reduce  compoMfid  quadratic  surds  tg  infinite^  series. 
Ruu.  Sxtraet  the  square  root,  (Art.  57.  Fart  3.  VoL  1.)  attd 
continue  the  work  until  the  law  of  the  series  be  discoveied  j  after 


Fait  ¥11.  INFINITE  SEBIES.  18$ 

^hich  the  root  may  be  carried  to  any  lengthy  as  in  tlie  preceding 
rule^  and  it  will  be  the  series  required. 

Examples.—!.  Convert  a«-h««|+  to  an  infinite  series. 
Opbration. 

ExpUauftum, 
2 '  ^  The  lawoCcontiniiation  it  not 

*^"q]/       *  obviou»  in  this  example,  bn^ 

the  f  eries  may  be  made  tome* 

2'  -L  ^  what  more  simple  by  dividtng 

4«*  all  the  tcrmi  after  the  first  b| 


.9 


2*       «*     ,       z*  -—,  it  win  then  become 

jf^ ^ —    ) 2* 


I-t: 


4x«     ar*     64r« 
8x^""64Jc« 


3.  Let  ^««— jf*  be  converted  mto  an  infinite  series.    Ans.  a— 

^""8a»      16a* ""'  *^' 

b       b' 


3.  Change  v^*  +  ^  into  an  infinite  series.    Ans,  a-f- 


2a    Sa* 

4*  SacjMress  1  +2e\-l-  in  an  infinite  series. 

IS.  SIR  ISAAC  NEWTON'S  BINOMIAL  THEOREM  *. 

For  readily  Jindir^  the  pomert  and  roots  of  binomial  quantities. 

Rule  I.  Let  P=the  first  term  of  any  given  binomial^  <?= 
the  quotient  arising  firom  the  second  term   being    divided 


*  This  theorem  was  first  discovered  by  Sir  I.  Newton  in  \&S9,  and  sent  (in 
the  above  form)  in  a  letter  dated  Jotte  13tb,  1G76,  to  Mr.  Oldenbnigh,  at  that 
time  Secretary  of  the  Royal  Society,  In  order  that  it  might  be  comnmiHcated 
to  M.  Leibnitz.  As  early  as  the  beginning  of  tho  l6th  centory,  Stifelins  and 
elbcn  knew  bow  to  determine  ti^e  integral  powers  of  a  biooisial»  not  menly  by 
continued  moltiplication  of  the  root,  but  also  by  means  of  a  table,  which 
Stifellns  bad  formed  by  addition,  wlierein  were  arranged  the  coefficients  of  the 
termtol  any  power  within  the  limits  of  the  table.    Victa  seems  also  to  have 


186  r     AUSmSA.  PamtVII. 

by  the  fixst;  then  will  PQ=the  second  term.    Let  ^sthe  in- 

n 

dex  of  the  iNiwer  CMT  nxit  ixqiiii^d  to  be  found,  viz.  m 


Qiidentood  the  law  of  tlie  coefidoits,  but  the  method  of  gtoentiog  them  soc 

cessivelj  one  from  another,  was  fixtt  taught  by  Mr.  Henry  Briggs,  Savitian 

ProfetMir  of  Geometry  at  Oxford,  about  the  year  1000 :  thns  the  theorem  as 

far  as  it  relates  to  powen,  appears  to  hare  been  complete,  wanth^  oaly  the 

algebraic  form ;  this  Newton  gave  it,  and  likewise  extended  its  appUcation  and 

use  to  the  extractioa  of  roots  of  every  description,  by  infinite  series,  which 

probably  nerer  was  thonght  of  before  his  time.    The  theorem  was  obtained  at 

first  by  induction,  and  for  some  time  no  demonstration  of  it  appaan  to  hare 

been  attempted  $  several  mathematicians  have  however  since  given  denon* 

ftrations,  of  which  the  following  is  perhaps  the  most  simple. 

Let  I+d««l+«r+f««+«r*+«jr*  +  ,&c.l 

r — i-      ,  .      y  each  to  II +1  terms. 

i+y)"«i+/»y+«y»+ry*-i-*r*+»*c  J .      

Then    by    subtraction    l  +  jr/«  —  1+^ ■  — i».jr— y  +  ^.jr*— y'-f. rje»  — ya  + , 


&c  to  « terms ;  wherefore 


1+jr— l+jf 


x—\ 


> 


that  is^  (by  actual  division  ;  see  the  preceding  note,) 

I-f4:]»~»  +  l+y.l+jr>  -  «+  ,&c  (to  « terms]  8j»-f-  f!]r7ir+ rJTT^Ty* 
+  #jr*  +*'y-h*y2,+y*  +9  9tcton  terms. 
Let  jr«y,  then  n,f+x\^  —  *  »p+Sq*+3rx'  +  4mx*  + ,  Ac. to n  terms, 

whence  j^r+3»« i»+2<jw+3r*»  +  4*r*+,  &cx  l"+i 

^p  +  2qx+3rx»  +  4sx»+,  StCl 


>+2j+j»jr+3r+2fjr»+4#+3rjr»+,&c.  (4).  Butbe<sanse  l+«|*«i^ 
p»-^qx'  +<"**  -h,  &c.  by  the  above  assumption,  therefore  fi.l~^fjr)*ssis^ 
fl^ + mqx*  +nnF*  + ,  &c.  (S)  wherefore  the  two  series  ^  and  j9  (being  each 
equal  to  nA-^x)  •)  are  equal  to  one  another^ and  consequently  the  coeflicients 
•f  the  same  powers  of  x  will  be  equal ;  that  is, 
1.  /»*», 

$.  gj+^»jy,  or  2f +»»»•,  V  2t=n*->^»nM^,  ind  ,«l!iZi 

"  «   «      «               -      — :r            *— 3.y    «•»— 1JI-.2 
3.  3  r  +  2  y  =r wjr,  or  3  r=«-2.y,  •.•  r=  -j-  = — ;  &c,  &c  &c. 

--  r— *-s  »4«— I  11.11.^  l.|t^3 

Hence  i^jr^-^Bl +jMr+-j— .*• +. ^ ^\+,&c(C) 

Now  since  «+»=.« x  1  +  V'  '•'  «+^"=«»  ><  1+-^!  -(by  subatitnting 

^  *" a 

h  b      «.«— 16' 

~  for  jr  in  the  series  C)  a'X  l  +  ».— +  -5^.-7+, &c.  =««+»ui«»-'i+ 


Past  Vlf .  INFINITE  SERIES.  IBT 

Thtor,  n=it8  denominator  5  then  P+PQ]^  will  expreM  the^vea 
binomial  with  the  index  of  the  required  power  or  root  plaoei 
over  it. 

II.  Let  each  of  the  letters  A,  B,  C,  D,  ^.  represent  theiraltte 
of  the  term  in  a  series^  which  immediately  precedes  the  term  in 
which  that  letter  stands. 

III.  Then  will  the  root  or  power  of  the  binomial  P+  PQl?  be 

expressed  by  the  following  series,  viz.  PIt  +  ^  ^Q-^  -5 —  BQ 

TO— 2n  ^^    III— 3n  ^^ 
+  -T— -  CQ+  — —  DQ+,  *c. 
*>n  4a 

IV.  If  the  terms  and  index  of  any  binomial,  with  their  proper 
signs,  be  substituted  respectively  for  those  in  the  above  general 
form,  then  will  {he  series  which  arises  express  the  power  or 
root  required. 

ExAMPLBs.— 1.  To  extract  the  square  root  of  (fi-^g^  in  an 

infinite  series. 

z»  1 

Here  Pssa\  Q= ,  and  (since  -—  %$  the  index  of  the 

tquare  root)  i»=l,  n=r2  j  then  P+PQtfssfl*— **1*>  «wJ 
P|v=(^=)  a^the first  term A. 

.  TO      ^^  1  ^  ««         1  2»  fl2*        .        «• 

n  ^2  a*      8  o»         2a*  2a 

<A«  second  term B. 

TO— «  «^     .1—2     „         «•  1  z*  2*     . 

+  Tr*'^=<— ^*^-?=— ^-«5'^"-^=>- 

- — zsthe  thkrd  term C:. 

8  a' 


***""  ut*  —•*•  +  ,  &o.  in  which,  il?i.»  be  tabttitated  for  «%  ^  for  — ,  and 
i^,  -B,  C,  &c.  for  the  preceding  teimt,  the  wries  will  become  if  »»  +  —  .AQ-^ 
-^ — J?P+  — = — . CQ+ ,  Ike.  at  above. 
Jf  the  index  - .  be  a  positive  whole  number,  the  series  will  terminaie  at  the 

"  +  IM  term ;  bat  if  it  be  negative,  or  fractional,  the  series  will  not  termi* 
nate :  all  which  is  maailest  from  the  above  cnmptes. 


Itt  ALGXBKA.  PauVII. 

+  — ^ C(}s=:(-^-xCx =— S-><-.S-7X 7=)- 

— — -sslAe/oar^A  ierm D. 

^--^--^^^the  fifth  term £. 

5n  ^10  a*         10         ISSa?         a' 

72»o 

""^>^  ^=^^  *"^^  *«»^ ^« 

S56ii» 


ifC.  8(Cm     Wherefore  the  square  root  of  the  gieen  binomial,  or 

-^ — -^,  z*      X*       z*         hz*        7z^ 

fl«— z«|T=sa— .rrr-^t  *c.  as  required^ 

'  2a    8a»     16a*     128  a'     256  a»  ^ 

2.  Find  o+Hf  in  an  infinite  aeries. 

Here  Pssa,  Q=— ,  m=S,  n=5,  oiui  P+JPQJv  =«+3t. 


a 
P)^=s«t  the  first  term  of  the  series A. 

H ifQ=(---x-^X — ss—xof  X — =^)  — 7 *«e second «er»  ^. 

n  5  a      5  a         5|jr 

+  -- — BQ=(-— •  xBx— =— — X— ^  X— =) ths 

^  2n  Mo  a  5     5<jf     a     '     35^ 

third  term .• C 

-f  — —  CQ=:(— —  xCx— =--^X rX— =) 

Sn  15  a  15         25(fi'     ^ 

'     rt«  fourth  term D. 

125  a^       -^ 

w-3n^_     3-15     _      6  3  _    76*         6      ^ 

4»  ^20  a  5      i25aiJL     a      < 

— — --  the  fifth  term E, 

€^ba^      ^ 

.,       ,      3  6      3  6*         76*  216* 

*c.  4c.  Wherefore  a+6»3-=tfr+ — --f - 

5«*    26rf     12607'     — 


+ 


6fl5av 

«     /   r.  -,     ^        V       3  36     3  6«       7  6»        216* 

+,4c,(icfcicA6yifr^9.)=fffxl+ — +-^-- — — — +, 

^         ''^  /         '^    "Tg^    25a*^125a»    625a*^ 

4c. 


P**»  Vfl.  INHNITB  SERIES.  1S9 

3.  To  find  the  value  of      '       in  an  Infinite  series, 

and  then  multiptt  **«.  retuUing  teria  by  y* ;  wherefore  in  the 
pretent  can  P=y»,  Qa=fl,  m=-l,  nsS,  and  j5+p^^_, 

y 

<erifi , ^^ 


'y    y 

the  3rd  *en» /•  .  C. 

the  4th  term • , j}^ 

+—7-— i)Q=( — -— xDx— S3— ^X— - — X— =) 
4»  ^8  y»  8  16y»     y'     ^ 

------  <^  5<A  <€rm R. 

«c.  *!!.     2%a  «ri«  ffua^ipiied  4y  y*,  according  to  what  was  pre* 

Vy  +^       '^       y     2y'^8y«     16y'^128y» 
4.  To  invoh^  1«,  or  hi  equal  ll+l,  to  theciibe. 

1  "  m 

Here  Ps=Il, Q= j^,  i»=3,  n=l  5  f^,  as  te/ore,  P+P(SF 
(+yXl3Slx^==)+863(+|.xS<J8x^=s)+3S(+lx 

r 

^X— ta;)+l,  where {nnee the  oo^fit^m^of^  next  term  wdl 

heo)  the  eeries  mu»t  emdmtfy  terminate.  VFkerefore  cotketmg 
the  «tettt  ierme,  (1331*f*dfi8-f  13-hls)  17«8  iitbewkei^  18, 
a$  wag  required. 


190  ALGEBRA..  :.  FAiTVa 

5.  Find  the  value  of  x+p  -r  in  an  infinite  series.    Ans.  xt+ 


Sjtt    9a4     81xT 
6.  To  find  r- —  in  an  infinite  series.  Jns.  -^x 


c     c*     c 


7.  Find  ^a*+6  in  an  infinite  series.    Ans,  04-5——^^^  + 
6' 


•,  *c. 


16a* 

8.  Ettract  the  5th  root  of  2488SS  by  infinite  series.  Ans.  12/ 

9.  Find  ==ra  ^  infinite  series.    Ans.-^-^ — 2I4.-? -i 

jf-jTy)*      "^  X'       X*        JC*         OE* 

r» 

10.  Ilnd in  an  infinite  series. 

1 1 .  Tb  find  *  ^x*  ^z*  in  an  infinite  series. 
\%  Find  y  x  y — 1;]  ^  in  an  infinite  series. 

14.  A  series  being  given,  to  find  the  several  orders  of  differences, 

RuLB  I.  Subtract  the  first  term  from  the  second^  the  second 
from  the  thirds  the  third  JErom  the  fourth,  and  so  on-;  the  seve- 
ral remainders  will  constitute  a  new  series,  called  the  first  order 
of  differences* 

II.  In  this  new  series,  take  the  first  term  from  the  second, 
the  second  from  the  third,  &c.  as  before,  and  the  remaindecs 
will  form  another  new  series,  called  the  second  order  of  differ^ 
ences. 

III.  Proceed  in  the  same  manner  for  the  third,  fourth,  fifth, 
Sfc.  orders,  until  either  the  difierences  become  O,  or  the  work  be 
Carried  as  &r  as  is  thought  necessary  *. 


*  Let  o,  b,  c,  d,  f,  S^e,  be  the  terms  of  a  given  series,  then  if  JD^tbe  first 
term  of  the  «ith  onlerof.diffiBr«iiGes,  the  foliowiiig  theorem  will  Exhibit  the  vaJae 

9i  jD:riZf  ±u-{-nb±n,'^A!+n.—n—M±n,'^.—,^.€'h,^c. 

(to  n-f.  1  terms) »/>,  where  the  upper  tfifos  aost.be  tihiii  when  •  it  «a  OTeii^ 
number,  and  the  lower  signs  when  ft  is  odd. 


Pax*  Vn.  INHNITE  $BBI£S.  191 

Examples.— 1.  Given  the  aeries  i,  4>  8, 13^  19«  26^  &c.  Xo 
find  the  several  orders  of  differences. 

Tkui  I,  4,  S,  13^  19j  26,  ^c.  the  given  seriei. 
Then  . . .  3j  4,  6,    6,   7»  ^c.  the  first  differeneee. 

And \»  I,    1>    Ij  ^c.  the  second  differencee, 

AUo 0,    0,    O,  iicthe  third  differences. 

where  the  work  evidently  must  termtnaie. 

9.  Given  the  series  I,  4,  S,  16,  S%  64, 19B,  &c.  to  find  the 
several  orders  of  di^renoes. 

J9ere  1,  4,  8,  16,  32,  64, 128,  4c.  given  series. 
And  ...  3,  4,  8,  16,  39,  64,  ftc.  Ut  diff. 
1,  4,  8,  16,  3«,  *c.  gnddi/. 
3,  4,    8,    16,  4rc.3rddj^. 
1,     4,     8,   SfC.4thdiff. 
3,    4,    8(C.6thdif. 
1,    «c.  6MdtJ.  «c. 

3.  Find  the  several  orders  of  differences  in  the  series  li  3,  3, 
4,  &c.  Ans.  First  differences  1,  1, 1^  1,  Sfc.  Second  diff.  0, 0, 0, 
*c.     . 

4.  To  find  the  several  orders  of  differences  in  the  series  1,  4, 
9,  16,  26|  &c.  Ans.  First  differences  3,  5,  7>  9,  4rc.  iSecond. 
8,  2,  2,  *c.     I%trd  0,  0,  *c. 

5.  Required  the  orders  of  difierences  in  the  series  1, 8, 97, 64; 
125,  &c. 

6.  Given  1,  6,  20,  60, 105,  &c.  to  find  the  several  orders  of 
dijflferences. 

7.  Given  the  series  1, 3,  7s  13»  21,  &c.  to  find  the  third  and 
fixnth  orders  of  differences. 

15.  To  find  any  term  of  a  given  $erie$. 

RvLS  I.  Let  a,  b,  c,  d,  e,  &c.  be  the  given  series ;  d^d^SdV", 
^,  &c«  respectively,  the  first  term  of  the  first,  second,  third, 
fourth,  &c.  order  of  differences,  as  found  by  the  preceding  arti* 
cle;  nsthe  number  denoting  the  place  of  the  term  required. 


If  the  dilferenccf  be  rery  gnat,  the  logarithms  «f  the  qnantttict  may  b*. 
used,  the  dUTereiicefl  of  which  will  be  much  smaller  than  those  of  the  quantities 
tlkmsehres;  and  at  the  close  of  the  operation  the  natural  number  answeribf 
to  the  logazitbmical  resnlt  will  be  the  auwtr.  See  JEmsTM*'*  JDigtrtnixai 
Mttkod,pTop.  1. 


Wt  AL6BBAA.  PaktVIT. 

.«'"+— j—.-^,-j-.-—^+lrc.asto  the  «*  tenn  leqmred  •. 

£xAMPLEs.-^l.  To  find  the  10th  term  of  the  series  8,  S,  9, 
U,  90,  he. 

Here  {Art.  12.)  %  5,  9,  14,  20,  *c.  ««rief . 

3,  4,  5,    6,  ^c.  IH  d^. 
1,  ],    1,  isc^rnddiff. 
O,    0,  Ssc,3rdd^. 
Where  <P=3,  cP'=l,  d"»=:0,  olfo  a=2,  nsslOj  vAerefore 
-  .  ""-^  ^  .  »— ^  «— 2  ^.     ^^  .  10—1     ^     10—1     10—2 

Iss)  2+27+36=:65slAe  10t&  lerm  reqiared. 
2.  To  find  the  20th  term  of  the  series  %  6, 12,  20, 30,  Bsc. 
Here  a=2,  »=20;  and  Art,  12. 
2,  6,  12,  20,  30,  «c.  MTter. 
4,  6,    8,  10,  *c.  Ill  diff. 

%   %    2,  Ac  2iid  di^.  or  d' =4,  d"  3=2)  wkemse 

+342=r  420=  the  90th  term  required. 

S.  Required  the  5th  term  of  the  series  1,  3,  6,  10,  &c. 
^tu.  15. 

4.  To  find  the  10th  term  of  the  series  1, 4,  8,  13, 10,  Ac 
Ans.6^ 

5.  To  find  the  14th  term  of  the  scries  3, 7, 1«>  la  25,  ftxu 
Ans.  133. 

6.  Required  the  20th  term  of  the  series  1,  8,  27.  64,  125. 
r  &c.    ^i».  sooa 

7.  To  find  the  60th  term  of  1,  4, 8,  13,  19,  &c. 

8.  To  find  the  10th  term  of  3,  f,  12, 18,  26,  &e. 

16.  If  the  succeeding  terms  of  a  given  series  be  at  an  wHts 
distance  from  each  other,  any  intermetUate  term  may  be  found  by 
mterpolaiUm,  asfaUows. 


•  For  ths  ioTMtifitioB  <f  this  twkt,  m  JEmenot^*  DjftnMlmf  MttMtf 


Part  VII.  INFINITE  SERIES.  193 

RvLE  I.  Let  y  be  the  term  to  be  interpolated^  x  its 
distance  from  the  beginning  of  the  series,  d*,  d",  d»",  dS  &c. 
the  first  terms  of  the  several  orders  of  d]£ferences. 

II.  Then  wina4-JdHj.^^.d"-har.^T'V^'r^.d"'-f*.^^> 

— ^- .  -7-  .d^+  *c,=y,  the  term  required  '. 

Examples. — 1.  Given  the  logarithms  of  105>  106, 107,  108, 
and  IQ9,  to  find  the  logarithm  of  107.5. 

Stries.        Logarithms.  XH  diff,  2nd  diff,  Zrddiff.    Mhdiff. 

105 0211893  .„^^ 

106 ... .  0253059  lii?^  -387  « 

lor  . . . .  0293838  ^^  -379  ""^    -0. 

108 ... .  0334238  ^^^  —373  ""^ 

10&  . . . .  0374265   **^^' 

5 

Here  a?5=  (107.5—105=2.5) -^=iAe  distance  of  the  term 

y,  o=.0211893,  d»=41166,  d"  =  — 387,  dM»  =  -.8,  d«'=-.2. 

iP  "~  1  X""^!  wT— 2  jC— 1 

rA€»   y=a-f«d*-f-a?.-— -.d^+x.— — -. .d"»+jp. . 

iS  «        3  2 

X— 2x— 3^5,       ,        5  _,      5      3       ,..      5       3       1 

5       3       1  1        -.  B  _,,     15^..      6  _,,,,       5    ^ 

■2  ^T^T>< -T^  ^'=^+-2 ^'+T^"  + i6^"^-l2s^'== 

j0211893+|-x41166+~X-387+^X-8-^X.-2  =  ) 

0211893+102915-725-2.5 -|-.078=.031407128,*^eZo^arU^w 
required, 

2.  Given  the  logarithmic  sines  of  3®  4\  3°  5',  S^  6^  3°  7\  and 
3<>  8S  to  find  the  sine  of  3°  6»  IS^*. 

Series.       Logarithms,        1st  diff'.        2nd  diff.       Zrddiff. 
3M»....  8.7283366       g^.,^ 
3  5 ....  8.7306882       '^J'J^         -126 
3  6.... 8.7330272      q^^^"        -127  t 

.  3  7 ....  8.7353535      ,^^*?r         -123         "^^ 
3  8....  8.7376675       ^^^ 

Herexsz(S^  6^  15"— 30  4»=a2oi5»=)-j-=fAedi«fa«ceo/</ie 
terwiy,  to  be  interpolated ;  a=8.7283366,  d'=23516,  d»i  =  — 126, 

'  This  rule  is  investigated  in  Eoierson's  Differential  Method,  prop.  5, 
VOL.  11^  O 


194  ALGEBBA.  Pakt  TIL 

ii«"=l,  and  y=fl+xJ«  4-^.^.4** -h*.^^.^^^"=(«+~ 

3  2        3  4 

45  15 

^' +^' +T^*"=)8  7«8a3W  •♦-.O05W11-.O0OO1771W5  + 

.0000000117=8.73300999996^  the  log.  sme  regvirvdL 

3.  GiTen  the  series  —-,  — p  --,  --,  --,  to  find  line  term  which 

50  51    5»   53   54 

stancb  in  the  middle,  between  rr  and  --.    .^nt.  •-*-• 

52  53  105 

4.  Given  the  Icgvithmic  sines  of  V  O',  V  V,  1»  2',  and  V  S\ 
to  find  the  logarithmic  sine  of  1^  i>  40>^    4ns.  8.2537533. 

6.  Given  the  series  — ,  —-,  -—-,  -—,  -—,  &c.  to  find  the  nuddk 

23450 

term  between  —  and  — . 

5  6 

17.  If  ihefcrit  differences  of  a  series  of  eqniMffkrent  terms  he 
snuUl,  any  intermediate  term  may  h^fownd  by  interpolation,  as 
follows. 

RuLK  1.  Let  a,  b,  c,  d,  e,  &a  repres^t  th^  given  series,  and 
fissthe  number  of  terms  given. 

II.  Then  will  a-^nb+n.—^^.c-^n.—-—. .d+n.— r— .— -— 

o  2  2        3  2       3 

.——.«+,  &C.SO,  fipom  whence,  by  transposition,  &c.  any  re-. 

ijuired  term  may  be  obtained  i. 

Examples.— 1.  Given  the  square  root  of  10,  11,  12,  13,  and 

15,  to  find  the  square  root  of  14. 

Here  ns5,  and  e  is  the  term  required. 

a=(Vl0=)3.1622776 

fc=(^U=)3^166248 

c=(^12=r)3.46410l6 

d=(v^l3=)3.6055512. 

/=(Vl5=)3.8729833 

And  since  n=s5,  the  series  must  be  continued  to  6  terms. 

^,       .  ,        n— 1  w— In— 2  ,       n^ln— 2 

Therefore  a^nb-jrfi* .c-^n.      ■  . M4-n.^ • — 

•^  ^  S323 

«— 3  9—1  n— 2  n— 3n— 4  - 

4  2        3        4        5    -^ 


f  For  the  investigation  of  tbb  rule,  sec  Emerton's  Difftreniial  Method^ 
prep*  ۥ 


Pa£t  VII.  INFINITE  SERIES.  195 

Whence,  hy  trampositian,  in  order  to  find  e,  we  thall  have 

n— 1  »-2  n-3  ,        n— 1     ,     n-in— «^ 

«.-~^.-^.— j-.e=: — a  +  n6— n.-^.c  +  n.—  - -3— •<*  +  «• 

—^ r — .  -— — .  —r-'fi  t"^  t«  numbers  becomet  5  c=  —3.1622776 

S3         4         5 

+  5  X  S.3166S48— 10  X  3.46410164-10  x  3.6055513+3.6729833 

= 56.5 1 16193 -37.8032936=  18.7083257,  (wd  c=  i?^^5???5Z, 

5 

3.74166514=^^6  root,  nearly, 

2.  Given  the  square  roots  of  37>  S8>  39,  41,  and  42,  to  find 
the  square  root  of  40.    Am,  6.32455532. 

3.  Given  the  cube  roots  of  45,  46,  47>  48,  and  49>  to  find  the 
cube  root  of  50.    Ans,  3.684033. 

4.  Given  the  logarithms  of  108, 109,  110,  111,  112,  and  114, 
to  find  the  logarithm  of  1 13.    Am.  2.0530784. 

18.  To  revert  a  given  series. 

When  the  powers  of  an  unknown  quantity  are  contained  in 
the  terms  of  a  series,  the  finding  the  value  of  the  unknown 
quantity  in  aootiier  series,  which  involves  the  powers  of  the 
quantity  to  which  the  given  series  is  equal,  and  known  quanti- 
ties only,  is  ddled  reverting  the  series  ^. ' 

Rule  I.  Assume  a  series  for  the  value  of  the  unknown  quan- 
tity, of  the  same  form  with  the  series  which  is  required  to  be  re- 
verted. 

II.  Substitute  this  series  and  its  powers,  for  the  unknown 
quantity  suid  its  powers,  in  the  given  series. 

III.  Make  the  resulting  terms  equal  to  the  corresponding 
terms  of  the  given  series,  whence  the  values  of  the  assumed  co- 
efficients will  be  obtained. 

Examples.— I.  Let  aa?+fc:c*-|-ca?^  +  da?*  +  ,  &c.=2  be  given, 
to  find  the  value  of  x  in  terms  of  z  and  known  quantities. 


^  Various  methods  of  rerersion  may  be  seen,  as  giren  by  Demoivre,  io  the 
Philosophical  Transactions,  No.  240.  in  Maclaorin's  Algebra,  p.263,&c.  Col- 
ton's  Comment  on  Newton's  Fluxions,  p.  219;  Uorsley's  Ed.  of  Newtoo's 
Worisa,  vol.  I.  p.  291,  &c.  Stuart's  ExpUaalion  of  Newton's  Analysis,  p.  455. 
Simpson's  Fluxions,  &c.  &c. 

O  2 


J96  ALGEBRA.  Past  VU. 

Lei  ^^x,  them  U  it  piam  tkai  tf  3^  amd  U9  pamten  he  99hUi' 
iutedinthegwemteriafarxoMdUsfomen,  the  mOees  rfzwnU 
he n,2n,Sn,  4m,  isc.  amd  1  -,  whemee  «s=l,  amd  the  diferauxt 
ofihete  imdkes  are O,  I,  %  3,  4,  4rc.  JFberefore  the  mdke^oftie 
serieg  to  he  astmmed,  must  hace  the  tame  differemces;  let  therrfare 
thisserie»heJz'^Bz*^&-^nz^'^,tse.=x.  Jmd  if  tkit  eeria 
be  mvohed,  amd  substituted  for  the  several  powen  of  x,im  thegivem 

series,  U  will  become 

aJz+aB2!^-^aC3^-^aDt*+,  tec. 

*  -\'bJ^7^'\'^bAB7?'^^bACi^-\',ke. 

*  *  *       +    6B«r*+,  ftc  >=rz. 

*  *  *      +    d^t*^,ke. 
Whence,  by  equating  the  terms  which  comtaim  Uke  powers  ofz. 


tte  obtain  {aAzt=z,  or)A=. — ;  (aB;^-f  6.A;*so,wAaice)B=3( — 

bA*  b 

=) ^,(aCz'+26JBz'+c^z»=o;  whemee)  C=(— 

a  a'  '  /         \ 

^bJB+cjP     ^aP^ac     ^     ^     ^bAC^rbB^-k-^cA^B-k-dJ^     , 

=)— ^r-  J  ^=(-— ;; =) 

habc^blP'-'C^d 

^ ^kc.  and  consequently  xsi^Az+Bsfi-^Cfi+ySse, 

,2      bz^    ^b^^ac    ,     Sfc'— 5a^+a'il 

=) T-H r — ^ jB*+,  9sc.  the  senes 

a      (^  a*  a^ 

required. 

This  oDDclusion  forms  a  general  theorem  for  every  similar  se- 

ries^  involvings  the  like  powers  of  the  unknown  quantity. 

2.  Let  the  scries  x—af2_^jj3^jj*^^  ^.==z,  be  pfoposed  for  re- 
version. 

Her^  az=tl,  5=— I^  c=l^  d==— I,  4rc;  tto^  rofoes  6dii^ 
substituted  in  the  theorem  derived  from  the  preceding  example,  we 
thence  obtain  x=2^z*  +a^'+z*-f•,  8(c,  the  answer  required. 

X*       X'       X* 

3,  Xet  X — ^'\'^ T-+>  &c.=y,  be  given  for  reversion- 

's     o      4 

Substituting  as  before,  we  have  a^l,  6=:~~-j  ^^T'  ^'''^^ 

it  3 

s  — 7->  4c.    These  values  being  substituted,  we  shall  have  x=:  jf+ 
4 

y>       «*       «* 

^+^+|--f,  SfCfrom  which  if  y  be  given,  and  sufficiently 
small  for  the  series  to  approximate,  the  value  of  x  wiU  be  known. 


Pa«t  VII.  INFINITE  SERIES.  197 

Let  2"=x,  then,  if  z  he  transposed,  the  indicis  will  be  I,  nm^ 
nm-^np,  nm-^^np,  nm+Snp,  *c.  where,  if  the  twe  least,  1  and 

nm,  be  made  equal  to  each  other,  we  shall  have  fi= — :  and  the 

m 

differences  are  -C.,  -£,  -X,  -£,  ^,     The  series  therefore  to  be 

m     m     m     m  *' 

I  l+p  l+2p  l+Sp 

assumed  for  xisAzln-i-Bz  m  ^Cz  m  ^Dz^^nT  +^  ^c.=ztf 
Mi*  series  being  involved,  and  the  like  terms  of  bath  compared  as 
before,  we  have  ^=1,  B=-.l,  c^l-^m^^pMb^o.mc ^   ^^ 


^"W^~"^"W 


»»  9mJ 


from,  whence  the  pfllue  of  x  being  found,  theorems  for  innumerable 
cases  may  thence  be  deduced. 

5.  Revert  the  series  z+--;-H 1 1-.  &c.=«.    Ans  zsix 

x^  x^ off 

1.2.3  "^1.2^.4.5     1.2^.4.5.6.7^'  *^* 

6.  Revert  the  /series  aj?  +  &jp»  +  ca?»+ilr*.-f,  Ac.  sr^+A«»-f 

19.  To/jid  t^  turn  ofn  terms  of  an  infinite  series. 

RuLB  I.'  I^t  a,  b,  c,  d,  e,  Slc.  be  the  jgiven  series^  .«s=the  sum 
of  « terms,  and  cf  ,  d",  d"',  d^  &c.  respectively  the  fi^t  jterms  of 
the  several  orders  of  differences,  found  by  Art.  12. 

II.  Th.u  win  na+n.'^.i+n.^!^.dr+n.!^.^, 

»— 3  ^„        n— 1  n— 2  n— 3  »— 4  ^ 

-_-.tf  -|-».-_..-,_._«.«^.iP^^  &c.ss#,  the  sum  of  n 

terms  of  the  series,  as  was  required'. 


'  XliM.nil6  i$  inveitigated  by  Mr.  Emenon,  Ui  bit  D^ertnHai  Meihod^ 
pmp.  3.  The  tOTettigations  of  this  aod  tome  of  the  foregoing^  raies,  aUhongh 
not  ^iBcvtt^  are  rather  prolix,  aod  require  too  qiveh  room  to  be  admitted 
witbiii  the  compass  of  notes ;  for  this  reason  they  are  omitted.  The  follow- 
ing problems  on  Ihe  siiB»mation  of  series,  which  afed  bat  a  very  imperfecj; 
specinea  of  timt  upble  biaodi,  wei»  taken  mostly  firoitti>M£ms'«Afo<A«ma<Ma/ 
RepMUcry^  voL  I.  where  a  great  Q«mber  of  problems  on  the  sabjept»  with  in>- 

O  3 


19B  ALGEBRA.  Fart  VII. 

Prob.  1.  To  find  the  sttm  of  n  tenns  of  the  series  1^  2,  d»  4, 

Firsts  bff  Art.  13.     I,  2,  S,  4,  5,  isc.  the  given  tenet. 

\,  \,  \,  \,  S(C,  first  differences. 
O,  O,  O,  4rc.  second  differences, 

Herea^l,  d'sl,  <«'»=oj  thenwiUna^n!^Xd^:sz 
( — : ,whichj  (smce  a  ana  d'  eacA  =1)= s:) 


2 


— - — =s«,  iAe  sum  required. 

The  sum  of  n  terms  of  this  series  may  likeivise  be  found  as 
follows. 

Let  1+2+S+4+5+,  Sfc.  ...  -f«=t   

Invert  this  series,  ondii-f-ji— 1+»— 3+«— 3+»+4-h,  S;c. 

•  •  •  •  T*  1 3Sa» 


■  MiiMM  ■■    M  M^ta^M^aaaM  •^^■MiM^^  ^M^wa^^n^  — 

^dd  bothseries  together,  and  »-f  1  -|-]|+  1-f  i>-hl-ffi+  1+n+l 

+  ,SfC. .  .  .  +w+l=r.2«;  that  is,  n.n+l=:2 s, whence  s=:^^ — , 

as  before. 

Examples.—-!.  Let  the  simi  of  20  terms  of  the  above  series 
be  required.  

Acre  )is20>  a»d  #=3— —at — - — sllO«  the  answer. 

2  2 

2.  Let  the  sum  of  1000  terms  be  required.    Ans.  500500. 

3.  Let  the  suiti  of  12345  terms  be  required. 

Prob.  2.  To  find  the  sum  of  n  terms  of  the  series  1,  3,  5,  7, 
9,  &c. 

Here  1,  3,  5,  7>  9,  Sfc.  the  given  series, 
2,%t2f  %,  Sec.  . ,  first  difference. 
O,  0,  0,  &ic.  .  .  second  difference, 

9t— 1 

Wherefore  a=xl,  d'=2,  i'z^o,  and  na+n.— — -.d'=(fifl+ 

.d'ss  (since  aal  and  dsrS)  n-f-B^^^-ns)  tt'szej^  lAesMm  f«- 

quired, 

genlooB  sdatiofis,  may  ^  feeen.  I'be  doctrine  of  iHArit*  Series  wUi  probaWy 
nerer  be  comptete ;  but  it  would  reqsire  a  very  large  treatise  to  do  anple  Jtu* 
tke  to  tbe  subject,  evea  ia  Hs  present 


^ 


pAiT  Vli.  INHNITB  SMlES.  1^ 

Or  thus,       

Let  1+3+5+7+9+,  *c.,. . .  .+2n— 1=5. 

This  inverted,  m*»— l+8»-3+3»— 5+2«— 7+2n— 9+,  *c. 
+1=«. 

TAe  smi  of  both  is  2n+2n-t-2n+2n+2n+^  ^c. . .  +2ftr=2 1. 
Whence  n  terms  of  this  sum  is  2  n.n=2  «^  or  <=sn%  (u  before. 

EXAM^LB^.-—!.  To  find  the  sum  t)f  10  tenn^  of  the  above  se- 
ries. 

Here  nve:lO,  and  sts(n^va)  100,  the  answer. 
52.  To  find  the  sum  of  50  terms.    Jns,  2500. 
3.  To  find  the  sum  of  1928  terms. 

PsoB.  3.  Td  find  the  sum  of  n  terms  of  the  series  l>f  squares 
I,  4,  9y  16,  25,  &c. 

Here  I,  4,  9,  18*  25,  ftc.  the  series. 

3,  5,  7,  9^  9!t 1st  ^. 

%  %  2,  *c 2nd  diff. 

O,  O,  4c 3rd  diff. 

V  n— 1 

Whence  a=:l>  rfar3»  d^=B2,  d*'s=o,  «»id  na+n.— — ^+n. 

»— In— 2  ^,  n— 1  .  ^    n— In— 2    3n*— n^ 

-^.-—.rf  ^(,+8  ».-^+g  „.^._^..-_^+ 

-  n'— 3n*H-2n    .n.n+1.2n+l         ,,  .    , 

-I -^ — : — ) : V — .ssf,  the  sum  required. 

3  0 

^     Examples. — 1.  Let  the  sum  of  30  terms  of  the  above  series 

be  required. 

^         ^      .     n.n+ 1.2  n+1     30x31x61 
Heren=305  wherefore — ^2— g — L-=: s9455> 

the  answer, 

2.  Let  the  sum  of  70  terms  be  required.  

Prob.  4.  lb  find  tie  sum  of  a  tenns  of  the  sories  a+«+d+ 

a+2d+a+8d+,  &c.  

«— 1               n.n— l.d 
Here,  6y  «Ac  rule,  na+n.---.d=sna+ — ^ =:«,  Wc  n^m 

Or  thus, 
Since  #^>gfiCTtt-fa-fd+a+2d+g-h3  d+,  *c. 

^  f +l-H4-l-f  l  +  l+>4c.xai      ^  *^ 

1  -^0+1+2+3+4+,  *c*>^d  J  ""  ' 
irefcaw«Ae«umo/**c/rUo/«fte»e>+l  +  l  +  l  +  l+>*<^(^o 

o  4 


800  ALGEBRA..  PiWRT  VIL 

«  terms)  =n.-  and  the  sum  of  (^  latter,  -fO+ 1+^+3  +  *  5fC, 
(to  n   terms)  sz^^^^^^,  (theor.   92.  Jrithmetkal  Progression,) 

-wherefore  na+  -^ .(i=«j  <w  before. 

Or  thus. 
Because      a      +       a+d      +  c+9d       +«   +     3<'+' 

*c -f  a4-»— l.d=«>  

if «d     a + »d— d-f  fl  -f  nd— 2d+  a  4-«d — 3<i+  a-f-«rf—  4d+, 
^c . . .  +a=#,  * 

ofbo^lT  }  ^  **+ nd— d+ 3  a+nd-^d+2  a+»d— d+2  a+?id— 4+, 

^c +2a+nd— d=2«. 

■     »    — .— — 

•m^  ,  .     ; — ;       ^  2a4-n— l.d.n      ' 

That  u,  aa+nd— d.«i=?2*,  or  «s=( — : — •—. z=i)na-l^ 

? 

n.n— 1 

— T — Ay  as  before. 

Prob.  5.  To  find  llie  sum  of  n  terms  of  the  serte  ]>  x,  47% 
a?»,  &c. 

Let  1  +ar-j-j?*  +a:*  +,  ^c.  (to  j:*— i)=s;  mM^pfy  <^w  serks 

ky  x,  and  x-f  ar'  +a?*  +J?*4-,  ^c.  (*o  3?*)=;:^;  subtracting  the  wp- 

^" 1 

per  from  the  lower,  we  feaue— l+a;"=«a:— «;  whence  5= -, 

<Ae  5ttm  required. 

When  JT  is  a  proper  fraction,  the  sum  of  the  series  in  mfinitum 
may  be  found  in  the  same  manner. 

Thus  l+x-\-x'-{'X^'\',  ^c.=«. 

^nd  x+x'  -\-x^  ■j-x'*  +,  8iC,=zsx;  whence,  subtracting  as  be- 

fore,  -^  Is^sx-^Si  md  s^ ,  the  sum  of  the  smes  in  mfinitum. 

Prob.  6.  To  find  the  sum  of. an  infinite  number  of  terms 
of  the  circulating  decimal  .99*99/ &c» 

First,  .99999,  *c.=— + ,—  +-^+  — ^  +,  *c.  5=*,  tha^ 

10^  100^  1000^  10000^ 

1        X  J      •      1  1 


i»ART  VTL. 


INFINITE  SERIES. 


SOU 


+ 


1 h ,  ^c.  =  — :  subtrchct  the  last  hut  one  from  ike  latt, 

100     1000  9 


andl=:( 


lOs 


S  Q  5 

— =)  ~,  or  «=  1,  the  sum  required. 


Hence, 

I' 
.1111,  8!C.or  — 

2 
.2222,  Ssc  or  — 

*f 

.3333,  *c.  or  — 


.4444,  fifc.  or  — 
Thesumof^  ^ 

.5555,  fire,  or  — 
9 

2 
.6666,  fifc.  or  ~ 

7 
.7777,  fifc.  or  ~ 

8 
.8888,  5rc.  or  ~  , 


9 

2^ 

9 

3" 
4 


>o/.9999,  5rc.=^  ^ 


5 

2^ 
3 

9 
£ 

9* 


Prob.  7*  To  find  the  sum  of  n  terms  of  the  series  a^+er+c/V 
+a+2dl*+fl+3tf]«+,  &c. 

i'trj^,  6y  actually  squaring  the  terms,  we  have 


a*  =ra« 


o+27p==a«+2x2ad+  4  d- 
a+3?l2=a*4-2x3ad-|-  9  cP 
a+4d]«=a*+2  x  4  ad+  16  d^ 
S(C.  fifc. 


Jff%€nce  l  +  l  +  l-fl  +  ^c-  {ton  terms)  x  a* 
-fO-hl-f^+3  +  fifC.  (^0  n  egrm) X 2 fld 
4ro-|-l4-4-f  9  +  *c.  (ton  e6rww)xd* 
l  +  l  +  l  +  l  +  *c. 


But    ^0+l+^+3  +  ^c.U^„^^^^^ 

the  sum  of  ]  i 

.0+l+4+9+*c.J 


1x2 


«.n — 1.2n— I 


1X2X3 


108  AU^EBRA*  pAkT  vn. 


Whence  (n.a^+n.^l.ad^'i^f'''^^^<^=) 
^  1x2x3 


n— 1.2n— 1 


n.a^+»— l.<ui-i .d'=x5,  the  ^m  required. 

Pkob.  8.  To  find  the  sum  of  the  infinite  series  H--^+-x+ 

« 


10 

First,  let  "7"+"«"+'x4-7r+>  ^c.  ad  infinitumxzs^ 
or,  which  is  the  same, 

which,  divided  by  ^,  becomes 

or^  which  is  the  same, 

that  is, 

1     1     1     1      p    I     T 

3       3       4      5'^ 

Whence  1=  ^  >  a«rf  therefore  5=3,  <Ac  «t»m  required. 

"Prob.  9.  To  find  the  sum  of  n  terms  of  the  above  series. 

1111  1 

Letz=:-+-+-+^^,8iC.to^. 

^^1      1      1     1     1     1      •    .     ^ 

And  z h =-;r  +  -:r4— r+-^+^  ^<^«  *o 


l^«+l""2^3^4^5"^^"'      n+1' 
Whence,  subtracting  the  third  from  the  first, 

1         1         1111.^1 

*=•7^"H"T^ 1 h  >  gfC.  to  "  ^ } 


1      n+1      2      6  '  12    20  n,n+l 

rr,,^     .   .  ^  1111  ,        ^  1 

That  w,_-=i~.+--.+--+--+,  %c.  *o  -=zr5 
«  +  i      2      o      12    20  n.n+1 


PAtTVH.  INFINITE  SiRIES.  90S 

This,  multiplied  by,  2>  becomes 

9fi       1       1       1       1        .     .      2 

=^+-Tr+— +:^+,  «fc.  to 


«+l      1   '   3      6  '10  n^fTfl' 

111  2n 

That  is,  the  sum  of  i-\ —  ■] 1 — +,  S^c,  to  n  terms  = 


3       6      10    '  n-fl 

pROB.  10.  To  find  the  sum  s  of  the  infinite  series  -r-+-r +-^ 

S     4     o 

4-,  &c. 

Let  x=—,  then  toiZ/ x+a:*+a?*-|-x*+a;*-f ,  fifC.=«; 

Substitute =(5=)x+x'+j:^+x*+a:*+,  ^c. 

1— X 


hy  actual  multiplication,  comes  out  =:x,  that  is,  :t:=z;  and  there* 
fore,  substituting  x  for  z  in  the\second  step,  it  becomes  x+a:*  -fx* 

X 

+r*-fx*ss— ^ — =»;  in  which,  by  restoring  the  value  of  x,  we 
1 — x 

quired. 

Pkob.  11.  To  find  the  sum  of  1000  terms  of  the  series  1  + 
5+9+13+ 17  +  ,  &c.    Ans.  1999000. 

Pkob.  12.  To  find  the  sum  of  20  terms  of  the  series  1+3  + 
9+27+81  +,  &c.    Jm.  174339220. 

Prob.  18.  To  find  the  smn  of  12  terms  of  the  series  4+9+ 
16+25+,  &c.     i^TW.  1562.  ; 

Prob.  14.  To  find  the  sum  of  n  terms  of  the  series  c^  +a+3i^ 

+a+2d]'+a+35)3  +  ,&c.    -rfn*.  «o»+ + 


2 


n.it-^  1.2  «— 1.3  ad'     n^^^n^+n\d^ 
^6  "*"  4  • 

Prob.  15.  To  find  the  sum  of  n  terms  of  the  series  1+3+^ 

7+15+31+,&c.    -4iM.  2"  +  »— 2  +  ». 

1       1 
Peob.  16.  Required  the  sum  of  the  infioite  series  i^^'^  "*" 

8      16^*  3 


804  ALGEBRA.  Past  VH. 

13      3 
Frob.  17.  To  find  the  sum  of  the  infinite  scries  -•  +  t'^ 

4        .  ^ 

-f— +,  &c.    An$,  2. 
lo 

Pbob.  18.  To  find  the  sum  of  — f 1-~+ --  -f ,  &c.  ad  ia- 

3     9    27    81 

finitum.    Jnt.lh 

Prob.  19.  To  find  the  sum  of  the  infinite  series  I       • 

1 .2.3    «.o.4 

pROB.  20.  To  find  the  sum  of  «  tenns  of  the  above  series. 

,      11+1.11+2—2 
Ans.  —     ■     - 

4.n+l.n+2 

1 

Prob.  21.  To  find  the  sum  of  the  infinite  series  ,  ^  ^  + 

1.2.3.4 

2.3.4.5^3.4.5.6^'  18 

Prob.  22.  To  find  the  sum  of  n  terms  of  the  above  series. 
1  1 


Am, 


^^    3.»+l.»+2.n+3 


20.  THE  INVESTIGATION  OF  LOGA- 
RITHMS. 

Let  there  be  given  &^=iN,  in  which  expression  x  is  the  loga- 
rithm of  a'3  it  is  required  to  find  the  value  of  x^  that  is^  the  loga* 
rithm  of  (a"=)  the  number  N. 

Let  a=l+*,  and  ^=sl+n;  then  foill  l  +  bY=sl+n,from 
tvhkh,  extracting  the  y*  root,  we  obtain  1  +  6)7=1 +«'y",  v 

^  *  J?  XX  b  ^       X  X  X  b^ 

(Art.  11.)  i+6l7=l+— .6+—. i._+_A-l. 2.- 

'  y       y  y       ^     y  y       y      2^ 

T:nJl^=i+JL.„+l.i_i.^+l.i_i.±.2.^+,^e. 

y       y  y       ^     y  y        y      «^ 

X 

Here,  if  y  be  assumed  indefinitely  great,  the  quantities  — , 


Part  VII.     INVESTIGATION  OF  LOGARITHMS.        *5 

— ,  may  be  considered  asszo,  since  they  will  in  that  case  be  inde* 

y 

finitely^  small  with  respect  to  the  numbers  \,  %  3,  4>  ^c. 

'^     y     ^         y  y  y 

-2,  ^c. 

These  values  being  substituted  in  the  above  series,  we  shall 

.i     ^'  X  X  b'      X  b' 

have  {lHh6)y=r+»t'=)  1  +  -.6 -^+---s~^  «fc.=  l  + 

^  y  y    2      y   3 

1  1    n?      1  ft'  X     I      '         .'  1 

y        y   2     y   3  y  y 

.  n— 4-n*-f4^n*— ,*c.     ,,       ...... 

n-^«  -|.4«3  -,  ^c.  or,  3^=^_T^,^  ,^3_^  ^^^=(^  st«6aWuti7ig 

for  n  and  b,  their  equals  JV—  1  and  a—  1) 

0-1— 4a^»+4a^'-,SfC. 

ci/to*  o/  ^Ac  iioo  toiter  fractions^  then  the  last  but  one  will  be- 
come X  {or  the  log.  o/ 1 + «)  =-j^^«— i»'  +i»'  — i«*  +*  ^c.  wAic^ 

imet^  w/^  n  i«  a  tofto/e  number,  does  not  converge,  and  therefore 
is  of  no  use;  but  we  may  obtain  by  means  of  it  a  series  which  will 
converge  sufficiently  fast  for  our  purpose,  as  follows: 


I    .      I    .      1.1 


21.  Since  log.  H-»=— .n-~n2+Y«'— j»*+yn'->*c. 
for  n  let  — n  be  substituted,  and  the  above  expression  becomes 
kg.  i«n=^.-n-^n«-^«'--«^--.«*-,  *c. 

And  if  the  lower  equation  be  subtracted  from  the  upper, 

^    ,        l+«     1 

the  remainder   is    {log.    1+n  — iog.  1— nss)  log.   YZI^^'^H- 

^  N 

he  substituted  for  n  in  this  equation,  and  it  will  become  j^^o^ 

r  I 

^  — I — ■         — h,   *c.    that  is,  log.  N—lof. 


**iyr-i^3.j?-il*    5.iv-i]» 


n 


V»  ALGXBRA.  Fait  VII. 

Whence,  by  transposition. 


2      1 


which  latter  is  a  very  convenient  series  for  finding  the  logarithm 
of  any  whole  number  N^  provided  N  be  greater  than  2^  and  the 
logarithm  of  N— 2  previously  known. 

22,  Since  a*=:N,  it  follows  from  the  nature  of  logarithms,  {see 
Vol  I.  P.  2.  Art.  18,  37.)  that  x  x  log.  a=  togf.  H;  hui  (AH.  20.) 

a 

x^log.  N:  wherefore  log.  «ssl  5  and  log.  — -^atlog.  a^log.  a^o. 

Wherefore,  {since  — =1,)  log.  1=0.   Having  therefore  the  Iqga- 

a 

rithm  of  1  given,  we  can  thence  find  the  logarithm  of  3 ;  for 

let  N=iS,  tJien  N— 2=1,  the  logarithm  of  which  is  o,  a»  we  have 

shewn ;  wherefore,  by  svhstitutii^  3  for  N  in  the  above  expression, 

we  shall  have  log.  3=]g^-"2-+32i+^5+>  8!e.^{log.  1=)  0. 

23.  Having  found  the  logarithm  of  3,  we  may  thence  find 
those  of  all  the  odd  numbers  in  succession ;  thus, 

2  T      i       I 

Let  ]\r=5i  then,  log.  5=^.---4-^-n[+r-:;T+>  S^c  +  log.S. 

M   4     3.4'     5.4* 


2    11  1 

Ut  N^7i  then,  log.  y=^.~+_+_^^-|-,«rc.-f  iogr.5, 


I*  ■  ■■ 


Let  JV=i9j  tbm,  log.  9=^—+  —+^+,  iic.+log.7- 


Let  N=n;   then,  log.  li=Z.^+-J-+-i_4..  *c.+ 
log.  9. 
24.  The  logarithm  of  the  number  2  is  thus  found. 


Los.  of  4  (by  what  has  been  shewn  above)  :=:^:rz. 1 1- 

— ^  -^      V  '  ^     M  3      3.3^ 


+-,  8fc.+log.2. 


5.3*  ^ 

But  log.  4=?og.  22=2xZo^.  2;  therefore  ^xlog,  2=-^. 


"^^  TTi  ■*■  TTb + '  *^-  +  %•  2  5  whence,  by  transposition, .  (2  X  ^. 


Because 


Faut  VII    INVESTIGATION  OP  LOGARITHMS.         207 

.  25.  Having  shewn  die  BieUiod  of  finding  the  kgaiithins  of  aU 
the  prime  numbers^  those  of  the  composite  numbeis  will  be  rea^ 
dily  obtained  by  addition  only  |  thus, 

flog.  4=^0^.2+ tog.  2. 
log.  6:=zlog.S  +  log.2. 
log.  S^log.4+log,2. 
log.  9=2og.  3  +  /og.  3. 
log.  lO=zlog.  b-^hg.  2. 
hg.  12stog.  6+tof.  2. 
Ike. 

^6.  Bat  before  we  can  apply  the  above  expressions  to  the  ac- 
tual construction  of  logarithms,  the  value  of  the  quantity  M. 
most  be  determined  5  it  is  called  the  modulus^  of  the  system,  and 
may  be  assumed  equal  to  any  number  whatever :  whence  it  is 
plain  that  (by  varying  the  value  of  AT)  innumerable  systems  of 
logarithms  may  be  formed  for  the  same  scale  of  numbers,  in 
eadk  of  which  the  magnitude  of  the  logarithm-  of  any  number 
tdli  depend  on  the  value  of  M;  moreover  ilf  depends  on  the  va« 

1   1    

loe  of  a,  (since  ikf=a— 1— —.a— ll«-f  — .a— IF— ,  &c.)  which 

•  3 

therefore  is  called  the  hose  of  the  system,  and  may  be  varied  at 
pleasure. 

If  jif=i,  then  win  log.  iv==Ar-I-4-l^-ri1*+4-^'^-^ 

«  3 

&c.  the  logarithms  of  this  system  are  denominated  Napier's  or 
hyperbolic  logarithms. 

Lei  N^~JP^\9^ I.^-Ol^-,  8fc.=p;  iken  if  M  he 
A  3 

P 
^  modidus,  we  shall  have  log.  Nsz  -^^  if  Jlfssl,   then  toill  hypu 

l^.  Nszp  ;  and  if  this  vdtue  efphe  suhstvtuied  in  the  preceding 

etiaation,  it  becomes  log,  ^= — ■'       — ,  whence  also  hyp.  log. 

N^Mx  log.  N. 
27.  Hence  hyperbolic  logarithms  are  changed  into  others, 

k  The  name  moduhu  was  lint  gi^eo  to  this  fiictor  by  Mr.  Cotes,  in  a  learned 
paper  on  the  nature  and  constaniction  of  logarithms,  printed  in  the  Philoso- 
pkieal  TramactioiUy  No.  888,  and  afterwards  in  a  tract  entitled  Logometria. 
The  modulus  is  a  fourth  proportional  to  the  fluxion  of  the  number,  the  fluxion 
of  the  logarithm,  and  the  number  itself ;  or  it  is  the  number  which  expresses  the 
sQhftaageiit  etf  the  l^gmrithmic  ob  iogiMtie  (afenra. 


WB  ALGEBRA.  J^astVH. 

whote  modoliB  is  ilf,  by  dividiiig  the  former  bj  M:  and  loga- 
rithms whose  modulus  is  M^  are  changed  into  hyperbolic  loga- 
rithms, by  multiidying  the  former  of  these  by  M. 

Lei  N=za,  then  s'mce  log.  N=^'!^' — ,  we  shall  have  iy 

M 

htm    Ijut     J\r 

subsiitutum^  log,  fl=r    ^'  ^' — ;  hut  it  has  heen  shewn  that  kg. 

aszl,  wherefore  bif  multiplication  (aM:=l  x  M:=)M=^hyp.  log. a. 

But  since  the  value  of  a  may  he  assumed  at  pleasure,  fef  a=lO  3 
substitute  this  value  for  a  in  the  above  equation,  and  M^rzhyp.  hg. 
10. 

Logarithms  derived  from  this  assumption  are  usually  called 
Briggs*s>  or  the  Common  Logarithms  9  and  to  construct  a  table 
of  them,  it  is  plain  we  must  first  find  the  hyperbolic  logarithm  of 
10,  which  has  been  shewn  to  be  the  modulus  of  that  system. 

Now  log,  lO=log,^xS=log,  2-hlog,  by  and  the  modulus  of 
the  system  of  hyperbolic  logarithms  is  unity,  or  M^l. 

Therefore,  {Art,  24.)  hyp.  log.  2=2x-j4-j^+g;p+»  *c,= 
.69314718. 


Hyp.  io^.  3=2  x—+  —  -f-^+,*c.=  1.09861228. 


Hyp.  log.  5=2x— +— 5+--+,  SiC-hlog.  3=1.6094379^1. 

d8.  Having  found  the  hyperbolic  logarithms  of  2  and  5,  we 
have  from  the  nature  of  logarithms,  hyp.  log.  I0=ihyp.  log.^ 
-^hyp.  log.  5=(.69314718+1.60943791=)2.S0258509=Af,  the 

2 
modulus  of  tfie  system  of  common  logarithms;  and  since  -rz^ 

=  .868588964,  thk  quotient  being  substituted  for  its 


2.30258509 


equal  —,  will  become  a  constant  multiplier  of  the  general  series, 
that  is,  com.  log.  2V^=  .868588964  x-r= h  _  -f +> 

^-1   3.]^=1]'  5.i^=il* 

^c.+to^.  IST— 2j  which  is  a  general  thcOTem  for  finding  the 
common  logarithms  of  all  the  prime  number  above  2; 
the  theorem  for  finding  the.  logarithm  of  the  number  2  being 


Paat  VII.    INVESTIGATION  OF  LOGARITHMS.         909 


.868588964  X  — H ;  H l  + ,  *c.  (Art.  «4.)  and  since  theloga^ 

^thms  of  the  composite  numbers  are  derived  from  those  of  the 
prime  numbers  by  addition  only^  we  are  now  in  possession  of  the 
means  of  constructing  a  complete  table  of  these  useful  numbers. 
29.  To  construct  a  table  of  common  logarithms. 

Let  A=r.86SB88964,  then  the  above  theoreni  for  finding  the 

logarithm  of^  wUl  become  --•  +  v^+"T&+*  *'^*  ^^^^^  **  ^ 

3      3.3      5.3' 

rived  the  following  practical  rule  for  finding  the  logarithm  of  the 

number  2. 

Rule  I.  Divide  the  factor  .868588964  by  3^  and  reserve  the 

quotient. 

II.  Divide  the  reserved  quotient  by  9,  and  in  like  manner 
reserve  the  quoticftit  |  divide  this  last  quotient  by  9,  and  reserve 
the  quotient ;  and  so  on,  continually  dividing  by  9>  as  long  as 
division  c^  be  made. 

III.  Set  the  reserved  quotients  in  order>  under  one  another^ 
and  divide  them  respectively  by  the  odd  numbers  1,  3^  5^  7,  9, 
&c.  placing  the  quotients  one  under  another  as  before. 

IV.  Add  the  last  mentioned  quotients  together^  and  the  sum 
will  be  the  logarithm  of  2^  as  was  required. 

Examples. — 1.  To  find  the  logarithm  of  the  number  2. 

Operation. 

1 )  .289529654(.289529654 

3)  32169962(  10723321 

5)  3574440(  714888 

7)  3971 60(  56737 

9)  44129(  4903 

11)  4903(  446 

13)  546(  42 

15)  60(  4 


3)  .868588964 
9). 289529654 
9)  32169962 
9)  3574440 
9)  397160 
9)  44129 
9)  4903 
9)  545 
9)  60 
6 


Ans.  log,  o/2=.30l029D95 

ExplanatioM. 

The  firit  (or  left  hand)  oolomo  cooUiiu  the  divitors  3, 9)  9,  &o.  the  scoond 
contains  the  dividend^  and  successive  quotients,  which  arise  by  dividing  each 
nnmber  in  it  by  the  opposite  divisor ;  the  third  contains  the  divisors,  Ij  3, 5, 7f 
Sec.  In  the  fourth  column  the  reserved  quotients  above  mentioned  are  arranged 
under  one  another  in  order,  each  opposite  its  respective  divisor.  The  fifth  con- 
siitft  of  the  quotients  arising  from  the  division  of  each  of  the  reserved  qnotieou 
by  its  proper  divisor ;  the  sum  of  these  latter,  subjoined  at  the  bottom^ 
is  the  logarithm  required. 

Note.  In  some  of  the  above  divisions,  where  the  reawinder  is  very  large,  the 

VOL,  !!•  P 


no 


AL6EBBA. 


Fabt  VII. 


faft  ifBotirat  figure  is  afnmied  gvcaCer  bj  out  tkao  it  oOf hi  ftfictly  to  be  ; 
tim,  w  it  iervrs  «d1/  tio  aake  19  for  other  nnaU  remaiaden  lett,  will  be  fio- 
dnctire  of  00  error  of  conseqaence  in  tbe  icsalt. 

2.  To  find  the  common  logarithm  of  the  number  3. 

Here,  by  assuming  A  as  before^  ike  general  theorem  for  find- 
tag  the  common  logarithms  ofaU  tmmbers  greater  than  %  will  he- 

come  -- — ^+  ■ -♦-■  4-.  8fC.+log»  JV— 2.  In  tltis ease  , 

JVs=3,  V  2^- 1^2,  ^—T^ ^-,2  X  4,  iV~T)*=:2  X  4  X  4,  TT^^ssi 
2x4x4x4,  i?— D»=2x4x4x4x4,  *c.  SiC  whence  it  '» 
plain,  that  the  first  column  of  divisors  ^ust  be  2,  4^  4,  4,  4,  &c. 
and  the  other  column  of  divisors,  in  this  and  eveiy  other  case, 
.will  be  tlie  odd  numbers,  1,  3,  5,  7,  &c.  and  proceeding  as  be- 
fore^ the  work  will  stand  thus : 


2).8685SS964 

l).434294482(.4342944a2 

4)  .434294482 

3).108573620{ 

36191207 

4).  108573620 

5) 

27143405( 

5428681 

4)  27143405 

7) 

67S585I( 

969407 

4)   67a5851 

9) 

1696463( 

188496 

4)   1696463 

H) 

4241 16( 

38556 

4)    424116 

13) 

106029( 

8156 

4)        106029 

15) 

26507( 

1767 

4)    26507 

ir) 

6627( 

389 

4)     6627 

19) 

1657( 

67 

4)     1657 

31) 

414( 

19 

4)      414 

23) 

103  ( 

4 

4)      103 

25) 

25( 

1 

25 

Sum  .477121252 

To  which  add  (log.  N'^2:si)log.  l=.OO0OO0Qao 
The  sum  is  tfte  log,  rf  3=. 477121252 

In  a  similar  manner  the  logarithms  of  the  other  prime  num- 
bers are  obtained,  and  by  means  of  them  those  of  the  compo- 
site numbers,  as  has  been  already  shewn. 

3.  To  find  the  logarithm  of  5.    Ans.  .698970004. 

4.  To  find  the  logarithm  of  7-     ^^fns,  .845098040. 

5.  To  find  the  logarithm  of  4.    jhu.  .602059991. 

6.  To  find  the  logarithms  of  8,  9,  10,  11,  12. 


PART  VIIL 


GEOMETRY. 


HISTORICAL  INTRODUCTION. 


fjrEOMETRY '  is  the  science  of  magnitude,  or  local  ex- 
tension ;  it  teaches  and  demonstrates  the  properties  of  lines, 
surfaces^  solids,  ratios,  and  proportions,  in  a  general  manner, 
and  with  the  most  unexceptionable  strictness  and  preeision. 
Geometry,  or  measuring,  must  have  been  practised  as  an  art 
at  the  commencement  of  society,  or  shortly  after,  when  men 
began  to  build,  and  to  mark  out  the  limits  of  their  respective 
territories.  That  thb  art  had  reached  a  considerables  degree 
of  perfection  at  the  time  of  the  general  deluge,  can  hardly 
be  doubted  from  that  stupendous  nonumenl  oi  human  folly, 
the  Tower  of  Babel,  which  was  begun  about  115  years  after* 
that  period :  Herodotus  informs  us,  that  this  vast  building 
had  a  squase  base^  each  side  of  which  W9s  a  furlong  in  length ; 
Strabo  affirms  that  its  height  was  likewise  a  furlong;  and 
Glycas  says^  that  the  constant  labour  of  fqr^y  years  was  con* 
sumed  in  erectiog  this  unfiaished  and  useless  fabric.  The 
Pyramids,'  Obelisks,  Temples,  and  other  public  edifices  with 
which  Egypt  abounded,  existed  prior  to  any  authentic  date  of 
profane  history  :  many  of  these  had  been  in  ruins  probably 

*  The  name  Gemtketry  is  derived  from  yn  the  earth,  and  fmr^  to  measare. 
The  iuYention  of  measaring  if  ascribed  to  the  JEgyptialis  by  UerodotHs, 
Diodomsy  Strabo,,  and  Proclas;  to  Mercury  by  others  among  the  ancients.^ 
and  to  the  Hebrews  by  Jo^ephns. 


212  INTRODUCTION.  PartVBI 

for  ages  before  the  earliest  historians  lived,  who  speakcrf  thek 
magnificence  as  surpassing  that  of  the  most  splendid  struc 
tures  in  Greece ''.  Can  it  be  supposed  possible,  that  buildings, 
whose  magnificent  remains  alone  were  sufficient  to  excite  the 
wonder  and  admiration  of  a  learned  and  polished  nation  like 
the  Greeks,  could  have  been  raised  without  the  assistance  of 
Geometry  } 

The  priest$  of  Memphis  informed  Herodotus,  that  their 
king  Sesostris  divided  the  lands  bordering  on  the  Nile  among 
his  subjects,  requiring  that  the  possessor  should  pay  an  an- 
nual tribute  proportionate  to  the  dimensions  of  the  land  he 
occupied;  and  if  the  overflowing  of  that  river  occasioned 
any  diminution,  the  king,  on  being  applied  to,  caused  the 
land  to  be  measured,  and  claimed  tribute  in  proportion  only 
to  what  remained.  "  I  believe,"  adds  Herodotus,  "  that  here 
Geometry  took  its  birib,  and  hence  it  was  transmitted  to  the 
Greeks."  On  the  strength  of  this  conjecture  we  frequently 
hear  it  affirmed,  that  ^^  Geometry  derived  its  origin  from  the 

annual  inundation  of  the  Nile  ^  but  it  is  plain  that  this  as- 

• 

,  ^  Sevvrai  inttaftces  of  lbi»  lamf  be  given.'  The  tomb  of  OsymandyM,  oife 
of  their  kings,  is  said  to  have  been  dnconunpnJy  nuigni.ficc«t ;  it  was  sumoad* 
ed  by  a  circle  of  gold,  365  cubits  in  circumference,  divided  into  as  many  eq^oal 
parts,  which  shewed  the  rising  and  setting  of  the  sun  for  every  day  in  the  year : 
fhift  circle  was  carried  away  by  Caabyses,  kis^  of  Bmia,  when  he  eoo^eMd 
Egypt,  A.X.  525.  Gognei  Orig.  des  Loi»,  ^-c.  T.  2.  /tv.  S.  MoUm'*  Anc  HiH^ 
vol,  /.  p.  3.  The  fEunous  Labyrinth  contained  12  palaces  surrounded,  by 
1500  rooms,  adorned  with  innumerabk  ornaments  and  statues  of  the  finest 
parble,  jind  most  exquisite  woskamiitbip ;.  tiierc  were  besides,  1500  tsfater> 
nuieous  apartments,  which  Herodotus  (who  surveyed  this  nobla  and  beautifid 
structure)  was  not  permitted  to  see,  because  the  sepulchres  of  their  king?  were 
there,  and  likewise  the  sacred  crocodiles  and  other  annuals,  which  a  nation  so 
wise  iu  other  reafMcts  worshipped  as  gods :  <'  Who**  (says  the  learned  and 
pious  Rollin)  <<  can  speak  this  without  confusion,  and  without  deploring  the 
blindness  of  man !"  Tbe  magnificent  city  of  Thebes,  with  its  numerous  and 
splendid  palaces  and  other  public  edifices,  which  was  ruined  by  Cambyses,  is 
the  last  instance  to  be  mentioned,  although  many  more  might  be  added.  It 
extended  above  23  miles,  had  an  hundred  gates;  and  could  send  oat  at  every 
gate  20,000  fighting  men,  and  SOO  chariots. 


PartVIIL  geometry,  2l3 

^ertion  deseives  little  credit ;  for  as  a  science,  Oeometry  never 
existed  in  Egypt  before  the  time  of  Alexander,  and  as  an  art 
it  must  have  been  known  there  (as  we  have  shewn  above)  long 
before  the  age  of  Sesostris;  for  according  to  tlie  very  pro- 
bable conclusions  of  our  most  accurate  and  best  informed 
chronologers,  Sesostris  was  the  Egyptian  king,  who  invaded 
Jerusalem,  A.  C.  971 ;  on  which  occasion  he  is  mentioned 
in  a  King^  ch«  xiv.  v.  25,  under  the  name  of  Shishak.:  now 
we  have  direct  proofs,  on  the  most  unquestionable  autliority, 
that  measuring  was  understood  by  the  Jews  who  came  from 
Egypt,  many  centuries  earlier  than  that  date;  see  Genesis, 
ch.  vl.  V.  15,  16.  Exodus,  ch.  xxv.  xxvi.  xxvii.  and  various 
other  parts  of  the  Mosaic 'history. 

Not  to  take  up  the  reader's  time  with  conjectures  about  the 
origin  of  Geometry,  which  at  best  must  be  vague  and  un-^ 
certltin,  we  hasten  to  inform  him,  that  the  Greeks,  to  whose 
taste  and  industry  almost  every  science  stands  indebted,  were 
the  first  people  who  collected  the  scattered  principles  and 
practices  of  Geometry,  which  .they  found  in  JEgypt  and  other 
easte^  countries,  and  moulded  them  into  a  form  and  con- 
sistence. Until  it  passed  through  their  masterly  hands. 
Geometry  could  not  by  any  accommodation  of  language  be 
properly  termed  a  science;  but  by  their  consummate  skill 
and  indefatigable  labours,  a  few  scanty  and  detached  princi- 
ples and  rules,  heretofore  chiefly  applied  to  the  measuring 
jof  land,  (as  the  name  Geometry  imports,)  at  length  grew 
into  a;Qd  became  the  most  complete  and  elegai^t  science  in 
the  .WiOii4*  We  .^dore  th^t  benign  Providence,  who  has 
repeatedly  condescended  to  make  even  wicked  and  idolatrous 
nations  useful  instruments  for  promoting  the  execution  of 
his  merciful  designs  to  man. 

Thales  ^  ranks  among  the  earliest  of  the  Grecian  philoso- 

«  Tbalesy  the  &ther  of  the  Greek  philotopfayy  and  the  first  of  the  seTen  wim 
;#WB  ^  GNpeece, was  boxfi  at  Milctum,  A. C.  €40 ;  alteT  acquiriof  the  besrt  leai^ 

p3 


iU  INTRODUCTION.  PabtVUL 

fhttSy  whe  travelled  into  foreign  comitries  m  quest  of  that 
knowledge  which  their  own  could  not  supply,  A.  C.  640*  He 
became  not  only  an  able  geometer,  but  was  likewise  very 
skilful  in  every  branch  of  Mathematics  and  Physics,  as  these 
Sciences  then  stood.  We  are  unacquainted  with  the  parti* 
ieulars  of  his  acquirements  and  discoveries  in  Geometry,  but 
he  is  mentioned  as  bnng  the  first  who  measured  the  height 
of  the  pyramids  at  Memphis,  by  means  of .  their*  shadows, 
and  who  applied  the  circumference  of  a  circle  to  the  Bieasur« 
ing  of  angles. 

Pythagoras'^ was-another  eminent  Grecian  philosopher,  who 

ing  his  own  country  aiforded,  be  trarelled  411  the  £ast,  aod  returned  with  a 
mind  enriched  with  the  knowledge  of  Geometry,  Astronomy,  Natural  Philoso- 
phy, &c.  which  he  improved  by  his  own  skill  and  application.  He  divided  the 
celestial  sphere  into  five  soncs ;  be  observed  the  apparent  diametcT  of  the  snn* 
making  it  half  a  degree ;  he  understood  the  cause  and  course  of  eclipsci »  cal- 
culated them  with  accuracy,  and  divided  the  year  into  365  days.  He  disliked 
taionarcby,  because  he  considered  it  as  little  better  than  tyranny,  to  every  spe- 
cies of  which  he  was  an  avowed  enemy.  One  evening  as  he  walked  out  to 
contemplate  the  stars,  be  bad  the  misfortune  to  fall  into  a  ditch,  on  which  an 
old  woman,  who  saw  him,  exclaimed,  *<  How  can  you  possibly  know  what  is 
doing  in  the  heavens,  when  yuu  cannot  see  what  is  even  at  your  feet  !'*  He  died 
at  the  Olympic  Games,  at  the  age  of  upwards  of  90  years^  Thales  was  the 
founder  of  the  Ionian  tect,  and  had  for  his  scholars  some  of  the  most  eminent 
philosophers  of  antiquity,  among  whom  are  mentioned  Anaxlroander,  Anaxi- 
menes,  and  Pythagoras.  It  is  uncertain  whether  he  left  any  writings  ;  Augus- 
tine mentions  some  books  on  Natural  Philosophy  ascribed  to  bim  ;  Simplicins, 
some  on  Nautic  Astrology ;  Laertius,  t^vo  treatises  on  the  Tropics  and  Equi- 
noxes j  and  Suidas,  a  work  on  Meteors,  written  in  verse. 

'  P^tbagoFBB,a  celebrated  philosopher  of  Samos.  He  was  early  instructed  i» 
music,  poetry,  astronomy,  and  gymnastic  exercise,  with  whatever  else  might 
tend  to  enlighten  his  mind,  and  invigorate  his  body.  At  the  age  of  eighteen 
he  resolved  to  travel  for  that  instruction,  which  the  ablest  philosophers  oi 
Samos  were  incompetent  to  supply :  be  spent  25  years  in  Egypt,  *here  havtug 
ingratiated  himself  with  the  priests,  he  became  acquainted  with  all  the  learn- 
ing of  that  country ;  having  travelled  through  Chaldea,  and  visited  Babylon, 
he  returned,  passing  through  Crete,  Sparta,  and  Peloponnesus,  from  whence  he 
crossed  over  into  Italy,  and  finally  fixed  hia  residence  at  Crotona.  Here  be  opened 
a  school,  which,  by  the  fame  of  bis  mental  and  personal  accomplishments,  was 
aoon  crowded  with  popils,  many  of  whom  came  from  distant  parts  of  Greece 
asd  Ualy*    Hit  icbcdJurs^  wbo  wwe  called  (be  Jtaliim  «crr«  were  fonned  bj 


FAvrVah  GEOMETRY.  815 

was  CBdowed  with  an  equal  tUrat  for  uaefol  kaowledge,  and 
employed  the  same  means  to  gmtify  it,  A.  C.  590*  The  32nd 
and  47Ui  paopeMtions  of  tbe  fiist  book  of  EocUd's  Elements 
are  ascribed  to  him ;  from  the  latter  of  which  be  was  led  to 
determine,  that  the  diagonal  of  a  square  is  incomioensurable 
to  its  side :  every  person  moderately  acquainted  with  Geo- 
metry will  adroowledge,  thai  the  useful  purposes  to  which 
these  important  proportions  maybe  applied  are  innumerable^ 
About  this  time,  or  shortly  after,  die  following  celebrated 

tlie  fwlct  Mfi/ud  fnm  tlie  B^^fplian  prieits  ;  •moog  other  aosteriticty  he  en- 
joined them  a  five  years  tUeiioe,  during  which  they  were  only  to  hear  ;  after 
this  they  were  allowed  to  propose  doabts,  ask  questions,  &c.  in  which  they 
w«re  permitted  to  say,  not  a  lUtk  in  mam^  w*nU,  but  wuiek  in  «f  few  wardt 
MpcsMk.  Qaery.  Might  not  the  prattling,  self-sufficient  young  gentlemen 
in  some  of  our  academies,  be  admirably  benefited  by  an  institution  of  this 
kind? 

Besides  the  propositioos  mentioned  above,  Pythagoras  wa9  the  author  of  the 
following,  vie.  only  three  rectilineal  figures  can  fill  up  the  space  about  a  point ; 
namely,  the  equilateral  triangle,  the  square,  and  tbe  hexagon.  He  invented  the 
multiplication  table ;  the  obliquity  of  the  ecliptic  was  first  discovered  by  him  ; 
he  called  the  world  tutfuty  and  asserted  that  it  was  made  in  musical  proportion  ; 
the  ann  he  called  tke  fiertf  globe  of  uttiiy,  and  maintained  that  the  seven 
planets  move  round  him  in  an  harmonious  motion  at  distances  corresponding 
to  the  musical  divisions  or  intervals  of  tbe  monocbord  :  he  taught  the  true 
aolar  system,  which  had  been  asserted  by  Phildans  of  Croiona,  hut  being  foiw 
gotten  and  lost  during  many  ages  after,  was  at  length  revived  by  Copernicus, 
and  demonstrated  by  the  illustrious  Newton. 

The  modesty  of  Pythagoras  was  not  less  conspicuous  than  his  attainments ; 
on  being  addressed  at  a  public  assembly  with  the  splendid  appellation  of  r«f  •(« 
wise  ffMtis,  he  disclaimed  tbe  title,  and  requested  that  they  would  rather  call 
him  ^tXMra^$f,  a  lover  of  wisdom  ;  a  circumstance  which  first  gave  rise  to  the 
terms  phUotophy  and  philosopher. 

Some  authors  affirm,  that  Pythagoras  offered  100  oxen  as  a  sacrifice  to  Apollo, 
in  gratitude  for  the  discovery  which  that  god  enabled  him  to  make  of  the  47th 
proposition  of  tbe  first  book  of  Euclid  ;  this  is  extremely  improbable,  as  he  was 
a  firm  believer  in  tbe  doctrine  of  the  transmigration  of  souls,  which  forbade 
taking  away  the  life  of  any  animal :  nor  is  it  much  more  credible  that  be  sub- 
ptitnted  little  oxen  made  of  flour,  clay,  or  wax ;  no,  this  would  doubtless  have 
been  considered  as  an  intolerable  aiTront,  which  the  meanest  heathen  god  in 
tbe  catalogue  would  disdain  to  put  up  with.  The  whole  story  is  perhaps  nothing 
better  than  a  fiction,  an  ingenious  sample  of  ancient  priest-craft. 

p4 


216  INTROireCTlON-  Part  VHI. 

problems  tMfgah  to  be  a^taled  among  the  learned ;  tmien^ff 
the  rectification  and  quadcatnre  of  the  chrde,  the  trisectioa 
of  an  angle,  the  findmg  two  mean  proportiqnak,  and  the 
duplication  of  the  tube  ^.  Some  of  the  ancients  mAveA  these 
problems,  but  their  solutions  were  either  meehanical,  hf 
approximatum,  or*  depended  on  the  properties  of  certain 
curres  njot  considered  as  geonwtrical;  consequendy  their 
mcihods  did  not  fulfil  the  necessary  condition,  requiring 
that  these  problems,  which  without  d^)ute  are  elementary, 
should  be  solved  by  pure  elementary  Geometry.  Some  of 
the  most  eminent  geometers  of  both  ancient  md  modern 
times  have  engaged  in  this  arduous  undertaking,  and  not  one 
among  them  all  has  succeeded :  no  solution  of  either  of 
these  famous  problems,  strictly  and  purely  geometrical,  has 
ever  yet  appeared.  What  a  useful  lesson  does  this  address 
to  the  noisy  advocates  for  the  omnipotency  of  reason  !  they 
may  hence  learn,  that  the  reasoning  powers  of  the  human 
mind,  although  unquestionably  great  and  excellent,  have  their 
limits,  narrower  perhaps  than  these  philosophers  have  been 
accustomed  or  are  willing  to  allow ;  and  consequently  that 
reason,  although  the  most  noble,  and  distinguishing  boon  that 
Heaven  has  ever  conferred  on  man,  was  not  given  him  to  be 
deified,   L^t  them  contemplate  with  becoming  attention  the 

*  The  rectification  of  a'circle  is  the  finding  a  right  line  equal  to  its  circom* 
ference,  and  its  quadrature  is  the  finding  a  square  equal  to  its  area.  !%€  find- 
ing two  mean  proportionals  consists  in  this ;  having  two  right  lines  given, 
thence  to  find  two  others,  such,  that  the  four  lines  will  be  continued  proper* 
tionals.  Tbe  duplication  of  the  cube  consists  in  finding  the  side  of  another  cubc^ 
which  cube  sl^all  be  in  magnitude  just  double  the  former :  the  two  latter  pro-  ' 
blems  depend  oq  each  other,  ^nd  form  but  one,  known  by  the  name  of  the  De^ 
liah  problem^  which  \\.  obtained  from  the  following  circumstance :  a  plague 
threatening  to  depopulate  /Vthetis,  the  oracle  of  Apollo  at  Delpho9  was  consulted^ 
and  returned  for  answer,  *'  Double  the  altar  and  the  plague  shall  cease."  The 
geopieters  immediately  set  to  work  to  find  the  side  of  a  cube  double  d  this 
altar>  vj^hich  was  likewise  cubical ;  but  after  much  labour  they  found  to  their 
great  iportification,  that  the  solution  could  not  be  effected  b^  auy  of  the  method^ 
then  ii>  use, 


pAar  VHI.  GEOMETltT.  ^17 

numerous  iiwannountable  oUtacIes  which  oppose  tbemselves 
at  the  very  threshold  of  almost  every  department  of  know- 
le4gey  and  candour  wiU  oblige  them  to  confess  that  the  men- 
tal powers  are  still  very  imperfect,  and  consequemly  that 
saperior  attainments  in  any  science  ought  ahvays  to  he 
accompanied  with  modesty,  diffidence,  and  humility. 

Of  those  who  engaged  with  ardour  in  theabove-^mentioned 
tlifiicult  researches,  Anaxagoras  of  Clazomene  wa^  one  of  the 
eaicliest,  A.  C.  500;  he  was  an  excellent  geometer,  and  com- 
posed a  treatise  expressly  on  the  quadrature  of  the  circle, 
which,  according  to  Plutarch^  was  written  during  hb  im« 
prisonment  at  Athens.  (Enopidus  of  Chios  and  Zenodorus 
flourished  about  A.  C.  480;  to  the  former  are  ascribed  the 
9tb,  11th,  1 2th,  and  23d  propositions  of  Euclid's  first  book 
of  Elements.  Zenodorus  proved,  that  figures  of  equal  areas 
are  not  necessarily  contained  by  equal  bpundacies,  as  some 
bad  asserted;  one  only  of  his  treatises  has  escaped  the 
ravages  f^  time;  it  has  been  preserved  by  Theon  in  his 
Commentaries,  and  is  the  earliest  piece  on  Geometry  at 
present  extant. 

The  school  of  Pythagoras  produced  a  great  number  of 
learned  geometricians :  with  the  names  of  some  of  them  we 
are  acquainted,  but  scarcely  any  thing  is  known  of  their 
discoveries  and  improvements;  as  most  of  their  writings, 
through  the  constant  .mutability  of  human  afiairs,  during  a 
long  lapse  of  ages,  have  been  destroyed  or  lost.  One  famous 
discovery  in  Geometry,  however,  remains  to  be  noticed  as 
originating  among  the  disciples  of  Pythagoras,  namely,  the 
ingenious  theory  of  the  five  regular  bodies  ^ 

f  Tbey  are  Vikewise  denominated  the  Platonic  bodies,  ^d  are  a«  follow. 
1.  The  THraidnfi,  or  regular  triangular  pyramid,  contained  by  four  equila- 
teral and  equal  triangular  faces.  2.  The  Hexaedron,  or  cube,  contained  by  six 
equal  square  faces.  3.  The  OetaSdron,  contained  by  eight  eqaal  equilateral 
triangular  facet.    4.  The  Dodecmidron,  contained  by  twelve  equal  and  regular 


1 


218  INTIOIIUCTION.  P4«T  Vin. 

Hffpocntes  '  of  Chios,  A.  C.  450.  distiQgiusbed  himaelf 
«8  the  ficst  who  squared  a  curvilineal  space  ^;  in  hb  attempts 
4o  aol^e  tfae .  oelebrated  problein  of  doubliiig  the  cuhe^  he 
discovered^  that  if  two  mean  proportioiials  between  the  side 
of  a  given  cube  and  double  tliat  side  be  found,  the.  least  of 
these  means  will  be  the  side  of  the  required  cube ;  the  same 
IS  demonstrated  in  Euclid  33.  1 1.  but  it  w^s  soon. discovered 
that  tlte  difficulty.  Instead  of  being  removed,  was  only  a  lit* 
tie  disguised;  for  the  two  mean  proportionals  themselves 
could  not  be  found  by  any  pure  geome^ical  process,  and  the 
problem  continues,  to  the  present  hour,  to  bid  defiance  to  the 
mnited  skill  and  labours  of  the  ablest  geometricians. 

Geometry  was  cultivated  with  the  greatest  attention  by 
Plato  * ;  his  school  was  a  school  of  geometers,  as  appears  from 

lientagonal  £ices ;  and  5«  The  IcosaSdren,  eontained  by  twentf  equal  and 
equilateral  triangular  faces,  These  iKre,  t<^geUier  with  the  i^Aov^  wludi  aiax 
be  considered  as  a  sixth,  are  all  the  regular  solids  that  can  possibly  be  made. 
The  following  are  called  mixed  solids,  each  being  compounded  of  two  of  tfte 
former:  viz,  1.  The  JSsoctoSdrott^  contained  by  fourteen  planes,  Tix.  six  equal 
•quarety  and  eight  equal  and  equilateral  triangles.  3.  The  leowUdecmSdnih 
contained  by  thirty-two  planes,  viz.  twelve  equal  and  regular  pentagons,  and 
twenty  equal  and  equilateral  triangles.  See  a  treatise  on  the  Regular  and 
Mixed  Solids,  by  FInssas,  subjoined  to  Bamiu^M  EueUd,  Ltmdu^  1751.  T%e 
five  flCfular  solids  may  be  constructed  with  pasteboard,  the  method  of  dohiy 
which  was  first  shewn  by  Albert  Darer,  an  ingenious  magistrate  of  Nuremberg, 
in  his  Imtitutumes  Geomefrictt,  Paris,  1533.  See  also  Hawney's  C&mpleie 
Meamrer,  9di  Ed.  p.  268.  Bonnycastle's  IfUroducHtm  to  Mauwtiiamg  4re. 
4th  Ed.  p.  181.  &c.'  Hutton's  Maik»  IHciumary,  vol.  I.  p.  215,  and  vol.  U. 
p.  355.  &c. 

r  I  am  equally  uncertain  whether  there  be  any  further  particulars  of  this 
geometrician  in  existence,  and  whether  the  above  date  be  correct:  he  must  not 
ht  confounded  with  a  learned  physician  of  the  same  name,  in  the  Island  of  Cos, 
who  was  much  esteemed  for  skill  and  fidelity  in  his  profession. 

*  This  curve  is  the  lunula :  if  three  semicircles  be  described  on  the  three 
sides  of  a  right  angled  triangle,  their  intersections  will  form  two  lunar  spaces, 
the  sum  of  which  is  equal  to  the  area  of  Che  triangle ;  the  proof  of  which  de- 
pends on  Euclid  47*  1  >  31 !  6,  and  2. 12.  Proclus  ascribes  the  lunula  to  (Eno- 
pidas. 

*  The  original  name  of  this  eminent  philosopher  was  Aristocles,  and  he 
)feceived  that  of  Plato  from  the  broadness  of  his  shoulders  j  be  was  bora  at 


paiitvui.         :^  GEcniBTinr. .  21s 

the  fcllowifig  mscriptioD  which  he  caused  to^be-fihoed  oter 
the  door;  let  no  ohb  pssschk  to  Bamut  BBftKr  WBo  it 
UNSKiixBD  IN- OBOMETRT.  Likc  hk  {Nnedceessovs,  Plato 
attempted  the  duplicatiaii  of  the  cube ;  for  this  purpose  he 
contrived  an  4B$tniineDt>  comirting  of  straight  roles,  moving 
in  grooves  perpendicularijr  to  each  other,  by  means  of  which 
he  was  enabled  to  find  two  mean  proportionab :  but  the  pro* 

Mhem  about  430  |ttMB  bclbn  Cbrift»  wd  «daca|cd  with  Um  gnmtert  atteatiMi 

|K>th  to  his  QU^vtol  and  corporeal  improTemcnts ;  having  in  his  early  years  ac- 
quired considerable  skill  in  music,  painting,  poetry,  philosophy,  gymnastic 
ezer^tMy  Sec.  he  at  SO  jcarf  old  becaoie  a  disciple  of  SocntM,  who  stilcd  him 
tJke  Swtm  1/  the  Academy,    Plato»  on  the  de^th  of  his  beloved  master,  retired 
to  Megara,  where  he  was  kindly  entertained  by  Euclid  the  philosopher :  from 
thence  he  passed  over  into  Italy,  where  he  perfected  himself  in  natural  philo- 
sophy oB^er  Arehytas  and  Philolaus ;  from  Italy  he  went  to  Cyrene,  where  ho 
received  kistmctions  in  geometry  from  Theodoras :  he  afterwards  travelled  into 
Egypt,  where  he  acquired  arithmetic,  astronomy,  and,  as  it  is  supposed,  an  ac- 
quaintance with  the  writings  of  Moses  ;  after  visiting  Persia,  he  returned  to  A- 
tbons*  where  he  opened  a  school,  and  taught  pbilotopby  in  the  Academia,  whonco 
his  disciples  were  called  Academic*,    Plato  afterwards  made  several  excursions 
abroad,  in  one  of  which  being  at  Syracuse,  he  had  the  misfortune  to  displease 
Dionysius,  and  uarrowly  escaped  with  his  life.   The  tyrant,  however,  delivered 
him  into  the  hands  of  an  envoy  from  liacedemon,  which  then  was  at  war  with 
Atben$,  a^d  he  was  sold  for  a  slave  to  a  Cyrenian  merchant,  who  immediately 
liberated  and  sent  him  to  Athens.  The  ancients  thought  more  highly  of  Plato 
than  of  all  their  philosophers,  calling  him  the  divine  Plato ;  the  mott  wise ; 
Oemogtsaereds  the  Hmner  ^  phUoBt^hersy  Hfc,  The  orator  Cicero  was  so  en- 
thasiastic  in  his  praise,  that  he  one  day  exclaimed,  <'  err  ate  tnehercule  malo  cum 
Platone,  quam  cum  istis  vera  sentire"    The  Platonic  philosophy  appears  to  be 
founded  chiefly  on  the  Mosaic  account  of  the  creation,  &c.  hence,  in  the  early 
9gcs  of  the  .church,  Platonism  and  Christiainity  were  incorporated  and  blended 
together  by  some  of  the  fathers  of  the  Eastern  church ;  but  this  union  is  severely 
and  justly  censured  by  Gisborne,  Milner,  and  others,  as  extremely  detrimental 
to  the  genuine  spirit  of  Christianity.  After  the  death  of  Plato,  which  happened 
A.  C.  348,  two  of  his  disciples,  Xenocrates  and  Aristotle,  succeeded  him :  the 
former  taught  -in  tlie  Academy,  and  his  disciples  were  called  Academics  ;  tlie 
latter  taught  in  the  Lycseam,  and  his  scholars  obtained  the  name  of  Peripntetic*, 
from  the  circumstance  of  their  receiving  their  instructions,  not  sitting,  as  is 
usual,  but  waUung.     The  works  of  PUto  are  numerous :  they  are  all,  except 
twelve  letters,  written  in  the  form  of  dialogue ;  the  best  editions  are  those 
nf  Lyons,  1588.  Frankfort,>/.  1602.  and  Deuxpontp^  12  vol.  8to,  17 1«. 


920  iNTtOBUenON.  Part  VIIL 

MSB  was  meekankalf  and  oonsequently  ccmld  not  be  admitted 
as  a  geomUrical  sdntiQii  of  the  probltiD. 

The  circle  was  the  only  curve  ifitberto  admitted  into 
Geometry,  but  Plato  introduced  into  that  science  the  theory 
of  the  conic  sections,  or  those  corves  which  are  formed  by  a 
plane  cutting  a  cone  in  various  directions.  The  numerotn 
properties  of  these  celebrated  curves,  and  their  usefulness  in 
Geometry,  soon  became  apparent,  and  excited  the  attention 
of  mathematicians,  who  considered  this  branch  of  Geometry 
of  a  distinct  and  more  exalted  nature  than  that  which  treated 
ei  the  circle  and  rectilineal  figures  only ;  and  hence  it  obtained 
the  name  of  the  higher  or  sublime  Geometry.  By  means 
of  the  properties  of  these  curves,  Archytas.  of  Tarentum  *, 
the  master  of  Plato,  taught  the  method  of  finding  two  mean 
proportionals,  and  thence  the  duplication  of  the  cube,  A.  C. 
400.  Menechmus  accomplished  the  same  thing  about  that 
period,  or  shortly  after :  they  both  effected  the  solution  by 
means  of  the  intersection  of  two  conic  sections ;  a  circum- 
stance which  merits  particular  notice,  as  being  the  origin  of 
the  celebrated  theory  of  geometrical  locif  of  which  so  many 
important  applications  have  been  made  by  both  ancient  and 
modern  geometrieians.  Were  it  possible  to  describe  the  conic 
sections  by  one  simple  continued  motkHi,  like  the  circle,  the 
above  solutions  would  possess  all  the  advantages  of  geometri* 
cal  construction,  according  to  the  sense  implied  to  the  term 
by  the  ancients;  but  failing  in  that  particular,  they  do  not 
fulfil  the  necessary  condition. 

The  great  problems  we  have  so  frequently  mentioned, 

^  Archytas  is  said  to  be  tbe  inventor  of  the  crane  and  screw ;  he  contrived 
also  a  wooden  pigeon,  which  could  fly :  the  ten  categories  of  Aristotle  are  a* 
scribed  to  him ;  as  are  also  several  works,  but  none  of  them  have  docendcd  to 
us.  He  was  a  wise  legislator,  and  a  skilful  and  valiant  general,  having  o«bi- 
manded  the  army  seven  times  without  having  been  once  defeated.  He  WR> 
at  last  shipwrecked  and  drowned  in  the  Adriatic  Sea. 


Part  VIII.  GEOMSIVr.  221 

aMioogh  now  given  up  as  impoiriUe  to  be  ilolved  by  the 
proposed  method,  were  stuped  by  the  aneteots  with  iBoenant 
ardour;  and  the  researdiea  to  which  speculations  of  this 
kind  gave  birth^  proved  a  fruitful  source  of  discoverMs  in 
Geometry. 

The  numerous  and  extensive  applications  of  Greometry  to 
other  branches  <rf  knowledge,  espedally  to  Astronomy,  made 
a  systematic  arrangement  of  its  principles  and  conclusions, 
according  to  their  logical  connexion  and  dependance,  indis- 
pensable.   Of  those  who  undertook  to  compos  Elements  of 
Geometry,  Hippocrates,  Eudoxus,  Leon,  Thaetetus,  Theu- 
dias,  and  Hermottnius,  were  the  chief,  and  the  usefulness  of 
their  labours  in  this  respect  was  apparent ;  but  their  treatises, 
of  which  scarcely  any  thing  is  known,  were  all  super- 
seded by  the  Elements  of  Euclid  ^,  which  have  maintain- 
ed their  supericvity  ov»  other  systems  of  the  kind  through 
every  succeediDg  age  to  the  present,  and  still  hold  their  rank 
as  the  only  classical  standard  of  elementary  Geometry.   Eu- 
did^s  Ekments,  as  we  now  have  them,  are  comprised  in  fif- , 
teeo  books,  and  the  subjects  they  treat  of  may  be  arranged 
in  three  divi«<ms;  of  which  the  first  includes  the  theory  of 
superficies,  the  second  that  of  numbers,  and  the  third  that 
of  solids :  the  first  four  books  explain  and  demonstrate  the 
properties  of  lines,  angles,  and  planes ;  the  fifth  treats  in  a 
general  manner  of  the  ratios  and  proportions  of  magnitudes ; 

1  Endid  was  one  of  the  mott  cclebratfd  ipattieiiMtiGiaiisof  tlie  Ale«uulriaA 
■chool ;  be  was  bom  at  Alexandria,  and  taiight  with  great  applause,  A.  C.  280. 
He  wrote  several  works,  as  mentioned  in  the  text,  of  which  the  Elements  is  tiie 
ddef.  Ill  -this  work  be  availed  himtelf  of  the  labmun  of  those  who  bad  gone 
before  bim,  collecting  and  properly  arranging  the  principles  and  propositions 
which  had  already  been  given  by  others,  supplying  the  deficiencies,  and  strength* 
ening  and  confirming  the  demonstrations.  The  particulars  of  his  life,  and  time 
of  liiadflBtb,  are  uakaolni :  it  it  said  that  King  PtolMiy  Lagtts,  on  examto- 
l|ig  tbe  Elamcirts,  asked  htm  if  it  was  not  .possible  to  arrive  at  the  same  oon- 
cluaions  by  a  shorter  method  ;  to  which  Bwclid  replied,  **  There  is  no  rojra/road 
to -Geometry." 


2Z2  INTKODUCnON.  Pabt  VIIl. 

tht  rixth  of  the'  propMrtmiB,  &c;  of  plane  figuries ;  .tke 
seventh,  e^th, and  ntntb, explain  and  prove  diefiundamenf 
tal  properties  of  nmnben  f  the  tenth  contains  the  theory  of 
commensurable  and  ineommensuraUe  lines. and  spaces ;. and 
the  remaining  five  books  unfold  the  doctrineof  solids* 

The  first  six  books^  with  the  eleventh  and  twelftli,  are.  all 
that  are  now  usually  studied ;  the  -modern  improvements  in 
analysis  having  furnished  much  shorter  and  more  conveDienk 
methods  of  attaining  to  an  adeqpute  knowledge  of  the  sub^ 
jects  contained  in  the  remaining  books,  than  those  given  in 
the  Elements. 

The  Elements  of  Euclid  furnish  all  thsu  is  necessary  for 
determining  the  perimeters  and  areas  tjf  rectilineal  figures^ 
the  superficies  and  solid  contents  of  bodieg  contained  by 
rectilineal  planes,  and  for  descrilHng  them  on  paper:  in 
them  it  is  proved,  &at  a  cone  is  equal  to  one*thirdof  its  cir^ 
cutnscriblng  cylinder ;  that  the  solid  content  of  a  cjplhkder  is 
found  by  multiplying  the  area  of  its  base  into  its  altitudes 
•we  are  likewise  taught,  what  ratio  similar  plane  figures,,  aid 
also  similar  solids,  have  to  one  another;  that  the  periphertor 
of  circles  are  as  their  diameters,  and  the  areas  as  the  squares 
of  their  diameters ;  that  angles  are  measured  and  compared 
by  means  of  the  intercepted  circumferences,  &e.  These  and 
several  other  properties  of  the  circle  are  given  in  the  Ele- 
ments, but  it  is  no  where  directly  sheivn  how  the  circum-' 
ference  (that  is,  its  ratio  to  the  given  diameter)  or  how  the 
area  of  a  circle  may  be  found :  it  is  true,  that  a  method  of 
^proximation  both  to  the  circumference  and  area  seems  to 
be  implied  in  the  sceoiid  proposition  of  the  twelfth  boak,.bul. 
no  further  notice  is  taken  of  it  in  any  of  the  subsequent 
propositions. 

In  hia  demonstiations^  Euclid  has  observed  for  the  most 
part  all  that  strictness,  for  which  the  ancients  were  so  distin- 
guished :  from  a  small  number  of  definitions  and  self-evident 


PaetVIII.  geometry.  223 


priaciplefs,  tie  ha0  deduced  with  moontestiible  evidence 
truth  of  all  the  proposilbns  which  he  proposed  for  proof.  ^n» 
rigorous  strictness  haS;  however,  sometimes  led  him  ueoessArily 
into  aa  indirect  and  complicated  chain  of  reasoning,  which' 
makes  hb  demonstrations  in  a  few  instances  tedions  and  dif-^ 
ficuk.  To  remedy  this  defect,  several  of  the  moderns  have 
undertaken  with  suceess  to  simplify  and  render  more  direct 
and  appropriate,  such  ot  the  demoDstrations  as  seemed  fio^ 
require  improvement ;  but  others,  who  have  lessened  the 
number  of  propositions  by  retrenching  those  which  they 
deemed  superfluous,  have  in  general  been  less  happy:  by 
removing  those  links,  which  appeared  to  them  unnecessary, 
the  chain  of  demonstration  has  in  many  cases  been  broken 
and  spoiled. 

The  Elements  have  been  translated  into  the  language  of 
evtry  country  where  learning  has  been  encouraged,  and  en^ 
riched  with  numerous  and  valuable  commentaiies*  The  Arabs 
were  the  first  people  who  engaged  in  tUi  way :  on  the  revival 
of  learning  ammig  them,  their  grand  eare  was  to  obtain  the 
mathematical  works  of  the  best  Greek  authors,  and  translate 
tlKminto  the  Ar^c  language.  There  wtre  probably  several 
translations  of  Euclid ;  one  in  particular  is  mentioned  as 
made  by  Honain  £bn  Ishak  al  Ebadi,  a  learned  physician^ 
who  flounced  in  the  reign  of  the  KhaUf  Al  Motawakkef, 
A.  D.  847.  Adelard,  a  monk  of  Bath,  in  the  twelfth  cen- 
tury, appears  to  have  been  the  first  who  made  %.  Latin  trans-^ 
lation  of  the  Elements,  which  he  did  firom  4ie  Arabic,  as  no 
Greek  copy  of  Euclid  had  then  been  discovered.  Carapanus 
of  Novaia  translated  and  commented  on  the  Elements  in 
1250,  which  work  Was  revised  and  further  commented  on  by 
Lucas  De  Burgo,  about  1470.  Orontius  Fln«us  published 
the  first  six  bodes  with  notes  in  1530,  which  is  said  to  have 
been  the  firirt  edition  that  appeared  in  print.  Pdetarius 
published  the  first  six  books  in  1 557,  and  about  the  same  time 
Tartalea  gave  a  commentary  on  the  whola  of  the  iBftfloi^bdoks.* 


ail  INTRODfUCnON.  Part  VIII. 

In  1670  BtlliDg8iey*s  Eiiclid  appeared,  with  a  very  plain 
and  useful  pw&ce  and  notes  by  the  learned  and  eccentric 
Dr.  John  Dee.  Candalla  published  the  Elements,  with  addi- 
tions and  improrements,  in  157^>  which  work  was  itfterwards 
reprinted  with  a  pnrfix  commentary  by  Clavius  the  Jesuit. 
Many  edilionsof  the  Elements  have  since  appeared,  the  chief 
of  which  are  those  of  De  Cfaales,  Tacquet^  Herigon,  Barrow, 
Ozanam,  Keill,  Whiston,  and  Stone ;  but  Dr.  Robert  Sim- 
son's  translation  of  the  first  six  and  the  eleventh  and  twelfth 
books^  with  the  Data,  first  publi^ed  in  the  year  1 7^6,  is 
that  now  most  generally  used  in  the  British  Empire. 
Playfair's  Euclid  is  an  improvement  on  Samson's ;  and  In- 
gram's edition  contains  some  particulars  chiefly  relating  to 
practical  Geometry,  which  are  not  to  be  found  in  either.  Be- 
fore we  conclude  this  enumeration^  it  will  be  necessary  to 
observe,  that  Dr.  David  Gregory  »,  the  Savitian  Professor  of 
Astronomy,  published  at  Oxford,  in  170^9  the  whole  of  tlie 
worics^  of  Euclid  in  Greek  and  Latin  $  this  he  b  said  to  have 
done  in  prosecution  of  a  design  of  Dr.  Bernard  *,  his  prede- 

"*  Dftvid  Oregiory  ww  bora  at  Aberdeen  in  laSl ;  here  and  at  Sdinbrn^  be 
received  bis  maUiematical  and  classical  education :  in  I6d4  he  was  elected 
Professor  of  Mathematics  in  the  University  of  Edinburgh  ;''and  it  deserves  to 
be  noticed,  that  he,  in  coi^nnction  with  bis  brother  James,  first  introduced  the 
Newtonian  phUoiophy  into  Scotland.  Tbrouf h  the  Inentty  inteifefenee  of 
Newton  and  Flamstead,  our  author  obtained  the  Saviliaa  Profesaorship  of 
Astronomy  at  Oxford,  where  he  was  honoured  with  the  degree  of  M.  D.  His 
works  are  EjtefcitaH&  Geometriea,  Stc;  4to.  Edinb.  1684.  Chtoptriem  et  IHtp" 
irie^  Sphmiem' JEkmenimfOxmo,  l^h*  jiHrwMntim,  P^fneaf^et  Gemmtrite 
Mkmemta,  and  some  others:  be  died  in  1710,  at  Maidenhead  in  Berkshire. 

B  Dr.  Edward  Bernard  rendered  himself  fieimous  by  being  the  first  who  un- 
dertook to  ec^ect  the  work»  of  the  ancient  mathematicians  for  puUicatioo ;  he 
likewise  tiioaght  to,  England  the  5tb,  ^h,  and  7th  books  of  the  Cooicsof 
A{i(ottoniu8,  being  a  c<9y  of  the  Arabic  Version  which  the  celebrated  Golios 
bad  obtained  in  the  East.  He  succeeded  Dr.  Wren  in  the  Professorship  in  1673, 
and  resigned  it  in  1^1,  on  being  presented  to  the  Rectory  of  Brightwell  m 
Berkshire.  He  died  in  I696>  in  the  SJStb  year  of  his  age.  His'  work»  on  ma- 
thematical subjects  are  mostly  inserted  in  the  Philosophical  Transactions:  they 
consist  of  Observations  on  the  Obliquity  of  the  £cliptic,  various  \/istr0nowdeat 
•ad  Cki»tt$gtgietd  TabUs,  ^^ 


PabtVIII.  INTftODUCTION.  ^25 

cesser,  and  in  obedience  to  a  precept  of  Sir  Henry  Saville  % 
the.  founder  of  the  Professorship,  reqiiiring  that  those  who 
fill  die  chairs  of  Geometry  and  Astronomy  should  publish  the 
mathematical  works  of  the  ancients.  Dr.  Gregory's  is  the 
completest  edition  of  Euclid  extant. 

According  to  Pappus  and  Proclus,  several  mathematical 
treatises,  brides  the  Elements,  were  written  by  Euclid :  hts 
Data,  a  work  still  extant,  is  calculated  to  facilitate  the  method 
of  resolution,  or  analysis,  shewing  from  certain  things  givf  n 
by  hypothesis,  what  other  things  may  thence  be  found.  His 
three  books  of  Porisms  are  said  to  have  been  a  curious  collec- 
tipa  of  important  particulars  relating  to  the  analysis  of  the 
ibore  diflScult  and  general  problems ;  but  no  part  of  this  wof  k, 
or  of  any  other  on  the  same  subject  written  by  the  ancients, 
had  been  preserved,  except  a  small  specimen  by  Pappus; 
from  whence  several  modern  geometricians,  particularly 
Fermat,  BuUiald,  Albert  Girard,  Halley,  Simson,  and  Play- 
&ir,  have  attempted  to  restore  either  completely,  or  in  part, 
what  the  ancients  are  supposed  to  have  delivered  on  the  sub- 
ject.    Euclid  wrote,  besides  these,  a  work  on  the  Division  of 

«  Henry  SaVille'was  bom  at  Bradley  in  Torkshire,  A^  D.  1549»  and  entered 
at  Merton  College,  Oxford,  in  1561,  of  which  college  he  was  chosen  a  fellow* 
and  took  his  degiree  of  M.  A.  in  1570.  In  1578  he  trarelled  through  different 
parts  of  Siuope  for  improvement,  and  on  his  retnm  was  appointed  Greek  Tutor 
to  Qaeen  Elizabeth.  In  1585  he  was  made  Warden  of  Merton  College*  over 
whkh  he  presided  36  years,  with  eqaM  credit  to  himself  and  advantage  to  that 
learned  body.  He  was  chosen  Provost  of  Eton -College  in  1596,  and  received 
tlie  bononr  of  knighthood  from  Sing  James  I.  in  1604,  after  declining  the 
most  flattering  offers  of  preferment  in  either  church  or  state.  Sir  Hanty 
Soiville  was  an  accomplished  gentleman,  a  profound  scholar,  and  a  munificent, 
patron  of  learning,  to  which  (on  the  death  of  his  mily  son)  he  devoted  his  wholef 
fortune.  In  1619  he  foanded  two  professorships  at  Oxford,  one  for  Geometry, 
apd  one  for  Astronomy,  each  of  which  he  endowed  with  estates.  In  addition 
to  tfaie  several  legacies  he  left  to  the  University,  he  bestowed  on  it  a  great 
i|«ABttty  of  mathematical  books,  rare  and  curious  manuscripts,  Greek  types, 
&c.  &c.  He  died  at  Eton  College  in  1722,  leaving  behind  him  several  works» 
.  of  which  the  only  one  pertaining  to  our  present  subject  is  his  CoUeciion  rf 
Mathematical  Lecturer  on  EucUd^t  Elements,  4to.  1621. 

VOL.  11  •  g 


226.  GEOMETOY.  PartVIII. 

Superficies ;  Loci  ad  Siiperficiem ;  four  books  on  Conic  Sec- 
tions ;  and  treatises  on  other  branches  of  the  Mathematics. 

Archimedes  ',  one  of  the  greatest  geometricians  of  anti- 
quity, was  the  first  who  approximated  to  the  ratio  of  the  cir- 

P  Archimedes  was  born  at  SyracQto,  and  related  fo  Hiero,  King  of  Sicily: 
lie  was  remarlcable  for  bis  extraordinary  application  to  mathematteal  studies, 
but  more  so  for  bis  skill  and  surprising  inventions  in  Mechanics.  He  excelled 
likewise  in  Hydrostatics,  Astronomy,  Optics,  and  almost  every  other  science ; 
he  exhibited  the  motions  of  the  heavenly  bodies  in  a«  pleasing  and  instructivs 
manner,  within  a  sphere  of  glass  of  his  own  contrivance  and  workmanship ;  he 
likewise  contrived  corions  and  powerful  machines  and  engines  for  raisiag 
weights,  hurling  stones,  darts,  &c.  launching  ships,  and  for  exhausting  the 
water  out  of  them,  draining  marshes,  &c.  Whdn  the  Roman  Consul,  Mar- 
cellus,  besieged  Syracuse,  the  machines  of  Archimedes  were  employed  t  these 
showered  upon  the  enemy  a  cloud  of  destructive  darts,  and  stones  of  vast 
weight  and  in  great  quantities  ;  their  ships  were  lifted  into  the  air  by  his 
cranes,  levers,  hooks,  &c.  and  dashed  against  the  rocks,  or  precipitated  to  the 
bottom  of  the  sea ;  nor  could  they  find  safety  in  retreat :  his  powerful  bnmiqg 
glasses  reflected  the  condensed  rays  of  the  sun  upon  them  with  such  effect^ 
that  many  of  them  were  burned.  Syracuse  was  however  at  last  taken  by  stormy 
and  Archimedes,  too  deeply  engaged  in  some  geometrical  speculations  to  be 
conscious  of  what  had  happened,  was  slain  by  a  Roman  soldier.  Maroellna  wa« 
grieved  at  his  death,  which  happened  A.  C.  210,  and  took  care  of  his  funeral. 
Cicero,  when  he  was  Questor  of  Sicily,  discovered  the  tomb  of  Archimedes 
overgrown  with  bashes  and  w^eeds,  having  the  sphere  and  cylinder  engraved  on 
it,  with  an  inscription  which  time  had  rendered  illegible. 

His  reply  to'  Hiero,  who  was  one  day  admiring  and  praising  bis  machines, 
can  be  regarded  only  as  an  empty  boast.  ^*  Give  me/'  said  the  ezultij^ 
philosopher, "  a  place  to  stand  on,  and  I  will  lift  the  eMrtV  (A»«  ^mi  r«  fw,  mu 
rifi^  ynf  *t9n^t*»)  This  however  may  be  easily  proved  to  be  impossible ;  for, 
granting  him  a  place,  with  the  simplest  machine,  it  would  re4|aire  a  man  to 
move  swifter  than  a  cannon  shot  during  the  space  of  100  years,  to  lift  the 
earth  only  &ne  inek  in  all  that  tinie«**— Hiero  ordered  a  golden  crown  to  be  made, 
but  suspecting  that  the  artists  bad  purloined  some  of  the  gi4d  and  substituted 
base  metal  in  its  stead,  be  employed  our  philosopher  to  detect  the  cheat  ^ 
Archimedes  tried  for  some  time  in  vain,  but  one  day  as  he  went  into  the  bath, 
he  observed  timt  his  body  exdvded  just  as  much  water  as  was  equal  to  its  bulk  ; 
the  th«mght  immediately  struck  htm  that  this  discovery  had  furnished  ampls 
data  for  solving  his  difliculty;  upon  which  be  leaped  out  of  the  bath,  and  ran 
through  the  streets  homewards,  crying  ont^  <«^»« !  tv^%m !  /  have  found  it  i 
J  have  /mmd  it  /—The  best  edition  of  bis  works  is  that  of  Torelli,  edited  at  the 
Clarendon  Press,  Oxford,  fol.  ITS^y  by  Pr.  Robertson,  Suviltan  Professor  sf 
Astronomv. 


l^AnrVni.  INTRODUCTION.  227 

tumference  of  a  circk  to  its  diameter,  A.C.  250:  this  he 
eflected  by  circumscribibg  about,  and  inscribing  in  the  circle 
Iregular  polygons  of  96  sides,  and  making  a  numerical  calcu^ 
lation  of  their  perimeters ;  by  means  of  this  process  he  made 
the  ratio  as  22  to  7j  which  is  a  determination  near  enough 
the  truth  for  common  practical  operations,  where  great  exact- 
ness is  not  required,  and  has  the  advantage  of  being  express- 
ed by  small  numbers.  He  was  the  next  after  Hippocrates, 
who  squared  a  curvilineal  space  3  he  applied  himself  with 
ardour  to  the  investigation  of  the  measures,  proportions,  and 
properties  of  the  conic  sections,  spirals,  cylinders,  cones, 
spheres,  conoids,  spheroids,  &c.  On  these  subjects  the  follow- 
ing works  of  his  are  still  extant,  viz.  two  books  on  the 
Sphere  and  Cylinder;  and  treatises  on  the  Dimensions  of  the 
Circle ;  on  Spirals ;  on  Conoids  and  Spheroids ;  and  on  the 
Centres  of  Gravity. 

The  next  geometer  of  note  after  Archimedes,  was  Apol- 
lonius  Pergsdus,  A.  C»  230 :  this  great  man  studied  for  a  long 
time  in  the  schools  of  Alexandria  under  the  disciples. of 
Buclid,  and  was  the  author  of  several  valuable  works  on 
Geometry,  which  were  so  much  esteemed,  that  they  procured 
him  the  honourable  title  of  the  great  Geometrician.  His 
principal  work,  and  the  most  perfect,  of  the  kind  among  the 
ancients^is  his  treatise  on  the  Conic  Sections,  in  eight  books ; 
seven  only  of  these  have  been  preserved,  the  four  first  in  the 
original  Greek,  and  the  5th,  6th,  and  7th  in  an  Aramc 
version  \ 

4  AceorStpg  to  l^ppUB  abd  Eutocitu,  the  following  works  were  likewise 
-Written  by  A|»dlloniQs,  viz.  1.  The  Section  of  a  Spa^e.  J2.  The  Section  of  a 
Ratio. '  3.  The  Determinate  Section.  4.  The  Inclinations.  5.  The  Tangen- 
cies,  and  6.  The  Plane  Loci ;  each  of  these  treatises  consisting  of  two  books. 
Pappus  has  left  us  some  particulars  of  the  abore  works,  which  are  all  concern- 
ing them  that  now  remain ;  but  from  the^  scanty  materials,  many  restorations 
liave  been  made^  ris.  by  Vieta,  SnelUus»  Ghetaldus,  Fermat,  Schooteu,  Alex. 
Andefioii,  HaUey»  Simaon^  Horsley^  Lawson,  Wales,  and  Barrow.  The  best 
edition  of  the  C«ntci  of  ApolloAios  is  that  by  Dr.  Halley,  foi,  Oxw.  1710* 

a  2' 


Xbeag^of  Arohimedes  ^d  AvQlhouhiS  hm  with  jusd«« 

th^  sci^i^ce  oever  acquired  so*  great  a  dtsgree  oi  brilliancy  at 
aqy  otber  p^uod  of  the  Greciao  history. 

XbeduglicsuioQ  of  the  oube,,qjuiadr«ture'Of  the  circloi  tri*- 
section.of  an^  aogle^  &c.  were  probleiss  of  which  the  ancients 
tu»ver  lost.^igbt;.  ijaaoy  of  the  proposilioiiSHin  tbe  Elements^ 
payiticuliurly.  piy^  27^  2S>  wd-  29' of.  th&  sixth  book,  are  inti- 
HM^tely  connected  with  the  aolution^.  and  probably  originated 
in,  the  atlm»pt(»  to.  obtain  it*    Thj»^  application  oft  the  conic 
sections  tQ  this  purpose  by  M en^hmus^  has  been,  fdready 
noticed :  about  ihe  aaoae  time  IKoostratus;  invented: the-qna^ 
di^triiS)  a  iQ€K;hanical  cuITC^po6sesl»()g  the  triple  adiwitits^ 
of  tfjsACting  and.  multiplying  aa  ang^e>.  and  squaring  the 
Qit€li»i    Tibet  conchoid  of  Nicomede^^  who-  flourished  A.  C« 
250,  has  been  applied  by  both  ancient  and*,  modern,  geome? 
tQra^ually.to  the  trisection,  finding:  two  mean  prc^rtiotials, 
2ind  tbeioonstruction  of  other  solid  probkni9^;.for  which  pur- 
poses, this. ourve  has  be^n  preferred  by  Archimedes,  Pappusi 
and  Nfiwton»  to  any  other.  (See  Newton- l^^rt^Ama^ca  Uni^ 
t€nalisi,p.  288).2H90  The  cissoid,  another  curve,  heie^  an 
unprovament  on  the  conchoid;  was  ini^nted  by  Diocles  about 
laOyearebefcore  Christ. 

Hero^ .  DosithfittSy  Eratosthe^eB^ .  and ;  Hypsides,  ^  who :  fbu^ 
liabed  in^  the^  second  century  befdre  Christ,  and  Geminius 
who  flourished  in  the  first,  were  all  eminent  for.  their  skill  in* 
Geometry:  indeed  the  science  continued  to  be  cultivated 
with  il^rdour  by.  a  numerousilistKotf  geometricians,  produced  by 
the  Alexandrian  school,*  until'thatfasnous  seat^pf  learning' fell 
a  prey  to  the  blind  and  merciless  bigotry  of  the  Arabs.  The 
fiffst:who  wrote  on.,  the  spberci  and.  its  circles  to  any  con»^ 

m 

Swiiloaof  tb«  GyUndeF  and  Cone,  prinM- fiieni  tht  ov^inai  GUseek)  witl|ia> 
LbtiQ  tvMMlatioQ, 


Tam  VIII.  INTRODUCTION.  2» 

derable  extent,  at  lealt  whose  works  have  been  preserve4^ 
was  Tbeodosius^  A.  C.  60 :  this  work,  in  which  the  Jproposi'- 
tions  are  demonstrated  with  equal  strictness  and  el^^nce, 
forms  the  basis  of  spherical  Trigononietryy  as  pntctiled  by  the 
BMxlems*  About  the  same  titne,  or  shortly  after^  Mehelaui 
wrote  lus  treatise  on  Chords,  which  b  lost ;  but  his  wblck  on 
Spherical  Triatigles,  containing  the  constmeiioti  and  tri« 
gonometrical  method  of  resolving  them,  accorditig  to  the 
ancient  practice,  is  still  extabt.  We  are  particulariy  indebted 
to  Pat^s,  A.  D.  380,  and  Procluis,  A.  D^  4dO>  fioT  their  kbo-^ 
Tious  researches ;  many  particulars  relating  to  the  scienees  df 
the  Greeks  would  have  been  lost  to  pcBterity,  but  for  their 
writings:  the  former  was  an  etninent  mathematiciaii  a£ 
Alexandria,  and  author  of  several  learned  and  useful  wdrks^ 
particularly  eight  books  of  Mathematical  Collections,  of  which 
the  first  and  part  of  the  second  are  wantit)^.  These  books  een- 
twn  a  great  variety  of  useful  information  relilting  to  Geome* 
try.  Arithmetic,  Mechanics,  &c.  with  the  sokitiori  of  proMcm^ 
oi  different  sorts.  Proclus  likewise  studied  at  Alexandria^ 
and  afterwards  presided  over  the  Platonic  school  at  Athens } 
be  wrote,  besides  many  otber  w<H*ks,  Commentaries  on  the 
fint  book  of  Euclid,  on  the  Mathematics,  on  Phil€isophyi 
also  a  treatbe  De  Splwrra,  wbieh  Was  published  by  Dr.  Bftia« 
bridge,  Savilian  Professor  of  Geometry  at  Oxford,  in  1690. 
The  writings  of  the  Greek  geometFieians  were  trfeslate4 
and  commented  on  by  several  learned  Arabians,  but  tfi^ 
improvements  they  introduced  were  chiefly  of  the  practH 
eal  kind ;  among  these  may  be  meotioned  the  fundamenUd 
propofiitiofM  of  Trigonometry,  in  wht€b,by  (be  substitution  of 
sines  instead  of  the  chords,  and  other  conveaknt  Abridge- 
ments, they  greatly  simpKfied  the  theory  and  solictiotis  of 
plane  and  spherical  triangles.  These  improvements  are  a-r 
jBieribed  to  MaiMMnet  Ebn  Mssa^  ft  geometer  of  whom  there 
ttill  exists  a  work  on  Plane  atid  Splrerical  Figtarres.  We  Bk«f«* 

as 


2S0  <5EOMETRY.  Part  VIII. 

wise  possess  a  work  on  Sarveying,  written  by  Mahomet  of 
Baghdad^  which  some  modern  authors  have  ascribed  to 
Euclid. 

A  few  learned  men,  famous  for  their  skill  in  Geometry, 
flourished  in  the  West  during  the  fifteenth  century.  Of  these 
the  chief  were  the  Cardinals  Bessarionand  Cusa ',  Purbach, 
Nicholas  Oresme,  Bianchini,  George  of  Trabezonde,  Lucar 
de  Burgo,  Schonerus,  Walther,  and  Regiomontanus;  the  latter 
wrote  a  treatise  on  Plane  and  Spherical  Trigonometry,  A.  D, 
1464 ;  in  which,  among  other  improvements,  he  introduced 
the  use  of  the  tangents,  and  applied  Algebra  to  the  solution 
of  geometrical  problems :  this,  is  the  more  surprising,  as  it 
occurred  several  years  before  the  publication  of  any  of  the 
worka^f  De  Burgo,  who  is  generally  supposed  to  have  be^a 
the  introducer  of  Algebra  into  Europe. . 

43il  the  revival  of  learning  iit  Europe  about  the  beginning 
of  the  sixteenth  c^tury,  the  study  of  Geometry  began  to 
be  cultivated  with  great  attention ;  the  works  of  the  Greek 
geometricians  were  eagerly  sought  after  and  translated  into 
Latin  or  Italian,  and  served  as  guides  to  those  who  had  a 
taste  for  that  correct  reasoning,  for  which  the  ancient  Geo- 
metry is  so  ji^tly  famed,  or  were  desirous  of  availing  thenn 
selves  of  the  knowledge  of  its  application  and  use,  as  ctm^ 
nected  with  the  necessary  business  of  life.  As  early  as  1522, 
John  Wenier,  a  celebrated  astronomer  of  Nuremberg,  pub- 
lished some  tracts  on  the  Conic  Sections,  and  on  other  geo^ 
metrical  subjects.  Tartalea  composed  a  treatise  on  Arith- 
metic, Algebra,  Geometry,  Mensuration,  &c.  entitled,  ^Tra^- 
tato  di  humeri  et  Misure,  155G,  being  the  first  modem  work 

'  Nicolas  De  Cusa  was  bom  of  poor  parents,  A.  D.  1401  ;  bis  application  to 
learning  and  bis  personal  merit,  boweyer,  raised  bim  to  tbe  rank  of  bisbop  and' 
cardinal",  bis  claim  to  tbe  honour  of  baving  squared  tbe  circle  was  ably  re- 
futed by  Begiomontanus  i  ne.veTtbelesiJ  be  was  a  man  of  very  extraordinaiy 
pattsy  and  excelled  in  tbe  knowlttdga  of  law,  divinity,  natural  pbilosopby,  aad 
feometiryi  on  wl(icb  8ob|eet9  he  i>  said  to  hare  written  some  eycdlent  trea^seif. 
He  died  in  1464.  ^ 


'v 


Part  VlIL  INTRODUCTION.  231 

which  teaches  how  to  find  the  area  of  a  triangle  by  means  of 
its  three  sides,  without  the  aid  of  a  perpendicular.  Mauro- 
licus  was  a  respectable  geometer,  ^and  wrote  on  various  sub- 
jects ;  his  treatise  on  the  Conic  Sections  is  remarkable  for 
Its  perspicuity  and  elegance.  Aurispa,  Batecombe,  Butes, 
Ramus,  Xylander,  Foilius,  Cardan,  Fregius,  Bombelli, 
Ficinus,  Durer,  Zeigler,  Fernel,  Ubaldi,  Clavius,  Barbaro, 
Byrgius,  Commandine,  Pelletier,  Dryander,  Nonius,  Lina- 
cre,  Sturmius,  Saville,  Ghetaldus,  R.  Snellius,  and  many 
others  who  flourished  at  this  period,  were  cultivators  of 
Geometry;  and  if  they  made  few  discoveries,  still  their 
labours  as  translators,  commentators,  or  teachers,  were  be- 
neficial in  diffusing  knowledge,  and  merit  our  grateful  ac- 
knowledgments. 

Vyious  approximations  to  the  ratio  of  the  circumference 
of  a  circle  to  its  diameter,  were  given  about  the  beginning 
of  the  1 7th  century,  approaching  much  nearer  the  truth  than 
any  that  had  hitherto  appeared ;  viz.  by  Adrian  Romanus, 
Willebrord  Snellius,  Peter  Metius,  and  Ludolph  Van  Ceu-* 
len ;  according  to  the  conclusion  of  M etius,  if  the  diameter 
be  113,  thcs  circumference  will  be  355,  which  is  very  near 
the  truth,  and  has  the  advantage  of  being  expressed  hy  small 
numbers.  By  continual  bisection  of  the  circumference.  Van 
Ceulcn  found,  that  if  the  diameter  be  1,  the  circumference 
will  be  3,14159,  &c.  to  3G  places  of  decimals;  which  dis- 
covery was  thought  so  curious,  that  the  numbers  were  en- 
graved on  his  tomb  in  St.  Peter's  Church- yard,  at  Leyden  •. 

*  The  simplest  (and  consequently  least  accarate)  ratio  of  the  diameter  to 

the  circumference  is  as  1  to  3  ;  a  ratio  somewhat  nearer  tl^tn  this,  is  as  6  to  19. 

We  have  noticed  before  that  Archimedes  determiqed  the  ratio  to  be  as  7  td 

22  nearfy,  which  is  nearer  than  the  above. 

A  nearer  approximation  is  as 106  to  333* 

That  of  Melius  is  still  nearer,  viz.  as    113  to  355  ' 

A  nearer  approximation  than  the  1^  j^^2  ^^  g^^^ 

last   is J  *  ' 

/  still  nearer  is : ...  as  1815  to  5702,  &c. 

Q  4 


m  GEOMETRY.  P^jit  VIIL 

Geometrical  problems  had  long  before  this  period  beeo 
solved  algebraically^  by  Cardan,  Tartalea,  Re^montaou^, 
and  BombelU ;  but  a  regular  and  general  method  of  apply- 
ing Algebra  to  Geometry,  was  first  given  by  Vieta,  about  the 
year  )580$  as  also  the  elements  of  angular  sections.  De^ 
Cartes  improved  the  dbcovery  of  Vieta,  by  introducing  a 
general  method  of  representing  the  nature  and  circumstances 
of  curve  lines  by  algebraic  equations,  distributing  curved 
into  classes,  corresponding  to  the  different  orders  of  equation^ 
by  which  they  are  expressed ;  A.  P.  IG37-  A  method  of 
Ixingents,  and  a  method  de  maximis  et  minimis,  nfUcb 
resembling  that  of  fluxions  or  increments,  owe  their  ori- 
gin to  Fermat,  a  learned  countryman  and  competitor  of 
Des  Cartes,  with  whom  he  disputed  the  honour  of  first  ap- 
plying Algebra  to  curve  lines,  and  to  the  geometrical  con- 
•  struction  of  equations,  secrets  of  which  he  was  in  posstssion 
before  Des  Cartes'  Geometry  appeared.  About  this  tim^,  q^ 
a  little  earlier,  Galileo  invented  the  cycloid ;  its  properties 
were  afterwards  demonstrated  by  Torricellius. 

The  improvement  of  Des  Cartes,  now  called  the  nm 
Ceqi^etrtfj  was  cultivated  with  ardoiu*  and  success  by  math^- 
nqatieians  in  almost  every  part  of  Europe;  his  work  w^ 
translated  out  of  French  into  I^atin,  and  published  by  Fran- 
cis SchoQten,  with  a  commentary  by  Schooten,  and  notes  by 
M-  de  Beaune,  16*49.  The  Indivisibles  of  Cavalerius,  pu^- 
lisbed  in  1635,  was  a  new  and  useful  invention,  applied  to 

Van  Cenlen's  nomben,  as  mentioned  above,  were  extended  to  72  places  of 
figures  by  Mr.  Abrabam  Sharp,  about  1706  ;  Mr.  Macbin  afterwards  extended 
the  same  to  100  places,  and  M.  De  Lagni  has  carried  them  to  the  amaxinc 
length  of  128  places:  thus,  if  the  diameter  be  1000,  &c.  (to  128  places}'  till 
circumference  will  be  31415,  92653,  58979,  32384,  62643,  38327,  95028, 
84197,  16939,  93751,  05820,^494,  45923,  07816,  40628^  62089,  98628, 
03482,53421,  17067,  98214,  80865,'  13272,  30664,  70938,446+,  or  7**. 
This  number  (which  includes  those  of  Vm  Ceulen,  Sharp,  and  Machtn)  is 
sufficiently  near  the  truth  for  any  purpose,  so  that  except  the  ratio  could  be 
completely  found,  we  need  not  wish  for  a  greater  de^^ree  of  accuracy. 


PfBT  VUl.  INTRQDyCTiON.  m 

ieteamne  th^  ttrea^^  of  cunre$,  tti^  soUdides  cf  hodics  n^f^r 
rated  by  their  reiBrolutiqi)  about  9,  fixed  lioej  &c.  Boberva)^  af 
/early  as  1634^  had  employed  a  ftimilar  metho^^  wbi^b  hf 
lipplied  to  the  cycloid,  a  eurve  at  thi^t  tin^e  jmd^  cel^brfi^f^ 
for  its  numerous  and  singular  properties;  be  likewise  i^r 
vented  a  general  method  for  tangents,  applicable  ^ike  t^ 
geometrical  and  mechanical  curves.  The  inverse  method  ^ 
tangents  derived  its  or^in  from  a  problem,  which  De  Beaune 
proposed  to  bis  friend  Des  Cartes,  in  1647*  In  1655  tl^ 
learned  Dr.  Wallis  published  his  Arithmetica  InfinitormQ ; 
being  either  a  new  method  of  reasonbg  on  quantities,  or 
else  a  great  improvement  on  the  Indivisibles  of  Cav^eriiif 
^bove  mentioned ;  peculations  which  led  the  wfiy  to  in^i)b| 
aeries,  the  binomiid  theorem,  and  the  method  of  fluxions :  thb 
work  treats  of  the  quadrature  of  curves  and  many  other  pro- 
blems, and  gives  the  first  ei^pression  known  for  thf  area  qf 
4  circle  by  an  infinite  series* 

One  of  the  greatest  discoveries  in  modern  Geometry  was 
the  theory  of  evolutes,  the  autluv  of  which  was  Christmn 
Huy^cns,  an  ingeniou$  Dutch  mathematician,  who  pjublished 
it  at  the  Hague  in  1658,  in  a  work  entitled^  Horeksgiufla 
Oscillatorium,  sive  de  Motu  Pendulorum,  &c. 

In  16G9  were  published  Dn  Barrow's  Optical  and  Geomcr 
trical  Lectures,  containing  many  v.ery  ingenioMS  and  proibujid 
researches  ojq  the  dimensions  and  properties  of  curves,  and 
i^pecially  a  method  pf  tangents,  by  %  mode  of  calculatioi^ 
differing  firom  that  of  fluxions  or  ioorements  in  scarcely  any 
particular,  except  the  notation.  About  this  time  the  use  of 
geometrical  loci  for  the  solution  of  eqviatioQs,  was  carried  to 
a  great  degree  of  perfection  by  Slusiua»  a  canon  of  Liege^ 
in  his  Mesolabium  ei  Problemaia  Solida:  he  likewise  in* 
isejted  in  the  Philosophical  Tram^actions,  a  short  and  easy 
method  of  drawing  tangents  to  all  geometrical  curves,  with 
a  demoqstratipq  of  the  same  \  and  likewise  a  tract  ovl  the 


HSi  GEOMETRY.  FabtYDL 

Optic  Angle  of  Alhazen.  Besides  those  we  hare  mendoDed, 
maojf  others  of  this  period  devoted  their  attention  to  the 
rectification  and  quadrature  of  curves,  &c.  of  whom  Van 
Heuraet,  Rolle,  Pascal,  Briggs^  Halley,  Lallou^re,  Tor-^ 
riceHtus,  Herigon,  Niell,.  Sir  Christopher  Wren,  Faher^ 
Lord  Brouncker,  Nicholas  Baker,  G.  St.  Vincent,  Mercator, 
Gregory,  and  Leibnitz,  ware  the  principal. 

Tlie  seventeenth  century  is  famed  for  giving  birth  to  two 
noble  discoveries;  namely,  that  of  logarithms  in  Hi  14  by 
Lord  Napier,  whereby  the  practical  applications  of  Geometry 
are  greatly  facilitated ;  and  that  of  fluxions,  to  which  pro- 
blems relating  to  infinite  series,  the  quadrature  and  properties 
of  curves,  and  other  geometrical  subjects  connected  with 
Astronomy,  Pliysics,  &c.  and  which  were  formerly  considered 
as  beyond  the  reach  of  human  sagacity,  readily  submit.  For 
this  subKme  discovery,  the  learned  are  indebted  either  to  the 
profound  and  penetrating  genius  of  Sir  Isaac  Newton  %  or 

<  Sir  I»aac  New^n,  one  of  the  greatest  mathematieiaqB  and  pfailosophert 
that  ever  lived,  was  born  in  Lincolnshire,  in  1643.  -Having  made  some  profi- 
ciency  in  the  classics,  &c.  at  the  gi-ammar  school  at  Gfantham,  he  (being  an 
•nlf  child)  waa  taken  home  hj  bts  mother  (who  was  a  widow)  to  be  her  com- 
panion, and  to  learn  the  management  of  his  paternal  estate :  but  the  Iotc  of 
books  and  stady  occasioned  his  farming  concerns  to  be  neglected.  In  1660 
he  was  sent  to  Trinity  College,  Cambridge ;  here  he  began  with  the  study  of 
Euclid,  bat  the  propoeitions  of  that  book  being  too  easy  to  arrest  his  atteolUott 
long,  he  passed  rapidly  on  to  the  Analysis  of  Pes  Cartes,  Kej^ler's  Optics,  &c. 
making  occasional  improvements  on  his  author,  and  entering  his  observations, 
&c.  on  the  maigin.  His  genius  and  attention  soon  attracted  the  favourable 
notice  of  Dr.  Barrow,  at  that  time  one  of  the  most  eminent  .mathematicians  in 
England,  who  soon  became  his  steady  patron  and  friesf).  In  1664  he  took  his  de- 
gree of  B.  A.  and  employed  himself  in  speculations  and  experiments  on  the  na^ 
ture  of  light  and  colours,  grinding  and  polishing  optic  gUwses,  and  opening  the 
way  for  his  new  method  of  .fluxioqs  and  infinite  series.  ^  The  next-  year,  the 
plague  which  raged  at  Cambridge  obliged  him  to,  retire  into  the  country  ;  here  he 
laid  the  foandatioii  of  his  universal  system  of  gravitation,  the  first  hint  of  which 
be  received  from  seeing  an  apple  fall  from  a  tree ;  and  subsequent  reasoning 
induced  him  to  conclude,  that  the  same  force  which  brought  down  the  apple 
might  possibly  extend  to  the  moon,  and  retain  her  in  her  orbit :  he  afterwards 
extended  the  doctrine  to  all  the  bodies  which  compose  the  solat  system,  and  . 


P4  RT  VIII,  INTRODUCTION.  ?85 

to  that  of  L^ibnite,  or  to  both,  for  both  laid  claim  to  the  in- 
vention.   No  sooner  was  the  method  made  public,  thail  9 

d€monstra;tc4  the  same  in  the  mo^'  ^dent  manner,  GeaGnniiif  the  laws  iriiidl 
Kepler  bad  discovered^  by  a  laborious  train  of  obseryation  and  reasoning  |( 
namely,  that  **  the  planets  move  in  elliptical  orbits ;"  that  "  they  describe 
equal  areas  in  equal  times ;"  and  that  "  the  squares  of  their  periodic  times  are 
as  the  cubes  of  their  distances."  Every  part  of  natural  philosophy  not  ooly 
T^eived  improvement  by  his*  inimitable  tpach,  ,bQt.  became  a  new  science 
nnder  bis  hands :  his  system  of  gravitation,  as  we  have  observed,  confirmed 
the  discoveries  of  Kepler,  explained  the  immutable  laws  of  nature,  changed 
the  system  of  Oopernicus  from  a  probable  hypothesis  to  a  plaib  and  demon- 
strated truth,  and  eflpectually  overturned  the  vortices  and  other  imaginary 
machinery  of  Des  Cartes,  with  all  the  improbable  epicycles,  deferents,  and 
islamsy  apparatus,  with  which  the  ancients  and  sdtaie  of  the  moderns',  had  en- 
cumbered the  universe.  In  fact,  his  PhUosophia  Naturalia  Principia  Matker 
matica  contains  ap  entirely  new  system  of  philosophy,  built  on  the  sol|d  basis  of 
experiment  and  observation,  and  demonstrated  by  the  most  sublime  Geometry  ; 
and  bis  treatises  and  papers  on  optics  supply  a  new  theory  of  lig^ht  and  colours. 
The  invention  of  the  reflecting  telescope,  which  is  due  to  Mr.  James  Greguryy 
would  in  all  probability  have  been  lost,  had  not  Newton  interposed,  and  by  his 
great  improvements  brought  it  forward  into  public  notice. 

In  1667  Newton  was  chosen  fellow  of  his  College,  and  took  his  degree  of 
M.  A.  Two  years  after,  his  friend  Dr.  Barrow  resigned  to  him  the  mathematical 
chair ;  he  became  a  Member  of  Parliament  in  1688,  and  through  the  interest 
of  Mr.  Montagu,  Chancellor  of  the  Exchequer,  who  had  been  educated  with 
him  at  Trinity  C>o]lege,  our  author  obtained  in  I696  the  appointment  of  War- 
den, and  three  years  after  that  of  Master,  of  the  Mii^ :  he  was  elected  in  1699 
member  of  the  Royal  Academy  of  Sciences  at  Paris ;  and  in  I7O8  President  of 
the  Royal  Society,  a  situation  which  he  filled  during  the  remainder  of  his  life, 
with  no  less  honour  t<^hiiuself  than  benefit  to  the  interests  of  science. 

In  1705|  in  consideration  of  his  superior  merit,  Queen  Anne  conferred  on 
him  the  hoQou]:  of  knighthood:  he  died  on  March  20th,  17^7,  in  the  85th 
year  of  his  age.  Virtue  is  the  brightest  ornament  of  sciience  :  Newton  is  in- 
debted to  this  for  the  bett  part  of  his  fame  ;  he  was  9k  great  man,  and  goodwm 
he  was  g^reat :  to  the  most  exemplary  candour,  moderation,  and  affability,  he 
added  every  virtue  necessary  to  constitute  a  truly  moral  character ;  above  all, 
he  felt  a  firm  conviction  qf  the  truth  of  Revelation,  and  studied  the  Bible  with 
the  greatest  application  and  diligence.  But  such  is  the  folly  bf  man, 
that  the  tribute,  which  is  due  to  the  gaeat  first  cause  alone,  we  trans- 
fer to  the  instrument;  Newton,  Marlborough,  Nelson,  Wellington,  &c. 
have  a// our  praise,  while  the  great  soujrce  of  knowledge,  stren^h,  victory, 
and  every  benefit  we  enjoy,  is  foigotten.  How  would  the  modest  Newton 
have  reddened  with  shame  and  indignation,  could  he  haive  heard  all  the  ex- 
travagant encomiums,  little  short  of  adoration,  which  have  with  foolish  and 


996  GBOUEm.  Part  VIII. 

sharp  und  virulent  contest  eosoed :  at  kngth  the  Rojni 
^ociely  was  appealed  to,  and  a  Committee  iqipoiiited  to  exa- 
mine letters,  papers,  and  other  documents,  and  thence  to 
£onn  a  decision  on  the  claim  of  each.  The  result  of  the 
inquiry  was,  *^  That  Sir  L  Newton  had  invented  hb  method 
hefore  the  year  1669,  and  eoosequeotly  fifteen  years  before 
M.  LeibnitK  had  given  any  thing  on  the  subject  in  the 
JLeipsic  Acts  :'*  the  same  Report  in  another  part  says,  **  that 
it  did  not  appear  that  M.  Leibnitz  knew  any  thing  of  the 
difierential  calculus,  before  his  letter  of  the  21st  of  June, 
1677-"  It  appears  however  that  this  decision,  which  con- 
firmed the  claim  of  our  illustrious  countryman,  did  not  give 
entire  satisfaction  to  the  continental  mathematicians  of  that 
period,  nor  are  their  successors  better  disposed  to  yield  the 
palm  to  Newton;  they  still  contend  that  Leibnitz,  ad- 
mitting that  he  was  not  theirs/  inventor,  (and  some  refuse 
to  concede  this  point,)  borrowed  nothing  of  his  method  from 
bis  rival;  a  fact  which  some  well  informed  Englishmen 
have  much  questioned. 

Other  tracts  containing  improvements  in  Geometry  were 
given  by  Newton;  as,  i.  EnumeratiQ  Idnearwn  Tertii 
Ordinis.  2.  Tract  at  us  Duo  de  Speciebus  et  Magnitudine 
Figurarum  Curvilinearum.  3.  GenesU  Curvitrum  per  Urn. 
bras:  in  these,  as  well  as  in  bis  Principia  and  other 
works,  he  has  for  the  most  part  employed  hb  own  new 
^naly$ii%  by  which  the  doctrine  of  curves  has  been  amaaongly 
extended  and  improved. 

Geometry  had  hitherto  consisted  of  two  kinds,  JElemea* 

Wyt  or  that  which  treats  of  right  lines,  cectilineal  figures, 

the  circle,  and  solids  terminated  by  these ;  and  Higher^  at 

Tramcendetit  Geometryy  which  treats  of  all  sorts  of  curves, 

* 

impious  yrofusioa  been  lavislied  on  bU  memory !  .  His  worHa»  collected  in  S 
▼oluraes  4to.  with  a  TsUuable  Commentary  by  Dr.  Horsley^  were  pubUsbed  ia 
1784. 


pABrr¥ia  INTROmJGTR)N.  28^ 

except  the  circle,  and  the  sdidir  gfeocratcd  by  their  revolu- 
tion :  to  these,  as  has  been  €A>serve€l,  the  diseoyeries  of  Sir 
Isaac  Newton  have  added  a  third,  viz.  the  Sublime  Geom^y^ 
#r  tke  doctrine  and  application  of  fluxions  ". 

Of  those  anthors,  who  have  since  applied  themselves  to 
the  evkiva^on  and  improvement  of  the  new  calculus,  (as  the 
doctrine  of  fluxions  was  called,)  and  to  the  extension  of  its 
applications,  the  following  are  the  names  of  some  of  the  chief; 
vir,  Agnesi,  IVAlembert,  Bossut,  the  Bernoulli's,  Cheyne, 
Cotes,  Craig,  Clairaut,  Colson,  Caifooli,  Condorcet,  Emerson^ 
Euler,  Fontaine,  Fagnanus,.Guisnee,  Le  Grange,  L'H<>pital, 
Hayes,  Hinl^on,  Harris,  Htttton,  Joites,  Jack,  Landen, 
Lorgna,  D^e  Lagni,  Manfred!,  Maseres,  NIaclaurin,  Nicole, 
Nieuwentyt,  Reyneau,  Riccati,^  Raphson,  Rowe,  Smith|» 
Sterling,  Saunderson,  Siuif»on,  Tirj^lor,  Vince,  Walmsley, 
Waring,  &c. 

The  IbllQwittg  inventions,  which  are  either  nearly  allied  to 
the  method,  of  fluxions  or-  capable  of  similar  application, 
have  been  already  noticed  in  the  Introduction  to  Part  III.  viz. 
Dr.  &xx>k  Taylor's  Methodus  Incrementorumy  17 15 ;  Kirk- 
by^s  Bdetriae  of  Ultimaton,  l^iS]  Landen's  Residual' 
Analysis,  1764;  and  Major  Glenie's  Doctrine  of  Vniversat 
Comiparisa»ylJS9f  and  his  Aftecedental  Calculus,  179S. 

It'  has  been  the  error'  and  misfortune  of  some  eminent' 


«  <*On  peat^tiser  kiG^Mtt^ie  de  dMRSr^ntes  flkaniires.  £n  ^I^mentair^,  et 
«» tTHnseendant^^  La  O^m^rie  ^Mnientaire  iw  consididTe  qtie  les  propri^s  d^ ' 
tijgnes  dtoSte*,  det  lignes  elreukiires,  et  dt%  sdltdes  ternilD^s  park;es  fibres:  Lq  ' 
oeff«le«H  &  teiik  fi^re  carviligne  doat  on^'p&rle  datis  les'^l^meos  de  G^o* 

<'<  lA  O^oin4t#«e-traiM0endafite  est  proprem^nfe  celle  qoi  a  pour  objet  toutes 
l4|»€oitflibsidiffi^Bie8*da  cercle,  comme  les  sections  coniqties,  et  Ics  coixrbes' 
dNm  genre  pltii»  ^iev4* 

«<  Far  \^  on  aaroit  trois  divistem  de  la'G^dm^fie :  G^ou^trie  ^l^mentaire, 
•V  de»  ligtRsilK^itei,  etda  cerele  ;  G^m^rie  traftsoendante,  oa  des  conrbes ; 
tx O^ouk^trie  sabUme,  oa  des  nouveaaxcalcah/^    IfAlemherty  EficpcUtpedie, 


258  GEOMETRY.  Part  Vllt 

and  otiierwke  deserving  characters,  to  direct  their  attentiod 
ahnost  exclimvely  to  malhetnatical  demonstration^  whereby 
they  have  been  induced  to,  deny  or  undervalue  the  force  and 
evidence  of  moral  certamty;  the  celebrated  Dr.  Edmund 
Halley  *  was  one  of  these*  Revelation  is  a  subject,  ^hich 
among  very  many  otlvers  does  not  admit  ci  mathematical 
proof;  and  therefore  he  affirmed  with  equal  rashness  and 
impiety,  that  ^'  the  doctrines  of  Christianity  are  incompre-* 
liensible,  and  the  religion  itself  is  a  cheat/*  This  hardy 
declaration  roused  the  iodignation  of  Dr.  Berkeley  %  the 

*  Edmund  Halley  was  born  in  London,  A.  D.  1656.  After  making  coo-' 
iiderable  progress  in  tfanicl^sica  at  St.  Pani*s  Sebool,  and  obtaining  some 
knowledge  of  tbe  mathematics,  he  was  sent  in  1673  to  Oxford,  where  be 
Applied  himself  closely  to  mathematics  and  astronomy.  Having  conceived  thf 
design  of  completing  the  catalogue  of  stars,  by  increasing  it  fi'om  his  own  ob* 
servation  by  those  in  tbe  southern  hemisphere,  he  embarked  for  St.  Helena  ia 
November,  1676;  he  returned  in  1678,  having  completed  his  catalogue,  oa 
which  occasion  the  University  of  Oxford  honoured  him  witb  the  degree  of 
M.  A.  and  tbe  Royal  Society  elected  him  one  of  their  Fellows.  In  1691  ktf 
applied  for  the  appointment  of  Savilian  Professor,  but  being  charged  witk 
infidelity  and  scepticism,  and  his  pride  scorning  to  disavow  the  charge,  be  did 
not  succeed  ;  however  in  1 703  h^  succeeded  Dr.  Wallis  as  Professor  of  Geo- 
metry  at  Oxford^  and  had  the  degree  of  LL.  D.  conferred  on  him.  Id  1713  be 
became  Secretary  to  tbe  Royal  Society,  an  office  which  six  years  after  he  tt* 
signed,  on  being  appointed  Astronomer  Royal :  in  prosecuting  tbe  duties  of  this 
office,  he  is  said  to  have  missed  scarcely  a  single  observation  duridg  eight  tea 
years  which  he  held  it ;  he  died  in  174;2*  ^r.  Halle/s  numerous  obsenrationft  on 
the  heavenly  bodies,  the  winds  and  tides,  the  variation  of  the  magnetic  needlci 
and  other  valuable  tracts  on  mathematical  subjects,  published  separately  or  in 
the  Philosophical  Transactions,  have  rendered  his  name  fomouS  all  over  Europe^ 
-  y  Gewge  Berkeley  was  born  at  ^ileriu  in  Ireland,  in  the  year  1684:  after 
receiving  tbe  first  part  of  his  education  at  Kilkenny  school^  he  became  a  Pen- 
sioner of  Trinity  College,  Dublin,  in  16999  "id  a  Fellow  in  1707  :  in  17S1  be 
took  the  degreesr  of  B.  D.  and  D.  D.  and  three  years  after  was  promoted-  te  tbe 
Deanery  of  Derry,  and  to  the  Bishopric  of  Cloyne  in  1733 ;  in  1753  he  removed 
with  his  family  to  Oxford,  where  be  died  the  following  year.  Besides  tbe 
ri^plies  and  rejoinders  to  which  the  above  dispute  gaVB  birth.  Dr.  B^rkel^ 
wrote  Arithmetica  absque  Algebra^  out  Eudide  Demonstraiaf  1707  ;  a  Muike* 
matical  MiaceHany^  inscribed  to  Mr.  Molineux ;  Theory  qf  Fitumt  1709 ; 
The  Principles  of  Human  Knowledge ,  1710;  Dialogues  between  Hylas  and 
P/tUonus,  1713.  In  tbe  two  latter  it  is  attempted  to  be  proved,  that  the  common 
notion  of  the  existence  of  matter  is  false*;  that  we  eannot  be  certain  that 


P4RTVXIL  INTRODUCTION.     .  tS9 


learned  and  virtuous  bishop  of  Cloyne^  who,  to 'aaseirt  the 
truth  and  honour  «f  injured  religion,  published  in  1734 
The  Analy^.  In  this  work,  whi<:h  is  addressed  to  Ha!ley  as 
an  infidel  mathematician,  he  shews  that  the  mysteries  in 
faith,  &c,  are  unjustly  objected  to,  especially  by  the  mathe* 
maticians,  who,  be  affirms,  admit  much  greater  mysteries, 
and  even  falsehoods,  into  science;  of  which,  he  says,  the 
doctrine  of  fluxions  furnishes  an  example.  This  avowed 
attack  on  a  new  branch  of  science,  the  principles  of  which 
had  not  then  in  every  particular  been  established  with 
sufficient  firmness,  called  forth  the  zeal  and  abilities  of  its 
admirers;  and  produced,  besides  a  direct  answer,  as  it  is 
supposed  by  Dr.  Jurin,  Robins'-s  Discourse  concerning  the 
Method  of  Fluxions,  &c.  1 735 ;  V^lton's  Vindication,  &c« 
1735 ;  and  Smith's  'New  Treatise  of  Fluxions,  with  answers 
to  the  principal  objections  in  the  Analyst,  1737:  but  the 
most  complete  vindication  of  the  method  of  fluxions  to  which 
this  contest  gave  rise,  together  with  a  firm  establishment  of 
its  principles,  &c.  are  to  be  found  in  Maclaurin's  Complete 
System  of  Fluxions,  with  their  application  to  tlie  most  con-- 
nderable  Problems  in  Geometry  and  Natural  Philosophy, 
In  2  vol.  4to.  published  at  Edinburgh,  in  17^2 :  this  is  indeed 
the  most  complete  and  comprehensive  work  on  the  science 
that  has  ever  yet  appeared. 

Of  the  modern  elementary  writers  on  Geometry,  who  have 
given  systems  of  their  own,  and  not  strictly  followed  Euclid, 
the  following  are  the  principal;  viz.  Borelli,  Pardies,  Wolfius, 

there  are  any  such  things  as  external  sensible  objects ;  and  that  they  are, 
as  far  as  we  can  know,  nothing  more  than  mere  impressions  made  upon  the 
mind  by  the  immediate  act  of  God,  according  to  certain  rules  called  laws  of 
nature.  He  was  a  truly  excellent  man,  and  the  line  by  which  Pope  has 
characterised  him,  by  ascribing  to  him  <<  every  virtue  under  heaven,"  is  said 
not  to  have  for  exceeded  the  truth.  In  addition  to  the  above  works,  h^ 
wrote  The  Minute  PhUoM/pher  ;  wn^  tracts  on  religious  and  political  subjects  % 
Siris,  or  the  Tirtaes  of  Tar  Water  -,  and  another  piece  on  the  same  subject. 


^m  GEOJifitRY.  Part  Vlli 

Stufrifttt%  IttMrfiMn^  Mttrch^fti^  Hfamilton,  Emerson,  Sinip- 
sbii,  Bonwycdstle,  and  Button^  those  of  the  three  last  are 
valuable  and  useftH  perferftitfn^s.  Those  who  have  writtcii 
dii  the  ^object  6(  pratotic^l  Geom«ry,  are  Bayer,  Bonny- 
eilstle,  CkVkid,  Gantd^rus,  Gregory,  Herigoto,  Hawneyy 
Hukius,  Kapler,  Ltgiitbody,  Le  Oerc,  Ikfallet,  Ozanam/ 
Ramutf,  Reinhold,  Scliwinterus^  Seheffelt^  Tacquet,  Voigtel,- 
Wolfiiis,  and  many  othei^. 


i ' 


PA^t  niL.     USEFULNESS  OF  GEOMETRY.  S4J 


ON  THE  USEFULNESS  OF  GEOMETRY. 

W  O  question  is  more  frequently  asked  by  beginners  in  Geome- 
iryj,  than  the  following:  Of  what  use,  u  the  study  of  EucluVs 
Elements  ?  The  industrious,  the  idle,  the  sensible,  and  the  dull, 
from  different  motives,  are  equally  concerned  in  the  inquiry : 
they  almost  daily  agitate  It  with  a  4egree  of  importunity, 
which  sometimes  proves  troublesome  to  the  Tutor*  because  he 
iSnds  himself  incapable  of  answering  'the  question  compktely  to 
his  own  or  their  satisfaction.  The  difficulty  hqpever  lies  not  in 
the  ignorance  of  the  Tutor,  or  the  want  of  usefulness  in  the 
science,  but  in  the  nature  of  things :  for  no  art  or  science  whatever 
can  teach  its  own  use ;  how  then  can  one,  who  is  learning  merel|F 
the  principles  of  Geometry,  expect  to  understand  fully  its  use- 
ftilness,  or  that  his  Tutor,  however  learned  he  may  be,  can  by 
any  explanation  do  justice  to  a  science,  of  which  the  various  and 
useful  applications  will  perhaps  never  be  completely  deter- 
mined ?  To  try  to  satisfy  alUthe  absurd  and  vexatious  scru[)]es, 
which  the  idle,  the  querulous,  or  the  captious,  please  to  stajt 
against  any  braflRrh  of  learning,  would  perhaps  be  a  vain  attempt ; 
but  it  will  be  proper  to  advise  the  diligent  and  well-disposed  stu^ 
dent,  (and  to  sucli  the  advice  can  hardly  be  needful,)  that  it  is  his 
duty,  and  will  be  to  his  advantage,  to  study  attentively  and  without 
scruple,  any  branch  of  learning  which  his  friends  may  think 
proper  to  recommend  to  him  as  useful,  and  which  the  experience 
of  wise  and  good  men  in  every  age  has  proved  to  be  so. 

But  in  the  present  instance,  an  implicit  reliance  on  authority 
is  not  at  all  necessary ;  the  obvious  uses  of  Geometry  are  suffi- 
cient to  recommend  it  to  the.  candid  and  impartial  inquirer  ^ 
some  of  these  we  shall  briefly  enumerate.  Gecnnetry  is  useful* 
as  it  4|)pliea  to  the  businesses  and  concerns  of  society,  and  as 
fua€laroental^  to  other  sciences  and  arts  connected  with  tKem. 
Whatever  relates  to  the  comparison,  estimation,  &c. of  distances, 
spaces,  and  bodies,  belongs  to  Geometry ;  and  consequently  on 
its  principles  and  conclusions  immediately  depend  Mensuration, 
Surveying,  Perspective,  Architecture,  Navigation,  Fortification, 
with  many  other  branches  equally  conducive  to  public  benefits 
ia  sfaort>  it  is  difficult  to  acquire  a  tolerable  degree  of  know- 
VOL.  11.  R 


242  GEOMETRY.  Part  VIIL 

ledge  in  philosophy^  or  any  art  or  science,  \tithoat  some  ac- 
quaintajQce  with  Geometry. 

In  addition  to  the  direct  and  practical  uses  of  the  science, 
there  is  another,  ivhich  Lord  Bacon  calls  "  collateral  and  inter- 
venient."  Geometry  strengthens,  corroborates,  and  otherwise 
improves  the  reasoning  faculties,  inuring  the  mind  to  patient 
labour,  teaching  it  method,  and  supplying  it  with  the  means  of 
contriving  and  adopting  proper  expedients  for  the  prosecution 
of  its  researches.  GeoAietry  may  then  be  justly  con^dered  as  a 
highly  valuable  science,  both  with  respect  to  its  practical  appli- 
cation, and  as  a  complete  model  of  strict  demonstration :  and 
in  the  latter  view  it  recommends  itself  to  the  diligent  attention 
ofevery  lover  of  truth. 

In  what  follows,  we  shall  treat  of  Geometry  in  the  two-fold 
tiew  abm'e  explsdned,  by  briefly  shewing  the  practical  applica- 
tion of  Euclid's  doctrine,  and  likewise  by  considering  it  purely 
as  a  system  of  demonstration. 

The  demonstration  of  a  proposition  does  not  depend  on  the 
correctness  of  the  diagram,  which  therefore  may  be  drawn  by 
hand 5  but  in  the  practical  uses  ot  the  propositions  which  we 
mean  to  exemplify,  accurate  figures  should  be  made,  and  for 
this  purpose  instruments  must  be  employed  :  we  will  therefore 
give  a  brief  description  of  such  instruments  as  are  necessary 
for  the  construction  of  figures,  and  explain  their  fitrther  uses 
hereafter,  repeating,  that  the  instruments  are  by  no  means  neces* 
sary  to  the  demonstration, 

DESiCRIPTION  OF  A  CASE  OF  MATHEMATICAL  IN- 
STRUMENTS. 

A  common  pocket  case  of  Mathematical  Instruments  cod^ 
tains,  1.  a  pair  of  Plain  Compasses  5  2.  ajiair  of  Drawing  Com" 
passes  >  to  the  latter  belong  3.  a  Port  Crayon,  4.  a  fiettiog 
Pen,  and  5.  a  Steel  Pen  :  6  *a  Drawing  Pen,  with  7.  a  Pointer  5 
8.  a  Protractor  3  9.  a  Plain  Scale  5  10.  a  Sector  j  11.  a  Parallel 
Uul«r;  and  12.  a  Black-lead  PenciP.  '         '     • 


'  •  Cases  of  Mathematical  Instrtimciits  may  be  had  at  all  prices,  £roBi  five  shil- 
ling) tQ^six  guineas ;  a  case  that  costs  tweDty-fiveor  thirty  sbittiDgv  will  be  suft 


Part  VIII.     MATHEMATICAL  INSTRUMENTS.         «43 

The  PLAIN  COMPASSES  are  used  for  the  following  piu:- 

poses: 

-    1.  To  draw  a  blank  or  obscure  line  by  the  edge  of  a  rulerj 
through  any  given  point  or  points. 

2.  To  take  the  distance  between  two  points,  and  apply  it  to 
any  line  or  scale  $  or  to  take  the  length  of  one  line,  and  apply 
it  to  another. 

3.  To  measure  any  line  by  taking  its  length  between  the 
points  of  the  compasses,  and  apply  them  to  the  divisions  of  a 
proper  scale. 

4.  To  set  off  any  proposed  distances  on  a  given  line. 

5.  To  describe  obscure  circles,  intersecting  ai'cs,  &c. 

G.  To  lay  off  any  propoeN^d  angle,  and  to  measure  a  given 
angle,  by  means  of  a  scale  of  chords,  &c. 
The  DRAWING  COMPASSES  ^  one  of  the  legs  is  filled 


ficiently  good  to  answer  the  leartier's  purpose,  and  be  should  not  go  tnueh  ttn« 
itt  that  price.  ^  M^gmines  or  ooaiplete  collection  of  every  kind  oi  aseful 
drawing  instrument,  will  cost  from  five  to  forty  guineas. 

lo  using  the  instruments,  lines  and  figures  should  be  drawn  as  fine,  neat, 
Md  exact  as  possible ;  the  paper  on  which  the  drawing  is  made  should,  if  pos- 
sible, not  be  pricked  through  or  deeply  scratched  with  the  compasses ;  i% 
should  be  laid  on  a  quire  of  blotting,  or  other  paper,  daring  the  operation  \ 
sod  the  drawer  should  sit  so  that  the  light  may  be  on  his  left,  and  not  by  any 
nwans  in  front.  The  drawing  pen  should  not  be  dipped  in  tlie  ink,  but  ink 
shonld  be  taken  from  the  stand  with  a  common  pen,  and  put  into  it.  The 
points  of  the  instruments  should  be  cleaned  and  wiped  quite  dry  after  they 
kave  been  used^  and  every  means  employed  to  guard  against  rust,  which  will 
otherwise  spoil  the  instruments. 

^  In  the  best  sort  of  compasses,  the  pin  or  axle  is  made  of  steel,  as  ako  half 
the  joint  itself,  as  the  opposite  metals  rnbbiug  on  each  other  are  found  to 
ivear  more  equally  ;  the  points  should  be  of  bard  well-'poliflbed  steel,  and  thii 
joint  work  with  a  smooth,  easy,  and  aaifoipii  motion.  In  the  dnMring  eom- 
psases,  the  shifting  point  is  sometimes  made  with  a  joint,  and  fusnished  with  a 
fine  spring  .and  screw  ;  so  that,  having  opened  the  compasses  jaeaW^  to  the  re- 
qatrod  extent,  by  turning  the  screw  the  point  will  be  moved  to  the  true  eiLtciit 
within  a  AatV^  hreadthy  for  which  reason  they  aie  named  Hair  Cow^MiMn. 

There  are  various  otlter  kinds  of  compasses  not  appertaining  to  a  common 
case  of  instruments,  which  are  noi  less  nseful  to  Ae  praatical  geometrician 
than  those  we  have  described;  vie. 

'  I.  Bom  CmnpaMiet^A  imall  sqrt  whtdi  sbat  np  in  a  hoop  ;  tbeir  use  is  to  it* 
scribe  the  circuniferenccs  and  arcs  of  very  small  circles. 

R  ^ 


244  GEOMETRY.  Pam  Vlll. 

with  a  triangular  socket  and  eerew,  to  receive  and  fi»ten  for 
use  the  following  supplementaiy  parts;  viz.  1.  a  STEEL 
POINT  j  which  being  fixed  in  the  toclDrt,  makes  the  com- 
passes a  plain  pair^  having  all  the  uses  above  described. 
S.  A  PORT  CRAYON,  with  a  short  piece  of  blade-lead  or 
slate  pencil,  finely  p<nnted  and  fitted  on  it  lor  drawing  circles 
and  arcs  on  paper,  or  on  a  slate,  3.  A  STEEL  P£N>  for 
drawing  lines  or  circles  with  ink;  the  small  adjusting  screw 
passing  through  the  sides  of  the  pen>  serves  to  open  or  close  them, 
for  the  purpose  of  drawing  lines  as  thick  or  fine  as  may  be  thought 
necessary.  4.  A  DOTTER  %  whidi  is  a  small  indented  wheel, 
fixed  at  the  end  of  a  common  steel  drawing  pen ;  from  which  it 
receives  ink  for  the  pui*pose  of  drawing  dotted  lines  or  cii'des. 

In  the  Port  Crayon,  Dotter,  and  Steel  Pen,  there  is  a  joint 
for  setting  the  lower  part  of  the  instrument  perpendicular  to  the 
paper,  which  must  be  done  in  order  to  draw  a  line  well. 

The  PRAWING  PEN  is  fixed  in  a  iHrass  handle,  and  its  use 
b  to  draw  straight  ink  lines  by  the  edge  of  a  ruler.  The  han- 
dle or  shaft  unscrews  near  the  middle^  and  in  the  end  of'  the 


2.  Spring  CompasKs,  or  IXviderst  made  of  hardened  steel,  haTjog  an  arched 
head,  which  by  itf  spring  opens  the  legs  ;  the  opening  being  directed  by  a  cir' 
eatar  screw,  and  worked  with  a  not. 

3.  Proportional  Covnptuses,  both  simple  and  compound;  their  nses  are  to  di- 
▼ide  a  given  line  Into  any  number  of  equal  parts  \  to  find  the  sides  of  similar 
planes  or  solids  in  any  given  ratio ;  to  divide  a  circle  into  any  number  of  equal 
parts,  &c. 

4.  Trisecting  Compasses,  invented  by  H.  Tarragon,  for  trisecting  arcs  and 
angles. 

5.  Trialtgular  Compasses  with  three  legs,  for  taking  three  points  at  Mioe. 

6.  Tharn'Up  Compasses  are  the  plain  compasses,  with  two  additional  points 
fixed  near  the  b«ttom  of  the  legs,  the  one  carrying  a  port  crayon,  and  the 
other  a  drawing  pen ;  these  are  made  with  fc  joint  to  torn  op,  so  as  to  be  oscd. 
or  not,  as  occasion  may  require. 

7.  Beam  Compassesfor  describing  very  large  circles. 

8.  BUiptieal  Con^Muwes  for  describing  ellipses. 

9.  Spiral  Compasses,  for  describing  spirals. 

10.  Cylindrical  and  SpkeHeal  Compassts,  or  Calt/wHr,  for  mcasariDg  the  dia- 
meters of  cylindrical  andnpherical  bodies,  &o.  && 

c  The  Btotting  Pen,  not  being  easily  cleaned,  soon  bectancs  rusty  and  use- 
less ;  the  best  way  to  draw  a  dotted  tine  is  fir^  to^nw  1^ Jiae^MLpeiK^^JUld 
then  to  dot  it  with  the  writing  o^  dnvwiBg  pen. 


P4itT  Vlll    MATHEMATICAL  INSTRUMENTS.  «46 

upper  part  is  fixed  a  fuie  SUel  Pint  or  POINTER,  for  making 
dots,  small,  neat,  and  with  the  greatest  exactness. 

The  PROTRACTOR «  is  a  brass  sesniciicle  divided  into  ISO 
degrees*  ^d  nuQibered  each  way  from  end  to  end ;  the  exterv 
pal  edge  of  the  Protractor's  diameter  is  called  th^  fiducial  edge^ 
and  IS  the  diameter  of  the  circle*  the  small  notch  in  the  mid^ 
die  of  the  fiducial  edge  being  the  centre.  The  use  of  the  Fro* 
tractor  is  to  measure  any  angle,  to  make  an  angle  of  any  pro* 
.   posed  qumber  of  degrees,  to  erect  perpendiculars,  8w, 

The  PLAIN  SCALE  is  used  for  measuring  and  laying  down 
distances :  it  contains  on  one  side,  a  line  of  6  inches,  a  line  of 
&0  equal  parts,  and  a  diagonal  scale.  On  the  other  side  it  lias 
a  line  of  chords  marked  C,  and  seven  decimal  scales  of  various 
sizes.  > 

The  line  or  scale  of  inches  has  each  inch  divided  into  10 
equal  parts,  and  is  used  for  taking  dimensions  in  inches  and 
tenth  parts  of  an  inch. 

The  line  of  50  equal  parts  being  6  inches  in  length,  is  pro- 
perly a  decimal  scale  of  a  foot>  for  by  it  the  foot  is  divided  into 
10  and  likewise  100  equal  parts.  By  this  line,  and  the  line  of 
inches  above  described,  any  given  decimal  of  a  foot  may  be  re- 
duced into  inches  ',  and  likewise  any  given  number  of  inches  to 
the  decimal  of  a  foot. 

Examples. — 1.  Reduce  .^  of  a  foot  into  inches. 

50      t 
Here,  opposite  30  i»  the  second  line  (for  M==T7^=o;:=-2) 

itmda  2tV  inches,  in  the  first:  therefore  ,^  foot  =^2-^0-  inches. 

2.  Reduce  5-^  inches  to  the  decimal  of  a  foot. 
Opposite  5-iV  in  the  first  line,  stands  45  in  the  second ;  tliere- 
,    fore  St^t  inchess^  .45  foot. 


^  A  Protractor  in  the  form  of  a  right  angled  paraHe)ogniii,i9  not  only  more 
conveai<:ot  for  the  case  than  the  Mmicircular  one,  but  likewise  measures  some 
angles  with  greater  exactness,  and  is  therefore  to  be  preferred.  The  Protractor, 
Scales,  and  Sector,  sbonld  be  made  of  either  iroiy,  steel,  or  silrer,  rather  than 
brass,  for  brass  ii^ttres  the  sight  when  nstd  long  together^  especially  by  candle- 
light. 

The  improTcd  Protractor.lips  an  index  moving  about  the  centre,  cutting  the 
circumference,  and  wiU  set  off  an  angle  tme  to  a  single  minute, 

b3 


246  GEOMETRY.  Pa»t  VllT. 

3,  To  find  tlie  value  of  3  inches.     Jtn.  .95  foot. 

4.  To  find  t!ie  value  of  .15  fixit.     Ans.  1  ineft  A- 

The  Diagonal  Scale  is  likewise  a  centesimal  scale,  for  by  it 
an  unit  is  divided  into  100  equal  parts ;  and  any  number  of 
tho^e  ]y.iTti  may  be  taken  in  the  compasses,  and  laid  down  <m 
pajier  nilh  sufficient  exactness  fbr  most  practical  purposes. 

To  explain  the  constnictinn  and  use  of  the  Diagooal  Scale, 
let  ABCD  be  a  section  of  the  scale,  which  b  equally  divided 
(siip|>ose  into  inches)  fiimi  B  (onards  A  in  the  paints  E,  1,  2,    ■ 

3,  &e.  Let  BC=.BE .-  and  let  each  of  these  be  divided  into  10 
eijiial  parts  in  the  points  marked  by  the  small  figures.  I,  3,  3, 

4 ,  &c.  I,  II,  Til,  IV,  &c.  also  divide  CF  in  the  same  manner  in 
the  points  a,  b,  e,  d,  &c.  and  let  the  lines  passing  through  B,'  E, 
1,  %  3,  be  perpendicular  to  AB,  and  the  lines  kl,  nil,  mill, 
olV,  &c.  parallel  to  it,  join  9  C,  8  a,  7  fc,  6  c,  5d,  Src, 


Since  9  B=B/=aC,  and  9 C  by  its  inclination  to  6C  meets  it 
inC,  if  the  dislaiice  of  yCandifCat  B,  that  is  9B,  be  called 
J,  then  will  their  distance  on  the  next  parallel  marked  /  be 

-'-,  and  at  the  next  parallel  marked  //,  it  will  be  —  :   at  the 
lo  *^  10' 

next  marked  ///,  it  will  be  — ;  at  the  next  marked  IV,  it  will 

be  —  ;  and  so  Od,  deeceasing  successively  by  — ,  down  to  the 

point  C,  whers  the  lines  meetj  and  consequently  the  dbtance  is 
nothing. 

]f  8Bbe  called  3,  then  will  the  distance  fh>m8ato£Coa 
the  parulltfl  marked  /,  be  1.^;  on  the  parallel  marked  II,  \-fji 
■  on  tbe  paiallel  markttti III,  IrVi  oa  the  parallet  marked  IV^ 
)  tV  i  and  the  like  for  other  divisions. 


Pait  Vllf .     MATHEMATlCAIi  ttWTRUMENTS.  24T 

ExAMrx.Bs.^1.  Ikit  it  be  i^uired  to  find  3.4.  on  thrscide. 
Here  it  wm  be  com^enieni  to  begm  at  £$  wkerefwre  if  the 
diiimee  of  itbe  lines  EFmdSfbe  iakm  in  the  compc^us  on  tftif 
jnrailel  marked  IF,  U  wUl  be  3.4,  the  number  required^ 

2.  To  find  7.8  on  the  scale. 

Ea^tend  the  eompassesfrom  ET  to  7  h  on  the'  parailel  marked 
VIII,  and  it  will  be  the  distance. 

3.  To  find  3.45  by  the  scale. 

In  this  ease  we  must  take  each  of  the  primary  divisions 
marked  with  the  large  figures,  I,  %  3,  SfC.for  unity,  and  then  the 
smaller  divisions,  E  I,  SfC.  will  each  represent  one  tenth,  and  the 
parallel  differences  each  one  hundredth;  wherefore  we  must  extend 
the  compasses  from  3  D  to  4e  on  the  parallel  marked  F,  and  it 
ioill  be  the  distance  required. 

fiooo"!  r  ^^ 

100  I  rp.  .„   j         10 

10    ^"^t!^  ^r         1  1  Aiid  eachsuc- 
jUachsub-  !      .1  leegsbe  paral- 
•1  I  f ""rL  K    I     -01  flel  difference 
.01     ^  ^^>  ^     .001 
&c.  J  L  &c. 

The^Dlagonal  Scale  has  the  decimal  and  centesimal  division 
At  each  end,  the  unit  of  one  being  double  that  ^f  the  other,  for 
the  convenience  of  drawing  figures  of  different  sizes  *. 

The  other  side  of  the  Plain  Scale  contains  seven  lines  deci<- 
naally  divided  and  subdivided ;  these  are  called  Plotting  Scales, 
and  serve  to  construct  the  same  figure  of  seven  different  sizes : 
by  the  help  of  these  we  can  accommodate  the  figure  to  the  dimen- 
sions of  the  page  or  sheet  on  which  it  is  required  to  be  drawn. 

The  number  at  the  beginning  of  each  of  these  lines  shews 
bow  many  of  its  subdivisions  make  an  inch. 

The  line  of  chords  marked  C  on  this  side  of  the  Plain  Scale^  is 
used  for  the  same  purposes  as  the  Protractor^  viz.  to  meltsure 
^'lay  down  angles^  ^c.  The  method  of  using  both  will  be 
explained  hereafter. 


*  Tbe.laethod  of  diagonals  was  invented  by  Richard  Chanseler,  an  Englisfi* 
Jaan,  aad  first  published  by  Thomas  Digges,  Esq.  in  his  Jl^,  seu  ScaUt  Mtt^ 
thmatictt,  London,  1573. 

R  4 


k 


«4*  GBOUETRV.  Paht  VIII. 

The  SIiCTOR '  is  nn  tDfltniiDent  coAftisltBg  of  two  fl&t  nilers 
or  legs,  moveabk  on  a  joint  or  9lxw,  Hm  rotddlc  point  of  which 
48  the  centre  *,  it  contaias  all  the  Ikiet  usually  set  on  the  Pkiii 
Scale,  and  several  others,  which  the  peculiar  conetruetioB  (if  thlB 
useful  instnirnent  renders  universal. 

The  hoes  on  the  Sector  are  distinguished  into  two  klads,  sin* 
gle  and  double. 

The  single  lines  on  the  best  Sectors  are  as  follow  i 

1.  A  line  of  Indies  decimally  divided. 

2.  A  line  of  a  Foot  centesiaially  divided  on  the  edge^ 

3.  Gunter*s  line  of  the  Logarithms  of  Numbers,  marked  n 

4.  Logarithmic  Sines s 

5.  Logarithmic  Tangents t 

6.  A  line  of  Chords Cho. 

7.  .     .     .  Sines Sin, 

8.  .     .    .  Tangents Tang, 

9.  .     .     .  Rhumbs Rhum. 

10 I^Oitude Lat 

11.    .     .     .  Hours >  Ifoa. 

13.    .     .     .  Longitude Lon. 

13.  .     .     .     Inclination  of  Meridians     ....       In.Mer. 

14.  .     .     .     Logarithmic  Versed  Sines     ^    .     .    ,    V.  Sin^"^ 
The  doubk  lines  are, 

1.  A  line  of  Lines^  or  equal  parts  ....      marked  Lin. 

2.  .  .  .    Chords Cko. 

3.  .  .  .     Sines Sin. 

4.  .  .  .     Tangents  to  45  degrees Tan, 

5.  .  .  .     Secants Sec. 

6.  .  .  .    Tangents  above  45  degrees Tang. 

7.  .  .  .    Polygons Pol. 

f  The  first  printed  account  of  the  Sector  appeared  at  Antwerp  in  1584,  by 
Gasper  Mordente,  who  says  that  bis  brother  TVtbrietus  invented  the  Sector  ia 
the  year  1554.  Soaic  ascribe  tha  invention  to  Guide  Uhal^Oy  A.  I>:  1568:. 
otbi^rs  again  to  Jnstus  Byrgias,  a  French  matbemattcal  initramcnt  maker,  who 
abo  flonrisbed  in  the  I6th  centary.  Daniel  Speckle  next  treated  of  the  Sector, 
TIC.  at  Strasbarg  in  1 58P,  and  Dr.  Thomas  Hood  wrote  on  the  same  mbject 
at  London  in  1^98,  as  did  Samuel  Foster,  in  a  postbamous  work  pnj^lisbed  at 
London  by  Leyboume,  in  1661.  Many  others  bare  sioee  explained  the  nature 
and  uses  of  this  instrnment ;  but  the  most  complete  account  of  any  will  be 
found  in  Mr.  Robert^'n's  Treatise  of  Mathematical  Instruments. 


PaktVih.  mathematical  instruments.        «4® 

The  scftles  of  Lines,  Glierda*  Skies,  TangcntSi  MmndMt,  Lati- 
tudes, Longitufte»>  Hoiifs,  and  Ind.  Mend,  being  set  on  one  leg 
oolkf,  may  be  u«ed  with  the  instnimaiit  either  ihut  or  Oftn, 
The  scales  of  Inches,  Decimals,  Log.  Numbers,  Log.  Sines, 
Log.  Versed  Sines,  and  Log.  Taofeats»  are  on  both  Ic^  and 
must  be  used  with  the  instrument  open  to  its  utmost  extent. 

The  double  lines  proceed  from  the  centre  or  joint  of  the  Sec- 
tor obliquely,  and  each  is  laid  twice  on  the  same  face  of  the  in- 
strument, viz.  once  on  each  leg.  To  perform  operations  pecu- 
liar to  the  Sector,  or,  as  it  is  called,  *'  to  resolve  proUenis  sector" 
wise,**  its  legs  must  be  set  in  an  angular  position,  and  then  dis- 
tances are  taken  with  the  compasses,  not  only  "  laterally,"  (or  in 
the  direction  of  its  length,)  but  '^  transversely,"  or  '*  parallel- 
wise,"  viz.  from  one  leg  to  the  other. 

The  PARALLEL  RULER  '  consists  of  two  straight  flat  rules, 
connected  by  two  equal  brass  bars,  which  turn  freely  on  four 
pins  or  axes,  fixed  two  on  each  rule  at  equal  distances,  so  that  the 
rules  being  opened,  or  separated  to'  any  distance  within  the  li- 
mits of  the  bars,  they  will  always  be  parallel,  and  consequently 
the  lines  drawn  by  them  will  be  parallel. 

The  BLACK  LEAD  PENCIL  should  be  made  of  the  best 
black  lead,  and  its  point  sci*aped  very  fine  and  smooth ;  it 
is  used  for  drawing  lines  by  the  edge  of  a  saile  or  ruler  where 
ink  lines  are  not  wanted.  Plans  and  figures  which  require 
exactness,  should  be  first  drawn  with  the  pencil,  and  then 
if  they  are  not  right,  it  will  be  easy  to  take  out  the  faulty  part 
with  a  piece  of  India  rubber,  and  make  the  necessary  correction  -, 
after  which  the  pencil  lines  may  be  drawn  over  with  ink.  The 
pencil  is  not  less  convenient  as  a  substitute  for  the  pen  in  writing, 
calculating,  &c.  A  piece  of  good  clean  India  nibber,  of  a  mode- 
rate size  and  thickness,  must  always  accompany  a  case  of  Mathe- 
matical  Instruments. 


ff  The  FdraUel  Ruler  qauiiUy  put.  into  a  case  of  (jastrameatft  is  onLy  six 
inches  long,  #Dd  too  small  for  most  purposes  ;  the  better  sorts  ar«  from  six 
ioches  to  two  feet  in  length,  and  sold  separate. 

Tbe  Double  Parallel  Ruler  consi^  of  three  rules,  so  connected  that  the  two 
exterior  rules  move  not  only  parallel,  but  likewise  opposite  to  eadk  other ; 
fur  some  account  of  its  constructtoQ  aad  use  see  Martm'n  Frincipie*  ^ 
Per^ctive^  p.  2a. 


S50  *  GBOUBTRY.  Past  VIII. 

The  §angomg  short  deflcriptioo  ww  deemed  necessaiy,  tmt 
the  uses  of  the  InBtmaieDts  must  be  deferredycSotil  the  learner 
has  acquired  suflbaent  skill  in  Geometiy  to  understaiid  them. 

OF  GEOMETRY,  CONSIDERED  AS  THE  SCIENCE  OF 

DEMONSTRATION. 

As  the  reader  is  supposed  to  be  unacquainted  with  logic^  it 
will  be  proper  in  this  place  to  introduce  a  few  particulars  taken 
from  that  ait^  which  may  serve  as  an  introduction. 

1.  The  uiind  becomes  conscious  of  the  existence  of  external 
objects  by  the  impressions  it  receives  from  them.  There  are 
five  inlets  or  channels^  called  the  organs  of  sense,  by  which  the 
mind  receives  all  its  original  information  5  namely,  the  eye,  the 
eaTj  the  nose^  the  palate,  and  the  touch :  hence  seeing,  hearing, 
smelling,  tasting,  and  feeling,  are  called  the  five  senses.  This 
great  source  of  knowledge,  comprehending  all  the  notices  con- 
veyed  to  the  mind  by  impulses  made  by  external  objects  on  the 
organs  of  sense,  is  called  sensation. 

^.  Pbrcbption  is  that  whereby  the  mind  becomes  conscious 
of  an  imtpression  -,  thus,  when  I  feel  cold,  I  hear  thunder,  I  see 
light,  &c.  and  am  conscious  of  these  eifects  on  my  mind,  tbis 
consciousness  is  called  perception, 

3.  An  idea  results  from  perception  3  it  is  the  representation 
or  impression  of  the  thing  perceived  on  the  mind,  and  which 
it  has  the  power  of  renewing  at  pleasure. 

4.  The  power  which  the  mind  possesses  of  retaining  its  ideas, 
and  renewing  the  perception  of  them,  is  called  memory  3  and 
the  act  of  calling  them  up,  examining,  and  reviewing  them,  is 

called  REFLECTION. 

5.  In  addition  to  the  numerous  class  of  ideas  derived  by  seu" 
sation  wholly  from  without,  the  mind  acquires  others  by  refiec" 
twn ;  thus  by  turning  our  thoughts  inward,  and  observing  what 
passes  in  our  own  minds,  we  gain  the  ideas  of  hope,  fear,  love, 
thought,  reason,  will,  &c.  The  ideas  derived  by  means  of  sen- 
sation are  called  sensible  ideas,  and  those  obtain^  by  reflec* 

tion,  INTELLECTUAL  IDEAS. 

6.  Erom  these, two  sources  alone  (viz.  sensation  and  reflec- 
tion) the  mind  is  furnished  with  ample  store  of  materials  for 
its  future  operations;  sensation  supplies  it  with  the  original 


Pajt  VIII.        PRINCIPLES  OP  BEASOMING.  «5l 

stock  derived  iVom  Without^  and  reflection  increases  that  stock> 
deriving  other  ideas  by  means  of  it  from  within.  ' 

7.  A  SIMPLE  IDEA  is  that  which  cannot  be  divided  into  two  or 
more  ideas  y  thus  the  ideas  of  green^  red^  hard,  96it,  sweety  &c. 
are  simple. 

8.  A  COMPLEX  IDEA  is  that  which  arises  from  joining  two  or 
more  simple  ideas  togettierj  thus  the  ideas  of  beer,  wine>  false* 
hood^  a  house,  a  square,  are  complex,  being  each  made  up  of 
the  ideas  of  the  several  ingredients  or  particulars  which  compose 
it^  together  with  that  of  their  manner  of  combination. 

9.  In  receiving  its  impressions,  the  mind  is  wholly  passive ; 
it  cannot  create  one  "new  simple  idea :  those  from  ^thout  ob- 
trude themselves  on  it  by  means  of  the  senses,  and  those  from 
within,  which  arise  from  the  mind's  contemplating  the  im- 
pressions it  has  already  received,  are  equally  spontaneous  and 
(with  respect  to  the  mind)  involuntary.  But  although  the  mind 
cannot  create  one  original  simple  impression,  yet  when  it  is 
stored  with  a  number  of  simple  ideas,  it  possesses  a  wonderful 
power  over  them :  it  can  combine  several  simple  ideas  together, 
so  as  to  form  a  complex  one,  and  vary  the  combinations  at  plea- 
sure }  it  can  compare  its  ideas,  and  readily  determine  in  what 
particulars  they  agree,  and  in  what  they  disagree.  Having 
combined  several  simple  ideas  so  as  to  form  a  complex  one, 
the  mind  can  again  separate  or  resolve  this  complex  Idea  into  its 
component  simple  ones :  this  it  can  do  both  completely,  and  in 
part ;  it  can  retain  just  as  many  of  the  simple  ideas  in  compo- 
sition (out  of  the  number  which  forms  the  entire  complex  one) 
as  it  chooses,  and  reject  the  rest ;  and  if  to  this  arbitrary  com- 
bination a  name  be  given,  whenever  we  hear  that  name  pro- 
nounced, the  idea  compounded  of  the  whole  of  the  parts  pre- 
scribed, and  no  more,  occurs  immediately  to  the  mind. 

10.  From  the  comparison  of  ideas  arises  what  is  called  bela- 
I'lON ;  and  among  other  relations  that  which  in  mathematics  is 
called  RATIO,  being  a  relation  arising  from  the  comparison 
of  quantities  in  respect  of  their  magnitude  only. 

11.  In  comparing  several  complex  ideas  together,  we  find, 
that  although  they  differ  with  respect  to  some  of  the  simple  ideas 
of  which  they  are  compounded,  yet  they  agree  in  sonw  general 
character :  thus,  a  triangle  and  a  square  differ  with  respeet  to 


262  GEOMCIT RY.  PArr  Ylil. 

tl^ir  fonPy  t)ie  number  of  their  sides*  and  the  niimbeF  aod  mag- 
nitude of  their  angles ;  but  they  agree  in  one  general  character, 
they  are  both  Jigures,  A  lion  and  a  sheep  differ  widely  from 
each  other  in  many  particulars ;  but  in  their  general  character 
they  agree^  viz.  they  are  both  animals. 

IS.  This  most  important  power  of  the  mind  over  its  oomples 
ideas  is  called  abstraction,  and  the  general  idea  produced  by 
its  operation  is  called  an  abstract  idra. 

13.  An  abstract  idea  then  comprehends  in  one  general  cl3ss> 
not  only  all  the  simple  ideas,  bi^t  all  the  complex  ones  &om 
which  it  is  abstracted :  thus  the  idea  of  beast  is  a  complex  idea, 
and  includes  the  ideas  of  lion,  horse,  bear,  wolf,  rabbit,  &c.  the 
idea  of  hnimal  is  likewise  complex,  including  those  of  man, 
beast,  birdj  fish,  insect,  &c. 

14.  Hence  an  abstract  or  general  idea  is  merely  a  creature  of 
the  mind»  and  can  have  no  existing  pattern  or  aixrhitype :  we 
can  form  in  the  mind  the  abstract  idea  of  a  triangle,  viz.  one 
that  shaU  include  the  ideas  of  all  particular  triangles ;  but  we 
cannot  describe  on  paper  any  figure  capable  of  representing  a 
triangle  in  general,  via.  all  the  varieties  of  triangles  that  can 
be  made. 

15.  Hence  also  whatever  is  true  of  an  abstract  idea  is  likewise 
true  of  every  particular  complex  or  simple  idea  included  under 
it  i  thu8»  if  it  be  pnn^ed  generally  tiiat  two  sides  of  a  triangle 
are  together  greater  than  the  third,  it  follows  that  the  same 
thing  is  thereby  pi'oved,  and  must  be  true  of  each  and  eveqf, 
individual  triangle:  in  like  manner  whatever  is  proved  of  plaSe 
rectilineal  figures  in  general,  will  necessarily  be  trUe  (not  only 
of  every  kind,  but)  of  every  particular  rectilineal  figure  that  can 
be  made ;  thus,  since  it  follows  from  prop.  32.  book  1.  of  foiclid, 
that  all  the  interior  angles  (taken  together)  of  every  rectilineal 
figure  are  equal  to  twice  as  many  right  angles,  wanting  four,  as 
the  figure  has  sides,  the  same  thing  must  be  true  of  each  parti- 
cular kind  of  such  figure  -,  as  of  squares,  triangles,  trapeziums, 
polygons,  &c.  and  likewise  of  every  particular  figure  included 
iwder  those  kinds. 

16.  Upon  an  examination  of  our  ideas  of  the  objects  that 
surround  us,  we  shall  find  that  several  of  them  resemble  each 
other,  except   in  one,  two,  oar  perhaps  more  circumstances  > 


Jabt  VUl.      PRINCIPLES  OP  REASONING.  5253 

now  if  We  leave  out  frDio  otir  consideration  the  particulari 
ill  whidi  they  disagree^  and  retain  those  only  in  which 
they  agree,  we  shall  obtain  the  abstract  idea  of  a  tracias, 
which,  as  it  id  supposed  to  arise  fit>m  the  lowest  possible  degree 
of  abfitraction,  is  called  tbk  inferior  species  ;  and  the  indi- 
viduals which  compose  it,  being  supposed  capable  of  no  subordi- 
nate arrangement,  are  called  farticulars.  If  this  idea  of 
species  be  compared  with  our  ideas  of  other  species,  we  shall  in 
lilce  manner  perceive  that  they  disagree  in  sofne  of  their  circum- 
stances only ;  wherefore  by  leaving  these  out  as  before,  we  shall 
obtain  tlie  idea  of  a  species  superior  to  the  former,  viz.  which  in- 
cludes the  former,  and  one,  two,  or  more  others.  In  like  man^ 
ner  by  continual  abstraction  we  pass  through  the  sticcessive 
gradations  of  species,  until  at  length  we  arrive  at  a  point  where 
no  further  abstraction  is  possible :  the  ultimate  idea  thus  obtain- 
ed, .as  including  the  ideas  of  all  the  several  species,  is  called  a 

GENUS. 

17*  Thus  by  successive  acts  of  abstraction,  a  guinea  is 
gold,  metal,  siitetanee,  being ;  a  herring  is  fish,  animal,  sub- 
stance, being ;  Tray  is  greyhound,  dog,  beast,  animal,  substance, 
being ;  ah  oak  is  tree,  vegetable,  substance,  being ;  James  is 
scholar,  man,  anhmai,  substance,  being,  &c.  In  the  examples  here 
proposed  it  mliy  be  observed,  that  aubgtance  is  common  to  them 
all  J  the^  Idea  of  substance  includes  therefore  those  of  metal, 
imimal,  and  vegetable,  and  consequently  the  subordinate  ideas 
of  guinea,  herring,  Tray,  oak,  and  James.  Substance  then  is 
to  be  considered  as  the  pRoXimatb  ^envs  of  these,  including 
them  a\\',  bring  is  the  highest  ch*  superior  oehus,  and  im-^ 
plies  merely  existence. 

18.  As  a  general  knowledge  of  the  operations  Of  the  mind  in 
componnding,  compaiing,  and  abstracting  its  ideas,  is  necessary 
to  those  who  would  folly  understand  the  plan  and  scope  of  Eu- 
elid,  so  it  will  be  equ«dly  profitable  to  shew,  in  as  plain  a  manner  as 
possible,  how  our  abstract  and  other  complex  ideas  are  nnlbkled, 
so  as  to  make  them  intelligible  by  words  (expressed  either  by 
the  voice  or  writing)  to  others. 

19.  And  first,  simple  ideas  are  expressed  by  words  arbitrarily 
assma^  as  their  repi^eseniatives ;  so  that  whenever  any  word  is 
read  or  proDoimced,  the  idea  it  stands  for  immediately  occurs  to 
the  mind  of  the  reader  or  hearer :  but  should  it  happen  in  any 


454  GBOMETRY.  Pabt  VUI. 

iDstance  otherwise,  tbe  object  whkli  jifoduoes  the  idea  must  be 
presented  to  him,  and  he  muBt  be  informed  that  suck  a  word  a 
the  sign  of  that  idea^  or  should  the  idea  have  two  or  thiee 
different  words  to  express  it,  these  should  all  be.prooouDoed, 
and  probably  the  idea  will  occur  to  him  &om  one  of  them : 
there  is  no  other  method  of  communicating  a  simple  idea  from 
one  mind  to  another.  I  point  a  person  to  the  object,  I  tdii  him 
its  name,  and  immediately  his  minil  associates  the  latter  with 
the  idea  of  the  former,  making  the  name  the  constant  reprc* 
sentative  of  the  idea. 

20.  But  although  simple  ideas  cannot  be  conveyed  to  tlie 
mind  by  any  verbal  descriptioo,  the  case  is  di0erent  with  respect 
to  complex  ideas ;  these  may  be  communicated  with  great  faci- 
lity :  for  since  a  complex  idea  is  composed  of  several  simple  otiss, 
if  the  names  of  the  latter  be  pronounced,  together  with  thei)* 
mode  of  connection,  the  complex  idea  will  immediately  occur  to 
the  hearer;  provided  his  mind  be  previously  furnished  with jts 
component  simple  ideas,  together  with  a  knowledge  of  the 
names  or  signs  by  which  th^  are  expressed* 

21.  It  has  been  shewn,  that  if  the  difference  betweea  indivi- 
duals, agreeing  in  their  general  and  noost  r^nai'kable  properties 
and  circuoistances,  (and  which  is  called  their  nuuekal  oipfeb- 
ENCB,)  be  rejected,  we  obtain  the  abstract  idea  of  a  species;  if 
the  di£ference  between  this  species  and  another  species  (called 
the  spec  I  FIG  dipfrrence)  be  rejected,  we  get  the  ide^of  a 
species,  which  includes  and  is  superior  to  the  former;  and  if  in 
like  manner  we  continually  drop  the  successive  specific  difkf' 
ences,  we  shall  at  length  arrive  at  the  genus,  or  srunmit  of  oar 
research. 

29.  Hence  an  easy  method  fHresents  itself  of  unfolding  a. com- 
plex idea,  or  of  communicating  our  con^lex  ideas  to  other  per- 
sons by  means  of  definitions,  namely  by  following  a  contrary 
order :  we  name  the  genus  or  kind,  to  this  name  we  jpin  that  of 
the  specific  difference,  and  both  together  will  convey  to  the  mind 
of  the  hearer  the  complex  idea  we  mean  to  describe.  Agflia,  if 
we  consider  this  specie?  as  a  genus,  and  join  to  it  the  next  lower 
specific  difference^  the  result  will  give  a  precise  idea  of  the  nest 
inferior  species  3  proceeding  in  this  manner  thrqiiigh  all  tbe  suc- 
cessive ranks  of  species  t«  the  lowest,  to  which  jpining  tbe 
numeral  difference;  We  at  leii^th  obtain  the  idea  of  a  particular 


PartVIII.      PRINCIFLBS  C»*  BSASONING.  S5& 

iDdKidual  t  thii  proteaa  U  exeoapUfied  in  th^  defioUions  prefixed 
to  the  Elements  of  Eudid. 

'  23.  It  tmj  be  noticed^  that  in  Imyiog  down  a  definitbn  there 
18- no  necessity  to- have  recourse  to  the  Kighesi  genus,  or  even  %o 
remote  species ;  the  proximate  superior  spmiies  may  in  ail  casea  be 
taken  for  the  genwf,  and  as  that  is  always  su]^M»ed  to  be  kuown» 
we  have  only  to  add  to  its  name  that  of  the  specific  differenee. 

94.  Thus,  in  defining  a  right  an§^ed  trangle^  I  describe  U  to 
be  a  triable  lisving  a  right  €mgle :  triangle  ia  the  species  or  liind 
to  which  the  figure  belongs,  and  its  having  a  right  angle  is  tiie 
circumstaai^  by  which  it  di^rs  firom  every  other  species  of  tri- 
angle. 1  do  not  say^  *'  a  right  angled  triangle  is  a  being,**  or  "  a 
figure,**  or  '^  a  plain  figure,**  these  species  are  too  remote  3  but 
1  Gall  it  a  '*  triangle,**  which  is  the  proximate  speciea  to  right 
angled  triangle :  now  the  idea  of  triangle  being  previously  known, 
that  of  aright  angled  triangle  will  likewise  be.  known  by  Rei- 
fying, that  it  has  a  right  otsgle, 

^5.  The  obvious  use  of  definitions  is  to  fix  our  ideas,  so  that 
wbenever  a  definition  is  repeated,  Ibe  precise  idea  intended  by 
it,  and  no  other,  may  immediately  occur  to  the  mind  >  and  when- 
ever, an  idea  m  present  to  the  mind>  its  definition  may  as  readily 
occur. 

S6.  Adequate  and  precise  definiti<M)s  may  then  be  considered 
as  the  true  foundation  of  every  sysl^em  of  instruction ;  for  when 
our  ideas  are  fitly  represented  by  words  whose  signification  is 
fixed,  there  can  be  no  danger  of  mistake  either  in  communis 
eating  or  receiving  knowledge. 

27.  There  are  some  ideas  of  which  the  mind  perceives  their 
^Igreement  or  disagreement  immediately,  without  the  necessity  of 
t^Pgmn&at  or  jtt^ooff  this  necessary  determination- of  the  mind 
IS'  called  a  jupoment,  and,  the  evidence  or  certainty  with 
which  it  spontaneously  acquiesces  in  tins  determination^  is  called 
INTUITION  3  also  the.  irresistible  force  with  which  the  mind  is 
impelled  to  its  determination,  is  called  intuitive  evipenc£. 

98.  The  feculty  by  which  we  pei'ceive  the  validity  of  self- 
evident  truth,  is  Called  common  senae  »,  which  signifies  "  that 
instinctive  persuasion  of  truth  which  arises  from  tHtuiiive  evi- 


»  

ff  See  An  Eiaay  on  th$  nature  qnd  iptmutahility  of  Truth,  by  James  Beattie, 
Llf,f^^p,  1.  c.  1. 


856  GBOMfifiatr.  pAiit  vni. 

dente:**  it  is  aoteeofolit  to  scienoe,  and  altko^gli  no  jMirt  of  it, 
yet  *^  it  is  the  foundation  of  all  reaaonteg.** 

39.  There  are  some,  ideas  of  which  the  mind  cannot  perceive 
the  agreement  or  di8agi«emeat»  withont  the  intervention  .4if 
others,  which  the  logicians  call  jaiddie  terms ;  the  proper  dioiee 
end  management  of  these  are  the  chief  hosiness  of  science. 

30.  These  midcye  temiEr  serve  as  a  Chain  to  connect  two  re- 
tnole  ideas,  that  is^  to  connect  the  subject  of  our  inqttlry  with 
some  self-evident  truth :  thus,  suppose  A  aad  D  to  be  two  ideas, 
of  whicli  the  truth  of  ^  is  self-evident^  but  that  of  JD  not  so; 
and  let  it  be  admitted  that  J  and  D  cannot  be  brought  toigetheo 
so  as  to  afibrd  the  requisite  means  of  comparison  fbr  determiniag 
their  relation ;  In  this  case  I  must  seek  for  some  idtermediite 
ideas,  the  first  of  which  is  Connected  with  A,  the  last  with  I>,  and 
the  succeesive  intervening  ohes  with  each  other :'  let  these  be  B 
and  C;  now  if  it  be  iHtuitivelif  certain,  that  B  agtces  with  Ay  that 
C  agrees  with  B,  and  that  D  agrees  with  C>  it  Mlows  with  no 
less  certainty  that  D  agrees  with  ^*  this  latter  certainty  is  how- 
ever not  intuitit>e,  btit  of  the  kind  which  is  called  denwnstrabk  \ 
and  the  process  by  which  the  mind  becomes  conscious  of  this  de- 
monstrable certainty  is  called  itSAsoNiKO,  or  demj>nstkation. 

31.  Every  well  ordered  system  of  science  will  therefore  con- 
sist of  DEFINITIONS  and  PROPOSITIONS :  defifdtims  are  used  to 
expfaiin  dbtinctly  the  meaning  of  tb^  terms  employed,  and  to 
limit  and  fix  our  ideas  rMpecting  them  with  absohite  precision. 
That  which  affirms  or  denies  any  thing,  is  called  a  proposition  : 
I  am  ).the  sun  shines ;  vice  will  inevitably  he  punished  5  two  and 
three  are  five,  &c.  are  prc^po^ions. 

39.  Propositions  are  either  self-evident,  or- demon^ral4e; 
and  since  thdre  cAn  be  no  evidence  tfupeHor  to  intukioni  it  !bi« 
lows  that  self-evident  propositions  not  only  requite  no  proof*  as 
some  have  said,  they  admit  of  none  '. 

-;  I  '  -    ■  —  -/    ^-  -  -  .^ 


^    «   »  I    ■  «        III  I    II  III  I  HI    It^J—fciAjl  ■       >     ■  I     ■»-»— i»— .<.^J»^>. 


^  Every  itep  of  a  cteSaonstAtion  mtist  follow  frtim  tnitli*  pmtvhuatp  kn^wn 
-with  inhtUive  ctttAnty  \  bnt  the  conclvuion  or  tbiof  to  ht  proved,  depeadiiig 
HA  a  ooDnected  Mries  of  intvitioas,  and  no  less  cettein  than  each  of  the  pre- 
ceding steps,  is  nevertheless  not  dignified  with  the  name  of  intuitioH;  It  is 
obtained  (as  we  have  noticed  above)  by  demonstration, 

i  For  every  proposition  is  proved  by  means  of  others  which  are  more  evident 
than  itself,  but  nothing  can  be  more  evident  than  that  which  is  setf-evideni ; 
wherefore  a  self-evident  proposition  can  admit  of  no  proof. 


Fart  WIL      PRINC1FZ4CI  <»  RBASONINQ.  85» 

33.  DtmoaAahle  propaaiHons  fti%  sudh  to  do  not^&dttiR  of  « 
determination  by  any  single  efot  of  the  niiad;  to  9Ttiv€  lit<  a 
consciousness,  of  ^Itveir^nilh^  We  ate.pUiged  fnequently.  (as  we 
kare  obssrrcd  akovo)  to  have  recourse  to  several  intermediate 
Btej^  the  first«of  which  resta  mVtk  intuitive  certainty  on  tome 
self-evident  trutb^llfe  r^  witli  tie  saiwe  intuitive  certainty  4^ 
pend  on  each  other  in  succession,  and  the  prifpsltiAi,  or  tpoth 
to  be  proved,  depends  with  Mke  intuitive  certainty  on  the  la^  of 
these ',  so  that  the  thing  to  be  pijved  must  evident^  be  true,  since  • 
it  depends  on  a  self-evident  troth,  which  d^ndance  is  consti-* 
toted  and  shifnrn  by  a  series  oft- truths  following  or  flowing  from 
each  other  witk  intuitive  certainty. 

34.  Propositions  are  likewise  dtvidyd  iotO  practical  a«i  tkeo^ 
T^ical.  A  practical  proposition  is  that  which  pfoposes  soma 
o|ieration^  or  is  immediately  directed  to,  and^rminites  in  pfac^ 
tie«;  thiS)  to  draw  a.  straight  line,  to  describe  a  cirde»  to  con« 
|(nicta'triai^le,  &c.*are  pfac^ical  profmtitiont^ 

35.  A  theoretical  propositioa  is  that  in  which  some  troth  is 
poposed  fbr  consideration,  and  which  terminates  in  theory: 
thus,  the  whole  is  greater  than  its  fiart  i  contentmelit  Is  better 
tlian  richcfs  i  two  sides  of  a  triangle  are  togel^r  greater  than 
the  thirds  4tc.  are.«^^oreitca2  proposi^isfis. 

^6.  ProposUioi^,  both  practical  and  the^|«ticai>  ar6  either 
H^'^midentOY  JtemfiMtrahle, 

37.  A  »elf^€videni  practical  prapoiition  is  named  by  Euclid  a 
I'OSTtJLAXE^^od  a  self-evident  iimnreHcal  proposition,  an  ax  mac. 

38.  A  demgHstrable  practical  propofition  is  called  a  PKOfiLBic^ 
^KdemonstraUe  theoretical  propgaition^  a  tdbosbm. 

38.  Hence,  postulates  and  axioms  being,  intuitive  truths  or 
nsixims  of  common  sense,  admit  of  no  demonstration  ^  hut4>ro- 
hlems  and  theorems  not  being  self>-evident5  therefore  require  to 
be  deoiohs^ppttei^.  ^ 

4Q,  Definitions,  postulates,  and  axioms,  m^  the  sole  principles 
on  which  demonstr^ion  is  founded ;  this  foundati^,  narrow  and 
sBght  as  it  niayseem^is  comiuually  extended  and  strengthened 
by^^l^Ml  eonstaot  accession  ol  new  materiab<j  for  every  truth, 
as  soon  as  it  is  demonstrated^^  hpooams  a  principle  of  equ«l 
force  and  validity  with  truths  whieh  are  8e)f*evident,  and  rear 
toniiig  may^be  btdlt  on  it  with  the  same  degree  of  certainty 
as  OB  Iheui:  thus  reasoning,  fay  its  p^igress,  continually  inir 

VOL«  II.  s 


«&  GBomerBY.  pam  vul 

creases  its  Iwb^iaid  the  powers  of-tiM  mind,  ampie  at  tiiey  are, 
must  lieAce  be  ioadequate  to  the  use  of  all  that  vaat  aocunm- 
lation  and  mrietyr  of  means,  provyed  for  tlieiK  employment. 

41.  When  ftrom  the  exanriaatmn  and  csmpaiison  of  two 
known  troths  a  thuti  follows  as  •&  evident  conseipieniee^-the 
known  truths  are  called  niHiiiafts^^ha-tnlth  derived  an  ikfs* 
B«»CB>  amk  the«st  of  deriving  it  finun  the.  f»reiiitset  is  called 

ORAWIHG,  or  MAKING  AN  INPBRENCB. 

Thus»  if  4«M^  and  two  be  equ^to  fopr,  and  three  and  one  be 
equal  to  four,  these  being  the  premitm*  it  follows  as  an  infereuoe 
that  two  and  two,  and  ihr^  and  «ae»  are  etpial  to  |)ie  same  ^vis, 
to  four) :  noW>  since  things  that  are  equal  to  thi^same  are  equal 
to  0ne  another,  it  follows  as  a  further 'ifi^cnctf,  that  tW0  and  two 
are  equal  to  three  and  one, 

42.  This  axamiAs  will  fiimish  a  general,  although  necessaoly 
an  imperfect,  notion  of  Euclid*s  method  of  proving  his  propo* 
sitions :  his  demoikitratiows  are  nothing  more  than  a  regular  a«$ 
well  connected  chain  of  suecessive  intuitive  inferences,  the  first 
of  which  is  drawn  from  self-evident  premises,  and  the  last  Hw 
thing  which  was  proposed  to  be  prored. 

43.  Hence,  although  demonstration  is  necessarily  founded  on 
self-evident  truth,  it  is  noi  at  all  necessary  in  every  case  that  Ave 
should  have  recoipse  to  first  principles^  for  this  woukl  make  de- 
monstration a  most  unwieldy  machine,  requiring  too  mueh  la- 
bour to  be  of  extensive  use :  every  inference  fiedrly  drawn  from 
self-evident  principles  is  of  equal  validity  with  inl^itive  truth, 
and  may  be  employed  for  the  same  purposes ;  thus  £nclid>  in 
his  demonstrations,  makes  U8#  of  the  truths  he  has  befose  do* 
m<Mistrated  with  a  confidence  as  weU  foiinded  as  though  Uiey 
were  self-evident,  and  merely  refers  you  to  the  proposition  wheit 
the  truth  in  question  is  proved.  This  saves  a.  great  deal  of  trou* 
ble,  for  truths  once  established  may  with  the  stnolwst  propriety 
be  employed  as  principles  for  the  proof  and  discovery  of  others. 

44.  It  frequently  happens  in  the  course  ;(if  a  demonstration, 
that  an  inference  presents  itself,  which  is  useful  in  other  cases^ 
although  not  imaawdiately  so  with  respect  to  ther  proptosillon 
snder  conskleration ;  when  such 'an  inlsrence  is  made,  it  is  ealled 
a  COROLLARY,  and  the  act  of  making  it  naouciNG  a  cobollart. 

45.  A  LEMMA  b  a  proposition  not  immediately  connected  with 
the  subject  in  hand,  but  is  assume4  for  the  sake  of  shorteniflig 


Pabt  WIL        on  EUCLID'S  JPIRST  BOOK.  86^ 

<]it  draiom^Mien  of  ofte  or  nunrt  of  the  isttoNiiig  pn^osi^ 

tiOBB. 

46.  A  SCBQUUM  lA  a  note  or  oUeryaUon,  aefving  to  coBfiroa» 
explahi,  illustrate,  or  apply  the  subject  to  which  it  refers. 

47.  Euclid  in  his  fifemeots  ewplo^FS  two  methods  for  establish* 
iog^  the  truth  off  what  he  intends  to  prove  f  namely,  direct  and 
imdirett,  both  proceeding  hp  a  series  of  inferences  in  the  manner 
explained  above.  Art.  41>  43« 

48.  A  DiftECT  DBMONSTKATioN  is  that  wiiich  proceeds  from 
intuUive  or  demomtrated  truths^  by  a  chain  of  successive  infe- 
rences»  the  last  of  which  is  the  thing  to  be  proved. 

49.  An  inoirbct  or  apologicai.  d&monstration,  or  as  it 
is  frequently  named,  aanucTio  ad  .  absukdum,  consists  in  as- 
suming as  true  a  proposition  which  directly  contradicts  the  one 
we  mean  to  prove  -,  and  proceeding  on  this  assumption  by  a  train 
of  reasoning  in  all  respects  like  that  employed  in  the  direct 
method^  we  at  leogtk  deduce  an  inference  which  contradicts 
seane  self-evident  or  demonstrated  truth,  and  is  therefore  absurd 
and  Mse ;  consequently  the  proposition  assumed  must  be  false, 
aoA  therefore  the  proposition  we  intended  to  prove  must  by  a 
necessary  consequence  be  true,  since  two  contrary  propositions 
cannot  be  both  true  or  both  false  at  the  same  time  ''. 

NOTES  AND  OBSETEIVATIONS  ON  SOME  PARTS  OF 
THE  FIRST  WyOK  OF  EUCLID'S  ELEMENTS. 

5<X  The  first  book  of  Euclid's  Elements  contains  the  princir 
pies  of  all  the  following  books ;  it  demonstrates  some  of  the  most 
general  properties  of  straight  lines,  angles,  triangles,  parallel 
lines,  parallelograms,  and  other  rectilineal  figures,  and  likewise 
the  possibility  and  method  of  drawing  those  lines,  angles,  and 
figures.  It  begins  with  definiiiom,  wherein  the  technical  terms 
necessarily  made  use  of  in  this  book  are  explained,  and  our  ideas 


k  Mathematical  demoostrations  "  are  notbing  more  than  series  of  entliy- 
meines;  «Tery  thing  is  concluded  by  force  of  syllo^sm,  only  omitting  the 
{[reinises,  which  either  occur  of  their  own  accord,  or  are  recollected  by  means 
of  quotations."  This  might  easily  be  shewn^  by  examples,'  but  the  necessary 
«xplaoatioii8,  &c«  w<^iihi  take  up  too  much  room.  See  on  tfali  subiect  The  Ble- 
menU  tf  Logic,  hy.  PTmsm  Jhtneemy  Professorvf  PhOmoplaf  in  thM  Maritchmi 
C^Utge  t^  ,M9riecM,  9tb  £4.  a  book  wbiob  ought  to  be  recommended  to  the 
pemsal  of-  students  in  Geometry. 

s  2 


«M  *€»OMETRY.  Past  Vlli 

respecting  tliein  Mcertained  and  fixed;  next  are  1^  down  tlie 
poitulates  and  axiomsy  or  those  self-evident  truths^  which  consti- 
tute the  basb  of  geometrical  reasoning  i  and  lastly,  the  propo- 
tUions  (whether  problems  or  theorems)  are  given  in  the  order  of 
their  connexion  and  dependance^  the  denionstcatkms  of  which 
depend  solely  on  the  definitions,  postulates,  and*  axioms,  previ. 
ously  laid  down  3  and  from  the  demAnstratiQas  uae^  corolla- 
ries ure  occasionally  derived. 

X)n  the  Dejinitions,  *  • 

51.  Definition  1.  The  first  definition,  as  given  by  Euclid, 
and  likewise  in  Dr.  Simson's  translation,  has  beerf  justly  com- 
plained of  as  defective  5  it  includes  no  positive  property  of  a 
point  5  we  learn  from  it  not  what  a  point  is,  but  what  it  is  not  5 
"  it  has  no  parts,  nor  magnitude  :**  now  since  every  adequate 
definition  admits  of  conversion,  let  us  try  the  experiment  on  this ; 
when  converted  it  will  stand  thus,  "  that  which  has  no  parts 
nor  magnitude  is  a  point;"  but  this  is' evidently  untrue,  for 
although  a  point  be  without  extension,  that  which  is  without 
extension  is  not  necessarily  a  point,  it  may  be  nothing. 

It  is  therefore  necessaiy  to  substitute  another  definition  of  a 
p6int,  which  shall  include  a  positive  property  as  well  as  the  ne- 
gative one  above  described;  this  will  help  the  student  over  a 
difiiculty,  which  (notwilhstanding  Dr.  Simson*s  illustration  in 
his  note  on  this  definition)  might  have  discouraged  him  in  his 
first  attempt  at  Geometry.  Instead  then  of  Dr.  Simson*s  defini- 
tion and  note,  let  the  following  be  substituted : 

52.  Def.  *'  A  point  is  that  which  has  position,  but  not 
magnitude  V* 

53.  The  idea  of  a  point  (as  above  defined)  is  evidently  an 
abstract  idea :  a  mathematical  point  cannot  therefore  be  made 
on  paper  or  exhibited  to  the  eye  ,•  we  may  indeed  represent  it 
by  a  dot,  but  this  dot,  make  it  as  small  as  you  possibly  can^  will 
have  lei)gth,  breadth,  and  thickness  too;  still  it  may  be  used  as 
a  m>ark  or  representation  of  position  or  situation,  shewing  simply 


1  This  impnvettetit  wat  probably  first  sn^sested  by  Dr.  Hooke,  who  say% 
that  *'  a  point  ba»  pMitiou,  and  a  relation  to  roa^aitade,  bat  has  itMlf  no 
magnitude  \"  his  id«at  on  this  snbject  have>been  adopted  by  both  Plfyfiiir  and 
Ingram.  ^ 


PWT  WII.       ON  EUCLID'S  raaST  BOOK.  ^1 

to  where,  ear  ftom  wheoee,  lines  a*e  to  be  drawn,  distanced  esti- 
mated, &c.  A  point  then,  as  made  on  paper,  is  to  be  considered 
as  a  mark  indkaiing  merely  position  }  this  mark  must  necessarilf 
have  magnitude,  but  it  is  made  the  representative  of  that  which 
has  Dot. 

54.  Dtf.^.  **  A  line  is  length  without  breadth."  The  obser- 
vatidDs  contained  in  the  foregoing  article  may  with  equal  pro- 
piiety  be  applied  to  this  defioi^o,.  To  repre§mt  a  mathematical 
line,  which  is  without  breadth  or  thiekness,  (or  rather  to  repre- 
sent the  idea  of  such^  line,)  we  are  obliged  td  have  recourse  t« 
Inline  which  has  ixith.  The  line  w«  draw  on  papejr  is  not  there*- 
fore  the  line  we  have  defined^  but  merdly  the  mark  by  which  the 
iito  of  such  a  line  is  represented.  Th» abstract  4dea  of  length 
(without  breadth  and  thickoess)  is  perfectly  familiar  to  every 
one;  thus,  if  it  be  asked,  "  what  is  the  length  (or  di&tance) 
from  hence  to  London  ?**  the  answer  is,  "  thirteen  miles :"  ^his 
would, as  we  might  suppose,  be  satisfactory;  but  should  the 
mquirer  farther  ask,  how  toide  9  or  Aoto  thick  ?  every  one  wqpld 
yiy  or  despise  him  for  his  stupidity.  « 

55.  I>e/l  3.  This  is  not  properly  a  definitwn,  but  an  inference 
from  the  two  former,  for  '^  that  which  terminates  a  line  can 
have  no  breadth,  since  the  line  in  which  it  is  has  none ;  and  it 
can  have  no  l^ngth)  for  in  that  case  it  would  not  be  a  termina* 
tien,  but  a  part  of  that  which  is  suppo^^d  to  be  terminate,"  and 
would  Gonsequeiatly  itself  have  terminations  or  extremities : 
wbenee  the  termination  of  a  line  can  have- no  magnitude,  and 
having  necessarily  position*  it  must  therefore  be  a  point,  by 
Art.  53* 

b^.Def.  4.  Wfi  bave  before  remarked,  (Art.  7,  19,  20.)  that 
a  simple  idea -admits  oi  no  definition ;  .no  definition  can  possibly 
be  gtieR  of  stfoightnessi  to  lie  ^'  evenly  between  its  extreme 
points**  is  a  very  awkward  paraphrasis  of  the  word  straight,  and 
will  not  perhaps  be  so  well  understood  by  a  learner,  as  the  defi- 
nition would  be  were  it  to  run  thus,  "  a  straight  line  is  that 
vhkh  i|>  not  erookedf* 

57.  Hence  it  follows,  that  "z  straight  line  is  (iie  shortcut 
dillance  betwaen  its  extreme  points/'  this  h^  bten  proposed 
instead  of  £uclid*s  defiaitioa  b^i^some,  but  it  haspheen  objected 
to  by  othefii^  Professor  Flayfi^ir  has^iven  the  foUowing*  l^hich 
is  Q«tfainly  an  itiprovemeatj  viz.  ''  lines  whieh  cannot  poi^cide 

s3 


202  GEOMETRY:  Pakt  VUI- 


in  two  points,  witliout  coinotding  altogvAher,  9Stt  caBed 
lines ;"  but  it  msKf  be  added,  tbaK  neither  of  the  two  krfter  defi- 
nitions is  suffidently  simple  and  perspicuous  to  stand  at  the  be- 
ginning of  a  system  of  Elements. 

58.  All  other  lines  besides  straight  Hnes  are  called  curve  Uim, 
or  simply  curvet ;  and  henoe  we  define  curves  to  be  '*  those  lines 
which  do  not  lie  evealy  iietween  their  extreme  points,"  or  **iprhich 
are  not  the  shot  test  distanee  between  thesr  extreme  points.** 

50.  Def.  5.  We  have  shewn  that  the  id«R  of  length  only  (or 
of  what  the  mathematician*  call  a  line)  is  perfectly  fuiuliar  to 
every  one ;  the  idea  of  a  superficies  (or  of  length  and  breadUl 
without  thickness)  may  be  shewn  to-  be  equdly  so :  in  calculating 
the  content  of  a  field,  it  is  well  known  thatt  the  superficial  c0k* 
tent  is  always  understood,  in  which  length  and  breadth  onjy  are 
concerned  5  thickness  does  not  enter  at  all  into  the  consideration. 

eo.  Oiir  ideas  of  a  geometrical  solid,  superficies,  line,  and 
point,  are  obtained  by  abstracti(Mi«  (See  Art;  1^ — 17.)  Thus 
in  ^ontemplatii^  any  material  body  that  first  offers  itself  to  our 
consideration,  we  shall  find  that  liesides  being  made  up  of  mUh 
ter,  it  has  extension,  or,  length,  breadth,  and  thickness ;  now, 
if  from  the  complex  idea  of  this  body,  we  exclude  the  idea  of 
matter,  there  will  remain  the  abstract  idea  of  extension,  or  of 
length,  breadth,  and  thickness  only,  namely,  of  that  which  in 
geometrical  language  is  called  a  solid.  If  from  the  complex 
idea  of  this  solid  we  exclude  the  idea  of  tbtfikness,  we  thence 
obtain  the  abstract  idea  of  length  and  breadth  only,  or  «f  a 
geometrical  superficies.  Again,  if  from  the  complex  idea  of  a 
superficies  we  exclude  the  idea  of  breadth,  the  result  will  fiu> 
nish  us  with  the  abstract  idea  of  length  only,  or  of  a  geometri- 
cal line.  And  lastly,  if  from  the  idea  of  line  we  exclude  that  of 
lefl^h,  "  we  get  the  very  abstract  idea  of  a  pmnt:  though  I 
confess,'*  says  Mr.  Ludlam,  ''  the  operation  of  the  mind  in  this 
case  is  so  very  subtile,  that  it  can  hardly  be  distinctly  and  clearly 
traced  out." 

61.  Def.  6.  To  this  definition  we  may  add,  that  if  thft  extre- 
mities or  boundaries  of  the  superficies  be  straight  lines,  it  hi 
called  a  rectilineal  superficies  -,  if  curves,  it  is  caBed  a  curviliimal 
superficies ;  and  if  some  of  the  boundaries  are  straight  lines, 
and  the  rest  curves,  it  is  tailed  m  mvatilineal  superficies. 

6%  The  defiokioa  of  a  plane  superficies,  a»  originally i^tveB 


PiitVIIL        ON  £UCLm*S  FIBIir  BOOK.  363 

by  Euclid,  is  as  faXtaw.  *'  A  plaM  tuperDcics  is  tliat  which  lie$ 
evenfy  between  its  extreme  lines  ;*'  the  term  '^  lies  evenly"  has 
already  been  objected  to  as  obsoure.  (Art.  56.>  Or.  tkoaon, 
convinced  of  its  impropriety^  has  subotiluted  another  definition, 
which  has  the  advantage  of  indudiag^  the  esMAtlaL  property  of 
a  |»lane,  and  consequeatiy  of  distiiq^i^ing  it  fiom  every  other 
knd  of  superficies :  for  be«des  a  plane,  theve  are  various  kinds 
of  superficies,  as  the  spherical,  cflmdricml,  amical,  and  many 
others.  Aooording  totliis  definilioB,  a  plane  superficies  "  is  lliat 
in  which  ami^  two  points  being  taken,  the  straight  line  between 
them  lies  wholly  in  that  superficies^"  the  term  '^ plane,"  in 
popular  language,  means  that  which  is  perfecilf  fiat,  or  kndh 
■owiftwo  points  be  taken  in  a  sifierficies  which  is  not  perfectly 
flat,  it  is  plain  that  the  intermediate  parts  of  the  straifht  line, 
which  joins  those  points,  will  foll-either  above  or  behw  the  super- 
ficies ^  we  see  moreover  not  only  the  propriety,  bat  the  absolute 
necessity^  of  the  distinction  ^'  any  «wo  points,"  for  two  points 
may  be  taken  (i»  one  partieolar  direction)  in  tha.  sucfcce  of  a 
•one  or  cylinder,  which  will  agree  with  the  definition,  but  not 
amf  two  pohuts. 

^.  Def,  8.  To  give  the  jj^arner  an  id^  of  what  is  h^ire  meant 
by  the  teni|^ ''  angle,"  or  ''  indinatloii  of  two  lines,"  it  will  not  be 
aimiss  to  have  recourse  to  a  fiuniliar  exampla :  let  a  pair  of  com- 
passes be  opened  to  several  different  extents,  these  will  be  so 
OKiny  different  angles  5  when  the  legs  are  opened  to  but  a  small 
distance^  this  opeuii^,  or  (as  it  is  here  called)  iocliQfltion  of 
the  legs,  will  be  a  small  angle  i  when  opened  wider,  the  legs  will 
form  at  their  meeting  a  larger  angle  than  before,  and  so  on. 

64.  The  two  lines  which  fbrm>4ftff  (as  it  is  usually  expressed) 
contain  an  angle,  are  sometimes  called  the  legs.  The  m^nitude 
of  any  angle  dpes  not  at  all  depend  on  the  length  of  the  legs, 
or  lines  which  contain  it  3  in  the  example  above  proposed,  the 
legs  of  the  eon^passes  may  |m^  an  in<^,  a  foot,  4^  any  other 
length,  or  one.  may  be  longer  than  the  other,  and  yet  the  o|{|»« 
i«g,  vni^inatiotti  m  an^  contained  by  them  may  still  remaui  the 
same. 

65.  De/.  d*."  The  object  of  the  eighth  definition  is  to  define 

*  <'  Tiie  fint  nine  defihitloiis  might  batir been  gittd  in  HA  form  of  an  inin^ 


4acyon,liir  nowof  then  are  9cometrical|.«Ke(t  the  u§M0  »  inended  by 

S  4 


264  ClK)MFraT.  BartVIIL 

in  general  every  anglB  wfaieh  caA  be  described  on  a  f^nei 
ivhether  such  angle  be  contained  by  straight  or  oorve  lines  j  but 
since  o^vilineal  angles-are  not  treated  of  in  the  EtementSi  that 
definition  might  itaVd  beep  omitted.  -  Ii|-  the  ninth,  where  *'  a 
plane  reotillti«al€»gie'*  is  defined,  the  word  *^  plane'*  is  a  redun* 
dancy ;  for  the  angular  point,  as  well  ar every  point  in  the  lines 
whitth  contain  any  vectilin^l  angle,  must  necessarily  be  all  in  one 
and  the  same  plane,  a»  is  -proved  in  the  second  propositicm  of  the 
eleventh  book.  The  note  subjoined  to  Def.  9.  in  th«  Klements 
is  merely  to  shew  how  we  .are  to  read,  write,  or  to  determine 
the  place  oP  an  angle  when  it  is  read  ta  us :  if  an  «ngle  be 
expressed,  by  three  letters,  as  is  usual,  the  anguh^  point  is  alwfiys 
lihderstQod  to  be  at  the  mt4d|?  letter ;  >  thus,  if  JBC  den^e  an 
angle,  4liis  angle  is  alwaye  understood  to  be  at  the  middle  letter 
A  and  not  at  either  Aor  C,     •  ^ 

^.  Def,  10^11,  19.  When  a  straight  toe  meets  another 
straight  line,  (without  crossing  or  tutting  it,)  two  angka  are 
ibrttied  at  the  point  where  they  meet;  if  the8»  angles  be  equal 
to  each  other,  they  aris  called  r^ht  angles:  but  if  one  be  greater 
than  the  other,  the  forcoer  (which  is  greater  than  a  right  angle) 
is  eelled^an  d)iu9e  angle;  and  the  fctter  (which  is  less  than  a 
right  angle,  see  prop.  IS\  boi^l,)  is  called  an  acute  mgle. 

'67.  Def,  13.  In  <he  sense  of  this  definition,  pdnts  are 'the 
boundaries  of  a  line,  lines  iof  a  superficies  and  superficies  of  a 
solid. 

68.  ii$f.  14.  Hence,  according  to  £u«elid>  neither  a  line  nor 
an  angle  can  be  called  a  figure,  because  they  are  not  either  of 
them  **  tndosed  by  one  or  more  boundaries." 


t)r.  Simsoo  ;'*  this  is  Mf.  Ingram's  opinion,  and  be  a^ds,  <'  The  t«rms  by  which 
a  line  and  a  super^cies  are  defined,  give  some  explanation  of  the  meaning  of 
the^e  words,  but  give  no  geometrical  criteria  by  which  to  "know  them  ;  and  the 
best  way  of  accfiHring  proper'ide^s  of  them,  is  by  coostderMig  their  relation  to 
a  BoUd  and  -to  one  another,  as  Dr.  Sims«n  has  done.*'  See  on  this  dnfaiject  the 
note  on  Def.  1 ,  iSknvm*9  Eutiid,  idth  £il.  p.  $80.  A  defiatfion  then  may  he  said 
to  be  geometrical,  when  it  furnishes  some  criterion  t«  which  we  may  refer,  and 
]|y  which  the  idea  of  the  thing  defined  m^^  be  completely  arrived  at  and  ob- 
tained, at  the  rMuIt  of  any  demonstration  where  it  is  concerned:  other  defini- 
tions are  usually  called  metaphysical;  they  are  employed  in  all  d^s  where 
gieometrical  4l«fiintion8  cannot  l)e  ^l#kn,  as  necessary  for  explaining  in  the  beet 
pnanner  poUibtc  Hie  aature  of  tlpt^thii^  defined,  the  meaatog  of  terma,  fto. 


PitT  Villi      ON  EUCLOrS  nSST  BOOK.  MS 

^.  B^.  ISff^We  have  here  a  complete  and  Mkkctorf  in« 
<tance  of  the  method  of  defining  a  species  by  means  of  the 
genus  imd  special  difference.  (Art.  23,  24.)  ''  A  cireU  i$  a 
plane  figurtt'  it  belongs  to  that  class  of  figures,  which  have  dSX 
tbeir  parts  in  the  same  plane,  a|id  consequently  agrees  in  this 
general  character  with  a  triangle,  a  square,  a  polygon,  an  ellip-^ 
sis,  &c.  it  is  *'  contmed  by  one  line  caUed  the  cvrcumftrmee}* 
kere  we  have  a  limitation  whereby  all  such  figures  as  are  con* 
taioed  by  more  than  one  line,  as  the  triangle,  square,  polygon, 
&c.are  excluded;  '^ and  u  such  that  all  straight  lines  dtaam 
ftom  a  eertam  point  within  the  figure'*  (called  in  the  next  follow- 
ing definition  '^  the  centre"*)  to  the  circumference,  are  equal  to 
Me  another :  this  latter  clause  operates  as  an  additional  limita* 
tion,  which  excludes  the  ellipsis  and  all  irregular  curvilineal 
figures  from  the  definition,  because  there  is  no  point  in  either 
of  those  figures,  from  whence  all  the  straight  lines  drawn  to  the 
circumference  are  equal.  Here  then  we  are  informed,  first,  to 
what  general  class  of  figures  a  circle  belongs,  and  secondly,  by 
what  it  dififers  from  every  other  figure  of  that  class;  whence  the 
definition  furnishes  us  with  an  adequate  and  precise  idea  of  the 
figure  called  a  circle. 

70.  Another  definition,  in  substance  the  same  as  Euclid's,  is 
this ;  f  A  circle  is  a  figure  generated  (or  formed)  by  a  straight 
line  revolving  (or  turning)  in  a  plane  about  one  of  its  extreme 
points,  which  remains  fixed,"  the  fixed  point  being  the  centre^ 
9od  the  line  described  by  the  revolving  point  the  circumference, 

71.  The  circumference  of  a  circle  is  likewise  called  the  peri* 
phery :  it  is  sometimes  improperly  named  the  circle  ;  a  circle,  in 
the  proper  acceptation  of  the  term,  means  the  space  included 
within  the  circumference,  and  not  the  circumference  exclusively, 

jl.  To  describe  a  circle  with  the  compasses^  you  have  only  to 
fix  one  foot  at  the  point  where  the  centre  is  intended  to  be,  and 
(the  compasses  being  opened  to  a  proper  extent)  turn  the  other 
(sot  quite  round,  and  it  will  trace  out  the  circumference. 

73-  After  Def.  IJ.  add  the  following,  which  is  in  continual 
use,  viz.  'f  a  radius,  or  semidiameter  of  a  circle,  is  a  straight-line 
drawn  from  the  centre  to  the  circumference."    - 

74.  Def,  18,  19.  Any  part  of  a  circle  cut  off  by  a  straight 
In}^,  is  called  a  segment  of  a  circle;  if  the  straight  line  pass 
throu^.  the  cen|^,  it  is  a  diameter,  (Deftpl7.)  and  divides  the 


9M  GEOMKTKS.  tA%r  VllL 

cirde  ioto  two  c^iia/  segaKiits,  criled  mmkr<Mk$:  hot  if  liie 
ftnig^t  line  wbieh  cuts  tlie  cirde  docs  not  pass  tfaroi^h  tke 
centre,  it  will  divide  the  drde  into  two  umeq^l  segments,  the 
greater  of  which  is  said  to  be''  a  $egmemt  grmtimr  than  a  sema- 
circfe,"  and  the  less '' a  MgfweiKl  len  than  a  mrnrdreUT  Bfthe 
terms  ''  segment  of  a  didr/*  and  *'  sCTn-drde,**  we  are  alwaqrs 
to  undeistand  the  tpaee  induded  between  a  port  of  the  circoni' 
ference  and  the  stnight  line  by  which  that  part  is  cot  off,  unless 
the  contrary  be  expressed. 

75.  Any  part  of  the  drcumfeienoe  is  catted  on  are^  and  the 
straight  line  which  joins  the  extremities  of  an  arc,  (or  which 
divides  the  drde  into  two  segments,)  is  called  a  chord,  Wz.  it  is 
the  common  chord  of  both  the  arcs  into  which  it  divides  the 
whole  drcumference. 

76.  Def.  23.  We  have  nothing  to  do  professedly  with  poty« 

gons  in  the  first  book,  yet  since  the  definition  is  introduoed,  it 

may  not  be  improper  to  observe,  that  a  polygon,  having  all  its 

sides  and  angles  respectively  equal,  is  called  an  equilateral,  «^t- 

angular,  or  regular  polygon.  These  figures  are  named  according 

to  their  number  of  sides }  thus, 

five,     1  pa  Pentagon, 

a  Hexagon, 
A  polygon  having  ^  seven,  y  sides,  is  called  ^  a  Heptagon, 

an  Octagon, 
&c. 

77.  Def.  24,  25,  26,  27,  28,  and  29.  Triangles  are  distin- 
guished  into  three  varieties  with  respect  to  their  sides,  and  three 
with  respect  to  their  angles :  the  three  varieties  denominated 
from  their  sides,  (as  laid  down  in  Def.  24, 25,  and  26.)  are  equi- 
lateral,  isosceles,  and  scalene;  the  latter,  although  defined  here, 
does  not  occur  under  that  name  in  any  other  part  of  the  Ele- 
ments, llie  three  varieties  which  respect  their  angles,  are  right* 
angled,  obtuse-angled,  and  acute^angled,  Def.  27,  28,  and  29. 

78.  Def,  SO.  A  square,  which  according  to  this  definition 
'^  has  all  its  sides  equal,  and  all  its  angles  right  angles/*  must 
evidently  be  just  as  wide  as  it  is  long ;  hence  there  can  be  no 
such  thing  as  a  long  square,  although  we  read  of  such  a  figure 
in  some  books  ^ 

'  ll.llll  ■!  Illll  II  11^  I.       Ill  I  I  ■■.»!  ■  1 

»  Euclid's  deSmtion  of  a  tqaare  may  be  coosidered  as  iaulty,  for  wHb  ithe 
essential  properties  of  a  square  be  has  incorporated  ao  iiifarcmce,  wbicb  is  tbe 


Pmt  VIIL        on  EUCLIVS  HBST  book.  967 

79.  Def.  31.  Since  the  wOfd  Mot^  does  not  cmce  occur  in 
mxf  rab6ei]uent  pmrt  of  tlw  Etements,  it  ehoukl  not  bave  found 
ft  place  here.  The  figure  defined  k  ft  species  of  that  which  Is 
called  in  the  second  bookt  and  elsewhere^  a  rectangle. 

80.  Def.  35.  In  the  definition  of  parallel  lines  as  here  bid 
down.  Dr.  Simson  has  iroprored  on  Euclid,  and  his  definition  is 
better  adapted  to  the  kamer's  comprehension  than  either  ttt 
those  approved  by  Wolfius>  BosooTich,  Thomas  Simpsony 
D*Alembert,  or  Newton ;  the  truth  is,  that  no  inferenoi  can  be 
dnmrn  from  any  definition  hitherto  given,  sufiicient  to  fix  the 
doctrine  of  paralki  lines  on  the  firm  basis  of  nniAjectiooabla 
evidence** 


tabject  of  the  cor.  to  prop.  46«  b.  I.  It  would  be  more  ttrictly  scientific 
to  doiiDc  a  tqvaro  to  be  <<  a  four-eMed  ignre  having  all  ite  tWiet  eqtuil)  and  eMt 
of  it#  anf let  a  dgfat  Juagk ;"  for  that  "  an  eqnilateral  foar-eiiled  figure  ie  a 
parallelogram,"  and  that "  erery  parallelogram  ha?iBg  one  right  angle  has 
all  its  angles  right  aiigles,"  are  plainly  inferences  from  the  definition  given  in 
this  note,  and  that  of  a  parallelogram,  prop.  34.  b.  I .  the  like  observations 
extend  to  Def.  33.  In  both  instances  Euclid  has  abandoned  his  own  plan, 
and  transgressed  a  rale  which  od^ht  never  to  be  violated  wHhont  absolute  nc- 
eesvity ;  the  d^g^rtnre  is  however  juttifiable  in  the  present  instaaoc,  as  Euclid's 
definition  wiU  he  more  easily  understood  by  a  beginner  than  that  which  we 
have  proposed. 

•  Having  explained  the  definitions  as  they  stand  in  Euclid,  we  may  bo 
allowed  to  remarh,  that  a  more  methodical  arrangement  of  them  would  be  a 
desirable  improvement;  should  any  future  Editor  think  this  hint  worth  his 
attention  and  adopt  it,  it  will  be  conducive  to  ckganee,  correctaess,  clear- 
ness, and  slmpKcity,  which  are  undoubtedly  points  of  importance,  especially 
at  the  beginning  of  the  Elements.  The  alteratioas  1  would  propose  are  as 
follow: 

Def.  18.  A  segment  of  a  circle  is  the  figure  contained  by  a  straight  line, 
and  the  circumference  it  cuts  off. 

19.  If  the  straight  line  be  a  diameter,  the  segment  is  oalkd  a  semicircle. 

From  the  20th  to  the  29th  inclusive,  may  stand  as  at  present. 

30.  Paraltel  straight  lines  are  such  as  are  in  the  same  plaae,  and  which, 
bein||L^ftfbdoced  ever  so  far  both  ways,  do  not  aoeet. 

31.  A  parallelogram  is  a  four-sided  figure,  of  which  the  opposite  sides  aaa 
parallel. 

39.  The  diameter  or  dkigonal  of  a  parallelogram  is  a  straight  line  wUflb 
joins  any  two  of  its  opposite  angles* 

33.  A  rhombus  is  a  parallelogram  whfch  has  all  iu  sides  eq««l^  but  its 
anglea  are  not  right  angles. 


M8  OBOMKnur.     .  PaktVUI. 

On  the  P^tMatet. 

81.  A  postulate,  as  we  have  befcx^  •bserv^ed,  is  a  self-«Ttdent 
practical  proposition :  on  this  subject  Mr.  Ludlam  very  justly 
remarks^  that  **  Euclid  does  not  here  require  a  practical  dexterity 
in  the  management  of  a  ruler  and  pencil^  but  that  the  postulates 
are  here  set  down  that  his  readers  may  admit  the  pasminUiy  of 
'what  he  may  hereafter  require  to  be  done.'*  On  this  we  remark, 
that  our  conviction  of  the  possibility  of  any  operation  depends 
on  our  having  actually  performed  it  in  some  particular  instance 
ourselves,  or  known  that  it  bto  been  performed  by  others  $  hav« 
ing  thus  satbfied  itself  of  the  possibility  in  particular  instances, 
the  mind  immediately  perceives  that  the  possibility  exteads  to 
every  instance^  or  that  the  operation  is  true  in  general.  On 
these  considerations  it  has  been  affirmed,  that  "  the  mathenu- 
tical  sciences  are  sciences  of  experiment  and^i)servatioQ,  founded 
solely  upon  the  induction  of  particular  fects,  as  much  so  as 
mechanics,  astronomy,  optics,  or  chemistry  ^"  T^s  doctrine,  to 
Its  fullest  extent,  it  would  perhaps  be  unsafe  to  adopt. 

82.  In  applying  the  postulates,  we  proceed  in  an  order  the 
converse  of  that  laid  down  in  the  preceding  article :  we  admit 
what  is  affirmed  in  the  postulate  to  ha  true  in  general,  1.  e.  in  all 
cases  $  and  since  it  is  true  in  all  cases,  it  follows  ai  a  necessary 
inference,  that  it  is  true  in  the  particular  case  under  considera- 
tion.    We  will  now  begin  to  exemplify  the  use  of  the  mathe- 


34.  A  rhomboid  is  a  parallelogram  of  which  all  its  sides  are  oot  equal,  nor 
any  of  its  angles  right  angles. 

35;  A  rectangle  is  a  parallelogram  which  has  all  its  angle#  right  angles  (or 
^icb  has  oae  of  its  angles  a  right  angle ;  see  the  foregoing  note,) 

36.  A  square  is  a  rectangle  which  has  all  its  sides  equal. 
>97.  All  other  four>sided  figures  besides  these  are  caUed  trapesiums. 

Note.  A  trapezinm  which  has  two  of  its  sides  parallel,  is  sometimes  called 
a  tiapesoid,  and  a  straight  line  joining  the  opposite  angles  of  a  trapesium 
is  ealled  its  diagonaL 

ThedefinitioM  preoeding  the  18th  might  stan4  as  they  do  at  present,  if 
instead  of  the  first  definition,  that  which  we  fawe  proposed  (see  Art.  £)^)i»were 
adopted. 

P  The  postulates  prefixed  to  the  Elements  are  in  number  (as  they  ought  to 
be)  the  fewest  possible;  for,  as  Sir  Isaac  NewtoQi observes,  «  postulates  are 
principiss  which  Geometry  borrows  from  the  arts,  and  its  excellence  consists 
in  the  paucity  of  them."  The  postidates  of  £nctid  are  all  problems  derivted 
from  the  mechanics.    Ingram. 


P4iT  VUl        ON  EUCXID3  FIRST  BOOK.  S6S> 

matfeal  instmiiients,  to  afford  the  student  an  oppoftunity  of 
practteal  as  well  as  mental  improvement. 

83.  Postulate  1.  If  it  be  granted,  that  ''  a  straight  line  may 
be  drawn  from  any  one  point  to  any  other  point,**  it  follows  as 
an  evident  consequence,  that  a  straight  line  can  be  drawn  from 
the  point  jti  to  the  point  B.  Lay  a  straight  scale  or  ruler,  so  that 
its  edge  nuiy  touch  the  two  proposed  points  A  and  B,  then  with 
a  pen  or  pencil  draw  along  the  edge  of  the  scale  or  ruler  a  line 
from  A  to  B,  and  what  was  granted  in  general  will  in  this  par- 
ticular instance  be  performed. 

84.  Post.  9.  To  produce  a  line  means  to  lengthen  it.  A 
straiglit  line  of  two  inches  in  length,  may  according  Xm  this 
postnkte  be  produced  until  it  is  three,  four,  five,  or  more  inches 
m  length.  Lay  the  edge  of  your  scale  touching  every  point  of 
the  ^ven  line,  and  with  th&  pencil  or  pen,  as  before,  draw  the 
line  to  the  length  proposed. 

85.  Post,  3.  Bsttend  the  points  of  the  compasses  to  the  re- 
hired distance,  then  with  one  foot  fixed  on  the  given  point  as 
a  centre,  let  the  other  be  turned  completely  round  on  the  paper, 
and  it  will  describe  the- circle  required. 

On  the  Axioms. 

86.  An  axiom  is  a  self-evident  theoretidfeil  proposition,  which 
neither  admits  of,  nor  requires  proof.  Axioms  evidently  depend 
in  the  firait  instance  oh  particular  observation,  from  whence  the 
nund  intuitively  perceives  their  truth  in  general :  Hke  the  pos* 
^tes,  these  ^neral  truths  being  previously  laid  down  and  ac- 
knowledged, are  applied  to  the  proof  c^  the  demonstrable  pro- 
poftitioDs  which  follow. 

87.  Axioms  I,  2,  3,  4,  5,  6,  7,  9,  and  10,  are  too  plain  to 
Kquire  illustration  ^  the  10th  is  what  is  usually  caUed  an  identi- 
cal proposition,  amounting  to  no  more  than  this,  namely,  that 
''all  right  angles  are  right  anglies.** 

88.  Ax,  8.  i&oald  the  learner  feel  disposed  to  hesitate  at  this 
^on,  he  may  be  informed,  that  every  one  readily  admits  its 
truth  in  practical  matters ;  a  farmer  who  has  two  quantities  of 
com,  eadii  of  which  exactly  fills  his  bushel,  would  be  surprised 
if  any  one  should  deny  that  these  two  quantities  areequid  to 
each  other. 

89.  The  Jf^th  apdom,  ios^t  is  called,  Is  not  propeiiy  an  axiom, 
but  a  jMpQposition  which  requires  proof ;  the  learner,  if  he  can- 


870  GEOMETRY.  FastVUL 

not  readily  uiKler9Ca»d  its  import,  may  pass  on  until  he  has  read 
the  2Sth  proposition :  it  mmt  then  be  resumed  as  necesBiny  to 
the  demonstration  of  the  99th. 

On  the  Propositions, 

90.  The  propositions  in  Euclidj  we  have  before  shewn^  are 
either  problems  or  theorems ;  the  problems  shew  how  to  per- 
form certain  things  proposed,  and  the  theorems  to  estaUish  and 
confirm  proposed  truths :  both  reipiire  demonstration,  and  the 
process  is  nearly  the  same  in  both )  indeed  proUuus  may  be 
changed  into  theorems,  and  theorems  into  pnoblems,  by  a  slight 
alterai^ion  in  the  wording.  The  demonstnition  of  the  first  ^irepo- 
sttion  depends  solely  on  the  definitions,  postulates,  and  axioms ; 
that  of  the  second  proposition  on  these  and  the  first,  and  so  on : 
the  truths  obtained  by  the  proof  of  propositions  being  always 
employed,  where  necessary,  in  succeeding  demonstrations. 

91.  Every  geometrical  [Ht)po6ition  may  be  considered  as  com- 
prehending three  particulars,  viz.  the  enunciation,  the  construe* 
tion,  and  the  demonstration.  The  enunciation  declares  in  gene« 
ral  terms  what  is  intended  to  be  done  or  proved.  The  con- 
struction teaches  to  draw  the  necessary  lines,  circles,  &c.  and 
applies  the  enunciation  to  the  figure  thus  constructed.  The 
demonstration  is  the  system  of  reasoning  which  follows,  whei^by 
what  was  enunciated  is  clearly  and  fully  made  out  and  proved, 

.  92.  Tlie  numbers  and  letters  in  the  margin  are  references  te 
the  proposition,  axiom,  postulate,  or  definition,  where  the  par^ 
ticular  cited  in  the  corresponding  part  of  the  demonstiutien  is 
to  be  found,  or  is  proved ;  thus  1  post,  means  the  firat  postulate  -, 
15  def.  the  15th  definition;  3  ax.  the  third  axiom;  2. 1.  means 
the  second  proposition  of  the  first  book,  &c.  the  first  number 
always  referring  to  the  proposition,  and  the  second  to  the  book. 
93.  Before  the  student  begins  to  learn  the  demonstration,  he 
mu^t  be  able  to  define  accurately  all  the  teems  of  science  which 
occur  in  the  proposition,  and  to  repeat  the  postulates,  axioms, 
enunciations,  &c.  referred  to  in  the  margin  -,  next,  the  enunci- 
ation of  the  proposition  must  be  well  understood  and  learned  by 
heart :  ali  this  will,  in  a  very  short  time,  become  perfectly  eiat^y. 
The  construction  of  the  figure  comes  next^.  th^  figure  should 
be  inade  solely  from  (he  directiona  wJbinh  immediately  follow  the 
^mmcifitlpai  i£  thia  be,  thmightdiffioidt  at  first,  the  figure  in 


Paut  VIIL        on  £UCUD*S  FIB8T  BOOK.  S7I 

£uclld  may  be  taken  as  a  guide :  every  part  of  tbe  figure  may  be 
drawn  by  hand,  avd  the  more  accurately  thia  b  done,  the  better 
will  it  assist  the  recollection ;  the  instruments  may  be  employed 
for  this  purpose,  but  they  are  not  ahtolutekf  necessary^  M  the  truth 
of  any  proposition  does  not  in  the  least  depend  on  the  accuracy 
of  the  construction :  letters  must  be  made  at  the  angles  and 
Qthor  prominent  parts  of  the  figure ;  these  liay  (at  first)  be 
copied  from  the  figure  in  Buclid.  Lastly,  in  order  to  prepare 
the  way  for  demonstrating  the  first  proposition,  as  well  as  some 
of  the  following  ones,  in  a  complete  and  satisfectory  manner,  it 
will  be  necessary  to  premise  the  three  following  axioms : 

94.  Axiom  1.  If  a  point  be  taken  nearer  the  centre  than  the 
drcumferenoe  is,  that  point  is  within  the  circle. 

95.  Axiom  2.  If  a  point  be  taken  more  distant  from  the 
centre  than  the  circumference  is,  that  point  is  without  the 
circle. 

96.  Axiom  3.  If  a  point  be  taken  within  the  circle,  and  ano* 
ther  point  without  it,  any  line  which  joins  these  two  points  will 
cut  the  circumference. 

97.  Previous  to  attempting  the  first  proposition,  the  student 
must  be  prepared  (agreeably  #o  what  has  been  said  in  Art.  93.) 
to  answer  the  following  questions :  viz.  what  is  a  proposition  ? 
(for  the  answer,  see  Art.  31.)  what  is  a  problem  ?  (see  Art.  38.) 
what  is  a  point  ?  (see  Art.  52.)  what  is  a  line  ?  (see  Def.  2.) 
mhat  is  a  straight  line  ?  (see  Def.  4.)  what  is  a  triangle  ?  (see 
Def.  2L)  what  is  an  equilateral  triangle  ?  (see  Def.  24.)  what  is 
a  circle  ?  (pte  Def.  15.)  what  is  the  firftt  postulate  ?  what  is  the 
third  poalmlate  ?  what  is  Euclid's  first  axiom  ? — We  will  now 
shew  how  the  first  proposition  ought  to  be  demonstrated. 

EnunciaMon, 

M.  PkioposiTioN  1.  Problem.    To  describe  an  equilateral 
triangle  upon  a  given  "^  straight  line.  {See  the  figure  in  Euclid,) 
JjbH  AB  be  the  given. straight  line ;  it  is  required  to  de* 
scribe  an  equilateral  triangle  upon  it. 


4  la  Eactid  it  is  <<  a  given  Jlnite  $trst|;bt  line  $"  here  tbe  word  "  finite*'  is 
•Qpcrflaoos,  for  whatever  is  given  must  of  necessity  be^nite;  a  line  is  said  to 
he  «  given,"  wMli  wwtb«r  lidi^eqQal  to  it  can  iM  aetnallf  dmwa ;  (see  EucUd's 
Basa,  ^>iir.  i .)  hat  who  man  diaw  a  line  equal  to  aa  tafioite  line  ? 


«W  "  GEOMETRY.  '       PartVIIL 

Construction. 

Sroitt  the  centre  J,  at  the  distance  JB,  describe*  the  circle 
BCD,  by  the  Sd  postulate '  j  and  from  the  centre  B,  at  the 
distance  BA,  describe  the  circle  ACE  by  the  3d  postulate;  these 
circles  mil  cut  one  another,  by  Art.  94,  95>  9(y ;  then  from  the 
point  C,  where  they  cut  one  another,  draw  thcr  straight  lines  CA, 
CB  to  the  points  A  and  B,  by  the  1st  postulate i  ABC  shall' be 
an  equilateral  triangle.  ^ 

Demdnstfatiort. 

Because  the  point  A  is  the  centre  of  the  circle  BCD,  AC 
is  equal  to  AB,  by  the  Ibth  definition^  and  becaiuse  the  point  B 
is  the  centre  of  the  circle  ACE,  BC  is  equal  to  BA,  by  the  nth 
definition:  therefore  CA,  CB  are  each  of  them  equal  to  AB; 
but  things  which  are  equal  to  the  same  are  equal  to  one  iano- 
ther,  by  the  1st  axiom;  wherefore  CA  and  CB  are  equal  to  one 
another,  being  each  equal  to  AB;  consequently  the  three  straight 
lines  CA,  AB,  and  BC  are  equal  to  one  another,  and  form  a 
triangle  ABC,  by  the  ^Ist  definition,  which  is  therefore  equilate- 
ral, by  the  24th  definition,  and  it  is  described  upon  the  given 
straight  line  AB,  because  AB  is  one  of  its  sides.  Which  was 
required  to  be  done* 

99.  With*  similar  accuracy  every  proposition  in  the  Elements 
ought  to  be  demonstrated ;  the  difficulty  of  acquiring  a  habit  of 
strict  and  close  reasoning  would  by  this  practice  very  soon  ht 
surmounted^  and  the  powers  of  the  mind  gradually  strengthened 
aod  enlarged. 

100.  Prop.  3.  Having  read  over  attentively  the  demonstra- 
tion, it  may  perhaps  be  objected,  that  in  drawing  the  straight 
line  from  A,  we  are  confined  by  Euclid's  figure  to  ofie  part icuW 
direction  AL ;  the  proposition  seems  at  first  sight  to  be  limited 
in  this  respect,  but  it  is  not  so  -,  for  if  from  ^  as  a  cen^,  with 
the  distance  AL,  a  circle  be  described,  straight  lines  may  be 
drawn  from  the  centre  A  to  the  circumference  in  everjf  direc- 
tion by  the  1st  postulate,  and  each  of  these  lines  will  be  ^ual  to  , 
AL  by  the  15th  definition. 

101.  Prop.  2.  and  3.  have  Oeen  objected  to  as  sufficiently  evt-^ 


'  Tbe  Mbteaces  ia- Italic  «rc  not  in  Euclid  b«t  thef^ife  n^cesaaff,  awl 
thonld  be  ioppUcd  by  tfaii'«t«de«t  m  m  pwai  that  be^iMi4ctt(a«dt  lis  flMbjecw  ' 


\ 


Part  VIIT.        ON  EUCLID'S  FIRST  BOOK.  S73 

dent  without  proof;  but  it  appears  to  have  been  the  design  <^ 
tlie  aneient  geometers  to  erect  a  oomplete  system  of  science  on 
as  harrow  a  basiB  as  possible :  hence  E^cUd  lays  down  aelf-evi* 
dent  principles  which  admit  of  no  demonstration^  and  of  these 
the  fewest  mimber  possible  that  can  be  talcen  to  efifect  his  pur- 
poae;  by  means  of  which  and  the  definitions  he  demonstrates 
ail  such  of  his  proposiHont  a$  am  $ittcepiible  ofproqf,  without  re- 
gard to  their  being  easy  or  difficult,  or  to  the  degree  of  ^videnoo 
inth  wfaidi  their  truth  may  «c  first  sight  appear. 

108.  The  third  propo^ion  being  mueh  less  difficult  than 
either  the  first  or  second,  iSL  may  be  asked,  why  was  it  not  pot 
first?  The  answer  is,  the  pvopf  ctf  this  proposition  d^iends  on 
tke  Moond,  and  that  of  the  second  depends  on  the  first,  and 
iifccemt^  depmAence  is  the  only  ord^r  that  can  possibly  be  at- 
tended to  in  any  connected  system  of  reasoning. 

lOS.  The  following  lemma  should  be  understood  before  the 
fourth  pn^tosition  is  attempted.  Lemma,  Let  LMN,  PQR  be 
two  equal  angles,  and  let  them  be  applied  to  (laid  upon)  e^ch 
other,  so  that  the  paimi  M  may  coincide  with  the  point  Q,  and 
Uie  straight  fine  q 

ML  with    the  ^  ^ 

straight  line 
QP;  then  ^ill 
MN  fall  upon 
Qft.      For      if 

LMN  be  appli-  j^/  "N  ^  J^     JSi 

«i  XQ  PQR  as 

above,  and  MN  do  not  fall  upon  QR,  let  it  fell  otherwise  as 
QT,  then  the  angle  litf  AT  becomes  PQT;  but  LMN  is  by  hypo- 
thesis equal  to  PQR,  therefore  the  angles  PQ  T  and  PQR  are 
equal  to  each  other,  the  greater  equal  to  the  less,  which  is 
akurd ;  ther^oi*e  MN  cannot  Ml  otherwise  than  on  QR,  which 
Wfts  to  be  shewn. 

104.  This  kind  of  proof,  we  have  already  observed,  is  what  is 
called  "  reductio  ad  absurdum."  The  method  of  proving  the 
equality  of  two  figures  by  laying  them  one  on  the  other,  and 
shewing  that  their  conrespondlng  parts  ooipcide,  is  called  supri^- 
poiUim,  and  has  been  ol^ected  to,  not  from  itsMrant  of  evidence^ 
but  beeause  it  has  been  considei^  Ungeometrical,  as  depending 
OB  BO  poslisllatei  indeed  we  are  no  more.boimd  to  admit  the 

VOL.  II.  T  * 


274  GEOMETRY.  Pakt  VIII. 

INMsilMlity  of  appljin^  one  figure  to  mntitber,  tiMm  we  wart  to 
admit  the  poasibtlity  of  joiamg  tifo  points,  ptwhiring  o  stniglit 
line,  or  describiiig;  a  drde:  henee  a  |M»flate  to  that  effect 
secoM  occttojuy  • 

'  105.  Prop.  4.  This  and  the  cightli  are  important  propDritions, 
as  on  them  depends  the  whole  doctrine  of  tiian^es;  thef  aie 
both  proved  by  sop^^lasit]ony  which  has  becQ  ex|daineQ  sbove. 
'"^  it  is  worth  while  to  reniarfc,"  sagfs  Mr.  Lndbm,  "  with  what 
cantion  and  aoooracy  all  fiodid  s  prapositioos  are  wonkaL  A 
earelws  writer  might  say,  tf  two  hima^tt  kmoe  two  nde$  tmd 
an  tmgU  equal,  then  the  third  side  of  lAe  one  will  ba  eqmmi  io  the 
third  tide  of  the  other,  Sfc,  But  Eiidid  cantioDB  yoa  not  oidy 
that  tlie  sides  most  lie  eqoal  each  to  each,  but  also  that  the  angier 
spoken  of  must  be  tliat  which  Is  eoalataed  Ay  itie  respectively 
equal  sides.  We  vnXi  shew  tint  two  triangfes  maj  have  (as  was 
mud)  two  sides  respectively  eqoal,  and  also  one  angle,  yet  neither 
their  thmi  sides  nor  the  figures  themselveB  wiU  be  equal.*' 

105.  «<  Let  JBC  be  an  isosceles  triai^,  J  the  vertex,  BC 
fhe  base«  -Produce  the  base  BC  to  A  and  join  jiD;  then  we 
shall  have  two  triangles  formed^  viz.  ABD  and  ACD,  having 
two  sides  and  an  angle  respectively  equal ;  that  iB«  the  side  JB 
in    the  tiiangle  ABD,  j. 

equal  to  the  side  AC  in 
the  triangle  ^CD;  also 
the  side  AD  common  to 
both  triangles.  The  an- 
gle ADC  is  also  common 
to  both   triangles  j    yet  ^^  ^^-      ^I^ 


'  «  Evclid/'  says  Mr.  Ingram,  **  never  supposed  any  thing  to  be  possible 
which  he  hat  not  before  shewn  to  be  possible ;  this  was  not  merely  to  avoid 
.  impottibilities,  ai  sone  allege,  bnt  to  secure  evidence,  and  to  nake  hit  reader 
as  certain  of  bis  coocliisions  as  he  himself  was."  Miem.  ^  JEttfRd^  p.  S8K  It 
mast  be  confessed  Ihit  it  Euclid's  general  rule,  to  which  the  Instance  ia  ques- 
tion is  undoubtedly  an  CMeption,  notwithstanding  the  great  difficulty  Mr. 
Playfair  finds  in  admitting  the  fact :  to  avoid  it,  the  learned  Professor  hat 
sfiewn  how  the  fourth  and  eighth  propositions  may  be  proved  withoqt  the  aid 
of  suprapotition ;  but  the  postulate  he  requires  for  that  purpote  cannot  coasts- 
tently  with  geometrical  correctness  be  gsanted,  becaate  it  is  a  demaOiUaMe 
propositiott.  CVmipare  hit  pot€«Ml*  {Skm.  of  Gfm.  p.  355:)  with  the  1 8th 
proposition  of  the  6th  book  of  Euclid}  Bn4^hey  wiU  be  found  to  bn  the 


Past  VIII.        ON  EUCLID'S  FWST  BOCK.  275 

tliie  third  side  BD^  in  the  famter  tHiMigte^  k  not  equal  to  the 
third  side  CD  io  the  latter ;  for  CD  by  the  construction  is  only 
a  part  of  BD  .*  nor  are  the  ftgures  ABD  and  JCD  equal,  for  the 
fiwner  contains  the  lattiT,  as  appears  from  the  Hg^ure  '." 

107.  Prop.  5.  Cor.  Every  eqnilateral  triangle  may  be  con- 
sidered  as  isosceles.  Let  ABC  be  such  a  triangle  -,  and  since  JB 
^AC,  the  angle  B=the  angle  C; 
and  since  BA:=BC,  the  angle  A:^ 
the  an^e  C,  both  by  the  proposition ; 
wherefore,  since  BtsC  and  AstC,  it 
follows  (from  axiom  1 )  thatBs^; 
wherefore  the  three  angles  A,  B,and 
C.  are  equal  to  each  other,  that  is, 
the  equilateral  triangle  ABC  is  also 
equiangular.  * 

106.  The  enunciation  of  every  theorem  consists  of  two  parls, 
viz.  the  SUBJECT  and  the  predicate.  The  subject  is  that  of 
which  something  is  affirmed  or  denied,  and  the  predicate  is  that 
which  is  affirmed  or  denied  of  the  subject :  thus,  in  prop.  4. 
two  triangies  having  two  sides  of  the  one  equal  to  two  sides  of  the 
other,  each'lfp  each,  and  the  included  angles  equal,  is  the  subject ; 
and  that  such  triangles  will  have  their  hoses  equul,  their  other 
oi^les  equal,  and  be  equal  in  all  respects,  is  the  predicate.  The 
subject  of  prop.  5.  is,  an  isosceles  triangle,  and  the  predicate  \<^ 
that  the  angles  at  its  base  are  equal  to  each  other,  and  likewise 
the  angles  under  the  base. 

109.  Two  propositions  are  said  to  be. the  con  ve ass  of  each 
other,  when  the  subject  of  one  is  made  the  predicate  oi  the  other, 
and  the  subject  of  the  latter  the  predicate  of  the  former.  Propo- 
sitions wherein  the  subject  and  predicate  thus  change  places,  are 
called  CONVBBSB  fkopositions  *. 

>  LodfauD^s  RudSmtnit  t^  Mtakematiet,  5th  £dL  p.  183,  184. 

"  Two  convene  pfoporitiona,  although  in  aiost  c«tr«  betk  true,  are  Dot  in 
ftU  case*  so ;  one  may  be  true,  and  the  other  feUe ;  thtt»,  the  proposition,  '<  If 
two  triangles  have  the  three  sides  ^f  the  one  respectively  equal  to  the  three  sides 
of  the  other,  the  three  angles  of  the  one  will  be  respectively  eqnal.to  the  three 
Uigles  of  the  other,"  may  be  proved  to  be  true ;  but  its  converse,  vis.  '*  li  the 
three  angles  of  one  triangle  be  respectively  eqaal  to  the  three  angles  of  ano- 
ther, then  vill  the  sides  of  the  first  tfiai^e  be  respectively  equal  to  those  of 
the  other,"  is  not  neceasarily  true  j  there  may  be  a  million  triangles  ciri:uBi- 

T  C 


216  QEQMESRY.  Fakt  VHi 

110.  Prop.  6.  16  the  oonverse  of  prop.  5.  and  its  prpoC  is  lay 
reductio  ad  absurdum ;  the  words  *'  the  base  DC  is  equal  to  ib» 
base  AB,  aad"  may  be  left  out  as. unnecessary,  and  instead  cf 
''  therefore  ^£  is  not  unequal  to  AC,  &c;*  it  will  be  more 
proper  to  read, ''  therefore  DB  is  not  equal  to  JC;  and  is.  the 
same  ttaaner  it  may  be  proved,  that  no  stiaaght  line,  either 
greater  or  less  than  AS,  can  be  equal  to  .^IC,  wharefore  AB  k 
equal  to  AC,  which  was  to  be  depaonstrated." 

111.  The  corollary  to  prop.  6.  may  be  thus  pjrotved:  (aee  the 
fig.  to  Art.  107)  because  the  angle  B=the cmgle  C^ ;.*  the  side 
^Cacthe  side  AB,  (by  the  prop.)  aad  because  the  angle  A^ 
the  angle  C,  •.-  the  aide  4C=;the  side  AB,  v  ACsiAB^BC, 
which  was  to  be  shewn.  This  and  the  corollary  to  prop.  &.  are 
the  converse  of  each  other. 

112.  Prop.  7.  Many  of  the  propositions  in  Euclid  ape  mccdf 
subsidiary,  that  is,  they  are  in  themselves  of  no  other  xme„  than 
as  necessary  to  the  (Mroof  of  otiier  propositions  that  are  uselul^  oi 
this  kind  are  prop.  7,  16,  and  17>  of  the  first  book  The  de^ 
monsUrati&B  of  this  proposition  i^  another  instance  of  reductio 
^d  absurdumi  we  here  suppose  aa  imposstbllity  to  be  possible, 
in  order  to  shew  the  absurdity  of  that  supposition :  a  figure  is 
hete  mode  to  represent  what  no  figure  con  represent*  L  a.  an  im* 
possibility  -,  Ibr  we  suppose  not  only  that  the  lines  AC  and  AD 
are  equal  to  one  another,  but  also  that  CB  aad  DB  are  ec^aal  to 
one  another,  which  the  demonstration  shews  cannot  be  tiue« 
unless  the  points  C  and  D  coincide,  and  then  the  two  triangles 

torib^d  aibeat  od«  aadhftr.,  which  have  tbtiar  mrsttpaadiiia  aag^eg  mM  eq«Ml  to 
each  otkier,  but  it  ia  ^laia  that  the  coixe^poDding  sides  of  no  two  of  the  tri- 
angles can  possibly  be  equal,  since  one  of  these  triangles  always  coBtains  the 
ether. 

Comferse  and  contrary  propositions  are  not  to  be  codfouoded,  they  are  alto- 
gether dtiferent ;  the  former  we  have  explained  above :  two  propositions  are 
contrary  to  one  another,  when  one  afiiiBia  what  the  other  denief^  or  d«aies  what 
it  afirms  ;  th«s,  if  it  be  olfimed  that  '*  two  and  three  «iw  five,"  the  eonirmry 
propoaition  is,  that  two  and  three  ar^  not  &ve.  Again,  *<  two  straight  iinst 
cfitmot.  incloee  a  ^paoe,?  and  ^*  two  straight  linee  can  inolo«e  a  space,*'  ara 
C0n/r«iy.  propi9sition»k  Two  contrary  propositions  caaikot  be  both  trne  w  false : 
thus,  A  is  equal  to  JB,  and  A  is  not  equal  to  B,  are  ooatrary^  pffof^tiona  ^  now 
^  it  is  evident,  that  if  the  fermer  of  these  be  true,  the  latter  caoaot;  and  if 
the  latter  be  true,  the  former,  caoaot »  in  the  same  maaiier  it.iAiiar  he  ahew^ 
that  they  cannot  be  both  false* 


Part  VUt       ON  EUCLID'S  FIHST  BOOK,  ^7 

will  altogetiier  coincide  and  form  but  one  triangle.  It  is 
possible  tbat  AC  and  AD  terminated  at  the  extremity  A  may  be 
equals  but  tben  €B  and  DB  terminated  at  the  extremity  B  can^ 
not  be  equal :  in  like  manner  CB  and  DB  may  be  equal,  but  if 
they  are,  AC  and  AD  cannot ;  and  this  is  all  that  vm  required 
to  be  proved.  The  enunciation  of  prop.  7-  which  in  the  oilgi- 
nftl  is  awkward  and  unintelligib]^  has  been  improved,  by  Dr. 
Simeon ;  he  has  likewise  added  the  second  case,  which  is  not-  \i^ 
the  Greek  fext  of  Euclid,  although  it  is  found  in  the  Arabic 
version  -,  this  case  is  demonstrated  by  means  of  the  latter  past 
of  prop.  5.  which  is  cited  in  no  ether  part  of  the  Elements. 

US.  Prop,  8.  The  7th  propositidn  is  of  no  other  use  than  as 
it  serves  to  demonstrate  this :  we  have  here  a  second  instance 
of  a  pro^  by  suprapoGJition ;  and  eince  it  is  shewn  that  the  tri- 
angles so  applied  completely  coincide,  it  fallows  fh}m  the  6th 
juciom,  that  the  trian^es  are  equal ;  that  the  s&ies  of  the  one  are 
respectively  equal  to  the  sides  of  the  other  3  and  the  angles  6^ 
the  one,  to  the  angles  of  the  other. 

114.  Cor.  Hence,  if  the  three  sides  of  one  tnangln  be  respec- 
tively equal  to  the  three  sides  of  another,  the  two  triangles  witt 
be  both  equal,  antd  equiangular  to  each  other  '. 

115.  Prop.  9.  If  the  angles  BAP,  C^F  be  bisected,  the  whole 
aagle  BAC  will  be  divided  into /our  equal  parts ;  and  if  each  of 
these  parts  be  bisected,  the  angle  BAC  will  be  divided  into  eight 
equal  parts ;  again,  if  each  of  these  parts  be  bisected^  the  whole 
angle  S^C  will  be<Kvided  into  sixteen  equal  psurts,  and  so  on. 
Hence  by  this  propoMtkHi,  an  angle  way  be  divided  into  any 
number  of  equal  jiarts^  provided  thut  number  bfi  some  power  of 
the  number  2. 

116.  Cor,  Hence,  if  a  straiglxt  line  bisect  an  angle  of  a^ 
equilateral  triangle^  or  if  it  bisect  the  angle  incluided  by  the 


*  The  terms  equUtngular  and  eqmangular  to  ench  taAer,  ma^t  oot  ^e  mis- 
understood or  confounded ;  a  figure  is  said  to  be  equiangular,  when  -iktt  ite 
angles  are  equal ;  and  two  figures  are  Mid  to  be  equfangMiar  49  each  other,  whea 
<Mb  of  the  angles  in  one  Bgofe  is  equal  to  its  correepondttig  angle  in  tlte 
<4hei^  alttoygh  neither  of  tbe^e  figqres  may  be  e<||iiai)giri«r  m  tbe  former 
sense :  a  similw  observation  applies  to  the  terms  epUtaterai  and  MuUatenUfif 
ncftoCAer. 

The  converse  of  vrop*  8* »  90^  necestafily  true,  as  is  shewn  in  the  note  0% 
Art.  109. 

T» 


578  GEOMETRY.  Part  VIIL 

equal  sides  of  an  isosceles  triangle,  it  shall  likewise  bisect  the 
base.  (See  the  (ig.  in  Euclid.) 

For  AC^BCy  and  CD  is  common 5  also  the  angle  ACD^ 
\he  angle  BCD,  therefore  (prop.  4.)  the  base  .^l>=the  base 
BD. 

117.  It  has  been'  shewn  in  prop.  9.  and  Art.  115.  that  any 
angle  may  be  bisected  geometncally^  but  the  geometrical  trisec- 
tion  of  an  angle  (except  in  one  particular  case>  see  the  note  on 
Art.  140.)  still  remaibs  among  the  desiderata  in  science;  no 
tkiethod  having  yet  been  discovered  whereby  any  section,  except 
the  bisection^  can  be  performed  by  the  Elements  of  Geometry '. 

118.  Prop.  10.   The  word  "  finite,"  as  used  in  this  place,  b  , 
redundant.    See  the  note  on  Art.  98.    The  method  of  bisecting 
a  given  straight  line  with  instruments  will  be  shewn  hereafter. 

119.  Prop.  11.  Drawing  a  straight  line  perpendicular  to  a 
given  straight  Ikle  from  a  given  point  in  the  latter,  is  called 
'^erecting  a  perpendicular .*' 

120.  From  the  corolbry  to  this  pro|)09ition  it  appears,  that 
two  straight  lines  can  meet  one  another  in  only  one  point ;  for 
if  they  meet  each  other  in  two  points  J  and  B,  (see  the  figure 
in  Euclid))  the  parts  inttircepted  between  A  apd  B  must  either 
coincide  or  inclose  a  space ;  bat  they  cannot  coincide,  otherwise 
the  two  "straight  lines  would  have  a  common  segment,  which  by 


y  A117  angle  may  be  tritected  oigcehraieaHy  as  follows : 

From  tile  angular  pbink.^  as  a  centre,  w^b  ikity  for  radius,  describe  tbe 
•ore  BC,  draw  the  cb«rd  BCmc,  and  let 
ar»tbe  dMHrd  of  Br,  one  third  the  arc 
BCi  then  will  jrS-^S ors  -«c,  which  solv- 
ed by  Cardan's  rule,  gives 


~1 

be  turned  into  a  number,  (by  restoring  C^ 

1^  value  of  c,  &fc.)  and  chords  be  drawn  from  .9  and  C  to  the  points  r  and  «, 

and  ^r  ^n  be  joined,  these  lines  will  trisect  the  given  angle  BAC,  as  wai 

rc%iiired. 

Several  methods  of  trisecting  anangle  may  be  found  in  the  works  of  thoee 
who  have  written  of  the  higher  Geometry,  as  Psappus,  Vi«ta,  Gnian^e,  L'H6pi- 
tal,  Simpson,  Macla«rin,  Emerson,  ITOmerique,  Waring,  &c. 


Past  VIU.        ON  EUCLID'S  FIRST  BOOK.  «;» 

the  coroUaiy  is  impossible ;  neither  can  they  inclose  a  space, 
(axiom  10.)  therefore  they  cannot  meet  each  <Hher  in  mure  tlian 
lOne  point. 

121.  Prop.  12,  Drawing  a  perpendicular  to  a  given  straigbt 
line,  from  a  given  point  wUh&ut  it,  is  called  **  letting  fall  a  per- 
pendicular." We  are  told  in  the  proposition  to  "  take  any  point  . 
D  upon  the  other  side  of  AB ;"  by  "  other  side,"  we  are  to  under- 
Aland  the  side  opposite  to  that  on  which  C  stands. 

122.  Prop,  13.  Leamen  are  generally  perplexed  with  de- 
jnoBstritions  of  which  they  cannot  previously  undewtand  some- 
thing of  the  plan  and  scope,,  and  with  none  more  frequently 
thsn  that  <if  prop.  13.  Let  such  as  find  it  difficult  observe, 
first,  (hat  CBE,  EBJ)  are  by  construction  two  right  angles; 
secondly,  that  the  three  angles  CBA,  J  BE,  BBD,  are  equal  to 
the  above  two,  consequently  to  two  right  angles ;  and  thirdly^ 
that  the  two  given  angles  DBA,  ABC  are  equal  to  the  last-men- 
tioned three,  conseqtiently  to  the  fore-mentioned  two,  and  con- 
sequently to  two  right  angles,  which  was  proposed  to  be  proved. 

123.  Cor,  Hence,  if  the  angles  ABD,  ABC  be  unequal,  the 
greater  is  obtuse,  and  the  less  acute  i  the  former  being  as  much 
greater  than  a  right  angle^  as  the  latter  is  less,  as  is  evident 
from  the  prc^position. 

124.  The  13th  and  14th,  the  18th  and  19th,  and  the  24th  and 
Vfttb,  are  converse  propositians ;  the  29th  is  the  converse  ci  the 
27th  and  28th,  and  the  48th  of  the  47th. 

125.  The  following  is  not  completely  the  converse  of  prop. 
15,  but  it  is  partly  so.  If  two  straight  lines  AE,  EB,  (see 
fiuclid*s  fig.  pr.  15.)  on  the  opposite  sides  of  CD,  meet  CD  in 
any  point  £,  so  as  to  make  the  vertical  angles  A  EC,  DEB  equal, 
.then  will^£  and  EB  be  in  the  same  straight  iine.  For  the  four 
ao^es  at  E  being  equal  to  fbur  right  angles  by^cor.  2,  and  the 
two  CEA,  AEDxth»  two  DEB,.  BEC,  each  of  these  equals  will 
be  the  half  of  four  right  angles,  that  is,  equal  to  two  right 
angles  j  whence  (prop.  14.)  AE  and  EB  are  in  the  same  straight 
line. 

126.  Prop.  20.  Dr.  Simson  remarks,  (from  Proclus,)  ^^t 
^'  the  Epicureans  derided  this  proposition  as  being  manifest  Xq 
Asses i*  some  of  tie  modems  have  done  the  same,  but  equally 
without  reason:  according  to  Ji)uclid*s  plan,  a  deipoqstjt^tioi^ 
was  necessary,  as  will  appear  by  referring  to  Art.  101. 

T  4    .    .     ' 


280 


GEOMETRY. 


Pakt  Vllt. 


127.  Prop.  21.  "  It  is  essentkl  to  tlie  tnitli  of  thk  propni- 
tkm,  that  the  straigfat  lines  drawn  to  the  point  within  the  urian- 
g;le,  be  drawn  from  the  two  extremities  of  the  base  "  omitting  this 
limitation^  there  are  cases  in  which  the  sUm  of  the  two  Unes 
drawn  from  the  base  to  a  point  within  the  triangle,  will  exceed 
the  sum  of  the  two  sides  of  the  triangle,  which  may  be  shewn 
as  follows : 

Let  ABC  be  a  triangle,  right  angled  at  A,  D  any  point  in 
JB,  let  CD  be  joined,  and  BA  produced  to  G ;  then  since  CAD 
is  a  right  «Dg\e,  CAG  is  also  a  right  angle,  (prop.  13.)  but  CAG 
is  greater  than  CDA,  (prop.  16.)  .*  CAD  is  likewise  greater  than 
CDA,    /   CD     is  ^Q 

greater  than  CA, 
(prop.  19.)  From 
CD  cut  off  DE^ 
AC,  (prop.  3.)  In« 
sect  CE  in  F, 
(prop.  10.)  and* 
join  BF;  then  will 
the  sum  of  the  two 
straight  lines  BF*^  I>  A  O 

and  FD  be  greater  than  the  sum  of  BC  anA  CA,  the  sides  df  the 
triangle. 

Because  CFsizFE  by  construction,  •.•  CP+FB^EF'\'FB, 
but  CF+FB  >  BC,  (prop.  17)  ••  EF+F»  >  BC;  to  these  ub- 
equals,  let  there  be  added  the  equals  .  . .  ED^AC 

and  we  shall  have  (by  axiom  4.) EF+FB-^^ED  ^  BC+  AC,, 

but  EF^ED=:FD  /  BF^FD  y  BC^AC.  Q.  E.  D.  and  the 
same  may  be  proved  if  the  angle  CAB  be  obtuse. 

128.  Prop.  22.  To  invalidate  the  force  of  an  objection  which 
has  been  made»to  the  demonifetration  c5f  this  proposition,  it  will 
be  necessary  to  prove  that  the  two  circles  (set  Simson's  figure) 
must  cut  each  other :  thus,  because  any  tWo  of  the  straight  lines 
DF,  FG,  GH,  are  together  greater  than  the  third  (by  hypo-' 
thesis),  •.•  FD  ^  (FG+  GH,  or)  FH,  •.•  the  circle  DKL  must 
meet  the  line  FE  somewhere  between  JF  and  H,  (see  Art.  95.) 
for  the  like  reason,  the  circle  KHL  must  meet  DG  between  D 
and  G ;  consequently  these  circumferences  wilt  pass  both  wiiliout 
and  within  each  other,  and  therefore  must  cut  leach  other.  SeQ 
Art.  96. 


IPakt  VIII.        ON  EUCLID*S  FIfiST  BOOK. 


SSI 


l^.  Pfjop.  ^.  It  in«i«t  be  ol»erred»  tkat  the  two  equal 
(viz.  ooe  in  each  triangle)  must  be  alike  situated  in  the  triangles  $ 
both  must  be  either  between  the  given  angles^  or  oppoeite  equal 
angles^  otherwise  the  triangles  will  not  necessarily  be  equal. 

Let  4^0  be  a  triangle^  right  angled  at  A,  from  whence 
let  AD  be  drawn  per- 
pendicular to  the  base 
BC,  (19.1.)  this  will 
divide  the  triangle 
into  two  others,  ^D^ 
and  ADC,  having  a 
right  a^gle  in  each^ 
(viz.  at  A)  and  the 
angles  ABD,  CAD  equal  %  and  also  the  side  AD  common  $  these 
triangles  therefore  have  two  angles  of  the  one  equal  to  two  an- 
gles of  the  other,  each  to  each,  but  the  common  side  AD  not 
lyin^  either  between  given*  or  opposite  equal  angles,  the  triangles 
are  therefore  not  necessarily  equal. 

129.  Prop,  29.  We  have  before  remarked^  tha*  this  proposi- 
tioD  is  the  converse  of  the  27th  and  2Sth.  It  has  given  the 
gemneten  of  tK>th  ancient  and  modem  times  more  trouble  than 
all  the  rest  of  Euclid's  propositions  put  together^  to  demon- 
strate it  the  18th  axiom  was  assumed  -,  but  this  axiom  is  by  no 
means  self-evident,  and  therefore  the  29th,  which  depends  on  it, 
cannpt  be  said  to  be  proved,  unless  the  axiom  itself  be  previously 
proved>  which  cannot  easily  be  done,  but  by  introducing  aa 
axiom  scarcely  less  exceptionable  than  that  which  was  to  be 
deoionstrated,  "  This  defect  in  Euclid,"  says  an  ingenuous  com- 
mentator,^' is  therefore  abundantly  evident,  but  the  manner  of 
correcting  it  is  by  no  means  obvious  -,"  the  methods  chiefly  em* 
ployed  for  that  purpose  are  the  following  three i  I. ''  A  aew  de*- 
fittition  of  parallel  lines  :*'  2. '*  A  new  manner  of  reasoning  on 
the  properties  of  straight  lines  without  a  new  axiom  :'*  and  3, 
''  The  introduction  of  a  new  axiom  less  objectionable  than  Eu- 


*  See  the  8th  prop.  b.  6.  al«o  Ludlam's  RodimeDts,  p.  18^. 

"W^^re  two  nwnbers  are  placed,  as  (12.  1.)  in  the  above  artfcte,  the.  ant 
tiQiAber  refers  to  the  proposition,  and  the  second  to  the  book  ia  £actid  ;  alto 
If  no  fii^nre  be  mentioned,  that  belonging  to  the  proposition  in  Euclid  which  $| 
under  consideration,  b  always  meant. 


282  GEOIUST&Y.  Pa&t  VIIL 

clid*8  13th  •.**  Omitdng  the  two  former  methods,  we  shall  qroil 
ourselves  of  the  laller^  by  introducing  an  axiom  which  Euclid 
himself  seems  to  have  tacitly  admitted,  (see  prop.  35,  36,  37i 
and  38,  book  1.)  although  he  has  not  formally  proposed  it.  The 
axiom  is  as  follows  : 

130.  Axiom.  If  two  straight  lines  be  drawn  through  the  same 
point,  they  are  not  both  parallel  to  the  same  straight  line. 

By  the  help  of  this  axiom  (if  it  be  admitted  as  such)  we 
may  demonstrate  the  29th  proposition  in  the  following  manner, 
without  the  aid  of  Euclid*s  12th  axiom. 

131.  If  AGH  be  not  equal  to  GHD,  one  of  them  must  be 
greater  than  the  other  i 
let  AGH  he  the  greater, 
4md  at  the  point  G  in  ^ .. ^P^"^"?"^!^ . ? 


K 


the  straight  line  GH 
make  the  angle  A^GH 
^GHD,  (23.  1.)  and 
produce  KG  to  L ;  then 
will  KL  be  parallel  to 

CD,    (27.   1.)   ...  two  ^^ 

straight  lines  passing  through  the  same  point  0  are  both  puuDel 
to  CD,  which  by  our  axiom  is  impossible.  The  alleles  AGH  and 
GHD  are  therefore  not  unequal^  that  is,  they  are  equal.  The 
latter  part  of  the  demonstration  may  proceed  as  in  Sipison,  be- 
ginning at  the  words,  but  the  angle  AGH  is  equul  to  the  fngle 
EGB,  kc. 

1^2.  Cor.  Hence,  if  two  straight  lines  KL  and  CD  make 


•  Boscoricb,  Thomas  Simpson,  Bezoiit,  -  Wolfius/  lyAlerobert,  Sturmios, 
VarigQon,  and  several  otben,  are  for  adoptini^  a  n^w  definition  of  parallel  lines ;, 
Ptolemy^FVanoescbiniSy&e.  have  endearoored  to  demoofltrate  the  prapertics 
of  parallel  lines  without  the  help  of  either  a  new  defiotUao  or  a  new  azi«a» 
bat  bave  fai^ :  Professor  Playfair  introduces  the  axiom  we  have  adopted  above, 
which  on  the  whole  seems  to  be  tlie  best,  and  preferable  in  several  respects  to 
Euclid's.  Clavius  has  be&towed  greater  attention  on  the  subject  than  any 
modem  geometer :  whether  he  considered  his  demonstration  as  founded  on  a 
newaxioai  or  not,  it  is  not  quite  certain,  but  it  appears  that  bis  reasoning 
dependa  on  a  proposition  which  ought  not  to  be  admitted  as  selfrevideot.  A 
further  elucidation  of  this  subject  may  be  found  in  the  notes  on  the  29th  prop, 
jn  Si$Mon*s  Euclid^  Ingram**  Euclid^  Pla^air'a  JSlemetiU  of  Geometry^ 
Simpwn's  Elements  ^  Geometry ^  &c. 


Part  Vllf.        ON  EUCLID'S  FIBST  BOOK.  MS 

with  another  straight  line  EF  the  ai^gles  KGH^  GHCtogether 
less  than  two  right  angles,  KL  and  CD  will  meet  towards  IT  and 
C,  or  on  that  side  of  EF  on  which  are  the  angles  which  are  less 
than  two  right  angles. 

For  if  not,  KL  and  CD  are  either  parallel,  or  meet  towards 
L  and  D;  but  they  are  not  parallel,  for  if  they  were,  the  angles 
KQH,  GHC  would  be  equal  to  two  right  angles  (by  prop.  29.) 
which  they  are  not:  neither  do  KL  and  CD  meet  towards 
L  and  A  for  if  they  did,  the  angles  LGti,  GHD,  being  in 
that  case  two  angles  of  a  tiiangle,  (17.  1.)  would  be  less  than 
two  right  angles;  but  this  is  impossible,  for  the  four  angles 
KGH,  LGH,  CHG,  DHG,  are  together  equal  to  four  right  an- 
gles, (IS.  1.)  of  which  the  two  KGH,  CHG  ar^  by  hypothesis 
less  than  two  right  angles  j  therefore  the  remattiing  two  LGH, 
J>HG  are  greater  than  two  right  angles.  Therefore,  since  KL 
and  CD  are  in  the  same  plane  and  not  parallel,  they  must  meet* 
somewhere  3  but  it  has  been  shewn  that  they  cannot  meet  to- 
wards L  and  D,  wherefore  they  must  meet  towards  K  and  C,  or 
on  that  side  of  £Fon  which  are  the  angles  KGH,  GHC,  which 
are  together  less  thsin  two  right,  angles.  Q.  £.  D.  Thus,  by  the 
assistance  of  our  axiom,  we  have  demonstral^pl  £uclid*s  13th, 
which  is  neither  self-evident,  nor  easily  understood  by  a  be* 
ginntib  * 

133.  Prop,  32.  This  proposition,  which  is  ascribed  to  l^ha- 
goras,  is  one  of  the  most  useful  in  the  whole  Elements,  as  will 
be  evident  in  some  sort  frotti  the  following  corollaries  derived 
immediately  from  it,  viz. 

134.  Cor.  1.  The  exterior  angle  is*  the  difference  between  the 
interior  and  adjacent  angle  and  two  right  angles,  and  each  of 
the  inteiior  angles  is  equal  to  the  difference  between  the  two 
remaining  interior  angles  and  two  right  angles. 

Thus,  let  R  represent  a  right  angle,  J,  B,  and  C  the  interior 
-angles  of  the  triangle :  (see£uclid*s  figure:)  then  wUl  the  exterior 
angU  JCDzrz^R-'C,  also  J^i^R-^B-^C,  B=z^R'^A^C,  and 

135.  Cor.  2.  The  difference  between  the  exterior  aagie  and 
either  of  the  two  interior  opposite  angles,  is  eqaal  to  the  other 
interior  opposite  angle. 

Th^is,  ACD'^JszB,  and  ACD-BzszJ, 

136.  Cor,  3.  If  one  angle  of  a  triangle  be  "jl  right  angle,  the 


M€  GBOMBTBT.  Part  Vin. 

other  two  ft^glcft  taken  togedier  neke  a  right  ang^  come- 
qoently  each  of  them  is  acute:  these  acote  angles  aie  calM 
comjdemmU  of  one  another  to  a  right  angle. 

ThuSf  if  C  be  a  right  angle,  thett  will  A  be  the  compUmaUmf 
B,  and  B  ^  compiemetU  of  ^. 

137.  Cor.  4.  If  one  a^e  be  obtuse*  tbe  reBttiniiig  two  wiH 
be  together  less  than  a  right  ai^gbj  and  cooseqaently  both 
acute. 

138.  Cor.  5.  If  the  sum  of  two  ang^  ia  cme  Iriangle  be  equal 
to  the  sum  of  two  angles  in  another^  the  leaiaioing  angle  In  the 
one  will  be  espial  to  the  reaaiaining  angle  in  the  other ;  and  if 
one  angle  in  one  triai^le  be  eqfual  to  one  angle  in  another,  the 
sum  of  the  two  remaining  aisles  in  the  fimaer  win  be  ei|ual  tQ 
the  sum  of  the  fwo  remainii^  angles  in  the  latter. 

139.  Cor.  6.  If  one  ai^le  at  the  base  of  an  isosc^es  tnan^ 
be  equal  to  one  aii^le  at  the  base  of  another  isosceks  triangtey 
the  two  remaining  angles  in  the  one  will  be  e^al  to  the  two 
remaining  angles  in  the  other,  each  to  each ;  and  if  the  vertical 
angle  of  one  isoscelies  triangle  be  equal  to  the  vertical  a^gle  of 
another,  ^^eh  of  the  angles  at  the  base  of  the  one  will  be  equal 
to  each  of  the  a|(gles  at  the  base  of  the  other. 

140.  Cor.  7.  Bach  angle  of  an  equilateral  triangle  is  one- 
third  of  two  right  angles^  or  two-thirds  of  oda  right  ang||p  ^. 

141.  Cor.  8.  "  AH  the  interior  angles,*'  &c.  as  Cor.  1.  in 
Simson. 

14^.  Cor.  9.  All  the  interior  angles  of  any  rectilineal  figure, 
are  equal  to  twice  as  many  right  anglesj  except  four,  as  the 
figure  has  sides. 

Thus,  let  n^the  number  of  sides,  Si=ihesum  of  the  interior 
engles  in  an^  rectUineal  figure,  then  wiU 

Cor.  8.  stand  thrn^  <S+4RaB^.jR. 

and  Cor.  9.  thus^     Szs^n^4.R. 


-r^ 


*  Hence,  if  the  angle  <tf  an  equilateral  triangle  be  bisected,  (9.  I.)  each  of 
the^narts  will  be  one-tbird  of  a  right  angi^  which  is  the  only  angle  that  can  be 
geowetrifisUy  trisected. 


Fa  rt  VIIL        ON  EUCLID'S  VlRfiT  BOOK. 


143.  Cor.  la  Hence,  tbm  interior  angles  of  the  kOowine^ 
rectilineal  figures  will^  be  as  below :  if  tbe  figure  kaTo 


Three 

Four 

Five 

Six 

Seven 

Eight 

Nine 

Ten 

Eleven 

Twelve 


L  sides,  the  sum  of  its  . 
interior  angles  wills 


8— 4=s4 
10—45=6 

14-4=10 
16-4=19 
IS— 4s  14 
20—4=  16 
22—4=18 
L  24- 4=20  J 


right 
ai^es. 


144.  The  converse  uf  the  former  part  of  prop.  34.  is  as 
follows :  "  If  the  opposite  sides  of  a  quadrilateral  figure  be 
equal>  the  figure  will  be  a  parallelogram.*' 

Let  ABCD  be  a  quadrilateral  figure,  having  its  opposite 
sides  equal,  viz.  AD^BC,  and  ^B=DC,  ^then  will  AD  be 
parallel  to  BC,  and  AB  to  DC,  Join  BD,      ^  j^ 

then  because  ADssBC,  and  AB^xiDC, 
also  BD  common,  •.•  the  angle  /rDB= 
the  angle  DBC,  and  ABD=BDC,  (8.  1. 
and  Art.  113.)  •.•  AD  is  parallel  to  BC,  B  C 

and  AB  to  DC  (27.  1.)  '.•  ABCD  is  a  parallelogram,  according 
to  the  definition,  prop.  34. 

14.5V'  CW.  ilence,  if  the  opposite  sides  of  a  quadrilateral  figure 
be  equal,  its  opposite  angles  will  likewise  be  equal  by  prop.  34. 

146.  The  converse  of  the  second  part  of  prop.  34.  is  this  -. 
^  If  the  opposite  angles  of  a  quadrilateral  figure  be  equal,  the 
figure  will  be  a  parallelogram.'*  Let  the  angle  BAD=iBCD, 
(see  the  above  figure,)  and  ADOszABC;  and  since  these  fouf 
angles  are  the  interior  angles  of  a  quadrilateral  figure,  they  are 
toother  equal  to  four  right  angles  3  (by  Art.  143.)  let  now  the 
above  equals  be  added  and  the  wholes  will  be  equal,  (Ax.  2.) 
that  is,  BAD+ADC^^BCD-^ABC,  •.•  the  former  two  angles, 
as  ip^ell  as  the  latter  two,  will  be  (half  of  four  right  angles, 
or)  two  right  angles,  *.*  (by  prop.  29.)  AD  is  parallel  to  BC,  and 
AB  to  DC;  that  is,  ABCD  is  a  parallelogram. 

146.  In  the  right  angled  parallelogram  ABCD,  if  the  side 
AB  be  supposed  to  move  along  the  line  BC,  and  perpendicular 


2sa 


GfiOMSTRY. 


Fart  Vllf, 


■— aMWM^H  ■      ■       ■■■    '  "  ■■■■       .      ■      ■■■■ 

■       ■"     — — -  — —  ■  -■ 

, ..  .  »  — — —  —  ■ 

■ 


«« 


to  Hi  wlien  ^HarriTes  at  C,  At  ■    T'     i r— i { )I> 

it  will  coincide  with  DC  and 
by  its  motion  it  wiH  h:i%'e 
described  or  generated  the 
parallelogram  A  BCD;  let 
AB  consist  of  suppose  4 
equal  parts,  each  of  which 
we  will  call  unity^  (or  1.) 
let  |}in= one  of  those  parts,  and  Br,  rs,  su,  &c.  each=J3iit; 
now  it  is  plain,  that  when  AB  arrives  at  r,  it  will  by  its  mcrtion 
have  described  the  four  rectangles  between  AB  and  jrr,  each  of 
which  will  be  the  square  of  {Bm,  that  is  of)  unity;  in  like 
manner,  when  AB  arrives  at  s,  u,  v»  z,  C,  it  will  have  described 
8,  12,  16,  20,  24  squares  of  {Bm,  or)  unity :  whence  it  appears, 
that  the  area  A  BCD  or  2^,  is  found  by  multiplying*  the  number 
of  equal  parts  (calfed  units)  contained  in  AB,  or  4,  by  the  num- 
ber of  like  parts  in  BC,  or  6.  In  like  manner,  if  AB  contaia 
n  units,  and  BC  m  units,  the  area  ABCD  will  contain  n  x  m=:nm 
units :  if  »=m,  the  figure  ABCD  will  be  a  square,  and  nm  will 
become  n'  or  m'.  Hence  the  area  of  a  rectangle  is  found  by 
multiplying  the  two  sides  about  one  of  its  angles  into  each 
other,  and  the  area  of  a  square  by  multiplying  the  side  into  itself. 

147.  Prop,  35.  fVom  this  proposition,  and  the  jnreceding 
article,  we  derive  a  method  of  finding  the  area  of  any  pai^e- 
logram  whatever :  for  let  ABCD  (see  Simson*s  first  figure)  be 
supposed  to  be  a  right  angled  parallelogram,  its  area  will  be 
ABxBC,  (by  Art.  146.)  or  the  perpendicular  ^£?,  drawn  into 
(or  multiplied  by)  the  base  BC;  but  DBCF^ABCD  by  the 
proposition,  •/  Di?Cf=:perp.  ^Bxbase  BC. 

148.  Fience  we  have  the  following  practical  rule  for  finding 


c  The  terms  muUipi^itt^  and  dividing^  do  ooi  occur  in  geometrical  laogoage ; 
thus,  ia  the  expression  AB  X  BC^ABCDy  AB  is  said  to  be  drawn  iMio  BC,. 
waA/iBCD  is  not  called  the  product  of  AB  and  BC,  but  their  rectangle;  and 

AB 

in  expressions  like  the  foUowiog  ~^^>  AB  is  not  said  to  be  divided  by  C,  but 

C  is  said  to  be  applied  to  AB,  The  old  writers  are  v^ry  particidar  in  this 
rf  spect,  but  the  moderns  are  less  so,  as  we  frequently  find  arithmetical  terms 
made  use  of  in  their  geometric«U  problems ;  but  this  abuse  should  as  mnch 
as  possible  be  avoided. 


PaxtVUL        on  fiUCLTD'S  first  book.  «87 

the  ai?a  of  a  panllelogram.  1.  Let  &n  a  perpeodicuhr  on  the 
faose  from  any  point  in  the  o|]|K)site  tide.  2.  Multiply  the  base 
aod  perpendicular  together^  and  the  product  will  be  the  area 
required. 

149.  Prop.  37.  Since  every  triangle  b  half  of  the  palallelo- 
gram  described  upon  the  same  base,  and  between  the  same 
parallels,  (see  abo  prop.  41.)  and  the  tOrea  of  the  parallelogram 
isszperp.  X  base,  (by  the  last  article^)  *.*  the  area  of  the  triangle 

will  be -^ J  that  is,  half  the  perpendicular  multiplied 

into  the  base,  or  half  the  base  multiplied  into  the  perpendicuHtTf 
will  give  the  area  of  the  triangle. 

150.  Prop.  38.  Cor.  Hence,  if  the  base  BC  be  greater  than 
the  base  EF,  the  triangle  JBC  wiH  be  greater  than  the  triangle 
£DF;  and  if  BC  be  less  than  ER  the  triangle  ABC  will  be  Ian 
tlum  the  triangle  EDF.  Also,  if  ABC  be  greater  than  EDF, 
then  IS  BC  greater  than  EF;  and  if  less,  less. 

151.  In  prop.  42.  we  are  taught  how  **  to  describe  a  paralle- 
logram that  shall  be  equal  to  a  given  triangle,  and  have  one  of 
its  angles  equal  to  a  given  rectilineal  angle.*'  In  prop.  44.  we' 
are  to  describe  a  parallelogram  with  the  two  former  conditions, 
ttd  also  one  more :  we  are  "  to  apply  a  parallelogram  to  a 
pvm  straight  line,  which  parallelogram  shall  be  equal  to  a  given 
triangle,  and  have  one  of  ks  angles  equal  to  a  given  rectilineal 
angle;'*  ta^'  t^PPfy  ^  parallelog^m  to  a  straight  line,"  means  to 
make  it  on  that  straight  line,  or  so  that  the  said  line  may  be  one 
of  its  sides. 

153.  Prop  45.  The  enunciation  of  this  proposition  is  general, 
if  by  <'  a  given  rectilineal  figure**  we  are  to  understand  *<  any 
given  rectilineal  figure :"  but  the  demonstration  applies  to  only 
a  partu^ular  case ;  for  it  extends  no  further  than  to  four-sided 
figures,  and  doe&  not  even  hint  at  any  thing  beyond ;  but  the 
defect  is  easily  supplied  as  follows ; 


sua 


GEOUETRY. 


Part  VIII. 


Let  ABCOND  be  any  rectilineal  figure  $  join  DB,  DC, 
CNj  then  having  made  the  parallelogram  FKML  equal  to  the 
quadrilateral  figure  ABCD^  as  in  the  proposition.  api>ly  the  pa* 
ndlelogram  LS=^DCN  to  the  straight  line  LM^  having  an  angk 
LMS^E,  then  it  may  be  prored  as  before^  that  ¥L  and  LP  are 
in  the  aame  straight  line  aa  are  KM  and  MS:  also  that  fS  is 
parallel  to  FK>  and  cpi^quently  that  FK8P  is  a  parallelogram 
and  equal  to  4BCND  j  and  applying  as  before  a  parallelogram 
PT^NCO,  having  the  angle  PST^E,  to  tiie  straight  line  PS, 
FKTR  may  in  like  manner  be  proved  ta  be  a  parallelogram 
equal  to  JBCONIK  and  hax-ing  an  angle  FKT=zEi  and  by  a 
similar  process  a  parallelogram  may  be  made  equal  to  any  ^vea 
rectilineal  figure  whateverj  and  having  an  angle  eqi^  to  any 
given  rectilineal  angle.  The  foregoing  illustration  being  under^ 
stood,  the  corollary  to  this  proposition  will  be  evident. 

Cor,  Hence  we  have  a  method  of  determining  the  difference  of 
any  two  rectilineal  figures.  Thus  AUCOND  exceeds  BOON  by 
the  parallelogram  FM, 

153.  Prop.  46.  Cor.  In  a  similar  manner  the  rectangle  con- 
tained by  any  two  given  straight  lines  may  be  described. 

154.  The  squares  of  equal  straight  lines  are  equal  to  one 
another. 

Let  the  straight  lines  AB  and  CD  be  equals  then  will  the 
squares  ABEF,  CDGH 
described  on  them  be 
equal.   For  since  AB=    M 
CD  by  hypothesis,  and 
HC^CD  (Def.  30.)  ••• 
HC^AB,hx3XFA^AB. 
(l>ef.-30.)  /  HC^FA;     ^  ^  ^ 

wherefore  if  the  square  FB  be  applied  to  the  square  HD,  so 
that  A  may  be  on  Cj.and  AB  on  CD,  B  shall  coincide  with  D 


X 


L 

■ 

Paht  Vni.        ON  EUCLID'S  FIRST  BOOK.  2i59 

lecause  AB^CD-,  and  AB  coinciding  with  CD,  ^F shall  coin- 
cide with  CH  because  the  angle  BAF=:DBn,  (Def.  30.  and 
Ax.  11.)  also  ^coinciding  with  C,  and  -^Fwith  CH,  the  point 
P shall  coincide  with  H,  because  AF=zC[I;  in  the  same  manner 
it  may  be  shewn,  that  FE  and  EB  coincide  respectively  with  HG 
and  GD,  therefore  the  two  figures  coincide,  and  consequently 
are  equal  by  Ax.  8.  Q.  E.  D/ 

Cor.  1.  Hence  two  sqimres  cannot  be  described  on  the  same 
straight  line  and  op  the  same  side  of  it. 

Cor,  2.  Hence  two  rectangles  which  are  equilateral  to  one 
another  will  likewise  be  equal. 

155.  If  two  squares  be  equal,  the  straight  lines  on  which  they 
stand  will  also  be  equal. 

Let  ABEF=zCDGH,  (see  the  preceding  figure)  then  will 
AB^CD',  for  if  not,  let  AB  be  the  greater,  and  from  it  cut  off 
AK^CD  (3.1)  and  on  AK  describe  the  square  AKLM,  (46.1) 
then  since  AK=z  CD,  the  square  ^L=the  square  CO,  (Art,  154.) 
but  AE:=CG  M|jypQthesis,  •/  AL^^AE  the  greater  to  the  less 
which  is  impossmie,  •••  AK  is  not  equal  to  CDy  and  in  like  man- 
ner it  may  be  shewn  that  no  straight  line,  either  greater  or  less 
than  AB,  can  be  equal  to  CD,  •••  AB=CD.   Q.  E.  D. 

156.  Prop,  47.  This  proposition,  which  is  known  by  the  nam<( 
of  the  PytJiagorean  Theorem,  because  the  philosopher  Pythagoras 
was  the  inventor  of  it,  is  of  very  extensive  application ;  its  pri- 
mary and  obvious  use  is  to  find  the  sum  and  difference  of  given 
squares,  th^  sides  of  right  angled  triangles,  &c.  as  is  shewn  in 
the  following  articles  ^, 

157.  To  find  a  square  equal  to  the  sum  of  any  number  of 
given  squares.  Let  A,  B,  C,  D,  &c.  be  any  number  of  given 
straight  lines ;  it  is  required  to  find  a  square  equal  to  the  sum 
of  the  squares  described  on  A,  B,  C,  D,  &c. 

Take  any  straight  line  EM,  and  from  any  point  £  in  it 
draw  EP  perpendicular  tq  EM  (11.1)  i  take  EFz==A,  EG:=iB 

'  This  proposition  has  been  proved  in  a  variety  of  ways  by  Ozanam,  Tac- 
^uet,  Stunhias,  Ludlaxn,  Mole,  and  others ;  it  supplies  the  foundation  for 
computing  the  tables  of  sines,  tangents,  &c.  on  which  the  practice  of  TrigoQo- 
metry  chiefly  depends,  and  was  considered  by  Pythagoras  of  such  prime  im- 
portance, that  (as  we  are  told)  he  offered  a  hecatomb,  or  sacrifice  of  100  oxen, 
to  the  gods  for  inspiring  him  with  the  discovery  of  so  remarkable  and  useful 
a  property. 

VOL.  II.  r 


290 


GEOMETRY. 


PAirVni 


(3.1),  join  FG,  make  EL=zFG,  jEH^C,  join  HL,  take  EN:^ 

HL,  EM=zD,  and  join 

MN;    the    square    of 

MN  win  be  equal  to 

the  mm  of  the  squares 

of  ^,  B,  Cy  and  D. 

Because    EF^A, 
andEG=B,  vFGl*=: 

(f!E)«-|-£G!«(47.  1.) 
=r)  ^-f  B*>  and  be- 
cause   EL=^FG,    and 

C«=)^  +  ^4.C;  and  because  EN-LH,  and   EJ*f=  A  v 
MiV)«=(EN|«+£itfl«=T5l«+D«=)  ^+B«+C«+1>*.   which 
was  to  be  shewn ,  and  in  the  same  manner  any  number  oC 
squares  may  be  added  together,  that  is,  a  square  may  be  found 
equal  to  their  sum.  .^. 

158.  To  find  a  square  equal  to  the  difiference  of  the  squares 
of  two  given  unequal  straight  lines. 

Let  A  and  B  be  two  unequal  straight  linesj  whereof  A  is 
the  greater;  it  is 
required  to  find 
a  sqviare  equal  to 
the  excess  of  the 
square  of  A  above 
the  square  of  B, 

In  any  straight 
Hne  CH  take  CD 

=:A,       DEz:zC, 

(3.  1.)  from  D  as 
a  centre  with  the 
distance  DC  describe  the  circle  CKF,  from  E  draw  £F  perpen- 
dicular to  CH  (11.1),  and  join  DF;  EF  wiU  be  the  side  of  the 
square  required. 

Because  FD=z  (DC=i )  A,  DE=:  B,  and  DEFis  a  right  angle, 
V  (47.  1.)  FB\''=(DEI^+EFi^=:)  B^+Wi^  that  is  ^=^JB«+ 
EF\^',  take  B«  from  each  of  these  equals,  and  ^-JB«=£J^^ 
that  is,  EF  is  the  side  of  the  square,  which  is  the  differenct 
required. 


A    B 


Part  VIII.        ON  EUCLID'S  FIRST  BOOK.  291 

169.  Hence*  if  any  two  sides  of  a  right  angled  triangle  b© 
given^  the  third  side  may  be  found.  (See  the  preceding  figure.) 

For  since  S£l«+£?^«=:5y''a,  v  ^DS)*+EFf^:szDF. 

Examples.— 1.  If  the  base  DE  of  a  right  angled  triangle  be. 
«  inches^  and  the  perpendicular  EF  8  inches^  required  the 
longest  side,  or  hypothenuse  DF  •  ? 

Here  ^J5£)H£?1«=  v^6«-h8«=  ^36+64=  ^100=10= 
DF. 

2.  Given  the  hypotl\enuse  =20,  and  the  base  =11,  to  find 
the  perpendicular  ? 

Thus  v^*— 111*=  ^400—121=  ^279=  16.703293= ^/^e 
perpendicular  required. 

3.  Given  the  hypothenuse  13,  and  the  perpendicular  10,  to 
fiod  the  base  ? 


Thus  v^i3)2— To)  2=  ^169—100=  ^6D=S.3066239=</ic 
hose  required, 

4.  Given  the  base  7»  and  the  perpendicular  4,  to  find  the 
hypothenuse  ?     Arts.  8.0622577. 

5.  Given  the  hypothenuse  12,  and  perpendicular  10,  to  find 
the  base  ?     Ans.  6.6332496. 

6.  Given  the  hypothenuse  123,  the  base  99,  to  find  the  per- 
pendicular ? 

ON  THE  SECOND  BOOK  OF  EUCLID'S  ELEMENTS. 

.  160.  The  second  Book  of  Euclid  treats  wholly  of  rectangles 
and  squares,  shevt^ing  that  the  squares  or  rectangles  of  the  parts 
of  aline,  divided  in  a  specified  manner,  are  equal  to  other  rectan- 
gles or  squares  of  the  parts  of  the  same  line,  differently  divided  : 
by  what  rectangle  the  square  of  any  side  of  a  triangle  exceeds. 


*  In  a  -right  aogled  triangle  the  longest  side,  (viz.  that  opposite  the  right 
angle)  is  called  the  hypothenuse,  the  other  two  sides  are  called  legs,  that 
on  wbidi  the  figure  stands  is  called  the  base,  and  the  remaining  leg  tiie 
perfendicuiar, 

u  2 


i92  eEOMETRT.  Paet  VUf . 

or  fidk  short  ti  tlie  torn  of  the  sqptures  of  the  other  two 
flides^  &c. 

161.  RecUn^es  and  squares  may  in  every  case  he  represented 
hy  numbers  or  letters,  as  well  as  by  gecmietrical  figures*  and 
frequently  with  greater  convenience ;  thus,  one  side  of  a  rec- 
tangle may  be  called  a,  and  its  adjacent  ade  h,  and  then  the 
rectangle  itself  will  he  expressed  by  ob  ;  if  the  side  of  a  square 
be  represented  by  a,  the  square  itself  will  be  represented  by  att 
or  a* ',  and  since  in  this  book,  the  magnitudes  and  comparisons 
only,  of  rectilineal  figures  are  considered,  its  object  may  be  at- 
tained by  algebraic  reasoning  with  no  less  certainty  and  with 
much  greater  &cility  than  by  the  geometrical  method  employed 
by  Euclid  -,  we  will  therefore  shew,  how  the  propositions  may 
be  algebraically  demonstrated. 

162.  Def,  1.  Euclid  tells  us  what  ''  every  right  angled 
parallelogram  is  said  to  be  contained  by***  but  he  has  not  in- 
formed us  either  here,  or  in  any  other  part  of  the  Elements,  what 
we  are  to  understand  by  the  word  rectangle,  although  this  seems 
to  be  the  sole  object  of  the  definition ;  instead  then  of  Euclid's 
definition,  let  the  following  be  substituted. 

'*  Every  right  angled  parallelogram  is  called  a  rectangle }  and 
this  rectangle  h  said  to  be  contained  by  any  two  of  the  straight 
lines  which  contain  one  of  its  angles  V' 

163.  Prop  1.  Let  the  divided  line  BCss.$,  its  paits  BD^zot 
DE=^b,  and  EC=c;  then  will  «=a+6-fc.  Let  tbe  undivided 
line  As^x,  then  if  the  above  equation  be  multiplied  by  x,  we 
shall  have  M?=(a-h6+c.x=)  ax+bX'\-cx',  "  that  is,  the  rectan* 
gle  sx  contained  by  the  entire  lines  s  and  x,  is  equal  to  the  seve- 
ral rectangles  ax,  bx,  and  ex,  contained  by  the  undivided  line  x^ 
and  the  several  parts  a,  h,  and  c«  of  the  divided  line  s"  Q. £. D. 

Cor,  Hence,  if  two  given  straight  lines  be  each  divided  into 
any  number  of  parts,  the  rectangle  contained  by  the  two  straight 
lines  wUl  be  equal  to  the  sum  of  the  rectangles  contained  by 
each  of  the  parts  of  the  one,  and  each  of  the  parts  of  thS  other. 
Thus,  let  s^za  +  b-^-c,  as  before, 
Andx^y-\-z. 
Then  sx=i{a-{-b-\'C.y-^z=,)ay'\'by-\-cy-^az-^bz'{-cz. 

*      •     m  I  ■  I  .    ■       .  ,     .1  'f 

*  The  rectangle  contained  by  two  straight  lines  AB,  BC,  i»  ft^fteiaUf 
ealled  <<  the  rectangle  under  JB,  BCs"  or  simply  "  tbe  rectangle  uiB^  BC" 


P4M  Vra .     ON  EUCLIDS  SECOND  BOOK.  29S 

164.  Prop.  9.  Let  ABsss,  ACsza,  and  CBszb. 

Then  a-^bszg,  multiply  these  equals  by  s,  and  as-^-hs^ss; 
that  is,  the  rectangle  contained  by  the  whole  line  s  and  the  part 
a«  together  with  that  contained  by  the  whole  line  «  and  the  other 
part  b,  are  equal  to  the  square  of  the  whole  line  s.   Q.  E.  D. 

This  proposition  is  merely  a  particular  case  of  the  former, 
m  which  if  the  line  «  be  divided  into  the  parts  a  and  b,  and  the 
undivided  line  x^::8,  we  shall  have  5J?=ax-f-&r,  become  sszsas-^ 
k,  as  in  this  proposition. 

165.  Prop.  3,  Let  ^B=*,  AC:=a,  and  CB=6,  then  will  s^ 
«+&«and  56=(a-|-6.6sr)  a5+^;  in  like  manner  £a=r(a-).6.a=) 
aa+a6;  that  is^  in-  either  case  the  rectangle  contained  by  the 
whole  s,  and  either  of  the  parts  a  or  b,  is  equal  to  the  rectangle 
(^  contained  by  the  two  parts  a  and  b,  together  with  the  square 
of  the  aforesaid  part  a,  or  6  as  the  case  may  be.   Q.  E.  D. 

This  proposition  is  likewise  a  particular  case  of  the  first,  in 
which  the  undivided  line  is  equal  to  one  of  the  parts  of  the 
divided  line. 

166.  Prop.  4.  «  Let  AB=:s,  AC:=za,  and  jBC=6,  then  will 
asa-^-b;  square  both  sides^  and  ««=s(a-|-6]*=)  aa-^Siab-^bb; 
that  is,  the  square  of  the  whole  line  s,  (viz.  ss)  is  equal  to  the 
siun  of  the  squa^res  of  the  parts  o^and  b,  (viz.  aa-^bb)  and  twice 
the  rectangle  or  product  of  the  naid  parts,  <viz.  2  ab,)   Q.  E.  D. 

167.  Prop.  6.  Let^C=CB=a,  CD^x,  then  will^2>=fl-h 
«,  and  DBi=^a^x,  and  their  rectangle  or  product  a-f-J^.a— g=s 
oa^xx;  to  each  of  these  equals  add  xx,  and  a-^-xM—x+xxs^aa^ 
tbat  is^  the  rectangle  contained  by  the  unequal  parts,  together 
with  the  square  of  (x)  the  line  between  the  points  of  section  is 
e^ual  ,to  the  square  of  (a)  half  the  line.   Q.  £.  D. 

In  the  corollary,  it  is  evident  that  CMG=the  difference  or 
excess  of  CF  above  JLG,  that  is,  of  the  square  of  (Cg,  or)  AC 
above  the  square  of  CD;  but  CMG  \&==iAa=z{AC+CDx 
^C-CD=:)  ADxDB,  therefore  (C5^*-Cl}l«,  that  is)  A^^^ 
cBi^^ADxDBjOT  as  we  have  shewn  above  au-^xxsaa-^-x. 
a— J?. 

nil"' .P  ■      I  !■  Ml  I  I    I  II  I  I  ■  '  !■    m,,,. 

*  In  Euclid's  demonstration  there  is  no  neeessity  to  prove  the  figure  CQKB 
rectangular  in  the  manner  he  has  done ;  it  may  be  jhewn  thus,  **  because 
COKB  is  a  parallelogram,  and  the  angle  CBK  (the  angle  of  a  square)  a  right 
ug^e,  therefore  all  the  angles  of  CQKB  are  right  angles  by  Cor.  46.  I. 

vs 


294  GEOMETRY.  Part  VIIL 

16S.  Prop.  6.  Let  AC^CB^a,  BD^x,  tben  will  AB:=s^a, 
at^d  AD^^ia-^-X',  then  the  rectangle  contained  by  AD  and  DB 
will  be  2  a-f  J7.x=9  ax-^-xx.  to  these  equals  let  aa  (the  square  of 
half  AB)  be  added^  and  2  a-f-ar.j:+aa=(aa+3  «rx+xr=)  a+J)*; 
that  is,  the  rectangle  contained  by  the  line  produced  and  part 
produced,  together  with  the  square  of  half  the  line  bisected^  is 
equal  to  the  square  of  the  line  made  up  of  the  half,  and  part 
produced.  Q.  E.  D. 

Cor.  Hence,  if  three  lines  x,  a-^  Xy  and  2a+x  be  arithmeti- 
cally proportional,  the  rectangle  contained  by  the  extremes 
(x.2tf -fx)  together  with  the  square  of  the  common  difference  a, 
(or  aa)  is  equal  to  (a  •fx]*)  the  squai*e  of  the  middle  term. 

169.  Prop,  7.  Let  AB=s,  AC=a,  CB^b,  then  s=ra-|-6,  and 
M=(a4-6l*=aa+2a6+66=)  ^ab-^bb-^aa,  to  these  equals  add 
bb,  and  m+6&=  (2a6+2  W+fla=2.a-f-6.ft+aa=)  2s6+a<l,• 
that  is,  the  square  of  the  whole  line,  (or  ss)  and  the  square  of 
one  part  6  (or  bb,)  is  equal  to  twice  the  rectangle  contained  by 
the  whole  5,  and  that  part  5,  (or  ^sb,)  together  with  (aa)  the 
square  of  the  other  part.    Q.  E.  D. 

Cor.  Hence,  becaifse  2«6+a«=5«+66,  by  taking  2«6  from 
both,  we  have  aa= w— 2  sb-^-  bb ;  that  is,  the  square  of  the  differ- 
ence of  two  lines  («)  AB  and  (5)  CB,  is  less  than  the  sum  of  the 
squares  of  («)  AB  and  (ft)  CB,  by  twice  the  rectangle  (2  sh) 
2.AB.CB  contained  by  those  lines. 

170.  Prop.  8.  Let  AB=s,  AC=:a,  CB=b,  then  «=o-f-ft,  or 
<t=s— ft,  •.•  aa=(s— ft]*=:=)  w— 2*ft-|-ftft,  to  each  of  these  equals 
add4sft,  and  4s6+aa=w4-SJsft+ftft=5-hft'*5  that  is,  f4  5ft,  or) 
four  times  the  rectangle  contained  by  the  whole  s,  and  one  part 
ft,  together  with  (aa)  the  square  of  the  other  part  a,  is  equal  to 
(5+TI|  ■  or)  the  square  of  the  straight  line  made  up  of  the  whole 
5,  and  the  part  ft.    Q.  E.  D. 

171.  Prop.  9.  Let  AC^CB^a,  CD=x,  then  will  the  greater 
segment  ^D=:a+a?,  and  the  less  segment  DB^=:a — x. 

Then  a-f  ^l*=««+2aa?+a:r 
And   a— x]  *  =rtfa— 2  ax'\-xx 


The  sum  of  both=2cra4-2xj?=2.aa4-xx 

That  is,  aH-x)*-f  fl— X '=2.aa4-xx,  or  the  sum  of  the 
squares  of  the  unequal  parts  (a+x  and  a— x)  is  equal  to  double 
the  square  of  the  half  a,  and  of  the  part  x  between  the  points 


AHT  Vlir.      ON  EUCLID'S  SECX>ND  BOOK.  «95 

of  section;  or,  which  is  the  same  thing,  *'  the  aggregate  of  the 
squares  of  the  sum  and  difference  of  two  straight  lines  a  and  x 
k  equal  to  double  the  squares  of  those  lines."    Q.  £.  D. 

172.  Prop.  10.  Let  JC=iCB=sa,  BDz=zx,  then  will  AD^ 
Sfl+T,  and  CDzsza+x. 

Now  *iJa-f-T)»=4aa+4ar+a:x 
Add  XX  to  this,  and  the  sum  is  4aa^4ax-\-2xx 
Also  a+x]*=aa+2aa:+xjr,  add  aa  to  this,  and  it  becomes 
2aa+^aa7-|-Tx;  now  the  former  of  these  sums  is  double  of 
the  latter,  that  is  4fla4-4ai:+2a?x=2.^aa+2ax+xx;  or,  the 
square  of  the  produced  line  Za-{-x,  together  with  the  square  of 
the  part  produced  x,  is  double  the  square  of  a  half  the  line,  and 
the  square  of  a4<d  the  line  made  up  of  the  half  and  the  part 
produced.   Q.  E.  D. 

173.  Prop.  11.  This  proposition  is  impossible  by  numbers,  for 
there  is  no  number  that  can  be  so  divided,  that  the  product  of 
tlie  whole  into  one  part,  shall  equal  the  square  of  the  other  part  ,* 
the  solution  may  however  be  approximated  to  as  follows : 

Let  ^£=2  a,  AHzzzx,  HB=:y,  then  by  the  problem  x-^y^ 
3.0,  and  ^exy^xx;  from  the  first  equation  y=2a— x^  this  value 
being  substituted  for  y  in  the  latter  equation^  we  shall  have 
iaa^^€a=^xx,  or  aRr+2a«=:4  aa,  this  solved  (by  Art.  97.  part.  3.) 
gives x=  -j-  ^5 aa~^a,  and  y=(2 a— j?=3  a—  >/5 aa^a=z)  3a— 
j^baa,  or  which  is  the  same  «=1.236068>  &c.  xa,  and  ysz 
^63931,  &c.  xa« 

174.  Prop.  12.  Let  ^jB=a,  BC=6,  CD=x,  and  AD=:z; 
Then  (47.  1.)  -i'B'l'  =  S5l»+S3)'=6TI)»+2z= 

bb-^^bx-^xx+zz 
And  CS)'  +  AC\'=:  bb  *  -^xx+zz 
(Subtract  the  latter  from  the  former,) 

Therefore  ^B> -7751'' +56]-= 2bx     *     * 

That  is,  the  square  of  AB/ihe  side  subtending  the  obtuse 
angle^  is  greater  than  the  sum  of  the  squares  of  CB  and  AC,  the 
sides  containing  the  obtuse  angle,  by  (2  bx)  twice  the  rectangle 
BC,CD.    Q,E.D. 

175.  Prop.  13.  Let  ABsia,  CB—b,AC^c,  AD=zd,  BD=zm, 
DC:=n;  then  the  first  case  of  this  proposition  is  proved  as 
follows : 

First,  66  -f  »^m=26OT + nn  (7. 2.)  To  each  of  these  equals  add 

V  4 


9»e  OEOHETRY.  Part  VUl. 

dd,  and  bb-jrtnm'^ddss^bm+dd'^'nn.  But  4xas mm +c((2>  and 
cc=dd-^nn  (47. 1.)  '•'  if  cw  and  cc  be  substituted  for  their  equab 
in  the  preceding  equation^  we  shall  have  fc6-hart=2  6ra-f  cc,  or 
cc=  6& + <za — 2  ^171 . 

Second  case.  Because  aa=cc+664*26n  (19.  2.)  add  66  to 
both  sides,  and  aa+66=cc+2  664-2  6n,  but  6m=6n+66  (3.2.) 
'.'  2  6m=2  6n4-2  66 ;  substitute  2  6m  for  its  equal  i^  the  preceding 
equation,  and  tfa  +  66=cc-|-2  6m,  or  cc::=:<ia-h66«^2  6m. 

Third  case.  Here  the  points  C  and  D  coincide,  *.*  6=m/ 
wherefore  since  cc+66=aa  (47.  1.)  to  each  of  these  equals  add 
66,  and  cc + 2  66  =±(za-p  66,  or  cc=aa+66— 266,  which  correfrr 
ponds  with  the  former  cases  since  2  66  here  answers  to  2  bm 
there.  Wherefore  cc  is  less  than  aa4-66  by  2  6m,  or  3C)»^ 
ifii)'4-5c)»  by  2,  Ca  BD.    Q.  E.  D. 

176.  Prop.  14.  By  help  of  this  problem  any  pure  quadratic 
equation  may  be  geometrically  constructed*  To  construct  an 
equation  is  to  exhibit  it  by  means  of  a  geometrical  figure,  m 
such  a  manner,  that  some  of  the  lines  may  exjMress  the  cour 
ditions^  and  others  the  roots  of  the  given  equation. 

Examples. — 1.  Let  x'ssab  be  given  to  find  a?  by  a  geome- 
trical construction.     See  Euclidts  figure. 

Make  BEi=ia^  EF^b,  then  if  BFbe  bisected  in  the  point  6, 
(10.  1.)  and  from  G^  as  a  centre,  with  the  distance  GF,  a  arcle 
he  described,  and  EH  be  drawn  perpendicular  to  BF  from  the 
point  Ej  (11.  1.)  it  is  plain  that  EH  will  be  the  value  of  x^  For 
by  the  proposition  EH]'szBExEF=iab,  but  by  hypothesis  x^zs 
ab,  *.'  JSH)»=ra?*,  and  ElJ^x;  which  was  to  be  shewn. 

But  the  root  of  x^is  either  +J7  or  —ar,  now  both  these  roots 
may  be  shewn  by  the  figure,  for  if  £H=: + J?,  and  EH  be  produced 
through  D  till  it  meet  the  circumference  below  BF,  the  line  inter" 
cepted  between  E  and  the  circumference  will  ^z^x,  for  in  this 
case  BE X  £jF=— a; x  — a:=  H-r%  as  before. 

2.  Let  x'  =s:36  be  given,  to  find  the  value  of  x. 

Here,  because  36=9x4,  7nake  JB£=:9,  £F=4;  then  pro^ 
ceeding  as  before,  eSI* =9X4=36,  and  EH^6. 

3.  Let  a:» =120=12  x  10  be  given. 

Make  i5£=l2,   JEF=J0,  then  JMB«=120,  0std   EHsi 
(^120=)  10.95445=3?. 

4.  Let  (r'=3  be  given. 


Pakt  Vra.       ON  EUCUD'S  THIB©  BOOK.  «^ 

Here  3=3x1;  make  BEszS,  EF=1,  then  EH)*=3,  and 
£J7=5l.73205=u 


ON  THE  THIRD  BOOK  OF  EUCLID'S  ELEMENTS. 

177*  This  book  demonstrates  the  fimdamenta]  properties  of 
circles^  teaching  many  particulars  relating  to  lines>  angles,  and 
figures  inscribed ;  lines  cutting  them ;  how  to  draw  tangents  i 
describe  or  cut  off  proposed  segments,  &c. 

178.  Def.  I,  **  This,"  as  Dr.  Simson  remarks,  *'  is  not  a  de- 
finition, but  a  theorem  5"  he  has  shewn  how  it  may  be  proved : 
and  it  may  be  added,  that  the  conv«*se  of  this  theorem  is  proved 
In  the  same  manner. 

179.  Def.  6  has  been  already  ^ven  in  the  first  book,  and 
might  have  been  omitted  here,  (see  Art.  74.)  Def.  7  is  of  no 
use  in  the  Elements,  and  might  likewise  have  been  omitted. 
Ia  the  figure  to  def.  10  there  is  a  line  drawn  from  one  radius  to 
the  other,  by  which  the  figure  intended  to  represent  a  sector 
of  a  circle  is  redundant :  that  line  should  be  taken  out. 

180.  Prop.  1.  Cor.  To  this  corollary  we  may  add,  that  if  the 
bisecting  line  itself  be  bisected,  the  point  of  bisection  will  be 
the  centre  of  the  circle. 

181.  Prop.  2.  X^is  proposition  is  proved  by  reductio  ad  ab« 
surdum.  The  figure  intended  to  represent  a  circle  is  so  very 
unlike  one,  that  it  will  hardly  be  understood,  the  part  AFB  of 
the  circumference  being  hent  in,  in  order  that  the  line  which 
joins  the  points  A  and  B  may  fall  (where  it  is  impossible  for 
that  line  to  fall)  without  the  circle. 

The  demonstration  given  by  Euclid  i^  by  reductio  ad  absur- 
dutn.  Commandine  has  proved  the  proposition  directly  ;  his  proof 
depends  on  the  following  axiom  which  we  have  already  given, 
viz.  '*  If  a  point  be  taken  nearer  the  centre  than  the  circum- 
ference is,  that  point  is  within  the  circle."    Thus, 

182.  Let  AB  be  two  points  in  the  circumference  ACB,  joip 
AB,  this  line  will  fall  wholly  within  the  cirde.   Find  the  centire 


£96 


GEOMSTRV. 


pajit  vni. 


D,  (Art.  179.)  m  JB  take  any  point 

E,  and  join  DA,  DE,  and  DB,  Be- 
cause DA=:DB,  •••  the  angles  DAB 
DBA  are  equal,  (5.  1.)  but  DEB  } 
than  jD-<^B  (16. 1.)  consequently  ^ 
than  JDJB^j  /  DB  >  DE  (19. 1.)  / 
by  the  axiom  the  point  £  is  within  the 
circle,  and  the  same  may  be  proved 
of  every  point  in  AB,  •/  AB  fells  within  the  circle.     Q.  E.  D. 

183.  Prop.  4.  It  is  shewn  in  prop.  3.  that  one  line  passing 
through  the  centre  may  bisect  another  which  does  not  pass 
through  the  centre ;  but  it  b  plain  that  the  latter  cannot  bisect 
the  former,  since  it  does  not  pass  through  the  centre,  which  is 
the  only  point  in  which  the  former  can  be  bisected. 

184.  Prop.  16.  A  direct  proof  may  here  be  given  as  in 
Art.  181.  prop.  2.  provided  the  corresponding  axiom  be  ad- 
mitted, namely,  '^  If  a  point  be  taken  ferther  fiom  the  centre 
than  the  circumference  is,  that  point  is  without  the  circle/ 
Thus, 

Let  BEA  be  a  circle,  D  its  centre,  BA  a  diameter,  and 
CAT  a  straight  line  at  right 
angles  to  the  diameter  BA 
at  the  extremity  A,  the  line 
C^r  shall  touch  the  circle 
in  A.  In  CT  take  any  point 
C,  and  join  DC  cutting  the 
circle  in  £,  then  because 
DAC  is  a  right  angle,  DCA 
is  less  than  a  nght  angle 
(17.1.) '.-J^C^D^  (19.1.) 
*.*  D  is  farther  from  the  cen- 
tre than  Ay  consequently  by  the  axiom  C  is  without  the  cirdc, 
and  the  same  may  be  shewn  of  every  point  in  CT,  -.-  CT  is 
without  the  circle.     Q.  E.  D. 

Cor.  Hence  it  appears  that  the  shortest  line  that  can  be  drawn 
from  a  given  point  to  a  given  straight  line,  is  that  which  is  per- 
pendicular to  the  latter. 

185.  In  the  enunciation  of  this  proposition  we  read,  that 
"  no  straight  line  can  be  drawn  between  that  straight  line  (i  e, 
the  touching  line,  or  tangent)  and  the  circumference  irom  the  ej(« 


Part  YIII.      ON  EUCLID'S  THIRD  BOOK.  <299 

tremity  (of  the  diameter)  so  as  not  to  cut  the  circle ;"  this  ap- 
pears to  be  an  absurdity,  for  how  can  a  line  be  said  to  be  between'^ 
the  tangent  and  circamference,  if  it  cut  the  latter  ?  and  how 
can  a  line  which  cuts  the  circumference  be  between  it  and  tlie 
tangent  ?  The  like  may  be  observed  of  the  sentence^  ^*  therefore 
no  straight  line  can  be  drawn  from  the  point  A  between  AE  and 
the  circumference^  which  does  not  cut  the  circle/*  It  was  for  the 
sake  of  the  latter  part  of  the  demonstration  that  the  seventh 
definition  of  this  book  was  introduced  ^  both  may  be  passed 
over,  as  they  do  not  properly  belong  to  the  Elements. 

186.  Prop,  24.  The  demonstration  of  this  proposition  is 
manifestly  imperfect  j  after  the  words  "  the  segment  AEB  must 
coincide  with  the  segment  CFD,**  let  there  be  added,  '^  for  if 
AEB  do  not  coincide  with  CFD,  it  must  fall  otherwise  (as  in 
the  figure  to  prop.  23.)  then  upon  the  same  base^  and  on  the 
same  side  of  it^  there  will  be  two  similar  segments  of  circles 
not  coinciding  with  one  another^  but  this  has  been  shewn  (in 
prop.  23.)  to  be  impossible  >  wherefore,  &c."  Without  this 
addition,  the  proposition  cannot  be  said  to  be  fairly  proved. 

18/.  Prop.  30.  It  is  of  importance  to  shew  that  DC  falls 
without  each  of  the  segments  AD  and  DB,  and  since  the  centre 
is  somewhere  in  DC  (cor.  1.3.)  it  must  be  likewise  without 
each  of  those  segments 3 .  wherefore  (by  the  latter  part  of  25.  3.) 
each  of  the  segments  ^D  and  DB  is  less  than  a  semicircle. 

188.  By  means  of  prop.  35.  and  36.  the .  geometrical  con- 
struction of  the  three  forms  of  affected  quadratic  equations  may 
be  performed. 

The  first  and  second  forms  are  thus  constructed  \ 


*  The  geometrical  construction  of  an  equation  is  the  redocing  it  to  a  geo- 
metrical figure,  wherein  the  conditions  of  the  pr»powd  equation  being  ex- 
hibited by  certain  lines  in  the  figure,  the  roots  are  determined  by  the  inter- 
sections which  necessarily  take  place  in  consequence  of  the  construction. 

The  ancients  made  great  use  of  geometrical  constructions,  which  is  probably 
owing  to  the  imperfect  state  of  their  analysis  ;  but  the  improvements  of  the 
rooderns,  particularly  of  Mercator,  Newton,  Leibnitz,  Wallis,  Sterling, 
Demoivre,  Taylor,  Cramer,  Euler,  Maclanrin,  and  others,  have  in  a  great 
itteasure  superseded  the  ancient  methods. 

Simple  equations  are  constructed  by  the  intersection  of  right  lines,  quadra- 
lies  by  means  of  right  lines  and  the  circle^  but  equations  of  higher  dimensions 
require  the  copic  sections^  or  curves  of  superior  kinds,  for  their  construction ; 


Sm  GEOlfBTRT.  Part  VUI. 

Tint  fotm      -xx+ax^he. 
*  Second  form  xx^axtsbc. 

fhxm  C  as  a  c^Dtre  with  a  dktanoe  2=4.4  describe  the 

circle  JGB,  then  (mippoBing  ft  ^  c,)  with  the  dintanrfi  6— c  ia 

the  compasfies  (taken  firom  any  convenient  scale)  from  any  pcMiA 

£  in  the  ciicamferenoe»  describe  a  small  arc  cutting  the  cireum- 

lerenoe  GB  in  F,  join  EF,  and 

produce  it  to  D,  making  FD 

s=c,  and  from  D  draw  DBCA 

passing  through  the  centre  C, 

then  will  DB  and  DA  be  the 

values  of  X  in  both  the  first 

and  second  forms,  viz.  x=s 

+DB  or— D^  in  the  firet 

formt   and   x:b+DA  or — DB*m.    the   second   form.     For 

since  ABssa  by  construction^  if  DB^x,  DA  will  be  «+«, 

but  if  DAszx,  then  DBssx^-a;  but  DA.DB=DE.DF  (37. 3.) 

or  (x+a.r=)  xx+ox^ftc  in  the  first  form>  and  {xjc — a=) 

xx'^cuo^bc,  in  the  second^  and  since  the  two  proposed  equatiood 

differ  only  in  the  sign  of  the  second  term>  it  is  plain  that  they 

will  have  the  same  roots  with  contrary  signs^  (see  Art.  30. 

part  5.) 

189.  If  we  suppose  (=c,  the  construction  will  be  still  more 
simple^  for  (6— c=)  EF=:o,  that  is  £F  will  vanish,  and  DFwill 
consequently  touch  the  circle  in  G,  and  become  DG,  and  we 
shall  then  have  DA,DB=zDG]9 ;  wherefore  if  a  right  angled 
triangle  DGC  be  constructed  having  GC^^,  and  DGszb, 

{BD=DC^CG  in  the  fast  form,  and  its  negative 
value -JDCTCG. 
DA^DC-k-CG  in  the  secondj  and  its  negative 
value  '^DC-'CG. 

190.  To  construct  the  third  -form  of  affected  quadratic  equa- 
tions, or  arr— ax  = — ab. 

From  the  centre  C  with  the  distance  CB  asj-a,  describe  tfae 
circle  AEF  as  before,  from  any  point  £  draw  EF^b-i-c,  make 

TUrions  methods  of  construction  eqnations  may  be  seen  in  tbe  writings  e# 
Slusios,  Vieta,  Albert  Oirard,  Schooten,  Fermat,  Des  Cartes,  Ghetaldos, 
De  la  Hire,  Barrow,  Robcrval,  Halley,  Newton,  Gregory,  Baker,  Hyac,  Star- 
mini,  De  I'Hdpitai,  Sterliog,  Madaarin,  Simpson,  Emerson,  and  others. 


Part  VIII.        ON  EUCLID'S  THIRD  BOOK.  301 

flDssb,  then  DF=c,  join  DC  and  produce  it  both  ways  to  A 
and  B. 

Since  ABsza,  if  AD  be  called  x,  then  wiU  DJBsa— x, 

but  JD.DBszED.DF  (35.3.) 
that  is,  (x.a — x=)  ox — xxssbc, 
or  which  is  the  sum  xx — ax= 
— be  as  was  proposed  to  be  shewn. 
The  like  conclusion  will  follow  by 
supposing  DFssx,  whence  the  two 
roots  of  the  given  equation  are  AD 
md  DB. 

191.  If  6=c,  then  will  EDssDF,  and  AB  will  be  perpen- 
dicular  to  EF  (3.  3.)  and  EC  being  joined^  we  shall  in  that  case 
have  a  right  angled  triangle^  the  hypothenuse  of  which  will 
=^a^  and  one  of  its  sides  ssb,  wherefore  the  sum  and  difference 
of  the  hypothenuse  and  the  other  side  will  be  the  two  roots  of 
the  equation  as  is  manifest. 

ON  THE  FOURTH  BOOK  OF  EUCLID'S  ELEMENTS. 

192.  This  book  will  be  found  of  great  use  to  the  practical 
geometrician^  it  treats  solely  on  the  inscription  of  regular  rec« 
tilineal  figures  in^  and  their  circumscription  about  a  circle; 
and  of  the  description  of  a  circle  in  and  about  such  rectilineal 
figures. 

193.  Prop.  1.  The  reason  why  the  straight  line  required  to 
be  placed  in  the  given  cirate  ifiust  not  be  greater  than  the  dia- 
meter, appears  from  the  1 5th  proposition  of  the  3rd  book, 
where  it  is  proved,  that  the  diameter  is  the  greatest  straight, 
line  that  can  be  placed  in  a  circle. 

194.  Prop.  4.  From  this  proposition  it  appears,  that  the 
three  lines  which  bisect  the  three  angles  of  a  triangle^  will  all 
meet  in  the  same  point  within  the  triangle.  Also  the  sides  of 
any  triangle  being  known,  the  segments  intercepted  between 
their  extremes,  and  the  points  of  contact,  may  be  found  K 

■   ■-■i^ii        iiiii         II  ■       ■  ■  ii« 

i  Thra,  Iet^fra»40,  ^C»30,  and  BC^30,  then  will  AB+  BC^SO;  horn 
ibis  lubtract  AC^AE+FC^m,  and  the  remainder  is  BB-hBF^SOi 
ther«foie  JR£«1W'«15,FC«C0«(^C— ^Z**)  5,  and  AQ^AE^iAC'^ 

CO»)  25, 


d04  GEOMETRY.         '  PARxVIIt 

195.  Prop.  5.  We  hence  learn  that  it  is  possible  to  describe  a 
circle  through  any  three  given  points,  provided  they  are  not 
placed  in  a  straight  line;  for  by  joining  every  two  points,  a 
triangle  v^ill  be  formed,  and  the  proof  will  be  the  same  as 
in  the  proposition.  Also  only  one  circle  can  pass  through 
the  same  three  points.  (10.  3.) 

196.  "  The  line  DF  is  called  the  locus  of  the  centres  of  all 
the  circles  that  will  pass  through  A  and  jB.  And  the  line  EFis 
the  locus  of  the  centres  of  all  the  circles  that  will  pass  through 
A  and  C.  And  this  method  of  solvit^  geometrical  problems, 
by  finding  the  locus  of  all  those  points  that  will  answer  the 
several  conditions  separately,  is  called  constructing  of  problem 
by  tlie  intersection  of  geometric  loci  V* 

197.  Prop,  6.  Hence  the  diameters  of  a  square  (being  each 
the  diameter  of  its  circumscribing  circle)  are  equal  to  each 
other  3  they  also  bisect  the  angles  of  the  square,  and  divide  it 
into  four  triangles,  which  are  equal  and  alike  in  all  respects : . 
and  since  the  square  of  jBD=the  sum  of  the  squares  of  BA 
and  AD  (47.  1.)  =2.^*,  it  follows  that  B5lH^'=3iJl*+ 

198.  Prop,  7.    Because  the  side  of  a  square  is    equal  to 

the  diameter  of  its  inscribed  circle  (for  GF==JBD,)  and  the 
square  of  the  diameter  is  equal  to  twice  the  inscribed  square> 
(see  the  preceding  article)  3  therefore  a  square  circumscribed 
about  a  circle  is  double  the  square  inscribed  in  it. 

199.  Prop,  10.  Since  the  interior  angles  of  ABD:=2  right 
angles  (32.  1.)  and  the  angle  B=iD=^9lA,  *.•  the  angles  at 
4,  £,  and  Dj  are  together  equal  to  (A-^-^A+^Assz)  bA,  that 


k  LudlcaxCs  Rudiments,  p.  207,  Loci  are  expressed  by  algebraic  equatioDS 
of  different  orders,  according  to  the  nature  of  the  locus.  If  the  equation  be 
constructed  by  a  right  line,  it  is  called  locus  ad  rectum;  if  by  a  circle,  loau 
ad  circulum  ;  if  by  a  parabala,  locus  ad  paraholam  /  if  by  an  ellipsis,  locus  ad 
eUipsim.  Th«  loci  of  such  equations  as  are  right  lines  or  circles  the  ancients 
called  plane  loci;  of  those  that  are  conic  sections,  solid  loci;  and  of  thos^ 
that  are  of  curves  of  a  higher  order,  sursolid  loci.  But  the  moderns  distin- 
guish the  loci  into  orders,  according  to  the  dimensions  of  the  equations  by 
which  they  are  expressed.—- fTu/Zon.  The  following  authors^  among  many 
others,  have  treated  of  this  subject,  viz.  Euclid,  ApoUonius,  Pappus,  AristaeoSy 
Viviani,  Fermat,  Des  Cartes,  Slusius,  Baker,  De  Witt^  Civg,  L'Hdpital^ 
Sterling,  Maclaunu^  Emerson^  and  Euler. 


Pabt  VIIL      ON  EUCLID'S  FOURTH  BOOK.  303 

is  5^=:2  right  angles^  and  A=r^  of  2  right  angles ;  wherefore 
if  ijf  be  bisected^  each  of  the  parts  will  be  -^  of  one  right  angle. 
Hence  by  this  proposition  a  right  angle  is  divided  into  live  equal 
parts,  and  if  each  of  these  parts  be  bisected,  and  the  latter 
again  bisected,  and  so  on,  the  right  angle  will  be  divided  into 
10,  20,  40,  60,  &c.  equal  parts  5  and  since  the  whole  circum- 
^rence  subtends  four  right  angles  (at  its  centre),  the  circum- 
ference will,  by  these  sections,  be  divided  into  (4x5,  4  x  10, 
4x20,  &c.  or)  20,  40,  80,  &c.  equal  parts;  and  by  joining  the 
points  of  section,  polygons  of  the  same  number  of  sides  will  be 
inscribed  in  the  circle. 

200.  Prop.  11.  Because  by  the  preceding  article,  CAD^s^^  of 
two  right  angles,  and  the  three  angles  at  Ay  which  form  the 
angle  BAE  of  the  pentagon,  are  equal  to  one  another  (being  in 
equal  segments  21.  3.)  '•*  BAE  =f  of  two  right  angles  or  4  of 
one  right  angle. 

201.  Prop.  13.  It  follows,  that  if  any  two  angles  of  an  equi- 
lateral and  eqaiangular  figmre  be  bisected,  and  straight  lines  be 
drawn  from  the  point  of  bisection  to  the  remaining  angles,  these 
•ball  likewise  be  bisected  5  and  if,  from  this  point  as  a  centre, 
with  the  distance  from  it  to  either  of  the  angles,  a  circle  be 
described,  this  circle  shall  pass  through  all  the  angles,  and  con- 
sequently circumscribe  the  given  equilateral  and  equiangular 
fi^e.    See  prop.  14. 

tBb.  Prop,  15.  Hence  the  angle  of  an  equilateral  and  equi- 
^gular  hexagon,  will  be  double  the  angle  of  an  equilateral  tri- 
angle, that  is,  4  of  2  right  angles,  or  4  of  one  right  angle.  This 
proposition  b  particularly  useful  in  trigonometry. 

203.  Pr^yp*,  16.  All  the  angles  of  a  quindecagon  (by  cor.  1. 

pr.32.b.l.)areequalto(2x  15— 4r=)  26  right  angles  5  wherefore 

26      11 

rr=  1 —  right  angle  =  one  angle  of  an  equilateral  and  equi- 
ps     15 

angular  quindecagon.  If  each  of  the  circumferences  be  bisected, 
each  of  the  halves  bisected,  and  so  on  continually,  the  whole  cir- 
cumference will  be  divided  into  15, 30,  60,  120,  &c.  equal  parts^ 
and  these  points  of  bisection  being  joined  as  before,  equilateral 
uid  equiangular  polygons  of  the  above  numbers  of  8ides>  will  be 
inscribed  as  is  manifest. 

204.  Hence,  by  inscribing  the  following  equilateral  and  equi- 
angular figures,  and  by  continual  bisection  of  the  circumferences 


304  GEOMETRY.  PaktVIII, 

subtended  by  their  sides^  the  circle  will  be  divided  into  the 
ffdiowing  numbers  of  equal  parts,  viz.  by  the 
Triangle,  into  3,  6,  12,  24,    4S,    96,  192,384,    &c^ 
Square  4,  S,  16,  32,    64,    128,  256>  512,    &c.  I  equal 

Pentagon         5,  10,  20,  40,    80,    160,  320,  640,    &c.  |  pots. 
Quindecagon  15, 30,  60,  120,  240,  480,  960,  1920*  &c.^ 

The  numbers  arising  from  inscribing,  bisecting,  &c.  an 
before,  of  the 

Hexagon,       ^  ^Trian^e, 

-uigun,  ■  ^^  included  in  those  of  the  <Z?^' 

Decagon,         |  |  Pentagon, 

Triaecmtagon,-^  ^Quindecagon, 

and  so  on  continually :  whence  it  appears  that  the  cirde  may 

be  geometrically  divided  into  2,  3,  5,  and  15,  equal  partSj  and 

likewise  into  a  number  which  is  the  product  of  any  power  of  2 

into  either  of  those  numbers  :  but  all  other  equal  divisions  of 

the  circumference  by  Geometry,  are  impossible. 

ON  THE  FIFTH  BOOK  OF  EUCLIIXS  ELEl^CEMTS. 

205.  In  the  fifth  book,  the  doctrine  of  ratio  and  proportion 
is  treated  of  and  demonstrated  in  the  most  general  manner, 
preparatory  to  its  application  in  the  following  books.  Some 
of  the  leading  propositions  are  of  no  other  use,  than  merely 
to  furnish  the  necessary  means  of  proving  those  of  whicMk 
use  is  obvious  K 

206.  Def,  1.  By  the  word  part  (as  it  is  used  here)  we  are  not 
to  understand  any  portion  wJiatever  of  a  magnitude  less  than 


I  Students  accustomed  to  algebra,  will  find  Professor  Playfair's  method  of 
demonstrating  the  propositions  of  the  fifth  book,  much  more  convenient  and 
easy,  than  that  of  Dr.  Simson.  There  are  those  who  would  entirely  omit  the 
fifth  book,  and  substitute  in  its  place  the  doctrine  of  ratio  and  piopcntion  as 
proved  algebraically  (p.  49 — 74.  of  this  volume;)  which  might  do  very  well,  if 
no  referenc&  were  made  to  the  fifth  book ;  or  if  the  sixth  might  be  allowed  to 
rest  its  evidence  on  algebraic,  instead  of  geometrical  demonstration  ;  but  if 
this  cannot  be  admitted,  it  will  be  advisable  to  read  the  fifth  book  at  least  once 
over,  in  order  folly  to  understand  the  sixth,  where  it  is  Heferred  to  not  less 
than  58  times ;  in  that  book  there  are  17  references  to  the  1 1th  piopo«tiaa» 
10  to  the  9th,  8  to  the  7th,  and  5  to  the  2^d  ;  these  four  may  therefore  be 
considered  as  the  most  useful  propositions  in  the  fifth  book. 


Pai^t  niL       ON  EUCLID'S  WITH  BOOK.  805 

the  whole  5  it  ioipliefi  that  part  cnly,  which  in  Arithmetic  is 
called  an  aliquot  part.  The  second  deiinitioa  is  the  converse  of 
the  first. 

207.  The  third  definiticm  will  be  easily  understood  from  what 
has  been  said  on  the  subject  in  part  4.  Art.  24.  &c. 

208.  Def.  4.  The  import  of  th^p  definition  is  to  restrain  the 
magnitudes^  which  '*  are  said  to  have  a  ratio  to  one  another,", 
to  such  as  are  of  the  same  kind :  now  of  any  two  magnitudes  of 
the  same  kind,  the  less  may  evidently  be  multiplied,  until  the 
product  exceed  the  greater :  thu8>  a  minute  may  be  multiplied 
till  it  exceeds  a  year,  a  pound  weight  until  it  exceeds  a  ton,  a 
yard  until  it  exceeds  a  mile,  &c.  these  magnitudes  then  have 
r^pectively  a  ratio  to  one  another  "'.  But  since  a  shilling  can- 
not be  multiplied  so  as  to  exceed  a  day,  nor  a  mile  so  as  to 
exceed  a  ton  weight,  these  magnitudes  have  not  a  ratio,  to  each 
other. 

209.  Def.  5.  '*  Ojie  of  the  chief  obstacles  to  the  ready  under- 
standing of  the  5th  book,  is  the  difficulty  most  people  find  in 
reconciling  the  idea  (^  proporticNoi^  which  they  have  already 
acquired,  with  that  given  in  the  fifth  definition  j"  this  obstacle 
b  increased  by  the  unavoidable  perplexity  of  diction,  prodiiced 
by  taking  the  equimultiples  of  the  aitemaie  magnitudes,  and 
imifiediately  after,  transferring  the  attention  to  the  multiples  of 
those  that  are  adjacent  5  operations,  which  cannot  easily  be  de- 
scribed in  a  few  words  with  sufficient  clearness;  besides,  the  de- 
finition is  en<nimbered  with  some  unnecessary  repetitions,  vi^ich 
aaight  be  left  out,  without  endangering  its  perspicuity  or  preci- 
sion. On  the  subject  of  this  definition,  as  it  appears  to  me,, 
much  more  has  been  said  than  is  necessary.  Euclid  here  lays* 
down  a  criterion  of  proportionality,  to  which  we  are  to  appeal 
in  all  cases,  whenever  it  is  necessary  to  determine  whether  mag- 


*  In  onicr  to  make  the  comparUon  implied  here,  it  is  bowetek-  -neceteary 
that  the  magnitmlefl  compared  should  be,  net  only  of  the  same  kind,  but  like- 
Wittj  0/  the  same  demmimtion:  properly  speaking,  we  cannot  compare  a 
minnte  with  a  year,  a  pound  weight  with  a  ton,  or  a  yard  with  a  mile ;  but  we 
can  compare  a  minute  with  the  number  of  minutes  in  a  year,  a'  pound  with 
the  number  of  pounds  in  a  ton,  and  a  yard  with  the  number  of  yards  in,  a. 
mile  5  the  ratio  of  a  guinea  to  a  pound  can  be  determined  only  after  they  are 
both  reduced  to  the  same  denomination ;  then,  and  not  before,  we  find  that 
tbey  have  a  ratio,  viz.  the  former  is  to  the  latter  as  21  to  20. 
VOL.  II.  .  2C 


905  GSmSTBY.  Paet  YUI. 


mtndes  are,  nr  «re  not  pssportioMils;  ani  k»  Im  ^vite  «  irt 
dik  bm^  BB  k»  lliu  tiMive  pin  Mii  eiflkit  cnnfln  of  ifei 
a^lkatkm;  so  that,  admitCiii^EDfdid's  criterion  to  be >it;te 
mode  of  itkreaea  is,  if  I  am  not  deceiicd,  a»  Mmple,  aai  tiie 
efidenee  as  aatisfiietflfy,  as  can  be  re^^md. 

2M>.  Bat  bow  ave  we  to  kiyvr,  whetkcr  Sadidrs  rtaadaidof 
pfoportiopality  be  just  or  not;  tfiat  is^  wbetber  It  doe»  or  daii 
not  s^ieeiiltli  our  raeehred  nolioBB  of  piopaftiMQlity,*  aa  didalail 
hj  eommen  sense?  we  wffi  eompgfe  Budiifs  doctrine,  m  \mi 
down  in  tiie  fiMi  de&iition>  wilb  tbe  notion  wU^  alt  ponons^ 
whefber  learned  or  not,  have  of  proportion,  and  they  witt  be 
foand  to  agree. 

1211.  Ask  any  man  wbat  be  meaBw  by  **  two  thiag»  b^i^  m 
the  saune  proportion  to  one  nnother,  that  twootber  thii^are  ?** 
and  he  win  immediatdy  answer,  *'  when  Hie  first  is  as  laige 
when  compared  with  the  aeeond,  as  the  third  is,  when  compared 
widi  the  ftfurtb."  Now,  the  obriovs  melbod  of  finding  ko»  i&rge 
one  magnitode  h,  when  compared  with  another,  ia  to  find  bMr 
often  it  eontains,  or  is  contained  in,  tbe  other;  or  inmnreeorrecf 
and  scientific  langm^,  to  find  what  nn^tl^le,  part,  or  parts  the 
former  magnitnde  is  of  tbe  kitar;  wbidi  is  ^ected,by  dbridiiig 
the  nmnbcr  representing  tbe  one,  by  that  r^iesentiog  Ae  ether. 
Wherefore,  tbe  common  notion  of  proportkmaliily  when  aden- 
rately  expressed,  wiU  be  as  foOows. 

21S.  '*  Two  megnitodes  are  proportional  to  two  Others,  wh^i 
the  first  is  the  same  mnKi]^,  pwt,  or  parts  of  tbe  second,  as  the 
thhd  is  of  the  fbnrth  5"  or,  when  tbe  ifiiotient  of  tbe  first  di- 
i4ded  by  tlie  second,  equals  the  quotieift  of  tbe  tlnrd  dkided  by^ 
thefottrth:  under  tbete  eircumstanoes  ^'tbe  foor  magnitudes 
aire  said  to  be  proportionals.'*  This  is  in  substance  tiie  snme  as 
def.  eo.  of  the  7th  book  of  Sodld's  Elements,  and  Mr.  Ludkm 
has  shewn  that  it  Mjpees  with  BocMd's  doc^ne  as  delivered,  in 
his  6th  booli,  that  is,  if  fonr  magnitudes  he  proportionals  ac- 
cording to  def.  5.  5.  they  tre  proporliOMds  nceonii^g  tn  tbb 
article ;  and  if  they  be  proportionals  ac<!onling  to  this  article, 
they  are  likewise  proportionals  according  to  def.  5. 5.    first,  if 

a:b  ::  c:  d  by  5.  def.  5.  book,  then  fiH31  ox  (l»^X  e,  and-^-s 


PauVBI.       ONEUCUD'aHITHBOOK-  307 

And  (15.  5.)k a  :  h  ::  ad  :  bd  ^ 

Atad  (II.  5.)  ..-.•...  c  :  4.  : :  ad  :  bd 
And  (15.  5.)  ........  c  :  d  ::  6c  :  2kI 

Whei^elore  (11.5.) . .  ad  ibd  ::  be  i  bd 

Consequently  (9. 5.)  . .  .  adz:zbc,  and  the  ---•  parts  of  these  equab^ ' 
will  likewise  evidently  be  equals  that  is  (<»^XTj=*^^Tj°'')*T" 

=— ,  so  that  if  four  magnitudes  «  :  6  : :  c  :  d  be  proportionals 

^ecQfding  to  £uclid*s  5th  definition^  they  are  also,  proportion's 
by  Art.  211.    Q.KD.    See  also  Art.  56.  p»rt  6. 

214.  It  remains  to  be  shewn  that  *«  if  ftmr  magnkudes  be 
profxnitionais  according  to  Art.  211.  they  are  afeo  proportionab 
to  def.  5.  5.  dBudid." 


0       c  . 

Let  -7-=-j*  then  will  adzzbc  agreeably  to  Art.  ?11,  ana 

if  ad=sbc,  then  will  a:  b  ::  e  :  d  agreeably  to  def.  5.  5.  Euclid. 
For  let  m  and  n  be  two  multii^^iers,  and  let  the  first  and 
tinisd,  (yisi,  a  9iid  c)  be  multiplied  by  m,  and  the  second  and 
fourth  (or  b  and  d)  by  «;  if  ma  be  greater  than  nb,  then  will 
n^  be  greater  than  nd,  and  if  tqual  equals  and  if  leas  less.  For 
since  a  x  ds6  x  c,  it  follows  that  nut  x nd^nb  x  ntc,  *.-  if  ma  be 
greater  than  96,  it  is  plain  that  mc  must  be  greater  than  nidp  if 
equal  e^iaL  and  if  less;  wherefore  Uy  def.  5.  5.  a,  6^  c^  fuod  d^ 
are  proporUonals.   Q.  £.  S. 

215.  k  will  be  readily  seen  that  tlie  d^finatium  (Axt  21 K)^ 
which  we  derive  from  the  popular  notloa  of  proporHonalsi  is 
restrained  to  magnitudes  which  can  be  expressed  by  cooHuen- 
fiurate  ni|inbei»«  Euclid^s  5th  definitioa  i^^plies  eq^iaUy  to  cqpi- 
mensurate  aqd  incommensurate  magnitudes  ^  this  capacity  of 
universal  a{^licalion  gives  it  a  d^dded  pi^&renoe  ta  the  defini* 
tkm  in  Art.  211.  and  we  have  ahewa  that  both  3gree  as  &r  as 
th^j  CM.be  comp^r^d  t^getber. 

21^.  JDrf*  6.  and  3.  properly  form  but  oxwe  definition^  which 
ma^  sUind  ap  fel(owB»  viz.  "  magnitudes  which  have  Ih^  saioe. 
ratio  are  .4»41ed  proportionals^  and  this  identity  of  ratios  >, 
called  proportion.'* 

217.  Tb^  loth  and  lUkdefinitioms  o^g^t  to  have  b^en  i^9<^d 

X  2 


308  geometry:  FartVUI. 

afbsr  def.  A,  since  duplicate,  triplicate,  quadnqdkate,  &c.  ntiot 
are  particular  species  of  compound  ratio  >  thus^  let  a,  h,  e,  dy  e, 
kc.  be  any  quantities  of  the  same  kind,  a  has  to  e  the  ratilycGin- 
pounded  of  the  ratios  of  a  to*  fr,  of  &  to  c,  of  c  to  d,  and  of  d 
to  e,  (see  Art.  40— 42.  part  4.)  and  if  these  ratios  be  equal  to 
one  another,  a  will  have  to  e  the  quadruplicate  ratio- of  a  ta  &> 
(or  o^  :  b*y  that  is,  the  ratio  compounded  of  four  ratios  each  of 
which  is  equal  to  that  of  a  to  6  ;  in  like  manner  a  will  have  to 
d  the  triplicate  ratio  (or  a?  :  ¥)  and  to  c  the  duplicate  ratio 
(or  €fi  :  b*)  of  a  to  b  ;  wherefore  it  is  pkia  that  each  is  a  parti- 
cular kind  of  compound  ratio. 

918.  Def,  i%  The  antecedents  of  several  ratios  are  said  to  be 
homologous  terms,  or  homoU^ous  to  one  another,  likewise  the 
consequents  are  homologous  terms^or  homologous  to  one  another  -, 
but  an*  antecedent  is  not  homologous  to  a  consequent,  nor  a 
consequent  to  an  aiUecedent ;  the  word  homologous  is  unneces- 
sary, we  may  use  instead  of  it  the  word  similar  or  like,  either  of 
these  sufficiently  expresses  its  meaning. 

ON  tri&  SIXTH  BOOK  OF  EUCLlD'S  ELEMENTS. 

219.  The  principal  object  of  the  sixth  book  is  to  apply  the 
dlDctrine  of  ratio  and  proportion  (as  delivered  iii  the  5th)  ta 
lines,  angles,  and  rectilfncral  figcfres  5  we  are  here  taught  ho^ 
to  divide  a  straight  line  into  its  aliquot  parts;  to  divide  it  simi- 
larly to  another  given  divided  straight  line  3  to  find  a  mean, 
third  and  fourth  proportional  to  given  straight  lines ;  to  deter- 
mine the  relative  magnitude  of  angles  by  means  of  their  inter* 
cepted  arcsi  and  the  converse ;  to  determine  the  ratio  of  similar 
xvctOineal  figures;  and  to  express  that  ratio  by  straight  lines 
with  many  other  useful  and  interesting  particulars. 

220.  Def,  1.  According  tx>  Euclid  *'  similar  rectilineal  figures 
are  (first,)  those  which  have  their  setefal  angles^  equal,  each  to 
each,  and  (secondly,)  the  sides  about  the  equal  angles  proportion 
nalsf  now  each  of  these  conditions  follows  from  the  other, 
and  therefin'e  both  are  not  necesssoy :  any  two  equiangular  rec- 
tiltjaeal  figures  wi&  always  have  the  sides  about  their  equal 
angles  proportionals  5  and  if  the  sides  about  each  of  the  angles 
of  two  rectlHnea]  figures  be  proportionals,  those  figures  will  be 
equiangular,  the  one  to  the  other.     See  prop.  18.  book  €, 

221.  De/;  2.  Instead  of  this  definition  which  is  of  no  use^ 


PahtVIH.        on  EUCLID'S  sixth  book.       ^  309 

Dr.  Simsoa  has  substituted  the  following.  '^  Two  magnitudes 
are  said  to  be  reciprocally  proportional  to  two  others,  when  one 
of  the  first  is  to  one  of  the  other  magnitudes,  as  the  remaining 
one  of  the  last  two  is  to  the  remaining  one  of  the  first,"  (see 
Simson^s  note  on  def.  2.  b.  6.)  this  is  perhaps  the  best  definition 
that  can  be  given  for  the  purpose. 

222.  Def.  3.  ITius  in  prop.  11.  b.  2.  the  line  ^B  is  cut  in 
extreme  and  mean  ratio  in  the  point  H,  fyt  BA  :  Mi  : :  4ff  • 
HB  as  will  be  shewn  farther  on. 

223.  Def,  4.  In  practical  X^eometry  and  other  branches 
depending  on  it,  the  Ui}ie  or  plane  .on  which  a  figure  is  supposed 
to  stand  is  denomjinated  the  hose;  Euclid  ma^es  either  side 
indififerei\tly  the  base,  and  a  perpendicular  let  fall  from  the  op* 
posite  ^ngle  (called  the  vertex)  to  the  base,  or  the  base  pro* 
dtK-ed,  is  called  the  altitude  of  the  figure  (for  an  example  see 
^e  Jtture^  JSgures  to  prop.  J3.  b.  2.) 

224.  Prop.  1.    Let  ^==the  altitude,   JB^the  base  of   oo^ 

parallelogram  or  triangle  3  a=the  altitude,  6==the  liase  of  anq- 

tfaeri  then  will  .^jB=the  first  parallelogram,  a^=the  second; 

AB  ah 

-r-=the  fi[rst  triangle,  and  —the second;  andif^:Ma,tlienwlU 

*  <& 

AB    a6  >  :  :  B  :  6;  and  if  B=:6,  then  will  J  ^B    06  >  : :  A^ 
2  '  T  J  I  ?    '  2  J 

c^  that  is,  parallelograms  and  triangles  of  equal  altitudes  are 
to  QQe  another  as  their  bases ;  and  if  they  have  equal  bases^ 
^ey  are  to  oi^e  another  as  their  altitudes.    Q.  £.  D. 

225.  Pr<yp,  2.  Hence,  because  the  angle  ADE^ABC,  and 
JED^ACB  (29.  1.)  and  the  angle  at  A  common,  the  triangle 
ADE  wiH  l>e  equiangular  to  the  triangle  ABCy  (32  1.)  And  if 
there  be  drawn  several  lines  parallel  to  one  side  of  a  triangl^ 
they  nvill  in  like  manner  cut  the  other  two  sides  into  jHX>portio- 
nal  segments  5  and  conversely,  if  several  straight  lines  cut  twp 
sides  of  a  triangle  proportionally,  they  will  be  paraflel  to  Ae  re- 
maining side,  and  to  one  another.  Hence  also  if  straight  lines  be 
^wn  parallel  to  one,  two,  or  three  sides  of  any  triangle,  another 
triangle  will,  in  each  case,  be  formed,  wbiph  i^  equiangular  tp 
the  ^ven  one. 

226.  Prop,  5.  Although  in  the  enunciation  it  is  expressly 
saud,  that  the  equal  angles  of  the  two  triangles  ABC,  DEF  ai-e 

X  3 


aio       •  GEOMETRY.  Paet  VIH. 

oppt)site  to  the  homologous  sides^  yet  this  circumstance  is  iiM 
bnce  ndtieed  in  the  demonstration  -,  and  hence  the  learner  will 
be  iieady  to  conclude^  that  the  proposition  is  not  completely 
proved;  hut  let  him  attentively  examine  the  demonstration^ 
and  he  will  find^  that  although  nothing  is  expressly  affirmed 
about  the  equality  of  the  angles  which  are  opposite  to  the  ho- 
mologous sides,  yet  the  thing  itself  is  incidentally  made  out ; 
thus  A^  atnd  DB  bemg  the  antecedents^  it  app^atrs  by  the  de- 
monstration that  the  angle  C  opposite  to  AB  is  equad  to  die 
angle  JT opposite  to  Dfi ;  and  BCund  EF  being  the  consequents, 
it  is  incidentally  shewn  that  the  angfe  A  opposite  to  J3C  is  equal 
to  the  angle  D  opposite  to  BP;  also  AC  and  DF  being  both 
aintecedents  or  both  consequents,  their  opposite  angles  B  and  £ 
are  in  like  manner  shewn  to  be  equal.  These  observations  are 
li&ewise  applicable  to  prop.  6. 

227*  Prop.  10.  By  this  proposition  a  straight  line  may  be 
divided  into  any  number  of  equal  parts  as  will  be  shewn  when 
'Wt  treat  of  the  practical  part  of  Geometry. 

22S.  Prop.  11.  A  third  proportional  to  two  given  straight 
lines  may  £^o  be  found  by  the  following  method,  (see  the  figure 
to  prop.  IS.)  Let  AB  and  BD  be  the  two  given  straight  lines, 
draw  BD  perpendicular  to  AB  (11.  1.)  join  ADs  at  the  point 
D  drav^BC  at  right  angles  to  AD  (11.  f.),  and  produce  AB  tiQ 
i€  cut  DC  in  C;  then  will  BC  be  the  third  proportional  to  AB 
and  BD.  For  since  ADC  is  a  triangle,  right  angled  at  !>,  from 
whence  DB  is  drawn  perpendicular  to  the  base,  by  cor.  to 
prop.  8.  A B  :BD  ::  Bb:  jBC,  that  is  BC  is  a  third  prOpottibnaJ 
toABanSiBD.  Q.£P. 

Let^B=:a,  ADscb,  then  a:  b  ::  b  :  — =:J3C  which  is  the 

a 

fiame  thii%  performed  algebrttieaVy. 

2^9.  Prop.  12.  Uet  o,  h,  and  c,  be  the  fhree  given  stral^t 

he 
linefl^hen  will  a:  b  ::  c  :  — z=HF,  the  fourth  pronortional  re- 

•  a 

quWed. 

^30.  Prop.  13.  libt  ABssa,  BCssb,  and  the  required  meaoa 
tkx,  thete  simife  «:«::%:  6,  we  facve  (by  moltiplyinjg  exUemn 
and  means)  xx:=zah,  and  x=:  ^^ab^szDB  °.. 


»  It  has  bfeien  asserted  in  the  introdcKstion  to  this  part,  Uutt  there  is  no 
knonrn  geometrical  method  of  finding  more  than  one  mean  proportional  be- 


pAWr  VOL        ON  EUCLUyS  SBOOl  BOOK.  dll 

£sAiiFi*Y«.-» 1*  To  t»i  a  ineAB  pe^^oitioiMi  betweieii  1 

Here  aitzl,  hsil€,  mid  d:ss^a&ssv^]#=34»  the  fneau  f|r 

Hr  To  find  a  mean  proporUooa!  betweeft  15  and  11. 
Hare  as=l$,   frsll,   and  ^ss^a^a^/lSxllsV^^'" 
ll8453dS57S,  <Ae  rc9»trtf<2  man. 

2^1.  Prop.  19.  By  the  help  of  this  useful  proposition  we  are 
enabled  to  construct  similar  triangles^  having  any  given  ratio  to 
each  others  thu8>  let  it  be  required  to  make  two  similar  trian^ 
gles^  one  of  whieh  sh2dl  be  to  the  other  as  m  to  n.  Make  BC 
s=m,  BG=:n,  and  between  BC  and  EG  find  a  mean  proportional 
EF  (IS.  6.)  upon  BCand  EJPmake  simflar  triangles  ABC,  DBF 
(18.  6.)  then  by  the^present  proposition  mm::  ABC :  DEF, 

SxAMPLBs.— 1.  Let  the  side  of  a  triangle  ABQ  viz.  BC:=ze, 
It  is  required  to  make  a  similar  triangle,  which  shall  be  only 
half  as  large  as  ABC. 

Bisect  BC  in  G  (10. 1.)  and  between  BC  and  BGfiud  a  mea^ 
proportional  EF  (13.  6,);  if  a  triangle  be  made  on  EF  similar  to 
ABC  it  will  be  half  of  ABC.  Thus  BC  being=:8,  EG  wiU=4, 
and  ^BCx  BG=  ^^8  x 4=  ^32=5.656854= JEF. 

tK.  Let  £F=S»  required  the  side  of  a  triangle  five  times  as 
hige  as  ni^y  and  simikr  to  itr  Ans.  v^x40srv'^20sa 
17^8854383  the  Me  rehired. 

232.  Prop.  20.  Hence,  if  the  homologous  sides  of  any  two 
similar  rectilineal  figures  be  known^  the  ratio  of  the  figures  to 
one  another  may  be  readily  obcainedj  nsunely,  by  finding  a  third 
{proportional  to  the  two  given  sides :  for  then,  the  first  line  will 
be  to  the  thirds  as  the  figure  on  tihe  first,  to  the  similar  and  simi- 
larly described  figure  on  the  second^  as  is  manif<»t  from  the 


^" 


tffttB  tw^hnm  ftraigibi  ham  a  «od  (/  tbitAajFhoWwrwlKiaoi^lfebnu* 

cally  )tf  the  feU^wiv^  theorems. 
One  mean  proportional  will  be  j^ab 

Two  means »  >/«*  J»  »  >/tfi* 

Three  means ,*^a»b,  «  v^<l»4»,  *  j/ab* 

Four  means ,  •  ^a*b,  •  js/a*b*f  iv^tf«6i,  » /t/«rft* 

nmeans ■+ V«»*. '+ V«""-*-ft'.'+ V*"""^* 

x4 


312  GEOMETRY.  Fart  VlU* 

tecond  cor.  to  the  profiosition.  Hence  also  any  rectilmeal  figure 
may  be  geometrically  increased,  or  decreased  in  any  as^gned 
ntio.  Thus,  let  it  be  required  to  find  the  side  of  a  pentagon 
one  fifth  as  large  as  ABCDE^  and  similar  to  it  5  find  a  mean 
proportional  between  AB  and  ■!■  AB  (13. 6.)  let  this  be  FG,  and 
upon  FG  describe  the  pentagon  FGHKL  similar  and  similarly 
situated  to  ABCDE  (IS.  6.)  then  will  the  former  be  i  of  the 
latter.  Again,  let  it  be  required  to  find  the  Me  of  a  polygon 
3  times  as  large  as  ABCDE,  and  similar  to  it  ? 
TTius  j^ABxiAB=:th€  side  required. 

233.  Prop,  22.  By  means  of  this  proposition,  the  reason  of 
the  algebraic  rule  for  multiplying  surd  quantities  together,  may 
be  readily  shewn.    Thus,  let  it  be  required  to  prove  that  ^a  x 
^b:sz  ^ahy     first,  since  unity  :  the  multiplier  : :  the  muUipli-' 
cand  :  the  product;  therefore,  in  the  present  case,  1  :  ^a  : :  jy/b  : 
^ox  ,/6=the  product,  but  by  the  proposition  (1*  :  ^a*  : ; 
^6*  :  j^a^X  ^h\  that  is)    1  :  o  : :  6  :  a6=the  square  of  the 
product^  wherefore  ^a5=the  product. 

234.  Prop,  23.  Hence,  if  two  triangles  have  one  angle  of  the 
one  equal  to  one  angle  of  the  other,  they  will  have  to  each  other 
the  ratio  coitipounded  of  the  ratios  of  the  sides  about  their 
equal  angles ;  this  will  appear  by  joining  DB  and  GE ;  for  the 
triangles  DBC,  GEC  have  the  same  ratio  to  one  another,  that 
the  parallelograms  DB  and  GE  have  (1.6.).  Also  it  appears 
from  hence,  that  parallelograms  and  triangles  have  to  one  ano- 
ther respectively,  the  ratio  compounded  of  the  ratios  of  theif 
bases  and  altitudes. 

235.  Prop.  30.  This  proposition  has  been  introduced  under  a 
different  form  in  another  part  of  the  Elements,  (viz.  11.2.)  there, 
we  have  merely  to  divide  a  straight  line,  so  that  the  rectangle  con- 
tained by  the  whole  and  the  less  segment^  may  equal  the  square 
of  the  greater }  we  have  to  determine  the  properties  of  a  figure, 
but  the  idea  of  ratio  does  not  occur  3  here  we  are  to  divide  a 
line,  so  that  the  whole  may  be  to  the  greater  segment,  as  the 
greater  segment  is  to  the  less,  and  the  idea  of  figure  has  no  place  } 
but  our  business  is  solely  with  the  agreement  of  certain  ratios. 
1  do  not  recollect  a  single  reference  to  this  proposition  in  any 
subsequent  part  of  the  Elements,  except  in  some  of  the  books 
which  are  omitted. 

236.  Prop.  31.  What  was  provied  of  squares  in  prop.  47.  b.  I. 


Part  VITL       ON  EUCLID'S  SIXTH  BOOK.  Sl$ 

is  here  shewn  to  be  true  of  rectUineal  figures  in  general ;  and  the 
same  property  belpngs  likewise  to  the  circle^  and  to  all  similar 
carvilineal  and  similar  mixed  figures^  with  respect  to  their  dia- 
meters or  similar  chords ;  but  the  six  former  books  of  Euclid  s 
Elements  do  not  furnish  us  with  sufficient  principles  to  extend 
the  doctrine  beyond  what  is  proved  in  this  proposition.  We  are 
here  taught  how  to  find  the  sum  and  difference  of  any  two  simi- 
lar rectilineal  figures,  that  is,  to  find  a  similar  figure  ecjual  to 
the  said  sum  or  difference.    See  the  observations  on  47.  1. 

237*  Prop,  33.  This  useful  proposition  is  the  foundation  of 
Goniometry,  or  the  method  of  measuring  angles.  If  about  the 
aagular  point  as  a  centre  with  any  radius,  a  circle  be  described, 
it  is  here  shewn,  that  the  arc  intercepted  between  the  legs  of 
the  angle  will  vary  as  the  angle  it  subtends  varies  3  thus,  if  the 
angle  be  a  right  angle,  the  subtending  arc  will  be  a  quadrant 
(or  quarter  of  a  circle)  3  if  it  be  half  a  right  angle,  the  sub- 
tending arc  will  be  half  a  quadrant }  if  it  be  equal  to  two  right 
aagles,  the  subtending  arc  will  be  a  semi-circle  3  and  if  it  equal- 
four  right  angles,  the  subtending  arc  will  be  the  whole  circum- 
ference. Now  if  two  things  vary  directly  as  each  other,  it  is  plain 
that  the  magnitude  of  one,  will  always  indicate  the  contemporary 
magnitude  of  the  other)  that  is,  it  will  be  a  proper  m^isure  of 
the  other.  Such  then  is  the  intercepted  arc  described  about  aa 
aagle,  to  that  angle  3  and  therefore  if  the  whole  circumference  be 
divided  into  any  number  of  eqiial  parts,  the  number  of  those 
parts  intercepted  between  the  legs  of  the  angle,  will  be  the  mea- 
sure of  that  angle.  It  is  usual  to  divide  the  whdle  circumference 
into  360  equal  parts  called  degrees,  to  subdivide  each  degree 
into  60  equal  parts  called  minutes,  and  each  minute  into  60 
equal  parts  called  seconds,  &c.  wherefore,  if  an  angle  at  the 
centre  be  subtended  by  an  arc  which  consists  of  suppose  30 
degrees,  that  angle  is  said  to  be  an  angle  of  30  degrees,  or  to 
meastire  30  degrees;  if  it  be  subtended  by  an  arc  of  45  deg. 
54  min.  the  angle  is  said  to  measure  45  deg.  54  min,  8sc. 

238.  Hence  the  whole  circumference  which  subtends  ^o«r 
right  angles  at  the  centre  (Cor.  1.  15. 1.)  being  divided  into  360 
degrees,  a  semicircle  which  subtends  two  right  angles  will  con- 
tain 180  degrees,  and  a  quadrant  which  subtends  one  right  angle 
wiU  contain  90  degrees,  wherefore  two  right  angles  are  said  to 
measure  180  degrees,  one  right  angle  90  degrees,  &c.  and  note« 


814 


GBOMfiTRY. 


pAftT  VID. 


degttm,  mintsteB,  «nd  necordt,  aM  thus  amked  ^  ^  '^  tinif  H 
degrees,  3  mintiles^  4  seocttds,  are  ufiual^  wrintea  1^«  3^  4^> 
&c. 

^8.  B.  Hence,  if  eiKmt eny sngiiter  fokti  C  sAaeiictm,  eevenH 
eODcentric  circles  be  dtesciibed,  ccftti^  CA  aiij  C#  ki  tlie  pdioU 
X,  Z,  A,  By  (he  ore  ^Bj  will  be  f 0  tlie  ivlkole  eiraxttifefettee  ef 
which  it  is  an  mt,  as  the  «fe  KZ  is  to  the  Whole  e(i«diirft»ettee 
of  ithkh  it  is  an  wc.  f^edhice  BC  to  I>»  aiid  ^hfoog^  <:4sim 
HK  at  fight  ttngles  to  DB  <11. 1.) ;  thm  BA  :  tBB:\  ftligi^ 
J?Cil :  ftttgle  BCH  (13.  6.)  */  Bil 
:  4K  BH  : :  angle  BCif  :  4K 
MUgle  BCA,  (13.  5.)i  that  is,  BA 
is  to  the  whole  circumference 
BBDK,  as  the  angle  BCA,  is  to 
four  right  angles;  in  the  same 
maimer  it  is  ih6wn>  that  XZ  is  to 
the  Whole  chx<uniference  ZXB  as 
the.siime  angle  BC^f  to  four  right 
angles ;  wherefore  AB  :  the  whole 
eircamfereoce  BHDK  : :  XZ :  the  whole  drcamferefice  ZXE, 
Q.  £.  D. 

1399.  Hence  also,  if  the  dreumferences  tt  Hiese  two  drcles  he 
eteh  dmded  into  36D  degrees,  as  above  (Art.  936.)  At  wOl 
^ontfain  as  many  degreed  of  the  chfcanifefence  BSDK,  as  XZ 
does  of  the  drcomilerence  ZXE. 


AN  APPENDIX  TO   THE   FIRST   SIX    BOC«S   OF 

EUCLID. 

Cmttainmg  some  useful  propositions  which  are  not  in  the 

Elements. 


240.  If  one  eide  of  a  triang^  be  ^iseoted»  the  sum  of  thb 
squares  of  the  two  remainii^  ndes  is  doable  the  square  id  hall 
tl^e  side  bisected^  and  of  the  square  of  the  line  dcawA  from  the 
point  of  bisection  to  the  opposite  angle. 

Let  ABC  be  a  triangie,  having  BC  bisected  in  D,  and  D^l 
drawn  from  D  to  the  opposite  angled;  then  will  BSl^-f  ^Q^ss 

fi.BS)HSS|». 

Let  AE  be  perpendicodar  to  BC,  ihm  foeciMtsc  BEA  k  a 
ri^t  angle,  2Zi|'aB^H  S3^  Md  ^es€£)«'f  iO|«,  (47- 1.) 


Part  Vltl.         APPENDIX  TO  EUCLID. 

+£C)H2.E3^.  But 

since  BC  is  divided  in« 
to  two  equal  parts  in 
D,  and  into  two  un- 
equal parts  in  £,  5£|« 


315 


=s  2 .  JBD>  + 

.2cl^=2.'55l«+^.f£5l^  But  Se|*+£51»=d31«,  (47. i.) 

SE^P  +  E2*=2.W  +  2.S31'=)  2.fiSl«+D7)«;  and  the 
same  may  be  proved  if  the  angle  at  C  be  obtuse,  by  using  the 
10th  proposition  of  the  second  book  instead  of  the  9th.    Q.  B.  D. 

241.  In  ai^  pandldogram,  the  sum  of  the  squares  of  the 
diameters,  is  equal  to  the  sum  of  the  squares  of  the  sides. 

Let  JBCD  be  a  parallelogram,  ^C  and  BD  its  diameter?, 

then  wm  2c1*+551^=^H5c]*-H'^'+S^'- 

Because  the  angle  AED^ 
CEB  (15. 1.)  and  EJD^ECB 
t59.L)  the  triangles  AEDy  CEB 
have  two  angles  of  the  one  =:two 
angles  of  tBie  other  each  to  each,  B  C 

and  a  side  opposite  to  the  equal  angles  in  each,  equal,  viz.  AD= 
Be (34. 1.)  •.•  -^E=ECand  D£==£B(g6. 1.);  and  because  BD 
is  bisected  in  E,  S2[i^^AS]^^2.BEi^+E2\\  and  5cl«+c5?* 
=:(2.Bll^-h£Cl^  (Art. 239.)  =)  2.B£)«+E^^  v  55l*+23l* 
+ 5(3^  e2J«= 4.S£)« +:B?«=(since4.1i9*=^BBlSand  4.e31» 

=:30«.by  4.2)B2S«+"5S|*.    Q.E.D. 
Cor,  Hence  the  diameters  of  a  paraBfelc^nrai  bisect  each 

otheir. 

242.  If  the  sum  of  any  two  opposite  angles  of  a  quadrilateral 
figure  be  eipial  to  two  right  angles,  its  four  angles  will  be  in 
the  circumference  of  a  circle. 

Let  ABCD  be  a  quadrilateral  figure,  having  the  sum  of 
any  two  of  its  opposite  angles  equal  to  two  right  angles,  and  let 
a  circle  be  described  passing  through  the  tlu-ee  points.  A,  B,  1>, 
(5. 4.  afld  A*t.  194.)  I  «ftf  the  ckcumfctence  sbdill  MkeWise  ptos 


316 


GEOMETRY. 


pa»t  vni. 


through  the  fourth  point  C; 
for  if  not^  let  the  fourth  point 
fidl  without  the  circumference 
at  a,  and  join  DC;  then  since 
-by  hypothesis  the  sum  of  any 
two  (^posite  angles  of  the  figure 
are  equal  to  two  right  angles, 
•/  B^D+B6D=tworightang. 
les,  but  B^D-f  BCl>=tworight 
angles  (29.3.)  v  BAD-^-BGD^zBAD+BCD,  take  away  the 
common  angle  BAD,  and  BGD^BCD^  the  interior  and  oppo- 
site equal  to  the  exterior  which  is  impossible  (16. 1.)  %*  the  fourth 
point  cannot  &11  without  the  circle,  in  the  same  manner  it  may 
be  ^ewn  that  it  cannot  fall  within  it,  '.*  it  must  fsdl  on  the  cir- 
cumference at.  C.    Q.  E.  D. 

Cor.  If  one  side  BCof  a  quadrilateral  figure  inscribed  in  a 
circle  be  produced,  the  exterior  angle  DCGssthe  interior  and 
opposite  BAD ;  for  DCG+DCB=two  right  angles  (13. 1.)  and 
B^D-f  I>CB=two  right  angles  (22.3.)  •/  DCG+DCBs^BJD 
+DCB,  take  away  DCB,  and  DCG:siBAD. 

243.  If  the  vertical  angles  of  se%'eral  triangles  described  on 
the  same  base,  be  equal  to  each  other,  and  the  circumference 

of  a  circle  pass  through  the  extremities  of  the  base,  and  one  of 
the  vertical  angles,  it  shall  likewise  pass  through  all  the  others. 
Let  ACB,  ADB,  AEB,  &c.  be  the  several  equal  vertical 
angles  of  triangles  described  on  the  common  base  AB,  if  a  cir- 
cle pass  through  A,  B,  and  C,  it  shall  likewise  pass  through  the 
remaining  points  D,  £,  &c.  Take 
any  point  IT  in  the  circumference 
on  the  other  side  of  AB,  and 
join  AK,  KB,  then  wiU  ACB-^ 
AKB:=:2  right  angles,  (22. 3.)  5 
but  ADB=AEB==ACB  by  hy- 
pothesis, '.*  each  of  the  angles 
AEB.ADB  together  with^JiTB 
=2  right  angles,  •.•  (Art.  241.) 
the  angles  E  and  D  are  in  the 
circimiference.     Q.  £.  D. 

243.  If  two  straight  lines  cut  one  another,  and  the  rectangle 


Part  VIII.  APPENDIX  TO  EUCLID.  317 

contained  by  the  segments  of  one  of  them^  be  equal  to  the  rec- 
tangle contained  by  the  segments  of  the  other,  the  circun^. 
fet^nce  which  passes  through  three  of  the  extremities  of  the 
two  given  straight  lines,  shall  likewise  pass  through  the  fourth. 

Let  AB  and  CD  cut  each  other  in  E,  so  that  AE  x  £5= 
CExED,  the  circumference  ACB,  which  passes  through  the 
three  points  A,  C,  and  B,  shall  likewise  pass  through  the 
fourth  D. 

For  if  not,  let  the  circumfe- 
rence, if  possible,  cut  CD  in  some 
other  point  G;  then  since  A,  C, 
B,  and  G,  are  in  the  circumfe- 
rence^ the  rectangle  AE  x  EB=s 
CExEG  (35,3.)  but  AExEBsz 
CExED  by  hypothesis  j  •/  CEx 
EG=:CExED,  V  EG^ED,  the 
lesssathe  greater,  which  is  ab- 
surd j  therefore  G  is  not  in  the 
circumference ;  and  in  the  same  tfiannei'  it  may  be  shewn,  that 
no  othft  point  in  CD,  except  D,  can  be  in  the  circumference. 
Q.  E.  D  •. 


"Join  CB,  and  through  K  draw  KP  parallel  to  Fd  then  since  the  ai^le 
^EC^ABC+  DCB  (S2. 1 .)  if  the  angtllar  point  E  were  in  the  circumference. 
It  if  plain  that  it  would  be  subtended  by  an  arc  equal  to  AC+  DB  ;  and  con- 
seqaently,  if  E  were  ai  the  centre*  it  would  be  subtended  by  an  arc  etfual  to 

^  "^1         («0«  3.)     Again,  if  JSrCbe  joined,  it  may  be  proted  (29.  l.  and 

3fi.8.)  that  CP  and  HK  are  equal,  but  the  arc  BDP^^CPB—CP^) 
^PB'-HKi  and  sin«e  the  angle  BKP^BFCi  and  BKP  is  subtended  by 
tfce  arc  BOP^  if  BKP  were  in  the  clmimference,  it  would  be  subtended  by 
an  arc  equal  to  BDP:  but  if  it  were  at  the  centre,  BKP  would  be  subtended 

,  BDP  CPB—HK 

by  an  arc  « (— —  (20. 3.)  that  48=) j by  what  has  been  shewn. 

And  since  an  angle  is  measured  by  the  subtending  arc  described  about  the 
angular  point  as  a  centre  (Art.  262.)  it  follows,  that  if  two  straight  lines  JB» 
CO  cut  one  another  within  m  circle^  the  angle  AEC  ie  measured  ^by  half  the 
*»tm  gfthe  subtending  arcs  AC  and  BDy  and  {hy  similar  reasoning)  the  angle 
^ED  is  measured  by  half  the  sum  of  the  arcs  APD,  CKB.  But  if  two 
straight  lines  CF,  FB  cut  one  another  without  the  circle,  the  angle  BFC  is 
measured  by  half  the  diference  of  the  intercepted  arcs  CPB  and  HK;  this  is 
«oiuieeted  with  Art.  261.  262. 


815 


ovmemY. 


PaatVDJ. 


5244.  hU  tkef^  bf  tiro  i^iiH^  lilies  CP  and  i^&  cii^^ 
drde  in  two  fdaii$,  mad  ««ch  <ilher  ia  a  pcunt  F  whboiat  the 
eirde;  aodletCf'cuttbeciceiinfei^aoeiii  CaiidH^ai^ 
it  in  Bi  i^u  iS  9^  ^9iBi  Klmtakenm  FB,9oibat  CFxFHrsi 
BFxFK,  1 9t^  the  point  JT is  in  the  drcumfeience. 

For  if  Dot^  kt  tba  ciieiuiifereiice  HJB  cut  FB  in  X^  then 
CFxFBssBFxFL  (3^.  3.  car.)  but  CFxFH^BFxFK  by 
hypcytbesis,  v  BFx  FLszBFx  FK  md  FL=:FK,  the  laos^cth^ 
greater^  which  is  absurd.  *.'  L  is  not  in  the  cireum&reaace }  and 
in  like  manner  it  may  be  shewn  that  no  other  point  in  £F, 
except  B  and  K,  can  be  in  the  cinnim£ereoce;  K  ia  therefore  in 
the  circumference.    Q.  £.  D. 

945.  If  a  straight  line  AB  be  drawn  from  the  eoOanemil^  A 
of  the  diameter  AC,  meeting  the  perpenjlictilar  ED  in  ^  then 
will  the  rectangle  BA  x  AE:n  CA  X  AD. 

Join  BC,  CE,  then  because  the  aiigle  ABC  in  a  aevuoird^ 
is  a  right  angle  (31. 3.)  CBE  is  also  a  rig^  angle  (13. 1.)  and  if 
a  circle  CDEB  be  described  on  C£  as  a  diamatev^tta  dffcufldb* 


rence  shall  pass  through  the  pcants  C,  B,  £>  and  D;  and  8uae& 
BE  and  CD  meet  in  the  point  A,  BAxAE^CAxAD  by  35^ 
oreor.  36.3    Q.  £.  D. 

Henc»  EA:AD::€A:ABQ3^  16.6.) 

246.  If  an  arc  of  a  circle  he  W^ctfid,  and  £rqm  the  egj^trf^mi* 
ties  qf  the  arc  and  the  point  of  i^isection^  straight  Unes  be  dtasm 
to  any  pcnnt  in  the  drcum&Kenee^  titon  wlK  the  som  of  the  two 
lines  drawn  from  t^  extremities  of  the  arc,  have  to  the  line 
drawn  from  the  point  of  bisection^  the  same  ratio  which  the 
chord  of  the  arc  has  to  the  chqrd  of  half  the  are. 

Let  AB  be  an  arc  bisected  in  C,  a)|d  D  Mff  point  ia  the 


Part  Vni.         APPENPOX  TO  EUCLID.  319 

caoamfeiwm,  'yAa  Al>,  CD,  BJ^,  4C  and  BC,  th^n  will  ifX>+ 

DB:DC::BJ:  AC. 

Bee^me  ACBX^Ib  a  quaAAiiteral 
fignre  inscribed  in  a  ctrck,  AM.CD 
{^JD.CB^DB.AC  (JD.6.)  nAicli. 
htOHMc    CBati#Q  otAB.AC^BD. 

AC,^AC,jm^EI>  (1.8.)  «m1  be- 
^soae  the  skies  of  eqiud  reetan^es 
are  reciprocally  proportional  (14.  6.) 
AD^BDiCDiiAB.AC.  Q.£.l>. 

247.  If  two  points  be  taken  in  the  semidiameter  of  a  circle, 
sacb,  that  tlie  rectangle  cc^tained  by  the  s^;inent8  between  them 
and  the  centre,  is  equal  to  the  square  of  the  semidittneter ;  the 
straight  lines  drawn  fixnn  these  points  to  any  point  in  the  circum* 
ference,  shall  have  the  same  ratio^  that  the  segments  of  the  dia- 
meter between  the  two  fore-mentioned  points  and  the  circum- 
ference* liave  to  one  anotluer. 

Let  I>  be  the  centre  of  the  drefe/  ABC  and  DF  the  semi- 
^Bameter  prodtteed,  in  whieh 
let  ^aod  Fbe  taken,  snefa,  that 
mOFx^AM^s  then  if  B»  5, 

•ad  n  be  drawn  irom  E  zxA    ^        ,        ^    ^ _^ 

^,  to  any  point  B  in  the  dreum-  F      ^a1        E    j>  j 

ferenee  ^B :  FB  : :  EA :  A¥. 

Join  AB,  ED ;  then  since 
*»f  hypothesis  £f>.l>Fs(:3S|« 
«)  BS\*',  M>F:DB: :  DB:  DE  (17. 6.)  j  that  is,  the  sides  about 
the  common  angle  D  of  the  triangles  FBDt  EBD  are  propor- 
tionals, *.*  these  triangles  are  equiangular  (6. 6,),  and  the  angle 
FBD=iBEI>^EAB^ABE  (32.1.)  >  but  EAB=ABD  (5.1.) 
••  {FBD^)  PBA-^ABD^sABD-k-ABE,  take  away  the  common 
angle  ABD,  ajad  FBA=:ABE,  •.'  B-4  bisects  the  angle  FBE,  •/ 
MB:BF::EA;  AF(3.6.)     Q.  E.  D. 

Cor.  Hence,  if  FB  lie  produced  to  G,  and  fiC  joined,  the  ex- 
terior angle  jiBG  will  be  bisected  by  ^c  For  since  ABC  is  ^l 
right  angle  (di.S.)  it  is  half  the  sum  of  the  angl^  fbe  ^nd 
JSBG  (13. 1.)  :  but  ABE^^  FBE, '.'  ^BCsz^EBG, 

248.  If  from  the  three  angles  of  any  triangle,  perpendicularly 


520 


GEOMETRY. 


Part  Vnt 


be  drawn  to  tlie  opposite  sides^  these  perpendiculars  shall  inter- 
sect one  another  in  the  same  point. 

Rrst,  In  the  acute  angldd  triangle  ^BC^  let  the  perpendicu- 
lars BD  and  CE  intersect  one  another  in  ^,  join  jfF,  and  pro* 
duce  it  to  G,  AG  is  perpendicular  to  BC. 

Join  DE,  and  let  a  circle  be  described  about  the  triangle 
AEF  (5. 4.)  then  since  by  hypothesis  AEF  is  a  right  angle,  AF 
will  be  the  diameter  of  the  eirde  (31.3.) ;  and  because  ADFs, 
ABF,  the  circumference  of 
the  same  circle  shall  pass 
through  the  point  D  (Art. 
242.)  and  the  points  A,  B, 
F,  D,  will  be  all  in  the  cb- 
cumference.  But  because  the 
angle  EFBslDFC  (15.1.) 
and  BEFsz  CDF  (by  hypothe- 
sis) '.*  the  triangles  BEFoxid 
CDF  are  equiangular  (32. 1.) 
V  BFiEF::  CF:  FD  (4.  6.) 
-.'  BF:  CF::  EF:  FD  (16. 5.)  and  since  the  an^e  BFCssEFD 
(15. 1.)  and  the  sides  about  these  equal  angles  are  proportionals, 
the  triangles  BFC  and  EFD  are  equiangular  (6. 6.)  •.•  the  ai^le 
FCB^BDF^EAF  (21.  3.)  /  EAP:=zFCGs  and  AFE^CFG 
(16. 1.)  •.•  AEF^FGC  (32. 1.)  j  but  AEF\&  a  right  angle  by  hy- 
pothesb^ '/  FGC  is  also  a  right  angle  and  AG  is  perpendicular 
to  JJC. 

Secondly,  In  the  right  angled  tiiangle  AFD»  draw  DH  perpen- 
dicular to  AF,  '.*  AD,  AD,  and  FD,  are  the  three  perpendiculars^ 
and  it  is  plain  that  they  all  meet  in  D, 

Thirdly,  In  the  obtuse  angled  triangle  BFC,  BE  ]^  perpendi- 
cular to  CF  produced,  CD  perpendicular  to  BF  produced,  and 
GF  perpendicular  to  ^C^  and  it  appears  by  the  foregoing  de- 
monstration, that  these  three  perpendiculars  BE,  CF,  and  CD 
intersect  each  other  in  the  same  point  J.    Q.  E.  D. 

249.  If  a  straight  line  tpuch  a  circle,  and  from  the  point  of 
contact  two  chords  be  drawn,  and  if  from  the  extremity  of  one 
of  them,  a  straight  line  be  drawn  parallel  to  the  tangent  meeting, 
the  other  chord  (produced,  if  necessary)^  then  wiU  the  two 
chords  and  the  segment  intercepted  between  the  parallels^  be 
proportionals. 


Tam  VSH. 


APPENDIX  TO  litCLljy. 


921 


T.- 


Let  TA  touch  the  circle 
m  A,  from  whence  let  the 
chords  AB  and  jiC  be  drawn^ 
and  from  C  the  extremity  of 
one  of  them  J  let  CD  be  drawn 
parallel  to  TJ  (31.1.)  meet* 
ing^B  in  D,  then  will  BA  : 
AC  II  AC',  AD.  Join  BC, 
then  because  the  angle  .^CBs 
TAD  (32 . 3.)  =  ADC  (29. 1 .) 
and  BAC  common,  the  tri- 
angles ACB,  ADC  are  equi- 
angnlar^  and  AB  :  AC  : :  AC 
:  AD  (4. 6.)  Q.  K  D. 


I 

Cor.  1.  Hence  BA.ADssAI!\2. 

2.  If  ^B  pass  through  the  centre,  then  will  ACS  be  a  right 
angle  (31.3.),  and  CD  will  be  perpendicular  to^^B  (18.3.  and 
29.  1.)  ;  and  since  AB  :  AC::  AC:  AD;  the  side  AC  of  th& 
triangle  ACB  is  a  mean  proportional  between  the  hypbthenuse 
AB  and  the  segment  of  it,  AD  adjacent  to  AC,  as  is  shewn  id 
cor.  8.  6. 

250.  If  a  perpendicular  be  diawn  from  the  vertitol  angle  of 
any  triangle  to  the  base,  (produced  if  necessary),  then  will  the 
rectangle  contuned  by  the  sum  and  difference  of  the  sides  of 
the  triangle^  be  equal  to  the  rectangle  contained  by  the  sum 
and  difference  of  the  segments  of  the  base. 

Let  ABC  be  a  triangle,  and  CD  a  perpendicular  drawn 
from  the  vertical  angle  C  to  the  base  AB,  meeting  it  (pro* 


VOL.  1I« 


1^ 


3j» 


eEQlfESBY. 


PmitVIBw 


dttced  if  necesaary^  as'  in  %  g.) 
in  D,  then  wiU  i^€-f  CB.AC-^  CB 
as^Z> + DB.JD'^  DB. 

Krom  C  as  a  centre,  with  the 
akort€8t  side  CB  for  a  distance, 
describe  a  circle,  catting  AC  pro- 
duced in  G  and  H,  and  ^B  (pro* 
duced  in  fig.  2.)  in  £  ^nd  B.  Then 
since  CGz^CH=CB,  •/  AH^AC 
+  C0=the  sum  of  the  sides,  and 
AG^IaC-^CG^)  AC^CB:=i 
their  difference  3  and  because  I>B 
ssD£  (3. 3.)  (AD+DB=z)  AB  is 
the  sum  of  the  segments,  and 
{AD  ^DB^AD'-DE::z)  AE 

their  difference  in  fig.  1 .  sdso  (AD     -A.  B^^^--.a — -^B 

+DBss^D+i>£=:^£s=:the  sum  of  the  segments  in  fig.  S. 
and  (^D—DB=)^Bstheir  difference.  Wl^ecefare,  (cor. 36.?.) 
AH.AG^ABJiE:  that  is*  the  rectangle  contained  by  the  sum 
and  difference  of  the  sides  AC  and  CJ6,  is  espial  to  the  r^taogle^ 
contained  by  the  sum  and  di&rence  of  the  segments.  Ap  an4. 
BD,  intercepted  between  the  extremities  A  and  jB  of  the  base*. 
(or  base  produced,)  and  the  perpendicular  CD.   Q.  £.  D« 

Cw.  1.  Hence  AB:AH::AG:AE  (16. 6.)  that  is,  tl|e  base 
of  a  triangte. :  is  to  the  sum  of  the  sides  : :  as  the  difference  of 
sides  to  the  sum  :  (fig.  2.),  or  difference  (fig.  1.),  of  the  s^ments 
of  the  bate,  according  as  the  perpendicular  CD  fsdis  without,  or 
within  the  triangle.  This  inference  is  particularly  useful  in 
trigonometry,  when  the  three  sides  of  a  triangle  are  given  to 
find  the  angles. 

2.  Because DBssPE,  and  B£=2  BD, '.'  AB^E=^(AB,ABT 
B£=:)  AB^Bt^ BPszASI^T^ABBD :  v  since  JCl^^VS* 
zsAC+CB.AC'^CB  (cor.  6.  2.)  ^21b\^+^AB.BD,  the  rec- 
tangle  contained  by  the  sum  and  difierence  of  two  sides  of  a 
triangle,  is  equal  to  the  square  of  the  base  minw  or  plus  twice 
the  rectangle  contained  by  the  base,  and  its  least  segment. 

3.  If  ABC  be  a  right  angle,  the  point  B  coincides  with  D, 
and  the  circle  described  from  C  with  the  distance  CB  will  touch 
the  base  AB  in  A  and  (36. 6.)  HA.A6si  (^*;  that  is^ 
since  B  coincides  with  Dzs)  aB^i  *.*  the  rectangle  contained 


rim  mi:  AFPENIXnC  TO  EUCLID.  Stt 

kf  the  sum  and  difiercace  o£  the  hypotlieniise,  and  one  of  the 
•ides  is  equal  to  the  square  of  the  other  nde. 

4>  Since  by  eor.e.  ^C+  CB.^C— eA=3:^75l«+2.ilB.BJ>,  and- 
M-h C3.dC^ CB^:^^^Cfff^  (5.  2.)  •/  A^^^aS^^AS\^^ 
^JB.BD,  and  ^«s±31bl«+e^«tf2.^B:BD.  Or  the  square* 
of  the  side  AC  is  less  or  greater  than  the  sum  of  the  squares  of 
AB  and  CB,  hj  twice  the  rectangle  contained  by  the  base,  and 
the  s^ment  Cft  according  as  the  angle  ABC  k  acute  or 
obtose.     This  is  the  same  as  1^  and  Id.  9  Euclid. 

250.  B.  The  chord  of  one  sixth  part  of  the  circumference  being 
given,  to  find  the  chord  of  half  that  arc,  and  thence  to  inscribe 
withm  the  circle  a  polygon  of  a  great  number  of  sides. 

Let  ABD  be  a  semicircle,  C  its  centre^ 
dmw  the  chord  DA^^AC  (1.  4 ),  bisect  the 
m  DA  in  E  (30. 3.).  and  join  EA;  EA  wiU 
be  the  side  of  a  regular  polygon  of  12 
sides.  Bisect  EA,  and  draw  a  straight  line 
ixDm  A  to  the  point  of  section,  and  it  will  be 
thesideof  a  polygon  of  24  equal  sides;  and  by 
continually  bisecting,  we  obtain  the  sides  of 
po^fgons  of  48,  96^  192, 884^  &c.  equal  sides. 

251.  To  find  the  circumftrence  and  area  of  a  circle,  having 
^  diomeler  given  p. 

RtJLB.  Eir»t.  Since  there  is  no  geometrical  method  for  deter- 
mining accurately,  the  length  of  the  whole,  w  any  part  of  the 
womference^  we  must  be  content  with  an  approximation; 
which  however,  may  be  obtained  to  such  a  d^pree  of  exactness, 
&9  to  differ  from  the  truth  by  a  line  less  than  any  given  line. 

Secondly,  If  two  similar  polygons  of  a  great  number  of  sides, 
be  doe  inscribed  in,  and  the  other  circunascribed  about «  cirde. 


.'  Hiii  prdblenuwill  serve  to  iliew  by  what  laborion9>  methodi  Wq^llis, 
Romairas,  Metins,  SneUins,  Vao  Cenlen,  and  others,  obtained  appraximatiuiis 
to  the  citdes  periphery ;  the  same  HAnf^  may  however  be  performed  with  madi 
nore  ezpaiUtion  and  ease,  by  the  method  of  fluions,  infinite  serie%  &c.  be« 
%^i<a  ilscfrtMe  and  Aj^pHcatim  n/  Flmeumi^  p«it.  1.  sect.  S, 


GSOHETBY. 


Pi«VHt 


the  circumference  wffl'be  greater  than  the  sum  of  the  sides  cf 
the  former,  but  less  than  the  sum  of  the  sides  of  the  latter  5 
aiKl  thtrefore^  if  the  numbers  expressing  these  sums  agree  in  a 
certain  number  of  figures,  those  figures  may  be  considered  as' 
expressing  (as  far  as  they  go)  the  length  of  the  drqiaafereiice 
which  lies  between  the  two  polygons  >  and  if  half  the  di£krenoa 
of  the  remaining  figures  be  added  to  the  less  number,  or  sub* 
tracted  from  the  greater,  the  result  will  afford  a  still  more 
accurate  expression  for  the  length  of  the  circumferenee. 

Draw  any  straight  line 
AC,  and  ofi  it  describe  the 
eqcfilateral  trian^e  ABC  (1.1.) 
fi'om  C  ^  a  centre,  with 
the  distance  CA  =  CB  describe 
the  ore  ^£jDB;  then  because 
ABa^AC^ihiQ  side  of  an  equi- 
lateral and  equiangular  hexa- 
gon inscribed  in  the  circle 
(15.  4.)  •/•  AEDB  Will  be  one 
sixth  of  the  whole  circum« 
ference. 

Let  f=^C=l,  c=^B=i,  the  arc  AE=ED=iDB,  and 
a:^AE=zihe  chord  of  one  third  of  the  arc  AB;  then  since  the 
arc  EB  ia  double  the  arc  AE,  the  angle  EAH=^ACE  (20.3.) 
and  AEC  is  common^  -.*  the  triangles  AEC  and  AEH  are  equi- 
angulat  (32. 1.)  and  CA:AE::AE:  EH  (4. 6,)  ;  that  is,  r  ; 

^EH;  alsQ  CEiAEi:  AH :  EH  •.*  AEssiAH,  in 


XX 


like  manner  it  is  shewn  that  BD=zBK,  ',-  AH=^BK,  •/  AH-^- 
BKzt9ix,  and  HK^iAB-^AH-^BK^)  c— 2x;  but  CE  :  ED 


XX 


: :  CH :  HK  (4. 6.)i  or  r  :  a: : :  r— . — :  c-^^-^  whence,  multij^y- 


fly- 


ing extremes  and  means,  cr'^irx^zrx--* — 5  which  bytransposi- 

r 

tion,  &c.  (since  c  and  r  each  =1,)  becomes  **— 3a?=— 1,  the 

root  of  which  is  the  chord  of  AE,  or  of  xt  part  of  the  whole 

circumference. 

Next  to  trisect  the  arc  AE,  let  3  y— ^ss^r,  the  chord  of  AB, 


Part  VHI. 


APPlENDIX  TO  EUCLID. 


335 


we  shall  hare  ap*=^fy»—27y*-|.9yr—y9  ^    ' 

and  — 3ar=— 9y+3y' 

and  + 1    = 4-1 

Their  sum  x*  — 3  a:+ 1  =  — 9y-h30y»  — 27y« >9y^— y  + 1 =o, 
<he  root  of  which  is  the  chord  of  ^V  pa^  of  the  whole  circurn* 
Terence. 

Again,  to  trisect  the  arc  of  which  y  is  the  chord  5  let  3  2— 
«»=y,  and  if  this  value  be  substituted  for  y  in  the  last  equation', 
we  shall  obtain  an  expression  in  which  the  Talue  of  z  will  be  the 
chord  of  the  -rW  part  of  the  whole  circumference.  Proceed- 
ing in  this  manner  after  sixteen  trisections,  the  chord  cff 
-nHulirsis  part  of  the  circumference  (the  radius  being  unity) 
will  be  found  to  be  .Oo6oOOOZ4SQ6979^S9SS^OSS,  which  num- 
i)er  being  multiplied  by  3582803^6  (or  the  number  of  sides  of 
the  polygon,  of  which  the  above  number  expresses  the  length 
of  one  side)  the  product  will  be  6.283  ia53d71795859684897'5l? 
=the  perimeter  of  the  inscribed  polygon.  "-   ' 

352.  Next,  we  arc  to  find  the  length  of  the  side  of  a  circuoir 
^bed  polygon  of  thf$  same  number  of  . 
sides,  in  order  to  which^  let  AB:^the  side  D^ 
of  the  inscribed  polygon  ,as  .found  above,  ^ 
DE  the  side  of  a  similar  circuinscribed 
polygon  5  bisect  AB  in  H,  join  '  CH  and 
produce  it  to  F.  Then 'c7p— 2^^=1151^ 
or  1«— .O00O006l2163499644916|^=  1- 
.000000000000000147950723611871658 
0846470516  =£  .99999999999999985204^! 
76388128342,  te.:?:CF|«,  the  square  root 
of  whicl^  number  is  .99999999999999999 

&c.=CH;  now  CHiHA::  CF,I^^ 

^DF   (4.    6.)    that    is    -0000QQ012163409644916016.X  l_. 

.^9909999999999999 
.000000012163499644916,  &c.  =  DF,  which  number  multiplied 
hy2give8  .O000OO0iJ432699929832,  &c.=  DE 
But  .00000002432699928983,  8iC.:^AB 

and  since:  these  two  numbers  agree  as  far  as  the  16th  place  of 
decimals,  and  the  arc  APB  lies  between  DE  and  A  By  it  follows, 
feat  those  16  decimal  placies  will  express  the  length  of  the  ar^ 

T  3 


^FB  very  nearly;  tbat  is,  tlie  above  number  will  difler  from 
the  troth  by  a  very  small  decimal,  whose  highest  |ilaoe  is  17  placo 
below  unity.  Whence  XXI0000(»4386999^9ssthe  length  of 
the  arc  JFB  or  of  the  matjairi  part  of  the  whole  circanh- 
ference  extremely  near.  Now  if  ihe  length  q£  the  arc  JFB  m 
above  determined  be  multiplied  into  the  denominator  of  tUi 
fraction,  the  product  will  be  6.^83185306ia9S4rS3sthe  cir- 
cumference of  a  circle  whose  diameter  is  3,  yery  nearfy. 

253.  Having  found  the  circumference  of  a  circle,  we  can 
readily  find  the  area»  if  not  with  strict  accuracy,  at  least  suffi- 
ciently near  the  truth  for  any  practical  purpose,  in  order  to 
which,  let  us  suppose  an  indefinite  numl^  of  straight  lines 
drawn  from  the  centre  to  the  circumference,  thesjs  will  divide 
.the  circle  into  as  many  sectojjps,  the  bases  of  which  will  he  infde- 
Anitely  small  orc*^  and  their  common  altitude  the  rsidius  of  the 
circle }  now  since  these  small  urcs  coiiicide  indefinitely  near  with 
U&e  sides  of  a  circumscribed  or  inscribed  polygon  of  the  saaae 
number  of  sides  as  there  are  sectors,  these  sectors  may  evidently 
be  considered  a^  triangles,  the  bases  of  which  are  the  above 
small  arcs,  and  their  oonmum  altitude  the  radius ;  but  half  the 
I)a8e  of  a  triangle,  multiplied  into  the  altitude,  will  give  the  area 
(42.  1.)  wherefore,  (half  the  sum  of  the  bases,  that  is)  half  the 
circumference  of  the  circle,  multiplied  into  the  radius,  will  give 
the  area  of  the  triangles,  that  is,  the  area  of  the  <»r^Ie  j  thus 

6.2831853,  &C.X1  ^  «  ,  .      .    ,        . 
;; ss3.1415<)26>  &c.=sthe  area  of  a  circle>  whose 

diameter  is  2. 

254.  Having  fbund:4lie  cireon^bienoe  of  a  cirde,  whose  dia- 
meter is  2,  we  are  by  means  of  it  enabled  to  find  the  Gircnm* 
ference  of  any  other  circle,  whatever  its  diameter  may  be ;  for 
let  the  inseribed  polygon  (whose  sides  coincide  indefinitely  near 
with  the  circumference)  have  n  sides,  the  length  of  each  being 
r ;  and  let  a  similar  polygon  be  inscribed  in  any  other  circle 
having  the  length  of  its  side  szs,  then  will  wr=the  periphery  of 
the  first  polygon,  and  n«=that  of  the  second.     Let  Issthe  ra* 
dius  of  the  former  circle,  ^ssthat  of  the  latter ;  then  if  linea 
be  drawn  from  each  centre  to  the  point*  of  division,  in  the  re- 
spective circumferences,  we  shall    have  1  :  r  : :  « :  ^,   (4.  6.) 
whence  (16. 5.)  1  :  *  :  •  r  : «,  and  consequently  (15. 5.)  l-.t:zm 
ft  ne,  that !«,  the  peripheries  of  the  similar  polygons  are  to 


»AHT  VIB.         PRAdWCAti  PItOBLEMS.  MT 

«tlttr  as  the  raiii  of  tJbeir  drctowcribed  erfcteti  but  theie 
iH^pOtts  ooineide  inAttivSMy  near  with  their  ciremoferenM: 
wherefore  the  chmnifereneeB  of  ckties«ait  «s  tHeir  radii. 

255.  The  aite  of  one  ctrde  bein^  known^  that  of  anoUier 
fiitle  having  ;a  given  diameter,  wulj  be  found ;  let  i>s:the  dia- 
flMler  of  a  circle,  ^sits  ^xesLj  aind  d»Uie  diameter  of  another 
^iirf^  whos^  area  4?  is  re^edj  then  .(«.  13.)  D^iO^iiAi  t^ 

Whence  ^=-^5-,  the  area  required. 

FIUCTIGAL  OEOBIfiTRY. 

SS5.  Practical  iGreometry  teaches  the  appli^tion  .<^  theoKs 
t!(jal  Geometry,  as  delivered  by  Euclid  and  other  inters,  to 
practical  uses  ^ 

256.  To  draw  a  straight  line  from  a  given  point  if,  to  repreeeni 
my  length;  in  yards,  feet,  inches,  Sfc. 

Rule.  I.  Let  each  of  the  divisions  on  any  convenient  scale 
of  equal  parts  represent  a  yard,  foot,  inch,  or  other  unit  of  the 
measure  proposed. 

II.  Extend  the  con^nsses  on  that  scale  untU  the  number  of 
spits  proposed  be  included  exactly  between  the  points. 

III.  With  this  distance  in  the  compasses,  and  one  foot  on  A, 
describe  a  small  arc  Bit  B  s  lay  the  edge  of  a  straight  scale  or 
nder  fhtfn  ^  to  B,  and  draw  the  Une  AB  with  a  pen  or  pencij^ 
ind  it  wUl  be  the  line  required. 

A — '—^- j  B 

iBxAMPLBs. — 1.  To  draw  a  straight  line  ftom  t)ie  point  4  to 
^present  13  inches. 


^  Hie  ^idlb|nii^  problems  are  intended  as  aa  introdoetion  to  the  practical 
^plication  of  fome  of  the  principal  propotitions  in  the  Elements  of  Euclid, 
ttd  likewtte  to  assist  the  stodent  in  acquiring  a  knowledge  of  the  use  of  a 
<tte  of  mathematical  instrameots.  From  a  great  rariety  of  problems  nsaally 
liven  by  writerf  on  Practical  Greometry,  we  have  selected  such  as  appear  most 
Becessary,  and  likewise  such  methods  of  solving  them  as  appear  most  simple 
ttd  obvious ;  to  a  learner  well  acifuatnted  with  Buefid,  other  methods  will 
occur,  and  he  should  be  encouraged  to  exereise  his  ingenuity  in  disooTering 
fod  applying  them.  The  best  Momentary  treatises  on  Practical  Geometry  a.o^ 
HciUQiation^  are  those  of  Mr.  Bonnycastie  and  Dr.  Hntton. 


MS  gbombtrt:  Faet  vni. 

mOi  01^  foot  en  O  txi€^  tke  44k^  io  Ae  l^ih  ^mtim  m 
ike  •oaU  yw  thoote  to  adap<»  and  ^ppiff  that  ditim$ee  from  A  m 
above  direeied,  md  U  mill  give  the  loigth  propoied, 

3.  Tb  draw  a  line  thai  shall  repreacnt  35  yards. 

Let  each  jniwumf  dvMtm  he  cmuidtnd  as  10  yards,  thm 
miU  each  subdiv Mum  represent  1  yard;  appliy  the  compares frem 
3  haekwardM  (to  the  left)  tothe&th  eabdimsum,  and  35  svbdieu 
sions  will  be  included  between  the  points:  apply  this  from  the 
given  point  and  draw  the  line  as  before. 

3.  To  draw  a  line  equal  to  9(S3. 

On  the  diagonal  scale,  lei  each  primary  dkrision  represent 
100,  then  will  each  subdivision  represent  10,  and  the  distance 
which  each  diagonal  slopes  on  the  first  parallel  wiU  be  1,  on  ths 
second  2,  on  the  third  3,  and  soon;  therefore  for  363  extend  from 
the  number  2  backwards  to  the  sixth  subdivision,  on  the  third 
partUlel,  {viz.  the  4th  line  downwards)  and  it  wiU  be  (he  distance 
required, 

S57'  To  measure  any  straight  Une  \ 

Rule.  Extend  the  compasses  from  one  extremity  of  the  given 
line  to  the  other,  and  apply  this  distance  to  any  convenient  scale 
of  equal  parts,  the  number  of  parts  intercepted  between  the 
points,  will  be  the  length  required. ' 

Note.  If  the  sides  of  a  rectilineal  figure  are  to  be  measured* 
the  sanie  scale  must  be  used  for  theto  all  5  and  one  scale  must  be 
used  for  each  of  two  or  more  lines,  when  their  relative  length 
is  required  to  be  ascertained  \ 

.   268*  To  bisect  a  given  strniiht  line  49.  # 


'  By  the  word  meature  it  meant  the  relative  measure  of  a  line,  thai  is,  tbe 
length  of  that  liae  compaied  witi)^  tl^e  ki^|[1^  of  aootlier  line,  both  bciof  mft- 
tured  from  tbe  same  scale ;  if  we  call  tpe  subdivisions  of  tbe  scal/s  feet  oc 
yards,  tbe  l^^e  will  represent  a  line  of  as  many  feet  or  yards  as  it  contains  such 
•ubdiyisions ;  to  find  tbe  abso^utf  ipeasare  of  a  line  in  yards  oj  feet,  w«  must 
evidently  apply  ^  scale  of  actual  yards  or  feet  to  it. 

•  Any  scale  of  equal  "parts  may  be  employed  for  tbis  purpose,  but  it  will  b« 
pjoper  to  cbvoae  one  tbat  will  bring  tbe  proposed  figure  witbin  the  limits  you 
intend  it  to  occupy ;  every  part  (f i«.  every  line)  of  tbe  figure  nmst  be  mea- 
ftured  by  one  scale,  and  not  one  line  of  tbe  figure  by  one  sc^e,  and  aootbef 
line  by  anotber. 


tlMT  VIII. 


FRACnCAL  PROBLEMS. 


xm 


AvLS  I.  With  any  dittance  in 
the  compasses  greater  tiian  lialf 
the  given  line,  let  arcs  be  de- 
scribed from  the  centres  A  and 
Bj  cutting  each  other  in  C  and 
D.  A- 

II.  Draw  a  straight  line  from 
C  to  X>,  and  it  will  bisect  the 
given  straight  line,  as  was  re- 
quired ». 


\°- 


/ 


\ 


■\ 


c/ 


^ 


\ 


./D 


l9 


^59.  From  a  given  point  J0$  in  ti  gwen  strtnghl  hne  JBi  toereet 
a  perpendicular  FD. 

Rule  I.  From  any  point  C 
(without  AB)  as  a  centre,  with 
the  distance  CD,  describe  the 
circle  £I>jP  cutting  AB  (pro- 
duced if  necessary)  in  E  and 
D,  and  draw  the  diameter 
ECF. 

^  II.  Join  FD,  and  it  will  be     i  ^ 

perpendicular  to  AB,  as  wias  "~" 

required  «. 

By  the  Peotractok. 

Lay  the  centre  of  the  protractor  on  A  and  let  the  90  on  itii 
cii'cumference  es^ctly  eoincide  with  the  given  line  3  draw  the 
Une  FD  along  the  radius,  and  it  will  be  the  perpendicular 
required. 

Si59.B.  From  a  given  point  F,  to  let  fall  a  perpendicular  to  a 
gwen  straight  line  AB»    See  the  preceding  figure. 

Rule  I.  In  4B  take  any  point  £,  join  FE,  and  bisect  it  in 
C,  (Art.  258.) 
II.  From  C  as  a  centre  with  the  distance  CF  or  CE,  describe 

*  If  the  points  jiC  and  J?C  be  joiaed,  ibis  rule  may  be  proved  by 
^lid  S.  I.- 

«  The  proof  of  this  role  depends  on  Euclid  31. 3.  Of  tb9  Tariont  methods 
fcr  erecting  a  perpendicular^  ^ven  by  writers  on  Practical  Geometry,  this  is 
tip  mo^t  simple  and  easy. 


^so  GEosoeneiT.  ^aetviJ. 

Die  circle  EDFj  jcin  FD,  tttd  it  w91  be  die  perpendictihr 
required*. 

260.  Through  a  given  point  B  to  draw  a  straight  line  paraUel 
to  a  given  straight  line  AB. 

RuLB  I.  Take  any  point.Fin  AS,  and  from  £  and  Fas  ce«« 
tres>  with  the  distance  EF,  describe  the  ares  EG,  FH. 

II.  Take  the  distance  £G  in  % 
the  compasses^  and  apply  it  fix)m             7L       "         ■     ^ — ^ 
Fto  Hon  the  arc  IV.                            /    """•••^.^            / 

III.  Through  E  and  Hdmw  [  -^^7 

the  stnught  line  CD,  and  it  ^^1^       ^  ^       5 

uriB  be  parallel  to  ^B  Bs  was  requh^d  ^ 

Bt   THB  PABAI.LBL  RVLBR. 

lAy  the  rukr  90,  that  the  edge  of  one  of  its  pamBeb  tasy 
exactly  coincide  with  the  line  AB.  Heldiog  it  steady  la  Uitt 
position,  move  the  other  parallel  up  or  down  untfl  it  cut  the 
point  E,  through  which  <hraw  a  line  CED,  and  it  will  be  panl* 
M  to  AB. 

If  £  be  too  near,  or  too  distant  for  the  extent  of  the  rukr, 
first  draw  a  line  parallel  to,  and  at  any  convehieiit  d^tsuace  from 
AB,  to  which  draw  a  parallel  through  £  u  before^  and  it  will 
be  parallel  to  AB. 

^61.  At  a  given  point  A,  in  a  given  stra^ht  line  AB,  to  make 
an  angle  BAC,  which  shall  measure  an^  gioen  number  ff  degifes. 

RuLB  I.  JBxtend  the  aompaases  from  the  beginning  Of  the 
scaleof  chords  (mark- 

ed  C,)  to  the  60tb  dcr  ^s 

gtee,  and  from  the 
given  ppint  A,  with 
,tbis  distance,  describe 
an  arc  cutting  AB 
(produced  if  neces- 
jBaiy)  in  £. 

11.  Extend  the  compasses  fro;m  the  beginning  ct  the  scale  of 


«  This  depends  on  Euclid  31.3. 

y  Since  the  arcs  EG,  HFaie  equal,  the  angles  JEFG,  F£lfai  the  ceot^ 
are  equal,  (Euclid  87. 8.)  and  therefore  A£  is  jMuraUel  to  CD^  EdcUd  S7. 1. 


Past  VI0.  FRACHCAL  PRCIBLElifS.  ^l 

idionfo,  to  tbe  aumber  deacrting  tlie  meftsore  of  the  |»D|Mfad 
angle,  and  from  JE  as  a  centre^  with  this  distaooe,  cut  4m  above 
arc  in  the  point  R 

III.  Through  F  draw  the  straight  line  JB,  jund  the  angle 
BACi^ill  be  the  an^^le  required  ". 

fixAMPLas'-^l,  Let  (he  angle  pi^pqsed  ooBBSMre  .30  dqprees. 
Bm>i$^g  described  the  arc  RFwUh  the  radim  W,  extend  the 
tmpaues  from  the  b^mofg  of  the  eeale  to  SOj  la^^  tkkti^ 
tent  from  E^  and  draw  a  line  through  the  point  mturked  with  the 
fiompasses,  and  the  angle  of  30P  wiU  be  made. 

.  S.  At  the  point  A  in  AB  make  an  angle  measuting  160 
degrees*. 

Here  the  proposed  angle  bemg  grmter  than  90,  ii  mil  be 
eon^^ient  to  take  ii  at  twice  $  ia^f  9/  BXP  first,  m  E¥$  ihemfnm 
i^>  %  <^  70^  tnore;  drm  a  Uft^  through  Jhe  extwmHyofike  70, 
and  it  wiU  make  wOh  AB  M^^nngle  ^  ISO  d^rem, 

By  tw«  (J{iip7fiAc:TAi. 
Laj  the  central  point  on  A,  and  the  fiducial  e4ge  of  tbe  ndiua 
along  AB,  so  that  they  exactly  coincide  3  then  with  the  pointer, 
make  a  fine  dot^  opposite  the  proposed  degree  (reckoning  from 
"tile  line  AB)  on  tbe  ci/cumferenee ;  through  A  and  this  dot,  draw 
a  straight  line,  and  it  will  make  with  AB  the  angle  required. 


iW«»4w«n^M*f*«saaMaiMMBi«fMi^ 


■  If  the  cireumference  of  a  circle  be  divided  into  860  equal  parts  called 
iegreet,  one  sixth  part  of  the  circamference  will  ueasnre  60  degrees,  and  its 
chard  wiH  be  equal  to  the  radias  of  the  circle  (EtKlid  15. 4.) ;  wherefore,  if  thfc 
4hit  60  degrees  on  any  scale  of  chorda  be  taken  in  tbe  compasses,  and  a  circle 
he  deBcri)>ed  with  that  distance  as  radius,  the  chords  on  the  scale,  wiU  be  tbe 
ffopcr  measure  for  the  chord  of  every  arc  ai  that  circnmference,  as  well  as  for 
the  circmnfereBce  itself;  and  since  the  arc  intercepted  between  tbe  legs  of  the 
^ligle,  (being  described  f^m  the  angular  point  as  a  centre,)  is  the  measure  of 
^  angle  it  subtends,  (Euclid  33.6.  Art.  236.)  the  rule  is  manifest.  By  this 
yvoUem  an  ang^e  inay  be  made,.eqaai  to  any  given  angle. 

*  1V>  measure,  or  lay  down,  an  angle  greater  than  90*,  the  arc  must  be  takea 
hk  tbe  compasses  at  twice;  thus  for  100%  take  60*  first,  and  then  40* ;  or  60* 
flrtt,  and  then  the  remaining  fit)*,  &c.  For  an  mv  of  170*  take  90»  and  80«, 
or  60*,  50*,  and  tO»,  vis.  at  three  times,  ftc.  &c.  If  two  straight  tines  cut 
Me  another  within  a  circle,  their  angle  of  inclination  is  measured  by  half  the 
^^A  of  tbe  int^Mepted  afeti  but  if  they  cut  without  the  citde,  their  angle  of 
hieUaaUon  is  meaMNd  by  half  te  Werenca  of  the  inlercepted  arcs.  See  thi 
Bote  on  Art*  S48.  ' 


.  <  I  ". 


i3S  QBOMBTRY.  Pa«t  VOT. 


X 


ExAMPLBs.  lltke  «t  given  poititf ,  ih,  given  straigfat  lines,  the 
/oUowing  angles,  via.  of  20^,  35S  45^  58«,  9a>,  160^,  and 

iri°i. 

262.  To  mefluure  a  given  angle  BAC.  See  the  preceding 
figure. 

RuLB  T.  Frcmi  the  angular  point  S  as  a  centre,  with  60^  from 
the  scale  of  chords  as  a  radius,  describe  the  arc  EF,  cutting  the 
legs  of  the  given  angle  (produced  if  necessary)  in  E  and  R 

IJ.  Extend  the  compasses  from  £  to  F,  and  apply  the  extent 
to  the  scale  of  chords,  so  that  one  point  of  the  compasses  be  on 
<the  beginning  of  the  scale  -,  then  the  number  to  which  the  other 
point  reaches  will  denote  the  measure  of  the  given  angle  \ 

ExAMPLB.'  To  measure  the  angle  BAC. 

Htming  with  the  radius  60^  described  the  arc  EF,  extend  the 
jcopipasses  JrcmE  ta-F;  then  ^ihis  extent  reaches  from  the  6«- 
ginning  of  theseale  toSS^,  the  awgle  BAC  measures  35  degrees. 

Bt   THE'  F)|{)t«ACTOK. 

Lay  the  fiducial  edge  on  ABl'so  that  the  central  notch  may 


^  The  reasoa  of  th^  rule  will  be  cvicknt  from  the  preceding  note.  Axt 
iogenious  method  of  measuring  angles,  by  means  of  an  undivided  semicircle, 
and  a  pair  of  compasses,  without  the  assistance  of  any  scale  wbateyer,  wu 
pTOpwed  by  M.  De  Lagni,  in  the  memoirs  of  the  French  Academy  of  Sciences ; 
some  account  of  bis  method  may  be  found  in  Dr.  Hutton's  Mathematical 
Dictionary,  under  the  word  Goniometry.  Thomas  Fajitet  De  Lagsi  was  bom 
at  Lyons  in  the  17th  century,  an4  died  in  1734  %  l^e  ,was  successively  professor 
royal  of  Hydrography  at  Rochford,  sub-4irector  of  .the  Generfd  Bank  at  Paris, 
and  associate  geometrician  and  pensioner  in  the  Ancient  Academy.  De  Lagni 
excelled  in  Arithmetic,  Algebr^,  s^nd  Geometry,  sciences  which  are  indebted 
to  him  for  improvements ;  he  invented  a  binary  Arithmetic,  re<|uiring  only 
two  ^gures  for  all  its  operations ;  likewise  some  convenient  approximating 
theorems  for  the  solution  of  higher  equations,  particularly  the  irreducible  case 
in  cubics.  He  gave  a  general  theorem  for  the  tangents  of  tmUiiple;-areSf  and 
determined  the  ratio  of  the  circumference  of  a  circle  to  its  diameter  to  120 
places,  which  is  the  nearest  approximation  for  the  purpose,  that  has  been  made. 
Our  author  was  particularly  foj^d  of  calculating,  and  It  may  be  truly  said  of 
jiiim,  that  **  He  felt  ^the  ruling  passion  strong  in  death  i"  for  on  his  death  bed, 
when  he  was  apparently  insensUilcv  one  of  his  friends  asked  him,  What  is  thf 
square  of  12  ?  to  which  he  immediately  replied,  144  i  we  regret,  that  the  last 
foments  of  this  ingenious  man,  were  not  emplo^d  on  subjects  of  iq$nitejy 
greater  importance. 


pabt  vnr;      pRAcrrcAL  problems.  Sss 

be  on  Ay  then  will  the  degrees  (on  the  circumference)  inter- 
cepted between  AB  and  AC^  be  the  measure  of  the  angle. 

'  Example.  I'd  measure  the  angle  BAG  by  the  protractor. 

Lei  the  centre  coincide  with  A,  and  the  fiducial  edge  with 
AB;  count  the  degrees  {on  the  circumference)  from  AB  to  AC, 
and  the  number  will  he  the  measure  of  BAC. 

263.  To  diofde  a  given  angle  ABC  into  any  number  of  equal 
parts. 

Rule  I.  From  the  angular  point  B  as  a  centre,  with  the 
radius  6(P  (from  the 
scale  of  chords,)  de- 
scribe the  arc  EF  as 
before,  and  find  the 
measure  of  the  angle' 
ABC. 

II.  Divide  the  num- 
ber of  degrees  in  this 
measure  by  the  num^ 
ber  denoting  the 
ntmaber  of  parts  in- 
to which  the  angle 
is  to  be  divided,  and 
the  quotient  will  be  the  degrees  each  part  will  measure. 

III.  Extend  the  compasses,  from  the  beginning  of  the  scale* 
of  chords,  to  the  degree  denoted  by  the  above  quotient,  and 
apply  this  extent  successively  along  the  arc  EF. 

IV.  Through  B  and  each  of  these  divisions,  draw  straight 
lines  Ba,  Bb,  Be,  Bd,  &c.  and  the  angle  ABC  will  be  divided, 
as  was  proposed  <•. 

Example.  To  divide  the  angle  ABC  into  5  equal  parts. 
Having  described  EF  with  the  radius  60°,  Ut  EF  measure 


•  If  either  of  the  lines  SC,  BA  be  less  than  the  proposed  radius,  (vi«.  the 
chord  of  60«»)  it  must  be  produced  to  the  circumference  EF\  likewise  BC,  BA 
may  be  either,  or  both,  so  long,  that  EF  cuts  them  ;  in  cither  case  the  rule  is 
the  same  as  is  plain.  See  the  note  on  Art.  261 .  So  to  measure  an  angle  with 
the  protractor,  it  will  sometimes  be  necessary  to  produce  the  line*  contain- 
ing the  angle,  until  they  meet  the  circumference  of  the  instrument ;  this  may 
be  done  with  a  lead  pencil,  and  the  produced  parts  may  be  rubbed  out,  after 
the  angle  is  measured. 


3M.  cfficof^miy.  fABTVin. 

€Ufpoie  55  degreay  them  — s=  1  V^thtnliiwJber  ofdegn^  m  each  of 

5 

the  parUf  take  11<»  (Jram  ihe  $caU  of  ebard$)  in  ike  eompasies, 

attd  apptjf  a  from  E  to  a,  from  a  to  b,jrom  btoc,  and  from  c  to 

df  and  Unroagh  the  jwnit  a»  b,  c,  and  d,  draw  Ba,  Bb,  Be,  and 

Bd,  and  ABC  wiU  be  dmded  into  S  equal  part9. 

864.  In  like  oiaiiner  the  whole  drcumference  muf  he  di?kled 
into  any  number  of  equal  parts,  and  by  joining  the  points  of 
di¥i^on>  polygonsof  any  number  of  sides  may  be  inscribed  in 
it }  and  if  straight  lines  be  drawn  perpendicular  to  the  several 
ladii  which  pass  through  the  points  of  divbion»  at  their  extremi- 
ties, polygons  of  the  same  number  of  sides  will  be  drcumscribed 
about  the  circle,  as  is  evident. 

Bt  tbb  Pbotractoh. 

Let  the  fiducial  edge  coincide  with  the  diameter  of  the  cirde^ 
and  the  oentral  notch  with  the  centre,  and  suppose  a  polygon  of 
36  equal  sides  be  required  to  be  inscribed  in  the  drde,  mark 
with  the  pointer  opposite  every  10th  degree  (on  the  protractor) ; 
draw  straight  lines  from  the  centre  to  these  points,  and  join  the 
points  where  they  cot  the  circumference  $  and  a  po^rgon  of  S6' 
sides  will  be  inscribed :  and  if  at  the  extremities  of  these  radii, 
and  perpendicular  to  them,  lines  be  drawn  meeting  each  other, 
a  polygon  will  be  circumscribed  about  the  circle,  similar  to  the 
former}  and  by  a  sunilar  method,  any  other  regular  polygon 
may  be  inscribed,  or  circumscribed. 

ExAMPLE8<»l.  To  inscribe  in,  and  circumscribe  about^  a 
given  circle,  an  equilateral  triangle,  and  a  square. 

2.  To  inscribe  in,  and  circumscribe  about,  a  circle,  regular 
polygons  of  10,  15,  30^  24,  and  30  sides,  respectively. 

.  S65.  To  divide  a  given  straight  Ime^AB  into  anj^  numher  of 
equal  parts. 

RvLB  I.  Draw  the  straight  UmAD  making  any  angle  with 
AB; 

II.  Beginning ati#,  wi^aqr extent in^tbe-companeBy  tdse at- 
many  equal  dirisions  (al,  12;  23,  3c,  &c.)  in  AD^BbAB  is  to 
lie  divided  into,  let  these  terminate  at  C,  and  johi  CB. 


\ 


9 


PiiitVBL         JfRAXmCAL  FKOBUSMS.  33& 

UI.TI«ougli  S 

these  divisions  C^x 

draw  8tiii%]|t 
Hnes    parallel 

to    CB,   and  » *- 

cutting -4B  in  ^  ..•••*''   \ 

the  points   a,  .-••"'t  » 

^  c,  &c.  these    X-^'^t^ j jf- 

will  divide  AB 

into  the  number  of  equal  parts  required  < 

ExAMpLss— '1.  It  is  required  to  divide  a  given  line  AB  into 
4  equal  parts. 

FtrsU  draw  an  indefinite  line  AD,  making  m^,  «f?gZe  {DAB} 
with  AB.  Secondly,  open  the  compasses  to  my  convenient  extent, 
(or  A\)  and  with  it  lay  off  the  equal  distances  A,!-,  1,2;  S,  3 
Old  3,  C.  Thirdly,  join  CB,  and  through  3,  2,  and  1,  draw  3  c,. 
26, 1 1^  each  paraUel  to  CB,  (^rt.  260,)  then  wiUAB  be  divided 
ifUo  4  equal  parts  in  a,  h,  and  c. 

2.  To  divide  a  line  of  44-  incliei  in  length  into  10  ^equal  parts. 

Note.  By  this  proUem  meif  itraight  line  may  be  divided  into 
parts  which  are  proportiond  to  thoaeof  a  given  <tivided  straight 
line*. 

266.  Tojind  a  third^ffoportUmal  to  two  given  straight  lines  4 

andB. 
RvLB  I.  Draw  two  indefinite  straight  lines  CD,  CF,  making 

anyan^eDCK 

IT.  In  these,  trite  CG  ^ 

equal  to  A,  CD  and  CJE  ' 

each  equal  to  B,  and  join        -----------^--•---— — • 

GE. 

III.  Through  jD  draw  c 
l)#pettai^  ta  G£  (Art 

2W.)  and  CF  wiii  be  the  _  

tkffd    proportional    re- 
quired; that  \s,  {CO  :  CE  ::  CD  :  CF,  or)  4  I  If :;  B  I  CFt. 

'  The  rcMoa  of  this  rale  will  appear  firom  EacUd  10.  S.  it  is  pretcsahle  t» 
the  oompiez  methodt  propoied  bj  tome  of  the  mgdern  writers, 
*  SeeSncUd  to.9. 
'  Thi»U  the  saac  with  Eudidll.S. 


ZS6 


GEOMETRY. 


Pak*  Vffl. 


267.  To  find  a  fourth  proportional  to  three  given. $tf0^[hi  Ima 
A,  B,  and  C. 

Rule  I.  Draw  two  indefinite  Unes  OD,  OF,  as  before. 

II.  Take  OD  equal 

to  J,  OF  equal  to  B,      j^ 
and  OG  equal  to  C 

III.  Join  DF,  and 
through  G  draw  GE 
parallel  to  DF  (Art. 
260.)  >  and  0£  will  be 
the  fourth  proportio- 
nal required  ^  for  ( DO 

:  OF  ::   GO  i  OE, 
that  i$)A:B::C:OEK 

26S.  To  find  a  mean  proportional  between  two  given  straight 
lines  A  and  B. 

Rule  I.  Draw  the  indefinite  straight  line  HK,  and  in  n  take 
HD  equal  to  A,  and  DK  equal  to  B. 

II.  Bisect  HK  in 
C  (Art.  258.),  and 
from  C  as   a  centre 


A.- 


.••" 


-•.,JE 


H^ 


Q 

-r 


-K 


with  the  distance  CM 
{^CK)  describe  the 
semicircle  HEK. 

III.  Through  D, 
draw  DE  perpendi* 
cular   to   HK,    (Art. 

259.)  and  it  will  be  the  mean  proportional  required;  for  (ED 
:DE::  DE:  DK,  that  h)  A  :  DE  ::  DE  :  BK 

269.  To  find  the  centre  of  a  given  circle  ABD. 

Rule  I.  Draw  any  straight  line  BD  in  the  given  circle,  and 
bisect  it  in  H,  (Art.  268.) 


f  This  is  tbe  same  with  Eadid  1ft,  G. 
k  Tbit  is  fincUd's  1^.  6. 


BkUlp  Vllf . 


PRACTICAL  PROBLEMS. 


Sd7 


IL   Throtigli    S  dnck  AS 
perpendicular  to  BD,  (Art.  259.) 
umL  produee  it  to  E. 
m.  Bi9Bct  JE  in  C,  (Art 
I    258.)  the  point  C  will  be  the 
i    ceatle,of  the  ^vea  circle  *. 


...•".«••»., 


270.  To  draw  a  tangent  to  a  circle  from  any  given  point,  either 
in  the  circumference,  or  without  the  circle, 

RtfLB  1.  Find  the  cehtre  C,  (Art. 
269.)  and  fot  T  be  a  given  point 
^thout  the  circle^  from  which  the 
taogent  is  required  to  be  drawn. 

II.  Jdin  CT,  and  on  it  describe 
the  eetnicircle  CAT. 

III.  Join^r^  and  it  will  touch  the 
circle  as  was  required. 

IV.  If  the  tangent  be  required  to 
be  drawn  ftom  any  point  itf  JM  the  eifiBunifNtn^e>  join  CAy  ahd 
dtttw  AT  perpendicular  to  it  (Art.  dA9.)  y  AT  mm  touch  th^ 

271 .  To  describe  a  triangle,  hatikg  its  Ihtee  sides  gibeki 
ItuLE  1.  Let  Ai  B,  and  C,  b6  the  thli^.i»ide8  of  the  i«f|^i#d 

triangle,  draw  a  straight 
line  DE  equal  to  one  of     A 
them,  suppose  A,  (Art. 
256.). 

II.  Take  the  length 
of  the  line  B  in  the 
compasses,  and  from  D 
as  a  centre,  with  this 
distance,  describe  an  arc. 

III.  From  E  as  a.cen- 


B 
C 


*  This  rule  depends  on  Eadid  1. 3.    Other  methods  xuay  be  derived  from 
Euclid  19,  3;  a\,3ySif8i  and  Tarious  other  parts  of  tlie  Elements. 
^  This  depends  on  Euclid  31.  3.  and  16.  3. 

VOL.  IJ.  Z 


S3§  GEOMETRY.  Fabt  VUh 

tre^  ^th  the  length  of  the  line  C  in  the  compo38e8>  describe  an 
are,  cutting  the  former  arc  in  F. 

IV/  Join  DR  EFi  and  D£F  wiU  be  a  trian^e,  having  its  ades 
respectively  equal  to  A,  B,  and  C  ^ 

Examples. — 1.  Desoibe  a  trisngle  of  w&ich  the  sides  aj<6 
4,  3,  ftnd  2,  respectively,  and  measure  the  angles.  Jns.  10$^^ 
AT,  Old  290^. 

2.  Describe  a  triangle,  the  sides  of  whiteh  afe  25,  36,  and  47, 
and  find  the  measure  of  its  angles. 

272.  To  describe  a  triangle  havit^  two  sides  and  the  i$icluded 
angle  given. 

HuLB  I.  Draw  a  straight  line  AB  equal  to  one  of  the  given 
sides. 

II.  At  the  point  A, 
make  the  angle  BAC 
equal  to  the  proposed 
angle,  (Art.  261.)  3  and 
make  AC  equal  to  the 
remaining  given  side. 

III.  Join  BC,  and  ^BC  will  be  the  triangle  required  ". 

Examples. — 1.  Given  ABssB,  AC^6,  and  the  angle  BAC=^ 
SCP ',  to  describe  the  triangle,  and  measure  the  remainij^  side 
CB,  and  likewise  each  of  the  angles  C  and  B.  Am.  side  CB^ 
4.25,  ang.  0=100°,  ang.  B=^hO. 

2.  Given  2  sides  equal  to  210  and  230  Ftspectively,  and  (he 
uicluded  angle  \0p^  to  find  the  rest. 

273.  To  describe  a  triangle  having  two  sides  ABj,  AC,  and  an 
smgle  ABC,  opposite  to  one  of  them,  given. 


.  I  H'  ■   «.  I  I  ■ 


>  Hm  proof  of  thit  rule  may  b«  found  in  £uclid  32.  T. 
*  This  rule  and  th*  t^o  next  are  sulSciently  ob^ou*. 


i*ART  VIII.  PRACTIfcAt  PROBLEMS. 


339 


Bulb  I.  Draw  the  side  AB,  and  at  its  extremity  B  make  an 
aogle^^Cequaltothepro- 
posed  angle  (Art  261.)  1 
and  produce  the  line  BC. 

IJ.  From  ^  as  a  centre, 
with  the  given  length  of 
AC  in  the  compasses,  de- 
scribe an  arc,  cutting  BC  in  C. 

Hi.  Join  AC,  and  ABC  will  be  the 
required  triangle. 

Note.  If  the  given  angle  be  (a  right 
angle,  or  obtuse,  viz.)  opposite  the 
greater  given  side  (as  in  fig.  1.),  the 
arc  will  cut  BC  (on  the  same  side  of  B), 
in  one  point  C  only;  but  if  the  given 
angle  be  (acute,  viz.)  opposite  the  tew 
side  (as  in  fig.  2.),  the  arc  will  cut  BC 
in  two  points  C,  D-,  and  either  of  th©  tri- 
angles ^JSC  or  ^BD  will  answer  the  proposed  conditions;  hence 
this  case  is  ambiguous* 

Examples. — 1.  Given  -^B=195,  -^C=291,  and  the  angle 
ABC^i^OP  (fig.  1.) ;  to  construct  the  triangle^  and  deteymtne 
(instrumentally)  the  remaining  side  and  angles.  Ans.  BCs^^iG, 
ang.  ^=48^  C=499. 

2.  Given ^JB=136,  ^C=53,  and  the  angle  Bss^^%  (fig.^) 
to  find  the  rest.  Ans.  BC^zUT.  axig,  BCA^s^^^,  ang.,  BAC^ 
58O4.,  or  J9i>=1834^  ang.  D=81o,  ang.  BAD^TG^^. 

274.  To  descrU>e  a  triangle,  having  two  angles,  and  the  adjacent 
fide,  given. 

Rule  I.  Draw  a  straight  line  AB,  ^{ual  \o  the  given  side. 

H.  At  A  and  B  respec- 
tively, make  angles  CAB, 
CBA  eqticd  to-  the  given 
angles  (Art.  ^61.);  and  pro- 
duce AC,  BC,  to  meet  in  C; 
i/BC  will  be  the  triangle  required. 

Examples.— 1.  Given  -4Bss:72,  ang.  B=322ot»  a^-  '^^^^0 
to  make  the  triangle,  and  find  the  rest  An$,  ^CsA9t/.  CBsk 
Sef,  ang.  CaBlS7°^. 

z2 


940 


GEOftlETRV. 


PitRT  VlII, 


2.  Given  ^BsfclO,  apg.  -rf=s45^  aog*  BssW,  to  canstroct 
the  triangle,  and  fiad  the  rest. 

275.  To  describe  a  triangle,  having  two  tingles  and  a  iide  offo^ 
site  one  of  them,  given, 

KuLB  I.  Add  the  two  given 
angles  tc^ether,  and  subtraet 
their  sum  from  180°  (see  Art. 
236.B). 

II.  Draw  AB  equal  to  the 
given  side^  and  at  the  point  A, 
make  the  angle  BAC  equal  to 
the  above  remainder  (Art.  261.). 

lii.  At  the  point  B,  make  the 
angle  ABC  equa],  to  one  of  the  given  angles ;  then  wiH  ACB  be 
the  other,  and  the  triangle  will  be  described  *. 

Note.  If  AB  be  opposite  the  less  an^e,  then  ABC  is  the  tn* 
apgle  3  but  if  AB  be  oppfosite  the  greater,  then  ABD  will  be 
the  triangle  required. 

Examples. — 1.  Given  -rfB=40,  the  angle -4BC=  80®^  and  the 
angle  -4CB=70^»  to  describe  the  triangle^  and  find  the  rest. 
Ans.  AC:=z86,  jBC=45,  ang.  ^^r30». 

2.  Given  ABss40,  and  two  angles=100^,  and  40°,  to  make 
the  triangle^  and  determine  the  rest. 

876.  To  describe  a  rectangle,  the  sides  of  which  are  giten. 

RiTLs  I.  Let  A  be  one  side  of  the  rectangle,  and  B  the  others 
draw  CD  equal  to  A, 

II.  At  the  point  C,  draw  CE 
perpendicular  ta  CD  (Art.  259.) ; 
and  make  it  equal  to  B. 

III.  Through  E  draw  EFpa- 
rallel  to  CD  (Art.  260.),  through 
D  draw  DF  parallel  to  CE,  and 
£CX>F  will  be  the  rectangle  con- 
tained by  A  and  By  as^  wa^  required  *• 


B 


-*•*- 


nr 


|.i      )y^»i>»i 


'    >>' 


■  1 1'l, 


n  The  three  angles  of  a  triangle  are  together  equal  to  two  right  angles 
(EiMlid  dft.  1.)  that  M,  to  }80«>;  wherefore  if  tiie  sum  of  two  angles  of  a 
tiMq(l9 1)0  ««fatfa#«ttd  fiBont  1 80»,  the  vcmahidsr  ^Ml  be  tbe  AM  angle. 

•  The  proof  wi  this  problem  may  be  inferred  from  Ettdid  4«;  t. 


Paut  vnt 


PRACTICAL  PROBLEMS. 


341 


In  like  manner  a  square  may  be  described  on  a  gifen  line  CD, 
by  making  CE  equal  to  CD  «*. 

^7*  To  make  a  figure^  similar  to  a  given  rectilineal  figure  "^ 
having  the  sides  of  the  former  greater,  or  less,  in  any  ratioy  tJian 
those  of  the  given  figure. 

Rule.  I.  Let  ABCDE  be  the  given  figure,  draw  the  lines 
EB,  EC,  &c.  from  any  one  of  the  angles  £,  to  the  other  angles 
B  and  C-,  and  first,  let  H: 

it  be  required  to  increase 
the  figure,  to  another 
whose  side  is  EF. 

II.  Produce  EJ,  EB,  ^ 
£C,  and  ED,  to  F,  G,  H, 
and  K;  and  draw  FG 
parallel  to  AB,  GH  tp 
BC,  and  HK  to  CD 
(Art.  260.);  EFGHK 
will  be  similar  to  the  given  figure  ABCDE, 

HI.  In  like  manner,  if  it  be  required  to  lessen  the  figure,  to 
another  whose  side  is  EL-,  through  L  draw  LM,  MN,  and 
W  respectively  parallel  to  AB,  BC,  and  CD  (Art.  260.)  -,  and 
LMNPE  wiU  be  similar  to  ABCDE  \ 

27S.  To  make  a  regttlar  polygon  of  any  number  of  sides,  on  a 
given  straight  line  AB, 

Rule  I.  Let  n=the  number  of  sides  of  the  polygon  to  be 


'  SeeEadid  46.  1. 

«  The  trath  of  this  oonstniction  is  evident,  for  the  triangles  ELM,  EAB, 
BFG^  beiog  efiaiangalar,  EL :  LM  :iEAi  AB  : :  £Ft  FO  (£noli<f  4.  6.) 
ia  like  maimer  it  may  be  shewn  that  the  sides  abdat  the  renuuaing  equal 
angles  of  the  figares  are  profoitionads,  wherefore  (Euclid  def.  1 . 6.)  the  tbre^ 
figures  are  iimilar. 


Z^ 


Mt  GEOMETRY.  Part  Vm 


i,  then  will  the  sum.  of  its 

interior  angles  be=:2n— 4  right  an- 

1        2n — 4  0 
eles,  and  each  of  its  angles  =—- —  ^^ 

right  angles  \ 

JI.  At  the  points  A  and  B  make  the 
angles  BAC,  ABC  each  equal  to  half 

the  above  angle>  that  is=— —^  (Art. 

261  •.). 

III.  From  the  point  C  where  these  lines  intersect^  with  the 
distance  CA's^  CB,  describe  a  circle. 

IV.  Take  the  distance  AB  in  the  compasses^  and  apply  it  to 
the  circumference  (as  AF,  PE,  ED,  &c.)>  which  will  contain  it 
as  many  times  exactly,  as  the  proposed  polygon  has  sides ;  draw 
the  straight  lines  AF,  FE,  ED,  &c.  and  the  polygon  will  he 
described. 

Examples. — 1.  To  make  a  regular  pentagon  on  AB. 

Here  n=5,  •/  ^^  =(-f.  of  a  right  angUz^^  of  90^=)  54^ 

n 

Make  BAC,  ABC  each  :^54^;  from  the  centre  C  with  the  radvu 

CB  or  CA  describe  the  circle  ^AB,  th^  AB  taken  in  the  com' 

passes,  and  applied  to  the  circumference,  will  meet  it  in  the  points 

ABDEF  and  A  ;  which  points  J>eing  joined,  the  pentagon  will  he 

described  as  proposed, 

%  To  make  a  hexagon,  and  a  heptagon  on  AB. 

n — 2 
For  the  hexagon,  «ss6  j  •.* =s(4  of  a  right  angle  =)  W 

n 

z=:BAC. 

71—2 

For  the  heptagon,  nss^  -,  •.•  — ^— = (j-ofa  right  angle  =)64''y 
zszBAC;  and  proceed  for  both  figures  as  before. 


'  This  depends  on  cor.  1.  32. 1.  of  Eaclid. 

•  That  the  lines  Cj^,  CB  drawn  from  the  centre  t6  the  angnlar  points  A 
and  B  bisect  the  angles  FJB,  AMD,  appears  from  Eaclid  book  4 ;  \ix.  in  the 
equilateral  triangle,  prop,  6 ;  in  the  square,  j>rop.  6  j  in  the  regular  pentagon, 
prop.  14 ;  ^"oA  in  the  regular  hexagon,  prop.  15 ;  and  the  same  may  be  proved 
of  any  regular  polygon  whatever. 


Paht  Vin.  PRACTICAL  PBOBLEMS, 


343 


n 


n 


(0 


™i^i 


/ 


/ 


279.  Tb  construct  a  scale  of  eqluU  parts. 

RfTLB  I.  Draw  three  lines  A,  B,  and  C,  at  convenient  dis- 
tances, and  parallel  to  one  another  (Art.  260.)  -,  and  in  C,  take 
the  pu-ts  .Ca,  ab,  he,  cd,  &c.  equal  to  one  another. 

II.  Through  C,  draw  DCE  perpendicular  to  Ca  (Art.  259.)  j 
and  through  a,  c,  d,  &c.  draw  lines  parallei  to  PCE,  cutting  the 
parallels  J,  B,  and  C;  the  distances  w 

Ca,  aby  be,  cd,  &c.  are  called  the  /  \^^, 

primary  divisions  of  the  scale.  /  /' 

III.  Divide  the  left  hand  pri-      /  v^     ^/y^ 

mary  divisions  Ca,  into  10  equal  '^ 

parts  (Art.  265.)  5  and  draw  lines 
through  these  points,  parallel  to 
DCE,  across  the  parallels  B  and  C; 
this  primary  division  will  he  divided  (• 

into  10  equal  parts,  called  subdivi- 
sions of  the  scale.  f      \P 

IV.  Number  the  primary  divi- 
sions from  left  to  right,  viz.  1, 
2,  S>  &c.  and  the  scale  will  be  com* 
plete. 

280.  To  make  a  scale  of  which 
any  number  of  its  subdivisiofis  will 
he  equal  to  an  inch. 

Rule  I.  Let  one  of  the  primary 
divisions  Ca,  of  the  scale  C,  be  an 
inch ;  and  let  it  be  divided  into  10 
equal  parts,  as  above. 

IT.  From  any  point  D  in  AD,  4.<;p 

draw  Da ;  draw  DS  making  any 
angle  with  DJ,  and  make  DS=  Ca. 

III.  Take  the  number  of  sub- 
divisions  (which  are  proposed  tp 
mak^  an  inch)   in  the  compasses  io» 
from  the  scale  C,  and  ^pply  this 

distance  from  D  to  E. 

IV.  Draw  ES,  and  through   C 

draw  CG  parallel  to  El$,  and  make 
DH-sDO. 

z4 


t<s 


-  1^ 


h 


SM  tSBOUKTBY.  Vau  VHt 

v.  Through  J7,  dnw  AL  iW9Ufll  to  C^  cittiiig  I>^ 
will  HK  be  one  of  the  primary  divisions,  containing  lO  of  the 
parts  proposed ', 

VI.  If  lines  be  drawn  thro^igh  D  to  each  of  the  subdivisioos 
in  Ca,  it  will  divide  the  line  HK  into  10  equal  parts  (Art.  3^1.)' 
which  will  be  tlus  subdivisions  of  the  scale  HL  ;  and  if  the  suo 
cessive  distances  Kl,  12, 23, 34,  &c.  be  taken  in  KL,  each  equal 
to  HK,  these  will  form  the  primary  divisions^  and  the  scale  HL 
will  be  constructed. 

£xAMPi.Es. — 1 .  To  construct  a  plane  scale,  having  20  of  its 
subdivisions  equal  to  an  inch. 

Take  the  distance  Cb  (=2  inches  =20  subjiivisions  of  Ca) 
in  the  compasses,  make  DE^Cb,  DS==Ca,  and  proceed  as  before, 

2.  To  construct  a  scale  of  which  35  subdivisions  make  an 
Inch. 

Extend  the  compasses  from  d  backwards  to  the  fith  subdivision 
between  C  and  a,  this  extent  ( =35  subdicisions  of  the  scale  Cd) 
being  applied  from  D  in  the  straight  line  DE,  proceed  as  before, 

3.  To  make  scales  of  which  15,  25 j  30f  an4  40  resfective 
subdivisions  will  equal  an  inch* 

2S1.  To  construct  scales  of  chords,  sines,  tangents,  secants,  Sgc, 
Rule  1.  With  any  convenient  radius  CA  describe  the  circle 
ABDE,  draw  two  diameters  AD,  BE,  perpencticular  to  each 
other  (Art.  259.),  produce  EB  indetoitely  towards  F,  draw  DT 
parallel  to  EF  (Art.  260.),  and  join  AB,  BD,  DE,  and  EA. 

II.  Divide  the  quadrant  BD  into  9  equal  part6^  (Art.  263.), 
and  from  the  centre  C,  through  each  of  the  dixisions^  ds%w, 
straight  lines  cutting  DT  in  10.  20i,  30«  40,  &c.  this  will  be  tlw 
scale  of  tangents. 

III.  From  D  as  a  centre,  through  each  of  the  di^fisions  of  the 
quadrant,  describe  arcs  cutting  BD  in  }0^  2(1,  30,  40>  &c.  thl^. 

will  be  the  scale  of  chords. 

■  ■  ■   >-   "  ■     '    ■    I .  -  I II  « -  ■  ■     . .     1     _  ■         I   ■ . ,, .    11.11 1. 1  1     I  >  ■«•» 

*  To  demonstrate  the  truth  of  this  construction,  let  the  number  of  subdivi- 
sions  of  HK  contained  in  Ca=Ba  be  called  n,  also  by  construction  Ca  con- 
tains 10  subdivisions  of  itself;  •.•  I)£=n,  T)S-10;  bat  DE  :  DS:i  DC: 
{DG^)  DH  (4, 6.j  and  DC  :  DH : :  Ca  :  HH;  •••  DE> xDSxxCax  HK,  or 

lOCa  C9 

n  :  10  : :  Gi :  HK, ','  HK^ ;  let  ««ao  (as  in  Ex.  1.)  than  J^ilT^-tr  ; 

2Ca 
let  «=35  (as  in  Ex.  2.)  then  HK-'-z-,  &c.    Q.  E.  D. 


Paet  vni.  PRACTICAL  PB<»LEMS. 


345 


2V.  Tbrough  the  dmuoMflof  thequadna^  dnw  Itaea  parallel 
to  BC,  cutting  CD  m  80, 70, 60, 50,  &e.  khia  wiJl  be  tke  acale  of 
SUMS  and  cosines. 

V.  If.  straight  lines  be  dravm  ftom  A  to  the  sewial  divisiona 
(io,  20, 30,  &c.)  of  DJ,  cutting  the  radios  in  10, 20k,  30, 40,  &c 
CB  will  be  a  scale  (^  semi-tangents. 


VI.  If  from  the  centre  C,  through  the  several  divisi^M  of  I>r, 
arcs  be  described,  cuttiog  BF  i»  ]iO>  Sp,  30,  te«  JHF  wiU  be  a 
acale  of  secants. 


346  GEOMETRY.  Past  VUI. 

Vn.  Divide  the  radios  AC  into  GO  equal  parts^  draw  straight 
lines  through  each  of  these  divisions  parallel  to  CB»  cutting  the 
arc  AB  \  and  from  ^  as  a  centre,  through  the  points  where  these 
parallels  cut  the  quadrant  AB^  describe  arc*  cutting  ^0  in  10, 
90,  30,  40,  &c.  AB  will  be  a  scale  of  longitudes. 

VIII.  Divide  the  quadrant  ^£  into  8  equal  parts,  and  through 
these,  from  £  as  a  centre,  describe  arcs  cutting  AE  in  1, 3, 3, 4,  &c. 
A¥»  will  be  a  scale  of  rhumbs, 

IX.  Draw  straight  lines  from  B,  through  the  several  divbions 
of  the  scale  of  sines  (CU),  these  will  cut  the  quadrant  £D  in 
as  many  points  >  from  A  as  a  centre,  through  each  of  these  pointt, 
describe  ara  cutting  £D  in  10,  SO,  30,  &c.  £D  will  be  a  scale 
of  latitudes. 

X.  If  the  above  constructions  be  aocorately  made,  with  a 
circle  the  radius  of  which  is  3  inches,  the  several  lines  will 
exactly  correspond  with  those  on  the  common  scales  ^  wherefore 
to  construct  a  scale,  we  have  only  to  take  the  several  lines  re- 
spectively in  the  compasses,  and  apply  them  (with  their  respective 
divisions)  to  a  flat  ruler;  and  what  was  required  will  be  done. 

9m,  To  find  the  area  of  a  parallelogram  ACDE. 

Rule.  Let  a=the  altitude 
AB,  6=the  base  CD:  then  will 
a6s=the  area  required  '. 

Examples.— 1.  To  find  the 
area  of  a  square  whose  side  is 
12  inches. 

Here  as=12,  fe=12,  and  a&sl2x  12^144  square  inckes=i 
ihe  area  required, 

2.  To  find  the  area  of  a  parallelogram,  the  base  of  which  is 
20  inches,  and  its  altitude  25.109. 

Here    a=25.109,    &s:20,    and    a6s=25.109x  20=502.18 
square  inches  =  the  area  required, 

3.  To  find  the  area  of  a  rhombus,  whose  base  is  42,  and 
altitude  23. 

4.  To  find  the  area  of  a  rhomboid,  whose  base  is  10,  and 
altitude  7-^. 


"  Every  paraUdogram,  is  eqnal  to  the  rectangle  contained  by  its  base  and 
^rpendicalar  altitude  (see  Eaclid  85. 1 ;  1, 9,&c.)  ;  whence  the  rale  is  phuii. 


Part  VIII. 


PRACTICAL  PROBLEMS. 


347 


283.  To  find  the  area  of  a  triangle  ABC. 

Rule.  Let  &1I  a  perpendi-  3 

cular  BD  from  the   vertical 

angle  B  to  the  base  JC,  and 

let  a:=BD,  b=AC,  then  will 

ab      , 

---=the  ai-ea  required  ». 

Examples. — 1.  The  perpendicular  height  of  a  triangle  is  2$ 
inches,  and  its  base  16  inches ;  what  is  the  area  ? 

fiere  a^^,  6=16,  and  —=z—^ — =224«oMarcincftc*,tAtf 

2  2 

area  required, 

j2.  1  ne  base  of  a  triangle  is  1.03,  and  its  perpendicular  alti- 
tude ^,11,  what  is  the  area  ?     Ans.  1.08665. 

3.  The  altitude  7A->  and  the  base  84.  being  given,  to  find  the 
area  of  the  triangle. 

284.  To  find  the  area  of  a  triangle,  Itaving  its  three  Hdes  given, 

JluLE.  Let  a,  h,  and  c,  represent  the  three  sides  respectively, 

a4-6-4-c  ■ " 

and  let  — ^: —  =p>  then  will  ^p.p— a.p— 6.p— c=the  area  of 


o 


the  triangle/. 


'  This  depends  on  Enclid  41.  1. 

y  LetABr^ayAC^hy  BC^^Cy  AD^Xy  then  /)C=6-x,  and  (Euc  47. 1.) 
c»  — 5— ;rl «  ^BHi  *  =a.«  — ar* ,  01  c»— 6»  +  26a:— ar»=a«—ar»,  whence a:= 

'""— .      But   BD)' -^aSI" -^ISi  ' ^AB-^AD.AB-'AD:^  (a-\^ 


26 


rt3+6a— c*       Sah+a'''  +  b'  —c      2fl6— o^—fta  +  ca 
2I         )  ><  (^^  '■      26         )  *"  26  ^  26 

^5^»— c»       c»— a— 6|»  1  ■ ~ 

and  thearea4.^CXfii>=iV("+^'— '^*)X(^''-o^O=-r 


_„____  ^ ■  a+b  +  ca+b — c c+O'—bC'^a+b 

V(«+  6+  c,a-k-  6— <?.<?+  o  — 6.C— «+  6)  =  v      ^ — • — 5 ^' 


2 


«+6+c 


this  expression,  by  putting  p^       ^ 
is  the  rule.     Q.  B.  D. 


,  becomes  ^pp-^cp^b.p'-'ay  which 


On-,  If  «»!!+  6,  and  d^b  c/>  c,  then  will  >/*«— fl« .  a»  — rf«  be  the  rule. 
J^omtycastk^s  MeMuratum,p,  47.^ 


348 


GEOMSniT. 


Faht  VUL 


fixAMFLEs.— 1.  To  tmd  die 
are  4,  5,  and  6. 


gf  a  ixiaa^,  wkoee  aides 


Here    a=4,    fc=5.  c=6,    p=(— :: =—=)  7-5.  <m(f 


2 


^p.^Z^.^Il6.p^=  V7.5  X  7^—4  X  7.5—5  x  75— C=      • 
V7.5X  3.5x3.5x1.5=  vd8.4375=9.S«15«l*e  orw  r«fwr«rf. 

2.  Required  the  area  of  a  tiiangle^  of  wluch  the  threeaides 
are  20,  30,  and  40,  respectively?     Ans.  290.4737>  &c. 

3.  The  sides  are  12,  20,  and  25,  required  the  area  of  the 
triai^le? 


285.   To  find  the  area  of  a  r^ular  foUfgon,  ha»mg 
and  also  the  number  of  sides  given. 

RujLB  I.  Let  ABDEF  be  any 

regular  polygon,  bisect  the  angles 
FJB,  JBD  by  the  lines  AC,  EC, 
and  from  the  point  of  intersection 
C  let  fall  the  perpendicular  CH, 

II.  Let  n=:the  number  of  eldea 
of  the  polygon,  a^s-CH,  and  6= 

nhn. 

AB,  then  will  — --  =the  area  of  the 

2 

polygon  •. 


enenie, 


Tbis  rule  is  g^ven,  without  a  dem<m8tratioOy  in  the  Geodrnt  «f  Hen>  thfr 
yoQiii^er ;  but  the  inventiiHi  is  snppoted  to  bekmg'  to  some  piecedingy  and  non 
profound  Geometer.  Tartalea  is  the  first  among  the  modems  who  introdoocs 
the  rule,  viz.  in  his  TraUaio  di  Numeri  et  Mkwe,  foL  Venice^  1959. 

ha 

*  Hiis  rale  is  evident,  for  the  area  of  each  of  the  triangles  wil(  be  ^e'*^ 

(Art.  283.]  ;  but  there  are  n  triangles,  where£Mre  the  area  of  their  sum,  (m. 

ha    nia 
of  the  giTea  polygon,}  will  be  «  X  -^^  "S"* 

if  tiie  side  of  each  of  the  following  figures  be  unity,  then  will  the  radias  of 
the  iQScribed  and  circamfcribed  circles  be  as  bdow : 


PlBT  VJIf , 


PRACTICAL  PROBLEMS. 


349 


£xAMn.E8.«-l.  The  aide  of  a  pentagon  is  4,  and  the  perpen- 
dicular from  the  centre  2,61,  required  the  area  ? 

„  ^     ,      ,  ^  .  nba     5x4x2.01 

Here  11=6,  6=4,  a=s2.0l,  and  —  = =20.1, 

the  area  required, 

2.  The  side  of  a  hexagon  is  7.3,  and  the  perpendicular  from 
the  oentre  6^2  required  the  area  ? 

Here  «»6.  6=7.3,  a=6.32,  and  !^^g X 7.3  X  6.39  ^ 

2  2 

138.408,  lAe  area  required, 

3.  To  find  the  area  of  an  octagon,  whose  side  is  9.941,  and 
perpendicular  12.    Ans,  477.168. 

4.  To  find  tlie  area  of  a  heptagon,  whose  side  is  4.845,  and 
perpendicular  5. 


Inscribed  cirtfle,  Ctremn.  cir.  Psrp.  keighi. 


Equilateral  triangle 

Square   

Pentagmi  ••«.<«.. 

Hexagon    

Octag«n 

Decagon •  • . . 

Dodec^gOQ   


0.57735027 
0.70710678 
0.  8506508 
1.00000000 
1.30656296 
1.61803398 
1.98185165 


0.86602540 


1.53884176 


0.28867513 
0.50000000 

0.68819096 

0.86602540 

1.80710678 

1.53884176 

1.8668201ff 

Hence  the  areas  of  thete  figures  may  be  readily  found,  and  likewise 
those  of  siauUr  figures,  whateyer  be  the  length  o£  the  given  side  ;  since  simi- 
lar polygons  are  to  one  another  as  the  squares  of  their  homologous  sides, 
(£ttcfid  20.6.)  or  as  tfa«  squares  of  the  diamet«r»  of  their  eircumscribiog 
circles  by  1. 12. 

If  the  square  of  the  side  of  any  regular  polygon  in  the  following  table,  be 
lAnHipUtd  into  the  number  ttandiflg  agaiiitt  its  name,  the  produot  will  be  the 

area.  « 

Ao.  qf  sides.  Names.  Multipliers, 

3  • . .  •  Trigon,  or  equilateral  triangle  0.43301 3— 

4  . . . .  Tetragon,  or  square 1 .000000 

5  . . . .  Pentagon 1.720477  + 

6  . . . .  Hexagon    2.598076  + 

7  ... . .  Heptagon 3.633912  -f 

8  . . . .  Octagon 4.828427  + 

9  ..^..Nonagon 6.181824 -i- 

10  ....Decagon 7-694209— 

11  ....  Dodecagon 9.365640-- 

12  ..  ..Dodecagon 11.196152  + 


550 


GEOMETRY. 


Part  VIII. 


S86.  To  find  th0  area  of  any  g'w^  rectilineal  figure  JBOUEE 

Rule  I.  Join  the 
opposite  angles  of 
the  figure,  viz.  AC, 
AD,  FD,  so  that  it 
may  be  divided  into 
triangles  ABC^ACD, 
ADF,  FDE. 

II.  Find  the  area 
of  each  of  the  tri- 
angles ABC,  ACD, 
^DF,    ADE,  (Art. 

283.),  and  add  these  __ 

areas  together,  the  sum  will  be  the  aoreaof  the  hgareABCDEF. 

ExAMPLBS.— 1.    Let   AC=zlO,   BH^4,   CL^S,  AD^li, 
CL=z6,  FD^S,  EN^3,  and  FK=:S. 
ACBH    10x4     40 


Then 


2  2 

AD.LCl^xe 
2      ■        ~ 

^D.FKVZx^ 

2      *■     2 


=  — =20=arca  of  ABC. 

72 

=^—=^S6:=area  of  ACD. 

eo 

=— =30=arca  of  AFD. 


FD.NE     8x3      24     .^  .  ^^^ 

2  2         2  *^ 

Their  sum  98=:area  of  ABCDER 

2.    Let  AC=z4t^,  BH=^10,  AD^bO,   Ci-=20>    fD=10a, 
£iNr=s2o,  and  FK^U,  to  find  the  aiea.    Am.  2076. 

287.  The  diameter  of  a  circle  being  given,  to  find  the  drcutn- 
ference;  or  the  circumference  being  given,  to  find  the  diameter. 

Rules  I.  As     7 :  22  -x  ^      ^, 

or,  as  113  :  355  \  ' ''  '^^  ^^^^^^'  ''  ^^^  '"" 

or,  as      1:3  1415927 /'"°'^""'^""\ 


•  The  first  of  these  prgportions  is  that  of  Archi®edes,  which  is  the  easiest, 
although  the  least  exact,  of  any  of  the  ruUs  Which  have  been  proposed  for  this 
purpose ;  the  second  proportion  is  that  of  Mctius ;  the  third  is  Van  CeolenV 
*  rule,  and  depends  on  Art.  252,  where  it  is  shewn,  that  if  the  diameter  be  «, 
the  circumference  will  be  6.2831853,  &c.  wherefore,  if  the  diameter  be  1,  the 
circumference  will  be  3.1415927  nearly,  which  is  the  same  as  therole. 


Pabt  VIIL  practical  PROBLEMS.  351 

...i.     ,,»  1    '. ••  the  circumference  :  the  diame- 

or,  as  355  :  113 


1    ::  the  ci: 


or,  as  3.1415937 

Examples. — 1.  The  diameter  of  a  circle  is  12,  required  the 

circumference  ? 

^  29  X  12     264 

Tkiu,  «  7  ;  22  : :  12  :  — 5— =^:r=37.714285    th^  cir^ 

7  7 

(umference  nearly. 

Or,  as  113  :  355  ; :  12  :  rri^==-_-=37.699ll5    the 

circumference  mare  nearly. 

Or,    as   I  I  3.1416927  : :  12  :  31415927  X  12=37.6991124 
the  circumference  very  nearly. 

2.  The  circumference  is  30,  required  the  diameter  ? 

SO  X  7     105 
Thus,  <w  22  :  7  : :  30  :  -—  =—=9.54545,  &c.  the  dia^ 

iMier, 

113x6     678* 

Or,  05  355  ;  113  ::  30  : —=-^=9.549295,  &c.  the 

71  71 

iiameter. 

30 
Or,    as    3.1415927  :  1  : :  30  :  ——_  =9.549296,    &c. 

^  diameter. 

3.  The  diameter  of  a  circle  is  6,  required  the  circumference  ? 
Ans.  18.8495562,  &c. 

4.  The  circumference  is  5,  required  the  diameter?     ^ns. 
1.5915493,  &€. 

5.  If  the  diameter  be  100,  what  is  the  circumference  ?     And 
if  the  circumference  be  100,  what  is  the  diameter  ? 

288.  Tojind  the  area  of  a  circle. 

Rule  I.  Let  c=the  circumference,  d=the  diameter,  then 

Will  -7-=the  area  of  the  circle. 
4 

Or,  2nd.  .7854d«=the  area.    Or,  3rd.  .07958  c»=the  area. 

Examples.— 1.  The  diameter  of  a  circle  is  4,  required  the 
circumference  and  area  ? 


These  proportions  are  the  conrerte  of  the  fonaer. 


3M  GfiOMETRY.  pAitr  Vffl» 

Tfti»  (JrL  25^.)  3.1415927  X4=]2.5d63706stibedrcian^ 

_,,      cd     1^.5663706x4  ,  , 

Then  — = =  12.5663708= <^  area,  by  rule 

4  4  . 

1.  (Jrt,  253.) 

Or,  .7854  <P=. 7854  X  16=s  12.5664  =  tft4?  area,  by  rule  2. 

Or,  .07958  c*= (.07958  x  12.566370b? '^=  .07953  x 
157.913675,  &c.=)  12.566769= ^Ae  area,  by  ruU^. 

2.  Required  the  area  of  a  circle,  whose  diameter  is  7,  and  its 
drcumference  22  ?     Jns.  38^ 

3.  What  is  the  area  of  a  circle,  whose  diam^er  k  1>  and  dr- 
camfeirence  3.1415927? 

289.  To  find  the  area  of  any  irregular  mixed  figure  JBCDEF, 

Rule  I.  Inscribe  the  greatest  possible  rectilineal  figure 
ACEF  in  the  proposed  figure,  and  let  ASCy  CDE  be  the  remain- 
ing irregularly  curved  boundaries. 

II.  From  as  many  points       JL^---^^^  S 
as    possible   in    the  curve 
ABC,  let  fall  perpendiculars 
(Art.  259),  to^C;  and  find 
their  sum. 

III.  Divide  this  sum  by 
the  number  of  perpendicu- 
lars taken,  and  multiply  the 

quotient  by  the  base  AC,  the  product  will  be  the  area  of  the 
curved  space  ABC. 

IV.  Proceed  in  like  manner^  to  find  the  area  of  tlie  space 
CDE. 

V.  Find  the  area  of  the  rectilineal  figure  ACEF  by  Art.  286. 
then  lastly,  add  the  three  aieas  together,  and  the  sum  will  be 
the  area  of  the  figure  ABCDEF  s 


«  This  method  of  approximatioa  is  used  for  measuriag  fields  and  other 
endosates,  which  bsve  very  cfoolied  and  ifreg^^la^  bonndaries ;  -the  ^eatef  the 
numbef  of  perpendiculars  be,  the  nearer  truth  will  th«  approximation  bc,.aa 
is  evident. 

To  find  the  area' of  a  regularly  tapering  board,  measure  across  the  two  ends, 
add  both  measures  together,  and  half  tfie  sum  multiplied  into  the  length  of 
the  board,  will  give  the  ar«a. 


iPikiVni.  PRACTICAL  t%6dL£MS«  363 

Examples. — 1.  Let  AE^^O*  the  perpendicular  FH=10^  the 
perpendicular  CK=9,  ACszl4,  C£=L1,  the  sum  of  9  perpen- 
diculars let  fall  on  AC,^S7i  ^^^  ^^^  ^^^  o^  7  perpendiculars 
let  fell  on  €E,  =25,  to  find  the  area  of  the  figure  ABCDEF. 

^       AExEH-^-KC    ,20x10+9     20x19     380  •      ,^ 
Brst, r—l =( p-J— = = =)  190 

*  »  ^2  2  2        '^ 

sKthe  area  of  the  rectilineal  space  ACElFi 

37 
iSecowd/y, —=4.1111,  &c.  then  -^Cx4.1111,  &c.=(14x 

4.1111^  &c.  =)  57.5555,  &c.  =<Ac  area  of  the  curved  space  ABC. 

-*.  25  * 

J%irdiy, —=3.571428,   &c.   then    CEx  3.571428,   &c.  = 

7 
(11  x  3.571428,  &c.=)  39.285708,  &c.  =*/rc  area  of  the  cui^ed 
space  CDE, 

Lastly^  these  added  together,  viz, 

190 =</ie  area  ACEP 

67.555565= ABC 

39 .28570^= ...CDE 

The  sum  286.841263= ABCDEF,  as  ivAs   re*  . 

quired. 

3.  het  AE^lOl,  fH=25,  CJK:=21,  -4C;i=87,  CJE=79>  the 
sum  of  20  perpendiculars  on  ^C=103,  and  the  sum  of  17.  on 
C£=72  5  to  find  the  area  of  the  figure  ABCDEF     . 

290.  To  find  the  solid  content  of  a  prism. 
Rule.  Find  the  area  of  its  base  by.  some  ot  the  preceding 
rules,  and  muUiply  this  area  into  the  perpendicular  height  o( 
the  prism,  the  product  will  be  the  solid  content  ^. 

Examples. — 1.  The  side  of  a  cube  is  13  inches,  required  its 
solidity  ? 

Thus  13  X  13= 169=arca  of  the  base  {Art  282.) 
Then  169  X  13=2197  cw6ic  inches  z=xthe  solidity  of  the  cube. 
Or  ^fciw,  13x13  X 13= (13)  3=)    2197  =  */*«   solidity,   as 
before. 

If  the  board  do  not  taper  regularly,  measure  the  breadth  in  several  places^ 
«dd  all  the  measures  together,  divide  the  sum  by  the  number  of  breadths  taken, 
and  multiply  the  quotient  by  the  length  of  the  board,  and  it  will  give  thtf 
area. 

^  This  rule  depends  on  Euclid  2  cor.  7.  IS. 

VOL,  IJ.  .  A  a 


354  GEOMETRY.  P^t  VUI. 

5^  The  skies  dbout  one  of  the  angles  of  the  base  of  a  rectan- 
gular prism  are  7  and  5  respectively^  and  the  altitude  of  the 
prism  20;  required  the  solidity  ? 

Thus  7x5si35=area  of  the  base;  then  35x20ss700  ike 
solidity. 

3.  The  sides  of  the  base  of  a  triangular  prism  are  2,  S>  and 

4,  respectively,  and  the  perpeqdicular  altitude  30;  requited  the 

soUdity? 

Q4.34.4 
Thus    {Art.    284.)    p=s.  ^    ^   =4.5,   and 


^415  X  4.5-2  X  4.5-3  X  4.5—4=3  v^.4S755s2.»47375=:anw 
of  the  base. 

Then  2.9047S75X  30^5 87. 1421250^  lAe  solidUy. 

4.  The  base  of  a  prism  is  a  regular  hexagon,  the  side  of 
which  is  8  inches,  and  the  altitude  oi  the  prism  is  4  feet ;  re- 
quired the  solidity  ?  

Here    {Art.  285.)   6=8,  «=  ^8«— 4«=(  ^48=)  6.9282, 

-         _  nba    6x8x6.9282      .^^^^^«  •    r       .l 

,n=s6,  and  -rr^ s =166.2768  square  uicto=<fcc 

2  2  ' 

area  of  the  base:  wherefore  by  the  rule  166.2768x48  {inches) 

=7981.2864  cti6ic  inches  =4  cubic  feet  1069.2864  cubic  inches. 

5.  The  length  of  a  parallelopiped  is  16  feet,  its  breadth  4^ 
feet,  and  thickness  6i  feet ;  required  the  solidity  ?  Ans.  486 
cubic  feet, 

6.  The  length  of  a  prism  is  5  feet,  and  its  base  an  equilatenl 
triangle,  the  side  of  which  is  2^  feet;  required  the  solidity? 
Aris.  13.5315  cubic  feet. 

7'  The  base  is  a  tegular  pentagon^  the  side  of  which  is  12 
inches,  and  the  length  d  feet  3  required  the  solidity  of  the  prism  ^ 

291.  To  find  the  solid  content  of  a  pyramid. 

Rule.  Find  the  solid  content  of  a  prism,  having  the  same 
base  and  altitude  as  the  pyramid,  by  the  last  rule ;  one  third  part 
of  this  prism  will  be  the  solid  content  of  the  pyramid  *. 

Examples. — 1.  The  altitude  of  a  pyramid  is  20  feet,  and  its 
base  is  a  square,  the  side  of  which  is  12  feet ;  required  the 
solidity  ? 


*  This  depends  on  cor.  U  7. 1?.  Eiidid. 


Part  Wir.  PRACtflCAt  PftOM^EMS.  3fci 

*  I 

»28SO=:5o/tdify  o/*  the  circuTUseribing  prism,  and  — ^«d60 

9 
euhic  feet  :a:  the  solid  content  of  the  pyrcanid. 

3.  The  altitude  of  a  pyramid  is  11  fytt,  and  iU  bade  a  i«gu1ar 
hexagon,  the  side  of  which  is  4  feet  5  what  is  the  solidity  ? 

Here   (^rf.  285.)   5=4,  a=  v'4«-2«=:  ^12=3.464101 6, 

«-/5  «  ^'*^<»     6x4x3.4641016 

«-6,fl«^.-5-«-7- *: — '-^  ^41.56^199»tfre<i   of  ike 

hase^  ako  41. 5692 Idftx  lias 457.^6 141 12= wZidi^y  0/  the  cir- 

M.r.o^'U'  ,A^     /^rv^X  457.2614112 

cumcnhmg  prism  {Art.  290.),  •/ -3:162.4204704 

cuhicfeet  :sxthe  solidity  of  the  pyramid, 

3.  What  is  the  solid  content  of  a  triangular  pyramid,  the 
height  of  which  is  10,  and  each  side  of  the  base  3  ?  Answer, 
12.99039. 

4.  What  is  the  solidity  of  a  Square  pyramid,  each  side  of  its 
base  being  IS,  and  the  altitude  25  ? 

292.  njbtd  the  selvi  i:(Ment  of  «  cylinder. 

RuL£.  Multiply  the  area  of  the  base  by  the  perpendicular 
altitude,  and  the  product  will  be  the  solidity  '. 


■»<»— i»i*»*«  I 


'This  ride  depends  on  £ttctid  1 1  and  14  <tf  book  l^i  The  eoiivex  »uper. 
ficies  of  a  cylinder  is  found  by  mnltiplying  the  circumference  of  the  base  by 
the  altitude  ol  the  cylinder ;  to  which,  if  the  areas  of  the  two  ends  be  added> 
the  sum  will  be  the  whole  external  superficies. 

To  find  ths  solidUy  (f  squared  timber.  1. 1^  the  stick  be  eiiualiy  broad 
and  thick  throughout,  find  the  area  of  a  section  any  where  taken,  and  multi- 
ply it  into  the  length,  the  product  will  be  the  Solidity.  S.  If  the  stifck  tapers 
regularly  from  one  end  to  the  other,  find  half  the  sum  of  the  areas  of  the  two 
cnds^  and  mnltipTy  it  inter  the  lengtlr.  3.  If  the  stick  dorrnot  taprr  regutarly, 
fiad  the  areas  of  seveval  different  sections,  add  them  together,  and  divide  thi$ 
tain  by  the  number  of  sections  taken,  this  quotient  multiplied  into  the  leugUii 
will  give  the  solidity. 

To  find  the  solidity  of  rough  or  unsquared  timber.  Multiply  the  square  of 
one  fifth  of  the  mean  girt  by  twice  the  length,  and  the  product  will  be  the 
solidity.  Or,  multiply  the  square  of  the  circumference  by  the  length,  take  ^ 
of  the  product,  and  from  this  last  number  subtract  ^  of  itself,  the  remainder 
^U  be  the  solidity.  See  on  this  subject  fftUton*s  and  BonnyeasHe'i  Tr^tise$ 
on  Mensuration, 

A  a  2 


VSt  ©BOMETRt:  pAETVni. 

£xAMPLB$*— 1.  The  altkude  of  a  cylinder  is  12  €eet>  and  the 
diameter  of  its  base  S  feet ;  required  the  solidity  ? 

First,  3  X  3,lAlB997^^M4776l:=zciramferenceofthe  hose. 

-aft.  387. 

Then,  i2i£i?^ZSl=7.o685836=afetf  of  the  hose.  Art. 

4 

388.  V  7«0685S36xi2s84.8230032  cubic  feet  ^the  ioMUy 
fefUxrei. 

9.  The  altitude  is  90  feet^  and  the  drcumference  of  the  base 
eo  feet ;  required  the  solid  content  of  the  cylinder  ?  Jns.  636.64 
feet, 

3.  The  diameter  of  the  base  is  4  feet,  and  the  altitude  9  feet  5 
required  the  solidity  of  the  cylinder  ? 

^3.  To  find  the  solid  content  of  a  cone. 

AtfLE.  Find  the  solidity  of  a  cylinder  of  the  same  base  and 
altitude  with  the  ^ven  cone^  by  the  last  rule  >  one  third  of 
this  will  be  the  solid  content  of  the  cone  K 

Examples  1.  The  circumference  of  the  base  of  a  cone  is  IS 

feet,  and  its  altitude  10  feet ;  requiml  the  solid  content  ? 

12 
ftf^^-  '     >  —  2=3.819718=: dtam.  of  the  base.   Art. 287. 
3«14159«7 

then^  —  x-^-—^ — s6xl.909859s:11.459154s:area   of 
2  2 

the  base.   Art  9SS. 

Whence  11.459154 x  10=  I14^69l54=5o/idi^  of  the  civ- 
cumscribing  cylinder.    Art.  292. 

114  59154 

lastly,  — 1-- —  =2:33.19718  cubic  feet  =ithe  solidity  of  the 

cone.  ' 


I  For  the  fouddatiou'of  the  rale  ait  EudM'  10. 13.  Let  fl«tfae  axU  of  a 
rcone^  <f~the  semidiameter  of  its  base,  then  (Euclid  47.  1.)  A/a^+^f^^rthe 
slant  height  of  the  cone ;  and  if  the  slant  height  be  multiplied  into  the  cir- 
cumference  of  the  base,  the  ptoduct  will  be  the  oonyex  superficies  of  the  cone, 
to  which^  adding  the  areaortbe  base,  the  sum  will  be  the  whcfle  extermd^ 
superficies.  Rules  for  finding  the  superficies  and  solidities  of  the  several 
sections  of  a  prism,  pyramid,  cone,  cylinder,  sphere,  &c.  may  befoudd  in  Mr. 
Bonnycastle^s  excellent  Introduction  to  Msnturatwn,  a  work  which,  cadadt  b^ 
too  highly  commended. 


fWtrVBI.         PRACTICAL  PROBLBBCS.  •     d57 

4 

9.  The  altitude  is  13>  and  the  diameter  of  the  base^i  re- 
,^ired ,  the  solidity  of  the  cone  ?    Am,  28.2743344. 

3.  The  area  pf  the  base  is  30>  and  the  altitude  14 }  required 
the  solid  content  (^  the  cone  ? 

S94.  To  find  the  solid  content  of  a  sphere.        ■' 

RuLB.  Find  the  solidity  of  a  cylinder^  of  which  the  altitude* 
and  the  diameter  of  its  base^  are  each  equal  to  the  diameter  of  the 
given  sphere  -,  two  thirds  of  this  will  be  the  solidity  of  the  sphere  K 


^  EacUd  has  proved  that "  spheres  are  to  each  other  in  the  ^plicate  ratio 
of^  their  diameters"  (18. 18.) ;  hat  this  Is  the  m^j  property  of  the  sphere 
to  be  found  in  the  Elements.    We  are  beholden  to  Archimedes  for  the  most 

9 

part  of  onr  orij^nal  information  on  this  salgeet ;  the  abenre  rnle,  which  was 
,taken  from  his  treatise  **  on  the  sphere  and  cylinjder/'  may  be  easily  denon- 
*  ttrated  by  **  indivisibles,*'  *f  the  metl^od  ef  mcremeots/'  <<  SioioDi,''  aid 
wme  other  modem  methods  of  computation  j  but  I  believe  it  cannot  be  effected 
by  elementary  Geometry. 

The  superficies  of  a  sphere  is  equal  to  the  convex  surface  of  its  cirenmsoibin)^ 
•j^der  s  it  is  likewise  equal  to  four  times  the  area  of  a  great  circle  of  the 
sphere. 
If  the  diameter  of  a  sphere  be  2,  then  will  tiia  cifcumlereBce  of  a  greal 

prcle  be 6.S8318 

llie  superficies  of  a  great  ciyclis  •  •  • .  • 3.14159 

The  superficies  of  a  sphere 1S.56637 

The  solidHy  of  the  sp^re    4.18790 

rite   Mde  ...•    1.62209 
And  of  the  inscribed  tetraedroa  i  superficies  •  * .  4.6188 

Vsolidity 0.15132 

{its  side  ..  ••  1.1547 
superficies  • . .  8.0000 
solidity 1.5396 

rits  side  •..•    1.41421 
The  inscribed  octaSdroti  J  superficies  ...  6.9382 

Isolidity 1.33333 

riU  side  ••».  0.71364 
The  iDMSribed  dodecaSdroa  <  superficies . . .  10.51462 

Isolidity 8.785l6r 

^lU  side  ••••   1.05146 
The  inscribed  icosa£droii  <  superficies...  9.57454 

Isolidity 2.53615 

Hence  the  superfiiiial  and  solid  content  of  a  soUd,  similajr  to  any  of  the 
jfbofe,  may  be  readily  obtained,  its  side  being  given  j  the  superficies  being 
u^  the  squares  (Euclid  20.  6,),  and  the  solidities  as  the  cubes  (cor.  a»  12.)  eC 
the  homologous  sides. 


9%l    .  CanMHTRT.  PuTVm. 

its  soUdi^  ? 

ito^'i  base.    Art.  287. 

Sfcoiii%  £2i£d^fZZ^87J9<»58S€84fce  ^ylinderV  tee. 

iln.  eas. 

7%tri^,  7.0685836x32=91.2057508=^  soU^  of  the 
e^Under.    Jrt  292. 

Lattly,  *  of  21.2057508=14.1371672  cu&ic/eee=<J^  sofi- 
dUy  of  the^here. 

9'  ThacUam^ter  of  a  sphere  u^  17  incbes^s  requiired  its  so|i» 
4ity?    jfM$.l.4»»e»qihicffieL 

3.  Jl  ^aeartii  be  a  pwfect  aphfliv  of  8000 mass  diameter> 
Wr  Mwmy  eabie  nriloi  of  wattiBr^tocfr it  contam? 


PART  IX. 


TRIGONOMETRY. 


HISTORICAL  INTRODUCTION. 

Trigonometry  •  is  a  sdejice  which  inches  how  tp 

determine  the  sides  and  angles  of  triangles,  by  means  of  the 
relations  and  .properties  cff  certain  right  lines  drawn  in  and 
about  the  circle ;  it  Is  divided  into  two  kinds,  plane  and  sphe^ 
rical,  the  former  of  which  applies  to  the  computation  of  plane 
rectilineal  triangles,  and  the  latter  to  triangles  formed  by  the 
intersections  of  great  circles,  on  the  surface  of  a  sphere. 

This  science  is  justly  considered  as  an  important  link  con- 
necting theoretical  Geometry  with  practical  utility,  and  mak- 
ing the  former  conducive,  and  subservient  to  the  latter.  Geo- 
graphy, Astronomy,  Dialling,  Navigation,  jSurveying,  Men- 
suration, Fortification,  &c.  are  indebted  to  It,  if  not  for  their 
existence,  at  least  for  their  distinguishing  perfections ;  and 
there  is  scarcely  any  branch  of  Natural  Philosophy,  which 
can  be  successfully  cultivated  without,  the  assistance  of 
Trigonometry. 

We  are  in  possession  of  no  documents  that  will  warrant 
us  even  to  guess  at  the  period  when  Trigonometry  took  its 
rise ;  but  there  can  be  do  doubt  that  it  must  have  been  in- 
herited not  very  long  after  the  flood.  The  earliest  inhabitants 
of  Chaldfea  and  I^pt  were  acquainted  with  Astronomy,  which 

"  The  Dftme  is  derived  from  v(ut  three,  yn^s  a  comer,  and  fur^w  to  measure. 
The  objects  of  Trigbnometry  are  the  sides  and  angles  only,  whateyer  respects 
the  areas  of  triangles  beloD|[8  to  Geometry. 

A  a4 


S60  TRIGONOMETRY.  Part  IX. 

(admitting  it  to  have  been  at  that  time  merely  an  art,  and  iii 
its  rudest  state)  would  still  require  the  aid  of  some  method 
similar  to  Trigonometry  to  make  it  of  any  benefit  to  mankind* 
We  may  reasonably  8uppo9e  that  the  anojent  Greeks 
eultivated  Trigonometry,  in  common  with  Geometry  and 
Astronomy  5  but  none  of  their  writings  on  the  subject  have 
been  preserved.  Theon  **,  in  his  Commentary  on  Ptole- 
my's Almagest,  mentions  a  work  consisting  of  twelve  books 
on  the  chords  of  circular  arcs,  written  by  Hipparchus,  an 
Astronomer  of  Rhodes,  A.C.  ISO  *.  This  work  is  believed 
by  the  learned  to  have* been  a  treatise  on  the  ancient  Trigo- 


.  ^  TlieoD,  a  respectable  mathematician  and  pbilosopliery  and  pr^ident  of  the 
Alexandrian  school,  ibnrished  A.  D.  370.  He  was  not  mbre  famous  for  his 
acqnirements  in  science,  -than  for  bis  veneration  of  the  DEriT,  and  his 
frm  belief  in'  the  constant  ^aperintendence  of  divine  providence;  .he  r^oom* 
inends  meditation  on  the  presenpe  of  .God^'  af  the  most  delightful  and  nseful 
'employment,  and  proposed,  tbaf  in  order  to  deter  the  profligate  from  committing 
crime*,  therer  should  be  written  at  the  corner  of  every  k'treet;  Remember  GoA 
8E^s  TBBE,  O  Sinner.  Dr.  Simson,  in  bis  notes  on  the  Elements  of  Euclid, 
has  ascribed  most  of  the  faults  in  that  book  to  Tbeoni  without  mentioning  oi^ 
what  authority  he  has  done  so. 

c  HipP<trchns  was  bom  at  Nice,  in  Bithynia:  here,  and  afterwards  at 
Kbodes  and  Alexandria,  bis  astronomical  observations  were  made.     He  dis- 
covered that  the  interval  between  the  vernal  and  autumnal  equinox  is  longer 
by  7  days  than  that  between  the  autumnal  and  vernal ;  he  was  the  first  who 
krranged  the  stars  into  -49  constellations,  and  determined  their  longitude* 
and  apparent  magnitudes ;  and  his  labours  in  this  respect  were  considered  so 
valaabie,  that  Ptolemy  has  inserted  his  -catalogue  of.  the  fixed  stars  in  his 
Almagest,  where  it  is  still  preserved!    He  also  di^lcovered  the  precession  of 
the  equinoxes,  and  the  parallax  of  the  planets ;  and,  after  the  example  of 
Thales,  and  Sulpicius  Gallus,  foretold  the  exact  time  of  eplipses,  of  which 
be  made  a  calculation  for  600  years.     He  determined'  the   latitude  and 
longitude,  and  fixed  the  first  meridian  at  the  'F&rtuhatdf  Ifuuke,  or  CetMfy 
Inlands;    in   which   particular   he  has   bee^  followed   by  most  succeeding 
geographers.  .  Astronomy  is  particularly  indebted  to  him ,  for  collecting  tbs[ 
detached    and    scattered    principles    and   observations  of  his  predecessors, 
arranging  them  in  a  system ;  thereby  laying  that  rational  and  solid  foundation, 
upon  which  succeeding  astronomers  have  built  a  most  sublime  and  magaificeBl 
superstructure.    Of  the  several  works  said  to  have  been  written  by  bim^ 
^is  Commentary   on    the    Pbcsnomena   of   Aratus    is    the  only  qd^    tba| 
remslins.  •  . 


JP4»T  IX,  INTIIODUCTION.  3dl 

wvfietTy,  and  Is  the  most  ancient  on  that  subject  of  which. 
^we  baye  any  account. 

The  Spherics  of  Tlieodosius  *  is  the  earliest  work  on  Tri- 
gonometry at  present  known.  It  was  written  about  80  years 
before  Christ,  and  consists  of  three  books,  "  containing  a 
variety  of  the  most  necessary  and  useful  propositions  relating 
to  th§  sphere,  arranged  and  demonstrated  with  great  perspi- 
^cuity  and  elegance,  after  the  manner  of  Euclid's  Elements/* 

We  are  in  possession  of  three  books  on  spherical  triangles 
by  Menelaus  *.  He  is  considered  as  the  next  Greek  writer 
wjio  tfeated  expressly  on  the  subject,  and  lived  about  a  hun- 
dred years  after  Cb"st.   This  work  of  Menelaus  was  greatly 


^  Theodosins  was  a  native  of  Tripoli,  in  Bithynia ;  and,  according  to  Stral?e, 
excelled  in  mathematical  knowledge.  The  work  above-mentioned  consists  oC 
Ihree  books  ;  the  first  oif  whicli  contains  23  propositions,  the  second  23,  and  the 
third  14.  It  was  translated  into  Arabic,  and  afterwards  from  the  Arabic  into 
Latin,  and  pub)isftd  at  Venice;  but  the  Arabic  edition  being  very  defective, 
a  complete  edition  was  obtained  by  Jean  Pena,  Regius  Professor  of  Astronomy 
at  Paris,  and  published  there  in  Greek  and  Latin,  A.  D.  1658.  Long  bef«re 
this  time,  a  good  Latin  translation  of  the  work  had  been  made  by  ViUltio,  a 
respectable  Polish  mathematician  of  the  13th  century,  and  the  first  of  the 
moderns  who  wrote  to  good  purpose  on  optics.  The  Spherics  of  Theodosius 
have  been  cnrichW  with  notes,  commentaries,  and  illustrations,  by  Clavins, 
Hdegan^us,  Gu^rinus,  and  De  Chalcs  ;  but  the  best  editions  are  those  of  Dr, 
Barrow,  8vo.  London,  1675 ;  and  Hunt,  8vo.  Oxon,  1707. 

There  are  still  in  existence  in  the  National  Library  at  Paris,  two  other  pieces 
by  Theodosius,  one  on  The  Ccel&tial  Hwaes,  and  the  other  on  Days  and 
Nights:  a  Latin  translation  of  which  was  published  by  Peter  Dasypody, 

A.D.  1572. 

«  Menelaus  was  a  respectable  mathematician  and  astronomer,  probably  of 
the  Alexandrian  school,  but  we  have  no  particulars  of  his  life  or  writings, 
except  that  he  is  said  to  have  written  six  hooks  on  the  chords  of  circular  arcs, 
which  is  supposed  to  have  been  a  treatise  on  the  ancient  method  of  construct- 
rag  trigonometrical  tables,  but  the  work  is  lost.  A  Latin  translation  of  the 
three  books  on  spherical  triangles  was  undertaken  by  RegiomonUnus,  but  wa» 
$rst  published  by  Maurolycus,  together  with  the  Spherics  <rf  Theodosius,  and 
his  own,  (Messanae,  1558,  fol.)  An  edition  of  this  work,  corrected  from  a 
Hebrew  manuscript,  was  prepared  for  the  press  by  Dr.  Halley,  and  published 
to  Costard,  the  author  of  the  History  of  Astronomy,  in  8v6.  1768, 


iG2  TRIGONOMETRY.  Pabt  K. 

improved  py  Pcoletny,  who,  m  the  fint  boc^  of  his  Almagest, 
has  introduced  a  table  of  arcs  and  their  chords,  to  every  half 
degree  of  the  semicircle ;  he  divides  the  radius,  and  also  the 
ate  equal  to  one  sixth  of  the  whole  circuBiference  (whose 
chord  is  the  radius)  each  into  60  equal  parts,  and  estimates 
all  other  ares  by  siirtieths  of  that  arc,  add  their  chords  l^ 
siiltieths  of  that  chord  (or  radius) ;  which  method  he  is  sup 
posed  to  have  derived  from  the  writings  of  Hipparchus,  and 
other  authors  of  antiquity. 

No  farther  progress  seems  to  have  been  made  in  the  sci- 
ence, until  some  time  after  the  revival  of  learning  among  the 
Arabians,  namely,  about  the  latter  part  of  the  eighth  century; 
when  the  ancient  method  of  computing  by  the  chords  of  arcs 
was  laid  aside  by  that  people,  and  the  more  convenient  me- 
thod of  coiAputing  by  the  sines,  substituted  in  its  stead.  This 
improvement  has  been  ascribed  by  some  to  Mahomed  Ebn 
Musa,  and  by  others  to  Arzachel,  a  Moor,  who  had  settled  ifi 
Spain,  about  the  year  1 100 :  Arzachel  is  the  nrst  we  read  of 
who  constructed  a  table  of  sines,  which  he  employed  in  his 
numerous  astronomical  calculations  instead  of  the  chords,  di- 
viding the  diameter  into  300  equal  parts,  and  computing  the 
magnitude  of  the  sines  in  those  parts.  We  are  indebted  to 
the  Arabs  for  the  introduction  of  those  axioms  and  theorems 
into  the  science,  which  are  considered  as  the  foundation 
of  modem  Trigonometry,  and  likewise  for  other  improve- 
ments. 

The  sexagesimal  division  of  the  radius,  according  to  the 
method  of  the  Greeks,  was  still  employed  by  the  Arabian^ 
althoug'h  they  had  long  been  in  possession  of  the  Indian,  or 
decimal  scale  of  notation.  But  shortly  after  the  diffusion  of 
science  in  the  west,  an  alteration  was  made  by  George 
Purbach,  Professor  of  Mathematics  at  Vienna,  who  wrote 
about  the  middle  of  the  15th  century;  he  divided  the  radium 
into  600000  equal  parts,  and  computed  a  table  of  sines  io 


Paw  IX.  INTRODUCTION.  d«9' 

these  part%  for  emy  ten  imnutes  of  the  quadmnt^  bjr  4it  de-* 
cimal  notation.  This  work  was  further  prosecuted  by  Regio* 
montanus^  the  disciple  and  friend  of  Purbaidi;  but  as  the  plan 
of  his  master  was  evidently  defective,  he  afterwards  changed 
it  altogether,  by  computing  anew  the  table  of  sines  for  every 
minute  of  the  quadrant,  to  the  radius  1000000.  Regiomon- 
t&aus  also  introduced  the  use  of  tangents  into  Trigonometry, 
the  table  of  which  he  named  Canon  Fecundusy  on  account  of 
the  numerous  advantages  arising  from  its  use.  He  likewise 
enriched  the  science  with  many  valuable  theorems  and  pre- 
eejpts;  so  that,  excepting  the  use  of  logarithms,  the  Trigo- 
nometry^^f  Regiomontanus  was  little  inferior  to  that  of  our 
own  times. 

About  this  period  the  mathematical  sciences^  began  id  be 
studied  with  ardour  in  several  parts  of  Italy  and  Germany, 
and  it  can  hardly  be  supposed  that  a  science  so  obviousiy 
useful  as  Trigonometry,  would  be  without  its  share  oi  admi-* 
rers  and  cultivators,  although  scarcely  any  of  their  writings 
on  the  subject .  have  been  comniitt)ed  to  the  press.  John 
Werner  of  Nuremburg,  (who  was  born  in  146B,  and  died  in 
158B,)  is  said  to  have  written  five  books  on  tiiangles;  but 
whether  the  woric  exists  at  present,  or  is  last,  we  are  not  in- 
fbrmed.  A  brief  treatise  on  plane  and  spherical  Trigono- 
metry was  written  about  the  year  1500,  by  Nicholas  Coper- 
nicus, the  celebrated  restorer  of  the  true  solar  system. 
This  tract  contains  the  description  and  construction  of 
the  canon  of  chords,  nearly  in  the  manner  of  Ptolemy; 
ttf  which  is  subjoined  a  table  of  sines  to  the  radius  lOOOOO 
with  their  differences,,  for.  every  ten  minutes  of  the  qua- 
dranty  the  whole  forming  a  part  of  the  first  book  of  his 
AMo&iunies  Orbium  CcBk^iumf  first  published  at  Nurem^- 
burg,  fol.  1543.  Ten  years  after,  Erasmus  Reinhold,  Pror* 
fessor  of  Mathematics  at  Wirtemburg,  published  his  Ca^ 
nan  Facundus,  ox  table  of  tangents;  and  about  the  same 


56^1  TRIGONOMETRY.  Part  IX. 

4iiiie  Fnmciscos  Maandjco%  Abbotof  Bfanna^in  ISdly^aiid 
one  of  the  best  Geometen  of  the  age,  published  his  Tabidm 
Benfficaj  or  canoo  of  secants. 

But  a  more  complete  work  on  the  subject  than  any  that 
had  hitherto  appeared^  was  a  treatise  in  two  parts  by  Viet^ 
one  qf  the  ablest  mathematicians  in  Europe,  published  at 
Paris,  in  157-^.    The  first  part,  entitled  Canon  Matkenmti^ 
cus  seu  ad  triangula,  cum  appendicibus,  contains  a  taUe  of 
sines,  tangents,  and  secants,  with  their  difl&rences  for  every 
minute  of  the  quadrant,  to  the  radius  100000.  The  tangents 
and  secants  Ufw^fis  the  end  of  the  quadrant  are  carried  to  8 
or  g  figures,  sind  tbe  arrangement  is  simibir  to  that  at  present 
in  use,  each  number  and  its  compliment  standing  ip  ^e  same 
line,  (^pposite  one  another.   The  second  part  of  this  volume, 
entitled  Vniversalium  Inspeethnum  ad  Cwonem  Matkemati" 
cum  liber  singularisy  contains  the  OHistnictioo  of  the  fore^* 
going  table,  a  complete  treatise  on  plain  and  sphc^eal  Tn-^ 
gonometry,  with  their  application  to  various  parts  of  A^ 
Mathematics;  particulars  relating  to  the  quadrature  of  the 
circle,  the  duplication  of  the  cube ;  with  a  variety  oi  other 
curious  and  interesting  problems  and  observations  of  a  mis- 
cellaneous nature  *•  Besides  the  above  masterly  performance, 
Vieta  w^  the  author  of  several  tracts  on  pli^ne  an4  spherical 
Trigonometry,  which  may  be  fouiid  in  tbe  cotlectioii  of  hisf 
works,  published  by  Schooten,  at  Leydep,  in  164& 

The  triangular  canon  was  next  underta]cen  by  George 
Joachim  Rheticus,  a  pupil  of  the  great  Cc^rnicus,  and  Pro^ 
fessor  of  Mathematics  at  Wirtemburg;  ^'  he  computed  the 


f  For  further  particulars  of  this  iDterestiog  volume,  see  The  History  tf 
Trigonometrical  Tablet,  p.  4, 5, 6, 7,  bj  Dr.  Hntton.  It  appears  tkatt  scafoely^ 
any  copies  of  this  ezcelleot  work  are  now  to  be  ioood  ;  for  tbe  Doctor  utji,  ia 
concluding  his  account  of  it,  **  I  never  saw  one  (copy)  besides  that  which  is 
in  my  own  possession,  nor  ever  met  with  any  other  person  at  all  aeqwuntecl 
with  such  a  book,"  p,  7. 


t^iJiTlX.  INTRODUCTION'.  36& 

i^on  of  sines  and  co-sines  for  every  ten  seconds  of  the 
^quadrantj  and  for  every  single  second  of  the  first  and  last 
degree ;"  he  had  proposed^  in  obedience  to  the  desire  of  his 
master,  to  complete  the  trigonometrical  canon,  and  extend 
it  ftirther  than  had  hitherto  been  done;  but,  dying  in 
iSjG,  the  completion  of  this  vast  design  was  at  his  re- 
quest consigned  to  his  pupil  and  friend  Valentine  Otho, 
mathematician  to  the  EHectoral  Prince  Palatine ;  who,  after 
several  years  of  indefetigable  labour  and  intense  application, 
accomplished  the  wcnrk,  and  it  wa&  printed  at  Heidelberg, 
in  1596,  under  the  title  of  0pm  Palatinum  de  Trianguiis,- 
We  have  here  an  entire  table  of  sines,  tangents,  and  secants, 
for  every  ten  seconds  of  the  quadrant  to  ten  place?  of  figures^ 
with  their  differences,  being  the  first  complete  eanon  of 
these  numbers  that  was  ever  published. 

But  notwithstanding  the  pains  th^  had  been  taken  in  the 
calculation,  the  tables  in  this  valuable  performance  were 
afterwards,  found  to  contain  a  considerable  number  of  errors, 
particularly  in  the  co-tangents  and  co-secants ;  the  correc- 
tion of  these  was  undertaken  by  Bartholomew  Pitiscus,  a 
skilful  mathematician  of  that  time,  who,  having  procured 
the  original  manuscript  of  Rheticus,  added  to  it  an  au3d- 
liary  table  of  sines  to  21  places,  for  the  purpose  of  supply- 
ing the  defect  of  the  former^  and  published  both  in  folio, 
at  Frankfort,  in  1613,  under  the  title  of  Thesaurus  Ma- 
them€Uicus^  &c.  Pitiscus  then  re-calculated  the  co-tan- 
gents and  co-secants  to  the  end  of  the  first  six  degrees  in 
Otho^s  worky  which  rendered  it  sufficiently  exact  for  alstrono- 
mical  purposes^  and  published  his  corrections  in  separate 
sheets,  making  in  the  whole  86  pages  in  folio. 

The  Geomeirica  Triangulorum  of  Philip  Lansbergius,  in 
four  books,  was  published  in  1591  >  a  brief,  but  very  elegant 
work,  containing  the  canon  of  sines,  tangents,  and  secants, 
with  their  construction  and  application  in  the  solution  pf 


SC6  TRIGONOMETRY.  Paiit  IX. 

plane  and  spherical  triUDgles;  the  whofe  betog  fully  aad 
dearly  explained.  This  is  the  first  work  in  which  the  tan- 
gents and  secants  are  carried  to  7  places  of  decimals  to  the 
last  degree  of  the  quadrant.  * 

A  comply  and  masterly  work  on  Trigonometry  by  Pids- 
cus,  was  published  at  FVankfort,  in  1500;  the- triangalar 
canon  is  here  given^  and  its  construction  and  use  clearly 
described,  together  with  the  application  of  Trigonometry  to 
problems  of  surveying,  altimetry,  architecture,  geography, 
djalUng,  and  astronomy ;  forming  the  most  commodious  and 
useful  treatise  on  the  sul]gect  at  that  time  extant. 

Several  other  writers  on  Trigonometry  appeared  towards 
the  close  of  the  16tb,  and  at  the  beginning  of  the  17th 
century,  of  whom  Christopher  Clavius,  a  Jesuit  of  Bamberg, 
may  be  considered  as  one  of  the  chief.  In  the  first 
volume  of  his  works,  (which  were  printed  at  Mentz,  in  5 
volumes,  folio,  16I2,)  he  has  given  an  ample  and  circum- 
stantial treatise  on  Trigonometry.  In  this  woric  the  caBon 
of  sines,  tangents,  and  i^ecants,  is  computed  for  every  minute 
to  7  places  of  decimals,  and  carried  forward  to  the  end  of 
the  quadrant,  the  sines  having  their  differences  computed  te 
every  second,  and  construction  of  the  tables  being  accom- 
panied with  clear  and  satisfactory  explanations,  chiefly  derived 
from  the  methods  of  Ptolemy,  Purbach,  and  Regiomontanos. 

Van  Ceulen,  in  his  celebrated  treatise  De  Circulo  tt  ad- 
scriptisy  first  published  about  the  year  1600,  treats  of  the 
chords,  sines,  and  other  lines  connected  with  the  circle; 
which  work,  with  some  other  of  Van  Ceulen's  pieces,  wss 
afterwards  translated  into  Latin,  and  published  at  Leydeu,  in 
16199  by  Willebrord  SneUhis,  who  has  also  himself  given  in 
li»  Doctfinm  Triangulorum  CamniciBy  the  construction  of 
sines^  tangents,  and  secants,  together  with  a  very  usefel 
synopsis  of  the  calculation  of  plane  and  spherical  triangles. 

A  eanoD  of  sines,  taageats^  and  secants^  to  every  mimte 


Paet  IX.  INTRODUCTIPN.  867 

of  the  quadrant,  was  published  in  1G27>  at  Aonsterdaiii,  by 
Francis  Van  Schootea,  the  ingenioiis  comm/eiilatar  oh  the 
Geometry  of  Des  Cartes,  His  assariVn),  that  bis  tiyi>le  was 
without  a  single  error,  has  been  since  found  to  h^  meonect ; 
some  of  his  numbers  have  been  discovered  to  err  in  the  last 

%usej  being  hot  always  calculated  to  the  nearest  unit '. ' 

-  ■  -  — ^ — 

9  In  tho  early  ages  of  Geometry  the  circamfcfcftee  of  the  circle  igms  divUled 
into  360  degrees^  each  degree  into  60  minutes,  each  minute  into  60  flecoods^ 
Sec.  ;  this  method  was  adopted  by  the  moderat,  and  still  prevails  among  the 
Bpglish,  and  most  other  nations  in  Bwofo }  but  the  Frensfa  aiathcmaticians 
have  introduced  an  improvement,  whkb,  when  it  is  generally  Q|ider8tood  and 
adopted,  will  be  of  the  greatest  advantage  to  Trigonometry.     Towards  the 
latter  part  of  the  eighteenth  century,  a  new  system  of  weights  and  measures 
was  instituted  iu  France,  in  which  they  were  decimfdly  divided  and  saiMHvided; 
this  was  followed  by  another  of  eq^al  importance,  a  new  division  of  the  qma- 
drant.    By  this  new  method,  the  whole  circumference  is  divided  into  400 
equal  parts  called  degvces,  r^w^h  degree  into  100  minutes,  each  minute  into 
10<>  seconds,  &c.  conseipiently  the  quadrant  will  contain  100  degrees.     One 
aidvantage  in  tbhi  method  is  its  convenient  identity  with  the  common  decimal 
scale  of  numbers,  for  !<>,  83',  45",  in  the  new  French  scale  will  be  expressed 
by  the  very  same  figures  in  common  deciaials,  viz.  by  1.9345^ ;  in  like  manner 
91«,  3',  4%  French,  is  expressed  by  S1.03O4»  common  decimals ;  ITO**,  1',  «", 
84"'  by  170.010234*;  5',  O",  11'"  by  .05001 1«;  12',  18",  14"'  by  .121814% 
Sec.    Among  the  works  on  this  plan  %t  present  in  use,  are  I^es  Tables  Porta' 
iimt  de  Callet,  2  Edit.  Paris,  1795 ;  the  Trigonometrical  Tables  of  Borda, 
improved  by  Delambre  j  4to.  an  IX. ;  a^  thm  taUes  lately  published  by  Hobert 
and  Ideler,  at  Berlin.   Likewise  tables  on  the  above  plan,  to  an  extent  hitherto 
unknown,  have  b«an  for  ipaAy  yesrs  under  the  hands  of  M.  Ptony,  assisted  by 
a  Qimbef  pf  fibU  mativKDUUticiaqt,  a  work  which,  ieaides  its  great  usefulness, 
will  be  the  most  ample  monument  existing,  of  human  industry,  in  the  provini^ 
of  calculation. 

To  reduce  degreeff  mumUis,  ^r.  i^  I4«  Ftvish  9wh>  t«to  degrees,  minutes, 
Sfc.  of  the  common  scale,  and  vice  versd» 

l^iacis  Ui«  qufkdfwt  is  4ivi|l«4  by  IhA  FTtmh  method  into  1 00%  and  by  the 
comvifin  q^ei^  intq  StQ%  '.*  \QQf>  Frmpk  ss»90^  csmman.}  '.-  To  udme  Freneh 
degrees,  minutes,  Sfc,  into  conunon. 

Rule.  Express  tjbe  Fkua^  nne^piTi  «(Mi|ii%,  mhtract  from  this  -rr  of 
itself ;  mark  off  the  pr^p^  decim*^  iB  the  re^uii^dar,  mtihipfy  these  by  60, 
xnark  off  the  decimals ;  multiply  these  agaia  by  60,  an^  mark  off  the  decimals 
AS  b^^ior^,-  %^. ;  the  resulting  ^ole  i|umbers  wiU  \»  the  degrees,  minutes, 
second^  &c.  te«mired,  a^oc^ing .  to  th^  ^t^gi^  $<^ak. 

£xAMPLE8.--ri.  In  %4%  ^',  SA"  |t<«Mil^>  Im»«  mdBy  4^¥«ei9  niontas, 
sccondsi  &c.  common  ? 


S6«  TftlGONOHilETRt.  Pa  ir  IX; 

The  invention  of  logarithms  by  Lord  Napier,  in  IGH^and 
their  subsequent  improvement  by  Mr.  Henry  Briggs,  greatly 
facilitated  the  pmctical  opei^tionk  of  Trigonometry.  Besides 
the  invention  of  logarithms,  we  are  indebted  to  Napier  for 
the  method  of  computing  spherical  triangles  by  means  of 
the  five  circular  parts,  and  other  valuable  improvements  in 
spherical  Trigonometry. 

The  docfrine  of  infinite  series,  introduced  about  tbe  year 
16^8,  by  Nicholas  Mercator,  and  improved  by  Newton, 
Leibnitz,  the  Bemouflis,  and  others,'  soos  found  its  applica- 
tion to  Trigonometry,  by  fun^ishinc;  expressions  for  the  sines, 
tangents,  &c.  for  which  purpose  the  exponential  formute  of 
Mr.  Demoivre  are  extremely  convenient.' 

But  the  gi-eatcst  aiid  most  useful  improvement  of  modem 
times  In  the  analysis  of  sines,  co-sines,  tangents,  &c.  which 

Fint,/roffn  d4S  56^,  32"  »34.56dS» 

Subtract  ^  of  the  same  s  8.45639 

The  remaimder  eai.lOSSS 

Multiply  the  decimals  by     fiO' 

6.41280 
Multiply  the  decimals  by  €0 

24,76800' 
Multiply  the  decimals  by  60 

46.08000 
Thereon  84S  56',  3S"  French  s^SlS  6',  ^4"^  46'"»  08  ctfMMum. 
S.  In  8%  12',  8"  French,  how  many  degfcet^  miiiatM,  &c.  common  ?    i^> 
7%  18',  88%  81'". 

8.  In  12*,  I',  9!*  French,  how  manyddgrees,  &c. common? 
4.  In  a*,  8',  7"  F^'eoch,  how  many  degrees,  &c.  eonmon  I 
To  reduce  common  degrees  into  French, 

RufcB.  Turn  the  minntes,  secomU,  See.  into  decimals,  to  the  whole  add  f  of 
itself;  then  the  integers  of  the  sum  will  be  degrees,  the  two  left  hand  decim^ 
minutes,  the  two  next  d<^imal8  seconds,  &c. 
ExAMnjBs.— 1.  To  redoce  34%  56^,  St"  commion,  to  French  measme. 
First,  to  34%  56',  32"  e  34 .942222%  3ec. 
^dd  ^  of  the  same^  8.882469 

The  sum  is  38.82469 1» 38%  82',  46",  Bl'^Freneb' 

2.  In  24%  44',  6"  common,  how  many  degrees  French  T    Ans.  24%  15^.^ 
a.  Turn  28%  27',  58"  common  into  jFremrA.    Am.  26%  17V35". 
4.  Turn  1%  2'^  34"  common  into  /^eiicA. 


Part  IX.  INTRODUCTION.  sGd 

we  owe  to  the  penetrating,  comprehensive,  and  indefatigable 
ttiind  of  the  venerable  Euler :  by  substituting  the  analytical 
mode  of  notation,  in  the  room  of  the  geometrical,  which  had 
hitherto  been  chiefly  used,  he  simplified  the  methods  of  pre- 
ceding writers,  investigated  a  great  variety  of  formulae,  ap- 
plicable to  the  most  difficult  cases,  and  made  the  trigonome- 
trical analysis  assume  the  form  of  a  new  and  interesting 
science* 

Admitting  that  the  Continental  mathematicians  are  out 
superiors  in  the  theory  of  Trigonometry,  as  well  as  in  their 
writings  on  the  science  *,  still  we  have  some  very  good  and 
useful  treatises  on  the  subject;  the  chief  of  which  arc 
those  of  Thomas  Simpson,  Emerson,  Maseres,  Horsley, 
Keith,  Vince,  and  Woodhouse ;  but  Mr,  Bonnycastle's  Trea^ 
Use  on  Plane  and  Spherical  Trigonometry^  is  the  most  com- 
plete work  on  the  subject  of  any  that  have  hitherto  appeared 
in  this  country* 


■MM 


*  See  the  Quarterly  Review  for  Nov£mber^  1810,  page  40). 


VOL.  II.  B  b 


/ 

f 


J 


TikT  iXi      DEFINITIONS  AND  PRINCIPLES.  87l 


PLANE  TRIGONOMETRY'. 

DEFINITIONS  AND  PRINCIPLES. 

i.  JrLANE  Trigonometry  teaches  how  to  determine^  ffooi 
proper  data,  the  sided  and  angles  of  plane  rectilineal  triangles^ 
by  means  of  the  analogies  of  certain  right  Hnesj  described  ini 
and  about  a  circle. 

2.  Every  triangle  contairm  6ix  parts^  viz.  three  sides^  and 
three  angles;  any  three  of  these^  whereof  one  (at  least)  is  a 
side,  being  given>  the  remaining  three  may  be  fbtlnd. 

3.  The  sides  of  place  rectilineal  triiSLngles  are  estimated  in 
feet^  yards^  ^hon»9^  chains^  &c.  or  by  abstract  numbers :  and 
each  of  the  angles,  by  the  arc  of  a  circle,  included  between  the 
two  legs  3  the  angular  point  being  the  centre. 

4.  It  has  already  been  observed  (Art.  237.  t>aft  H.),  that  the 
whole  circumference  is  supposed  to  be  divided  into  360  degrees, 
each  degree  into  60  minutes^  each  minute  into  60  seconds,  &c.  -, 
as  many  degrees^  minutes,  and  seconds  therefore,  as  are  con- 
tained ih  the  arc  intercepted  between  the  legs  (^  an  angle,  so 
many  degrees,  minutes,*  and  seconds,  that  angle  is  said  to  mea- 
sure ',  and,  note,  in  the  following  definitions,  whatever  is  affirm- 
ed of  an  arc,  is  likewise  affirmed  of  the  angle  (at  the  centre,) 
which  stands  on  that  arc. 

5.  Draw  any  straight  line  JC^  from  C  as  a  centre  With  the 
distance  CA,  describe  the  circle  JEN*  produce  AC  to  L,  and 
through  the  centre  Cdraw  £CK  perpendicular  to  AL;  in  the 
arc  EA  take  any  point  By  join  BA,  BE,  and  BCy  and  produce  th6 
latter  to  ^;  through  A  and  B  draw  AT^  BD  each  parallel  to 
CEi,  and  produce  them  to  S  and  G;  join  CG,  and  produce  it  to 
R  and  5,  produce  CB  to  T,  through  E  and  B  draw  REM,  MFB, 
each  parallel  to  CA,  and  join  J5L,  MN;  then  since  TA,  J^D  are 
both  parallel  to  EC,  they  are  parallel  to  one  another  (30. 1.),  and 
both  perpendicular  to  CA  (39. 1.) }  for  a  like  reason  EH  and  FjB 


*  An  easy  tract  on  Plane  Trigonometry  maj^lie  found  in  Lndtam's  Rudt- 
nenl*  of  MathemtUks,  Mr.  Bridge's  le<iHit«s  on  the  same  subject,  publisbad^ 
ia  1810,  is  likewise  a  neat  and  useful  work. 

B  b  2 


87d 


PLAKB  TEaGONOHBTRT. 


PikT IX, 


are  parallel,  and  both  perpendicular  to  EC,  and  BD^FC,  and 
FB  ^CD  (34. 1.) 

6.  Because  the  four 
right  angles  ACE,  ECU 
LCK,  KCA  are  sub- 
tended  by  the  whole 
circumference,  each  of 
these  angles  will  be  sub* 
tended  by  one  fourth 
part  of  the  wIk^  cir- 
cumference, which  is 
called  a  auADbAKT  j  the 
arc  ABE  is  therefore  a 
quadrant. 

7.  The  difiference  of 
any  arc  firom  a  quadrant, 
or  90^,  or  of  any  angle 
from  a  right  angle,  is 

called  THE  COMiaEMBNT 

of  that  arc  or  angle. 

Thusy  the  arc  BE  is  the  complement  of  the  arc  AB;  and  the 
angle  BCE  is  the  complement  of  the  angle  ACB  K 

8.  The  difference  of  any  arc  from  a  semicircle,  or  \S(P,  or  of 
any  angle  from  two  right  angles,  is  called  the  supplement  of 
that  arc  or  angle. 

Thus,  the  arc  BL  is  the  supplement  of  the  arc  AB,  and  the 
angle  BCL  of  the  angle  ACB  ^ 

0.  The  chord  of  an  arc  is  a  straight  line  drawn  from  one 
end  of  the  arc  to  the  other. 


y 


b  Id  li&e  manner  AB  is  the  complement  of  BE^  and  the  angle  ACB  of  the 
angle  BCE,  The  name  complemeni  likewise  applies  to  the  excels  of  an  dre 
Bboye  a  quAdrant,  or  of  an  angle  aborc  a  rfght  angle  ;  thus  EB  Is  the  cwkkp^ 
nent  of  the  arc  BML,  and  of  the  angle  BCL  ;  but  in  most  practical  qveitiotis 
it  is  usoally  restrained  to  what  an  arc  or  acute  an|^]e  wants  of  90«. 

«  The  arc  AB  is  likewise  the  supplement  of  the  arc  BML,  and  the  angle 
ACB  of  the  angle  BCL,  The  term  supplement  means  also  the  excess  of  air 
arc  abote  a  semicircle,  thus  the  arc  AB  is  the  supplement  of  the  arc  AMN., 
The  difference  of  aa  arc  from  the  whole  circumference  i»  tenned  it»  swfglc 
ment  to  a  circle. 


f  AKf  CL      DEFINITIONS  AN©  PBINCIPLES.  873 

Thu^f  %  straight  line  JB  U  the  ck9r4  of  the  wtc  AB,  or  of 
thfi  qngk  ACSl, 

C^.  The  chord  o(  9QP  Is  e^ual  to  the  raitius  (cor.  15. 4.) ; 
and  the  chord  of  180^  is  the  diameter. 

10.  Ths  co-chord  of  an  arc,  is  the  chord  of  the  complement 
4if  that  arc. 

Thus,  the  stra^ht  line  BE  (or  the  chord  of  the  arc  BE)  is 
iifl  co-chord  of  the  arc  AB,  or  of  the  angle  ACS. 

11.  Thb  supplemental  chord  of  an  arc,  is  the  chord  of  its 
supplement. 

Thus,  BL  {or  the  chord  of  the  arc  BML)  is  the  supplemeri-' 
tal  chord  of  the  arc  AB,  or  of  the  angle  ACB: 

Cor.  Hence  it  appears  tluit  the  diord  of  any  arc,  is  likewise 
the  chord  of  its  supplement  to  a  whole  circle  i  also  that  the 
chord  can  never  exceed  the  diameter  (15. 3.) 

Thus,  BL  is  not  onty  the  chord  of  the  art  BML,  but  also  <\f 
the  arc  BKL. 

12.  The  sine  of  an  are^  is  a  straight  line  drawn  from  one  end 
of  ^he  arc,  perpendicular  to  the  diameter  which  passes  through 
the  other  end  of  the  arc. 

Thus,  BD  is  the  sine  of  the  arc  AB,  and  of  the  angle  ACB* 
Cor,  Hence  the  sine  of  an  arc,  is  the  same  as  the  sine  of  it^ 
silpplementj  for  BD  is  not  only  the  sine  of  the  arc  AB,  but  also 
of  the  are  BML  ;  for  it  is  drawn  from  one  extremity  B,  (of  the 
arc  BML\)  perpendicular  to  the  diameter  AL,  passing  through 
the  other  extremity  L, 

13.  The  co-sine  of  an  arc,  la  that  part  of  the  diam^te^ 
(passing  through  the  beginning  of  the  arc,)  which  is  intercepted 
between  the  sine  and  the  centre^  and  is  equ?d  to  the  ji^e  of  th^ 
complement  of  that  arc. 

Thus,  CD  is  the  co-sine  of  the  arc  AB,  and  of  the  anglf 
ACB ;  and  it  is  equal  to  BF  (34. 1)  the  sine  of  BE,  which  is  the 
jcmplevf^t  of  AB. 

Cor.  Hence  the  sine  of  a  quadrant^  or  of  a  right  angle  Qa 
opt  qxdj  e^qual  to,  but)  is  the  radius  ^  and  the  co-sine  of  a  quadr 
r^i^t  or  riglit  angle  is  nothing. 

Thus,  if  the  pqint  B  be  supposed  to  move  to  E,  the  arc  AB 
^\ll  beofJim^  4Ej  the,  sine  of  which  is  EC;  and  thp  point  D  coin^ 
dding  with  C$  the  co^sine  CD  will  vanish, 

BbS 


W4 


PLANE  TRIGONOHBTRr. 


pAmr  IX. 


Hence  also  the  sine  or  co-sine  can  never  exceed  the  nuiias, 

14.  The  vbrsbd  siwb  of  an  arc,  is  that  part  of  the  diameter 

which  is  intercepted  between  the  beginning  of  the  arc  and  its 

sine. 

Thus,  DA  if  the  vprsed  sine  of  ifu  arc  AB,  and  of  the  angle 

ACB;  and  AP  is  the  versed  sine  of  the  arc  ABM,  and  of  the 

pngle  ACM. 

Cor,  Hence  the  versed  sine  of  an  ore  lets  than  a  quadrant^  is 
equ^lto  the  difference;  and  of  an  arc  ^eater  than  ^  qt^i^rant,  to 
the  sum  of  the  co-sine  and  radius. 

Thw,  4D  (the  versed  sine  ofAB)  ^CA—CD,  and  AP  {the 
versed  sine  of  ABJif)  rpCA+  CP. 

Hence  also  the  versed  sine  (being  alwajrs  within  the  qrcle,) 
can  nerer  e^^ceed  the  diameter,  (15.  S.) 

15.  The    co-versed 

91NB  <vf   an  arc,  is  the  ^ 

versed  sine  of  its  com- 
plement.    . 

Thus,  EP  is  the  co- 
versed  sine  of  the  arc  AB, 
and  of  the  angle  ACB. 

Cor,  Hence  the  co- 
versed  sine  is  equal  to  the 
excess  of  the  Radius,  above 
the  sine. 

16.  The  tangent  of 
an  arCf  is  a  straight  line 
at  right  angles  to  the  dia- 
meter, passing  through 
one  end  of  the  arc,  and 
meeting  a  diameter  pro.  ^ 
duced  through  the  other 

end  of  the  arc. 

Thus,  AT  is  the  tangent  of  the  arc  AB,  and  of  the  angle 
ACB, 

Cor,  Hence  a  tangent  may  be  of  any  magnitude  (according 
to  the  magnitude  of  its  arc)  from  nothing  to  infinity.  Hence 
also  the  tangent  of  45^  is  equal  to  the  radius  (6. 1.) 

17.  The  co-tangent  of  an  arc,  is  the^  tangent  of  the  coow 
plement  of  that  arc,     ' 

I  V.      4  (I 


PiBT  IX.     DEFINITIONS  AND  FRINCIFLE8.  378 

Thus,  EH  (the  tangent  of  EB)  U  the  co-tangent  of. the  arc 
AB,  and  of  the  angle  ACB. 

18.  Tub  secant  of  an  arc,  is  a  straight  liae  diawn  from  the 
centre,  through  the  end  of  the  arc,  and  produced  till  it  meet 
the  tangeivt. 

Thus,  €T  %8  the  secant  of  the  arc  AB,  and  of  the  angle  ACB. 
Cor.  Hence  a  secant  can  never  be  less  than  the  radius>  but  it 
increases  (as  4he  are  increases)  from  the  ra^^us  to  infinity. 

19.  Thb  co-^bbcant  of  an  arc  is  the  secant  of  its  complex 
ment. 

Thus,  CH  {the  secant  of  EB,)  is  the  co-secani  of  the  arc 

AB,  and  of  the  angle  ACB  ^ 

THE  VARIATIONS,  AND  ALGEBRAIC  SIGNS,  OF  THE 
TRIGONOMETRICAL  LINES  IN  THE  FOUR  QUAD- 
RANTS. 

SO.  If  the  sine,  co^ine^  tangent,  co-tangent,  secant,  co-secaiit, 
versed  sine,  and  co- versed  sine  for  every  aix  in  the  first  quadrant 
AE  be  drawn,  they  will  serve  for  the  three  remaining  quadrants 
EL,  LKy  KAt  that  is,  for  the  whole  circle,  as  will  be  shewn 
forther  on  -,  but  previous  to  this,  it  will  be  necessary  to  suppose 
the  point  B  to  coincide  with  A,  and  to  move  ^om  thence  roun4 
the  whole  circumference,  and  this  will  lead  us  to  explain  the 
manner  of  applying  the  algebraic  signs  tH  smd  —  to  the  Unas 
peculiar  to  Trigonometry. 

21.  When  the  point  B  coincides  with  A,  the  arc  AB  wil)  =a» 
and  the  points  D  and  T  wjU  coincide  with  A-,  wherefore 
AT=zo,  BI>sso,  DA=o,  CB  and  CD  each  s  radius ;  that  is, 
the  tang^at,  sine,  and  versed  sine,  (of  o  degfteea,  .or)  at  the  be- 
ginning of  the  quadrant  will  be  nothing,  and  the  secant  and  cck 
sine  will  be  radius. 

*  Some  of  the  trigonometrical  lines  reoeived  their  nunct  from^he  parts  of 
an  archer's  bowj  to  which  they  bear  a  similitade;  thns,  arc  oomea  froiti  arcus, 
>  bow}  CHORD  from  chorda^  ihe  string  of  a  bow;  saoitta  (now  generally 
called  the  versed  sine)  from  sagitta,  an  arrow ;  sine  from  sinus,  the  bosom, 
alliiding  to  that  part  of  the  chorda  or  string,  which  is  held  near  the  breast  in 
the  act  of  shooting,  the  sine  being  half  the  chord  of  double  the  are.  The 
prefix  CO  is  an  abbreviation  of  the  word  complemeni;  thns  co-sine,  eo^tamgent, 
ftc.  imply  con^lemeni  sine,  c^mplemeni  tangent^  &c.  or  the  sine,  tangent,  Sec. 
ef  the  coBplement  of  a  given  are, 

B  b  4 


5y« 


PLANE  TBIC30N0METRY. 


Pabt  is. 


32.  The  sine  BD  increases  (with  the  motioii  of  B)  from  o, 
during  the  first  quadrant  AE;  when  the  point  B  coincides  with 
E,  the  sine  BD  will  evidenfly  ciHndde  with  EC,  and  beeome 
radius  f  it  then  decreases  during  the  second  qoadcant,  at  tkeeod 
of  which^  (when  B  is  supposed  to  arrive  at  L,)  it  is  iigain  s9. 
Puring  the  progress  of  B^  through  the  third  quadrant  LK,  the 
sine  again  increases  from  o,  and  on  the  arrival  of  B  at  the  point 
K,  it  again  becomes  radius ;  after  which  it  graduafly  decreases 
through  the  fourth  quadrant  KJ,  at  the  end  of  whidi  (where 
the  arc  is  360  di^gree^j)  it  is  =;o,  after  which  it  again  increases 
as  before. 

23.  The  sines  are  con- 
sidered 9s  affirmative  or 
oegative  with  respect  to 
their  direction  from  the 
diameter  LA,  to  which 
they  are  referred  5  those 
on  one  side  that  diame- 
ter being  eonsidered  as 
affirmative/ those  on  the 
Other  side,  and  in  a  con-  ^ 
trary  directionj  will  be 
negative  5  fbr  instance, 
the  sipes  of  the  first  and 
second  quadrants  which 
are  on  one  side  the  dia« 
meter  being  reckoned 
-h,  those  of  the  third 
and  fourth  quadrants^ 
being  on  the  other  side  will  be  — . 

^4.  The  co-sine  at  the  beginning  of  the  first  quadrant  is 
radius,  and  decreases  wi^h  the  motion  of  the  point  B  through 
the  arc  AE  to  o ;  when  B  arrives  at  E,  D  coincides  with  C; 
that  is,  the  co-sine  of  a  quadjaixt  (or  90^)  is  =0.  It  afterwards 
increases  from  0  to.  th^  ^nd  Z-  of  the  secon4  quadrant,  where  it 
ifi  again  radius  j  i^  the  third*  it  co^i^tinually  clecreases^  a(  the  ^ 
(K)  of  which  it  is  again  nothing ;  (ifterwards,  during  the  fourth 
quadrant  KA,  it  again  increases,  at  the  end  of  which  (viz.  at 
the  point  A)  it  is  again  radius. 

^5,.  The  co-sines  originate  at  the  centre  C;  consequently  if 


» 

^ 

•] 

F 

• 
• 
• 
• 

• 

• 

/c 

\^ 

•[*  * 

• 

•• 

D 

N 


Pakt  IX.  ALGEBRAIC  &lGm.  37T 

those  in  the  direeticm  CA  be  considered  as  affirmfttive,  those  in 
the  opposite  direction  CL  will  be  negative.  The  co-sines  then  of 
the  first  and  fourth  quadrants  will  be  alike>  viz.  -f  3  those  of  the 
second  and  third  will  also  be  alike,  but  contrary  to  the  former> 
viz.—. 

26.  At  the  beginning  of  the  first  quadrant  (at  A)  the  tangent 
is  nothing;  from  o  it  increases  continually^  until  the  point  B 
coincides  with  E,  when  it  becomes  parallel  to  the  secant^  (which 
will  then  coincide  with  CE)  and  is  therefore  infinite.  When  the 
point  B  has  passed  £.  the  tangent  will  change  its  direction^  and 
(with  the  motion  of  B}  will  continually  decrease,  until  B  arrives 
at  L,  or  the  end  of  the  second  quadrant,  when  the  tangent  will 
ag^n  become  nothing}  from  0  it  changes  its  direction  to  AT, 
and  increases  until  B  arrives  at  K,  the  end  of  the  third  quadrant  ^ 
when-  it  is  again  infinite,  it  decreases  from  infinite  during  the 
fourth  quadrant,  at  the  end  of  which  it  is  again  nothing, 

9J.  The  tangent  originates  at  the  point  A ;  consequently,  if 
tlie  tangent  in  the  direction  of  ^ The  called  affirmative,  that  in 
the  direction  of  AS  will  be  negative )  but  we  have  shewn  that 
the  tangents  of  the  first  and  third  quadrants  are  in  the  direction 
of  AT 9  wherefore  they  are  both  + ;  whence  the  tangents  of  the 
second  and  fourth  quadrants  being  in  the  direction  of  AS  will, 
ibr  the  reason  given  above,  be  both  — . 

28.  The  secant  at  the  point  A  is  equal  to  radius,  and  it  in- 
creases (by  the  motion  of  B)  with  the  tangent^  and  with  it  be- 
comes infinite  at  £,  the  end  of  the  first  quadrant.  In  the 
second  quadrant  £L,  the  secant  changes  its  direction  from  CT 
to  CS,  and  decreases  from  infinity  to  radius ;  in  the  third  qua- 
drant LKj  it  increases  again  in  the  direction  CT,  from  radius  to 
infinity :  in  the  fourth  quadrant  KA,  the  secant  once  more 
change^  iU  directioa  to  CS,  a,pd  decreases  from  infinity  tp 
radius, 

29.  Theaeoaat  has  its  origin  at  the  centre  C  from  whence  its 
length  is  computed^  and  it  will  change  its  aiga  09  often  as  the 
revolving  radius  CB  passes  the  diameter  ^K;  having  the  same 
algebraic  sign  as  the  co-sine  5  whence  it  appears  that  the  secants 
of  the  first  and  fourth  quadrants  will  be  +^  those  of  the  second 
and  third  — . 

30.  The  changes' which  take  place  in  the  magnitudes  and 
directions  of  the  co-tangent  EH,  and  the  co-secant  CH,  may  be 


S7S 


PLANE  TRIGONOMETRY. 


Fakt  IX. 


explained  in  the  same  manner;  the  co-tangent  being  computed 
from  the  point  £,  will  change  its  direction^  and  consequently  its 
algebraic  sign  every  quadrant^  the  first  and  third  being  +>the 
second  and  fourth  will  be  — .  The  co-secant  at  the  point  A  is 
infinite^  at  the  point  £  it 

is  radius,  at  the  point  L  T 

it  is  infinite,  and  at  K  it 
is  again  radius.  In  the 
first  and  second  quad- 
rants its  sign  will  be 
+>  in  the  third  and 
fourth  —  ^  being  the  same 
as  the  sine. 

31.  The  versed  sine  at 
^  is  s=  Of  at  £  it  is  radius  ; 
at  L  it  is  the  diameter; 
at  K  it  has  decreased  to 
radiust  and  continues  it;s 
decrease  to  A,  where  it 
is  nothing.  This  line  being 
computed  from  Ay  will 
be  always  affirmative. 

39.  It  may  be  remarked,  in  general,  of  the  above  lines»  that  as 
oft  as  they  become  ir^nite  or  nothings  they  change  their  direction, 
and  consequently  change  their  algebraic  sign  3  these  changesi 
may  be  exhibited  in  one  point  of  view,  as  follows  * : 


<  It  is  Bometimes  necessary  in  analytical  oompatations  to  employ  am 
l^reater  than  the  whole  circumference,  which  ara  will  faU  in  the  5th,  eth,  7th, 
&c.  quadrant  (counting  the  quadrants  again  ronnd  the  circle)  ;  in  these  cases, 
the  proper  sign  of  the  arc  in  question  most  be  particilarly  attended  to;  it 
may  be  readily  found  from  the  above  table. 

Let  a  S3  any  arc,  its  sine,  tangent,  &e.  may  be  fonHd  in  tennt  of  the  rat 
from  the  foregoing  figure,  by  means  of  similar  triangles :  thus, 

r.cpsft      co-sec  a.  tan  a 


I.    Sine    of   a  s=  /y/r'^—coa'ass 
r.  tana  r* 


co-tan  a 


r 
r.  tana 


cosa.seca 


V'r'+tan'a       ^v^r*  +  co-tan  »  a 
tana,  co-tan  g     r,y8ec»o—r> 


co-sec  a 


sec  a 


CO- sec  a 


co-sec  a 


sec  a 


/ 


Part  iXp 


AL0JSBRA1C  SIGNS. 


S7» 


1st  2nd  3rd 

quad.  quad.  quad. 
Sine  and  co-secant     +          +  — 

Cp-sine  and  secant     -|-  —  — 

Tangent  and  co-tan.  +         —  + 

Versed  sine  +  +  + 


4th 

quad. 

+ 


9.  Co- sine  of  a«  V*"*— mo**  ~ 
r.  co-tana  r' 


r.  sin  a      sin  a.  co-tang^ 


tan  a  r 

r'       r.  co-tana      sin  a.  co-sec  a 


^r'  +  co-tan « a       ^^a  -f-tan'a 

tan  a.  co-tan  o      r^co-sec'a  — r* 
sec  a. 


1     sec  a       co-sec  a 


sec  a 


co-sec  a 


3.  Tangent  of  «=-;;j:j;;^- 
r>/r'»— cos^a     j^nWa-^r»  = 


r.  sin  a 


r^tona 


r.  SID  a 


cos  a    *sina.co-tan»a     >/r»-8in«a 
r.  sec  a  cos  a,  sec  a 


cosa 

sia  a.  co-sec  a 

CO- tan  a 


co-sec  a 


co-tan  a 


^co-sec^a— r« 

ra  r.  cos  a 

4.  Co-tangent  of  a» 


rs.sin  a 


r.  cos  a 


tan  a 


^r Vra—sin^  ^  ^co-sec^a  — ra  = 


sin  a  cos  a.  tan »  a       ^r*  —  cos«a 

r.  co-sec  a     cos  a,  sec  a 


sec  a 


tana 


sin  a 
sin  a.  co-sec  a 


>v/sec»a— r' 


»',^r»  4- co-tan 'tf 


co-tana 
sin  a.  CO*  sec  a 


'  sin  c.  co-tan  a 
r.  CO- sec  a 


r.  tan  a     eo-tan  a.  tan  a 
COS  a  sin  a  cos  a 

r.  co-sec  a      tana,  caseca^ 
co-tan  a  3* 


cosa 


^co-sec  a  a— r« 


6.  Cosecant  of  a=  V**'  +  co-tan*  « 


sin  a 


tana. CO- tang  __y \fr*  -f  tan«a ^'^ 

sin  a  tana  cos  a.  tan  a 


r.  co-tan  a  __ 
COS  a 

r.  seca     cos  a.  sec  a 


tan  a        sin.  a 


co>tan  a.  sec  a 


r.  sec  a 


»•  Vsecaa— r» 

And  since  the  versed  sine  of  a=r-cosa;  the  co-versed  sine  ^r-sin 
fi  the  Mpplemental  versed  sine  ^r  4- cos  a;  thechord  =  V^ar.r-cosa^- 
thc  co.chord  =  V2r.r-8ina;   and  the  supplemental  ehard  « 


dao 


PLANS  TBIOOMQHSTRY. 


FAmT]J^ 


At  the  hefftmifig  aa4  end  of  each  i|i|adrant^  the  values  of 
these  lines  will  he  as  follow : 


(fi         SCR 

18a> 

270P 

• 

Sine                 O     +  rad. 

•     • 

.     0 

7- rad. 

0 

Co-sine      +  rod.       O 

•5- rad. 

O 

rhrad. 

Tangent          0         inf. 

0 

inf. 

O 

Co-tangent    inf.        O 

inf. 

O 

wt/. 

Secant       •+•  rad.      inf. 

-^rad. 

inf. 

-hrod. 

Co-secant      inf  -f  rod. 

inf. 

-^rod, 

w/ 

Versed  sine     Q    +  rad.  • 

4-  diam. 

4- rad. 

O 

INTRODUCTORY  PROPOSITIONS. 

33.  The  sine^  co-sine, 
tangent,  and  secant  of 
any  arc^  are  re9pecti?ely 
equal  to  the  sine,  co-sine, 
tangent,  and  secant  of 
the  supplement  of  that 
arc. 

Let  the  arcs  AB 
and  AM  be  supplements 
of  each  other,  viz.  AB 
less  than  a  quadrant,  and 
AM  greater^  then  will 
the  sine  BD  of  the  arc 
AB,  be  equal  to  the  sine 
MP  of  the  are  AM,  and 
also  the  co-sine  CD  to 
the  co-sine  CP. 

For  since  AM-^-  AB 


^2r,r+coBa;  ^|tb^r  of  these  latter  may  be  fowad  in  terms  of  any  of  the 
above  by  proper  sabstitution,  regard  being  bad  in  every  case  to  the  c^iapgc  of 
signs,  when  the  arc  a  is  greater  than  a  quadrant.  From  these  expressions  iiar 
the  trigoDometrical  lines  belonging  to  a  single  wecy  others  may  be  derired 
which  are  applicable  to  a  great  variety  of  cases,  viz.  far  the  sums,  diffcMDces, 
multiples,  sub-multiples,  &c.  of  given  arcM  ;  but  the  pro^cution  of  this  bssIJb) 
part  of  Trigonometry  further  than  is  necessary  for  constructing  the  sin^,  tan- 
gepts,  &c.  would  require  piore  room  tha|i  c^  conveniently  be  spared ;  w^  must 
therefore  refer  the  inquisitive  student  for  the  gratification  of  his  wishes,  to  the 
writings  of  £u1er,  Cagnoli,  Vince,  Woodhouse,  BooDycgstle,  and  tome  othei^ 
who  have  treated  expressly  on  tte  tubject. 


PaitIX.    INTRODUCrrORY  PROPOSITIONS.  Ml 

^ISOPssAM+ML',  taking  AM  from  both,  the  arc  ABzsML, 
*.♦  the  angle  BCA^MCL  (27.3.)  j  also  tlie  angles  niPC,  BDC 
are  right  angles,  and  the  side  MCzszBC,  /  (26. 1.)  MP=tiD' 
and  CPzsCDi  that  is,  the  sine  and  co-sine  of  any  arc  or  angle, 
are  Respectively  equal  to  the  sine  and  co-sine  of  the  supplement 
of  that  arc  or  angle,  observing  that  the  sines  MP  and  BD  will 
be  both  -t-,  but  the  co-sines  will  have  different  signs,  viz.  CD 
will  be +,  and  CP  — 

Likewise  AS  the  tangent,  and  CS  the  secant  of  the  arc  AM 
are  respectively  equal  to  AT  the  tangent,  and  CT  the  secant  of 
the  arc  AB. 

For  the  angle  TCA^MCL  (as  shewn  above)r=^CS  (15. 1.), 
the  angles  at  A  right  angles,  and  the  side  CA  common,  *.*  (26. 1.) 
AS=zAT,  and  CS^CT;  that  is,  the  tangent  and  secant  of  any 
arc  or  angle,  are  respectively  equal  to  the  tangent  and  secant  of 
the  supplement  of  that  arc  or  angle. 

In  like  manner  the  sine*  co-sine,  tangent,  and  secant  of  an 
arc  terminating  in  the  third  qnadrant  LK,  will  be  thode  of  an 
arc  which  is  the  excess  of  the  proposed  arc  above  a  semicircle. 

Thus  the  sine  of  the  arc  AMN  is  PN=:PM  (3.3.)  =  BD, 
the  sine  of  the  arc  AB)  and  the  co-sine  PC^CD,  the  co-sin6 
of  AB;  only  this  ^e  and  co-sine  (PN  and  PC)  will  be  nega- 
tive. AT  will  likewise  be  the  tangent,  and  CT  the  secant  of  the 
arc  AMN»  (as  appears  from  Art.  16  and  18)  j  the  former  of 
whkh  will  be  -f,  and  the  latter  — . 

The  sine,  co-sine,  tangent,  and  secant  of  an  arc  terminating 
in  the  fourth  quadrant  KA  wiM  be  respectively  the  same  with 
those  of  an  arc  which  is  the  supplement  of  the  proposed  arc  to 
the  whole  chrcle. 

Thus  the  sine  of  the  arc  AMNG  is  GD,  which  is=:BX> 
(3.3.)  the  sine  of  the  arc  AB,  only  GD  is  negative  -,  the  co-sine 
CD  is  the  very  same  as  the  co-sine  of  the  arc  AB, 

AS  is  the  tangent  of  AMNG,  which  is  ^AT;  and  CS  the 
secant,  which  is  =Cr;  AS  will  be  — ,  CS+  -,  see  Art.  32. 

'I^h6  Versed  sine  AP  of  any  arc  AM,  terminating  in  the 
second  quadrant,  is  fequal  to  the  difference  of  the  versed  sine  of 
Its  supplement  and  the  diameter,  or  to  the  sum  of  the  co-sine 
and  radius. 

Thus,  (6ihce  Ato=:LP)  AP:=z{AL-^LP=z)  AL-AD^ 
PC-f-  CA.    ^The  versed  sine  of  any  arc,  terminating  in  the  third 


^8^ 


PLANB  TRIGOKOMETRY. 


pA%i  nf/ 


and  fourth  quadrants,  is  the  same  with  the  versed  sine  of  its 
supplement  to  the  whole  circle :  thus  AP  is  the  versed  sine  oP 
the  arc  AMN,  and  also  of  the  arc  NGA  ;  also  AD  is  the  versed 
iine  of  the  arc  AMNG,  and  likewise  of  AG  its  supplement  to 
the  whole  drde.  It  has  been  already  observed  that  all  the 
ver^  sines  are  affirmative  or  -f- . 

Thus  we  have  shewn  tliat  the  sine,  co^sine,  tangent,  and 
secant  of  any  arc  AB^  will  be  respectively  equal  in  magpaitude  to 
the  sine,  co-sine,  tangent,  and  secant  of  its  supplement  to  either 
a  semicircle,  or  to  a  whole  circle,  diffinring  only  in  the  algebrai<! 
signs;  and  therefore  if  the  sine,  co-sine,  tangent,  and  secant 
for  every  degree  and  minute  of  the  first  quadrant  be  computed, 
and  the  whole  arraoged  in  a  table,  this  table  wiH  serve  for  tte 
whole  circle. 
34.  The  sine  of  any 

nrc  is  equal  to  half  the 

chord  of  double  that  arc: 

and  conversely,  the  chord 

is  double  the  sine  of  half 

the  arc. 

Because  CA  cuts  B  G 

2k  right  angles  BD=DG 

(3.  3.)   V   BDzzzxBG; 

also  the  arc  B^=the  arc 

AG  (30.  3.)  •.•  the  arc 

BA:=i^  the  arc  BG;  that 

i$,  BD  the  sine  of  the 

arc  BA  is  half  the  chord 

BG  of   (the  arc  BAG, 

which  is)  double  the  arc 

BA.  Q  E.  D.    The  con- 

verse  is  sufficiently  evi- 
dent from  the  preceding  demonstration. 

Car.  Hence,  because  the  chord  of  60o=:the  radius  (Art.  9.  cor.) 

•/  the  sine  of  30°=  (4-  the  chord  of  60°=:)  ^  radius.    Hence  also 

the  co-sine  of  60o=  (sine  of  30°=)  ^  radius;  and  the  versed 

sine  of  60°=  (radius  —  co-sine  =)  4-  radius. 
35.  The  sine  or  co-sine  of  any  arc,  together  with  the  radius 

being  given,  we  may  thence  determine  the  rest  of  the  trigono* 

metrical  lines  belonging  to  that  arc,  as  follows  : 


l*ABt  JX       INTRODUCTORY  PROPOSITIONS.  3S3 

Rwt,  Let  CB  the  radius,  and  BD  the  sine  of  the  arc  BA, 
he  given,  to  find  the  cosine  CD;  then  (47. 1.)  CB)«=gDl'-h 

eS:«,  and  C5)V^*=CSl«,  •/  CD=  ^CB|«-B5I«  j  that  is, 

the  ohsine  of  an  arc  is  equal  to  the  square  root  of  the  difference 
of  the  squares  of  the  radius  and  sine. 

Secondly.  Let  CB  the  radius,  and  CD  the  co»sine  be  given,, 

to  find  BD  the  sinej  thus,  (as  ^bove)  -BDrav'CB)'— Cd)'; 
that  is,  the  sine  of  an  arc  is  equal  to  the  square  root  pf  the  differ- 
ence of  the  squares  of  the  radius  and  co-sitie. 

Thirdly.  Since -rfD=C^— CD,  and  ^1*=^C4-CP;  there- 
fore the  tersed  sine  of  any  arc  less  than  a  quadrant,  is  equal  to 
the  difference  of  the  radius  and  co^sine;  but  of  any  arc  greater 
than  a  quadrant,  it  is  equal  to  the  sum  of  the  radius  and  co-sine. 

Fourthly.   Because  -S?t«=55)»+"S5]«  (47.1.)  '.•  B^= 

V5B)*+D3)*5  that  is,  the  ehordof  any  arc  is  equal  to  the 
tquare  root  of  the  sum  of  the  squares  of  the  sine  and  versed 


tme,^. 


Fifthly.    Because    £5]«=  (gf)*-!-'^^  (47.  1.)  ;=  DC^+ 

CErc3«=:)5C|«+C£^irftD]*  •••  EB=  ^Dtl^+  CE^BDY ; 
that  is,  the  co^chord  of  an  arc  is  equal  to  the  square  root  of  the 
sum  of  the  squares  of  the  co^sine  and  the  excess  of  the  radius 
ffbotfe  the  sine. 

Sixthly.  Because  the  right  angled  triangles  BCD,  TCA^ 
fiCP,  and  HCE  have  the  acute  angle  TCA  which  is  common  to 
the  two  former,  equal  to  each  of  the  acute  angles  CPF,  CHE  in 
the  two  latter  (by '129. 1.)  5  these  four  triangles  are  equiangular 
(3S.  1.)^  and  have  the  sides  about  their  equal  angles  proportionals 
(4. 6.)  5  whence  we  have  the  following  analogies. 
if  CD        iDB'.iCA        :  AT         \  DB.CA 

*■  co'sine  :  sine  : :  radius  :  tangent  ^  '         ~*    CD    ' 
sine  X  radius     ^.-     ,.         ,.    sine 

W  TANGENT  = : =  (if  radlUS  =  1) ; —  g. 

co-sme  ^  co-sme 


'InHkemanner  it  is  shewn  that  XJ»=  (>v/5S]«  +^SZl  •  =)  ^BD)  a  +Z^^ 
<v>  The  tupplemental  chord  is  equal  to  the  square  root  of  the  sum  of  the  squares 
ff  the  sine  and  suppiemental  versed  sine. 

(  Hence  it  appears,  that  when  the  sine  and  co-sine  have  like  algebraic  signs, 
^«  tangenl  will  be  + ,  tmt  when  they  have  unlike  »igns,  the  tangent  wiU 


384  PLANE  TRIGOMCMBrRY.  PAftt  iS« 

o     f^^        '^^        ''<^         '^'^        1      -Clf^^^l^s: 

i  co-sine  :  radius  : :  radius  :  secant  ^    '  CD 

CB]^  radiuslft  1 

■—  ,  or  SECANT  as  ■  ■    '.      =g(if  r?id.=sl) : —  ^ 

CD  co-sine     ^  ^cosine 

cDBiCB      II  EC     :CH  1  . .  r£f==:£M?= 

^sine  t  radius  : :  radiua  :  co-secant-'  DB 

jl -,  ,  or  co-sECAMTza: — ,      "S=(if  rad.ssl) ' 

DB  sme 


sine 


^     cDB:DC       ::EC      i  EB  \  . .  £^-.^^£1^ 

^  sine  t  co-sine  : :  radius  :  co-tangent  ^   '  DB  * 

co-sine  X  radius     ,.^      ,      ^.co-sine. 

or  CO-TAHGEWT  a- — ■ : —       ■       it=(lf  rad.ssl)-^^- *. 

sine  sine 

r  r^  I  AC      ::CE       :  EH  \  ^_ 

'-  tangent  :  radius  : :  radius  :  co-tangent  ^   '        "" 
ACCE    ^^  radius]*     ,,^      .      ,^ 

-;;l  v>-  =-=-;-,  Or  CO-TAHTBtNT  = •.={if  rai.sal) 

TJ        TA  '  tangent    ^  ^ 

1       . 

tangent* 

\.     cTA         :TC      t:CE      :  CH  \      ^,,    TC.CE 

*-  tangent :  secant : :  radius  :  to-sfecant  ^  TA 

secant  x  radius     ,,_      _     ,  .    secant  , 

or  CO-SEC ANT=z ' — ' s(if  rad.si.)  — — ^ '. 

tangent  tangent 

Cor.  Hence  the  radius  is  a  mean  proportional  between  the 

co-sine  and  secant  5  between  the  sine  a&d  co-secant,  and  between 

the  tangent  and  co-tatigent. 

36.  The  secant  of  60°  is  equal  to  the  diameter. 

For  since  the  co -sine  of  60°=-i^  radius  (cor.  Art.  34*) =t 
CB,  if  this  value  be  substituted  for  CD  in  the  secoxid  analogy 

(given  above),  we  shall  have  Cr=:(— -L—)  _-=-_  =-3  CB; 
that  is^  the  secant  of  60^  is  equal  to  the  diameter.    Q.  E.  D. 


*»  Hence  the  secant  will  always  have  £he  same  algebraic  sign  with  the  co-sine. 

*  Hence  the  co-secant  will  bare  the  same  algebraic  sign  with  the  sine. 

k  Hence  the  co- tangent  will  be  +  when  the  sine  and  co.sine  have  ]ike«^« 
and  -*•  when  they  have  nnlike,  viz.  it  will  always  have  the  same  sign  as  tke 
tangent  (see  the  1st  analogy.) 

1  Hence,  when  the  tangent  and  secant  have  like  sign«,  tfcie  00  secant  will  be 
-f  >  bnt  when  they  have  nolike,  -«-. 


PaatIJL    INTKMDUCTOaY  PAOFOSITIONS.  985 

Cor,  Hence  the  tangent  of  60^=s'twice  the  sine;  for  since. 
CBiCTiiBD:  TA  (4. 6.  and  16. 5.)  and  Cr?=2  CB  •/  TA:st 
2  BD  (cor.  4. 5.) 

Z7'  From  what  has  been  ddiivered>  we  can  readily  determine 
the  arithmetical  values  of  the  chords  co-ehord^  supplemental 
chords  sine,  co-sine,  tangent,  co-tangent,  .secant,  -co-secant, 
versed  sine;,  co-versed  sine,  and  supplemental  versed  sine  of  the 
arcs  of  30°,  45®,  60°,  and  90°  to  any  given  raAus  ^  thus,  let  the 
radius  =1,  then 
Art  36.  secant  of  60° 

Art  19.  co-secant 


It  of  30°       I 
jineof  180°  j 


] 


^  .  „,  J,   .       ^,«^  ^-^thediameter=5:2.00Q000a 

Art,  31.  versed  sine 

Aft.  9.  cor.  chord  of  180° 

Art  9.  cor.  chord  of  60^ 

Art  10.  co-chord  of  30° 

Art\6,  cor,  tangent  of  45° 

Art  17.  co-tang,  of  45°       >'=the  i-adius  =     1.0000000. 

Art  13.  cor,  sine  of  90** 

Art  31.  versed  sine  of  90° 

Art  24.  co-tine  of  180° 

/-sine  of  30°  ^ 

-rfrf.34.  cor.  J  co-sine  of  60°  i       ,    « 

I  J    •        e  j^f'=^  i  ^^  radrus=:0.500000p. 

^versed  sine  of  60^  I 

Art  15.  cor.  co-versedsineof  30°^ 

Art  34.  cor. 

Art  13. 

Art.  35.  tangent  of  30°  1  .      sine  30°. 

-4re.  17.  co-tang,  of  60°  ->  co-s;ine  30° 

5 

;^^    -    ,  =g O.6773503. 

.8660254 

^r^  35.      versed  sine  of  30°     1  =rad.— co-sinp  30°= 
Art  15.      co-versed  sine  of  60°  J  1  -  .8660254==:       0.1339746. 
Art  35.      chord  of  30P  1  ^  ^sili^^of30°+;^^«of30° 

At,  10.      co-chord  of  60P         ^=:=  ^.25  +  . 0179492= 

0.5176380. 

ArtZ^.      secant  of  30°  x_     rad?]^     _        1        _ 

Art  19.      co-secatit  of  60°       /     co-sine  30°     .8660264. 

31.154700$, 

VOL.  11.  P  C 


■''•  r^^'°'}=^'-f=^-=''»- 


0.8660254. 


SM 


riuANS  TRfGOSOUKTRY. 


pAiT  1X» 


Att  84.  »i«»  of  450 

Art,  13.  cO^^hM  df  4&. 

Jrt.  35.  versed  sine  of  45*     1  =srad.—co*8in^3s  1-^.7071066 

^ri.  25.  convened  sine  of  45^ 


Jkt.  35.  Meant  of  45^ 

Art  19.  co-secant  of  45^ 

Art,  85.  cfiord  of  45* 

Art.  10.  co-chord  of  48*' 

Art.  35.  tangent  of  (5d^ 

^r^  17.  co-tangent  of  30^ 


}==:t*^<»a  erf  SK>«=iv^.fai:i« 

-^  =s:*t  V^«= O.7d710^a 

}  . 


.a^928982« 


eo-tiwt 


.7©71068 

1.4'14S18tf 


\  =  y/sinel^  4-  V-  sine^^s=^ 


0.7653668' 


}8inc 
=rad.  X— -r 
oo-si 


sine  of  §0^ 


ooHMneof  60^ 


.8660954 
.5 


1.7320508. 


In  like  manner  (Art.  35.)  tlie  chor4  of  the  suippleoieni  of    . 
©O^..  90^  ^1.414^135^ 

^^1       ,.     A    .ri20ol  .■-^,. =^;=^     rW66366S 

450  h^h-^^1  «f^  1350  H  ^slS;i^+«"P^^^^;^^=i  1.8477591 


300 


1500- 


1.93l851j^ 


38.  The  sine,  co-sine,  tangent,  seoaiH^  &c*  of  any  ore  AB  of 
a  circle,  vrha^  radius  is  Crf,  is  to  the  sine,  co-sine,  tangent, 
secant,  &c.  of  a  similar  arc  DE,  whose  radius  is  CD,  as  €A  t^ 
CD. 

From  the  point  B  let  fall  JBF  perpendicular  to  CD  (12.  l.)| 
and  through  A,  £,  and  D, 
draw  AK,  EG,  and  DT,  paral- 
lel to  BF  (31.  1.),  then  will 
BFhe  the  sine  of  the  arc  BA, 
CF  its  co-sine,  AK  its  tangent ; 
EQ  the  sine  of  ED,  CG  its 
co-sine,  and  DT  its  tangent 
(Art.  12.  16'.) ;  and  since  AB 
and  DE  each  subtend  the  com- 
mon angle  at  the  centre  C,  they  are  similar,  that  is,  they  contain 
each  the  same  number  of  degrees  (part  8.  Art.  239.)  3  now  siocc 
the  angles  at  F,  A,  G,  and  D,  are  right  angles,  and  the  angle  at 
CcommoD,  the  triangles  BCF,  KCA,  ECG,  and  TCD,  are  similar 


fa^tix  iNrR«Knn«Y  pi»posith)ns.         am 

<32.  l.)>  and  liave  the  sides  about  tbeir eqiiftl  aQglel  proportioaab 
(4. 6.)  J  that  is. 

First,  FB  :  BC::  Gf  :  £C,  and  ^teroatdy  (16.6.)  FB  : 
€E  :iBC:  ECi  that  is,  «ne  of  arc  BA  i  sine  of  arc  ED  : ; 
rod.  (BC)  of  the  former  arc  :  rad.  (EC)  of  the  latter. 

Secondly,  FC:  CB::  GC:  CE,  and  alternately  FC  i  GC:: 
CB  :  C£;  that  is,  cosine  of  arc  BA  :  o^sine  of  arc  ED  : :  rad. 
(CB)  of  the  former  :  rad.  (CE)  of  the  latter. 

Thirdly,  KA  :  AC : :  TD  :  DC,  and  alternately  KA:TD:: 
AC :  DC;  that  is,  tang,  arc  BA  :  tamg^.  arc  ED  : :  rod.  of  BA  : 
rad.  of  ED. 

Fourthly,  KC  zCAz:  TC :  CD  •/  altenirt^  ^C.TCix 
CA  :  QDi  that  is,  aeoant  of  arc  BA  :  secant  arc  fiD  .: ;  rod,  o^ 
fiA  :  rad.  of  JSJJ. 

Fifthly,  Because  BC :  CF::  EC-,  CG  /  fcy  conversion 
(prop.  B.5.)  BC'.FAiiECt  GD,  /  inversely  (prop.  B.5.), 
fA  I  (BC^)  AC ::  GD:  (ECz?:)  DC.'  alternately  FA  :  GD  : : 
AC :  DC;  that  is,  verud  sine  of  arc  BA  :  versed  sine  of  are 
ED : :  rad.  of  BA  ;  rod.  of  ED.  Wherefore  the  sines,  co-sine^ 
taogents,  socants,  and  versed  sines  of  ^  given  angle  in  different 
circles,  are  respectively  as  the  radii  of  those  drdes.    Q.  E.  D. 

Hence,  if  sines,  co^siaes,  tangents,  &c.  be  computed  tq 
a  given  radius,  thej  may  be  Ibiind  to  any  other  radius,  by  th^ 
above  proportions. 

S9.  The  co-sine  of  any  arc,  is  equal  to  half  the  chord  of  the 
Supplement  of  double  that  are. 

Let  AE  be  an  arc,  C  the  leentre,  join  CE,  and  from  4  <^^ 
4L  perpendicular  to  CE  (19. 1.),  and  produce  it  to  S,  join  BD, 
^nd  froai  the  centre  Cdraw  CJf  peipendicular  to  BD,  '.*  DF=9 


CO? 


88S 


HiANE  TRIGONOMSTRY. 


Pa&t  IX. 


FB  (S,S.)',  afsoCX  is  the 
co-sine  of  AE  (Art.  13.) » 
BD  the  suppleaiental  chord 
of  (AEB^)  double  of  AE 
(Art  11.),  and  FB=balf  the 
said  supplemental  chord. 

Because  DBA  is  a 
right  angle  (13. 3;),  and 
VLB,  CFB  right  angles  (by 
construction),  •.*  FB  is  pji- 
rallel  to  CL,  and  BL  to  PC 
(58. 1.),  •/  FBLC  is  a  paral- 
lelogram, and  CL:=xFB  (34.  l.)5  that  i8»  the  cosine  of  them 
AE  is  equal  to  half  the  supplemental  choid  of  (^H)  double  of 
AE.    Q.  E.  D. 

40.  The  chord  of  an  arc  Is  a  mean  prdporlional  between  its 
Tersed  sine  and  the  diameter. 

Draw  BK  at  right  angles  to  DA  (12  1 ),  then  because 
DBA  is  a  right  angle  (31.3.),  DA:  AB::  AB:  AK  (cor.  8.6.); 
that  is,  the  diameter  is  to  the  chord  of  the  arc  AEB,  as  the 
same  chord  is  to  the  versed  sine  of  AEB.    Q.  E.  D. 

41.  The  sum  of  the  tangent  and  secant  of  any  arc,  is  equal 
to  the  co-tangent  of  half  the  complement  of  that  arc. 

Draw  CH  at  right  angles  to  DA  (12.  1.),  and  let  AEhe 
any  arc,  AS  its  tangent,  CS  its  secant,  and  the  arc  EH  its  com- 
plement. Bisect  EH  in  B  (30.  3.),  and  di-aw  CBT  meeting  AS 
produced  in  T. 

Then  AT  is  the  tangent  of  the  arc  AEB  (Art.  16.)  that  is, 
the  CO' tangent  of  HB  (Art.  17.)  which  is  half  the  conipleuient 
of  AE. 

Because  AT  and  CH  are  parallel,  the  angle  HCB=CT4 
(^9.1);  but  HCB=zBCE  \'  BCE^CTA  \-  5C=6T  (6.  1) 
AS+SC=AT ;  that  is,  the  sum  of  the  tangent  and  secapt  of 
the  arc  AE\s  equal  to  (AT)  the  co-tangent  of  (HB)  lialf  the 
complement  of  AE.    Q.  E.  D. 

42.  The  radius  is  to  the  co-sine  of  an  arc,  as  twice  the  sine  to 
the  sine  of  double  that  arc. 

Because  the  right  angled  triangles  ALC,  AKB  have  ihe 
apgle  at   A  common,   they  are   equiangular   (32.  l.)> '•*  ^P- 


Partjx,     investigation  of  formula. 


S89 


CL  : :  AB  :  BK,  that  is  radius  :  co-sine  of  JE  : :  twice  the  sine 
f^AE  :  sine  of  double  of  JE.    Q.  E.  D. 


THE  INVESTIGATION  OF  FORMULA,  NECESSARY 
FOR  THE  CONSTRUCTION  OF  THE  TRIGONOME- 
TRICAL CANON. 

4S.  The  sines  and  co-sines  of  two  unequal  arcs  being  given 
to  determine  the  sine  and  co-sine  of  their  sum  and  difference. 

Let  KFy  FE  be  two  unequal  arcs  of  which  the  sines  and 
co-sines  are  given^  and  let  KF  be  the  greater^  from  which  cut 
M  FD=:FE  the  less  (34.«.)*  Jo»n  ED,  and  from  the  centre  C 
draw  CF  perpendicular  to  ED  (12. 1.)  '.•  EL^ID  (3. 3.) ;  draw 
DHt  FG,  LO,  EM,  each  perpendicular  to  the  diameter  €K,  and 
DS,  LN  each  parallel  to  it  (31. 1.)  meeting  LO,  EM  in  the 
points  S  and  N. 

Because  EL^zLD  EF=iFD,  •/  (30.3)5  and  because  LN 
is  parallel  to  DS,  the  angle  ELN=^LDS  (29. 1.),  •••  the  right 
angled  triangles  ELN,  LDS  having  all  their  angles  equal,  and 
the  homologous  sides  EL,  LD  equal,  are  equal  and  similar 
(26. 1,  and  def.  1.6.),  •.*  EN=lS  and  NL=:SD;  also  in  the 
parallelograms  NMOL,  SOHD,  we  have  NM=zLO,  NL=zMO, 
DH=SO,  and  SD=^OH  (34. 1.).  /  NL=zMOz=SD=iOH.  Let 
the  arc  KF=A,  the  arc  FEz=B,  and  the  radius  CF=:R',  then 
will  the  arc  /CJE=(^F+FE=)  A-j-B,  and  the  arc  KDs^{KF^ 


K  M. 


OG 


M  O  G        H C 

FD^KF^FE:s:)A^B; 

sdso  FG  is  the  sine  1  ^^  ^     and  EM  is  the  sine  *>    «  ^    -p 
CG  , . .  co-sine  /  CM . . .  co-sine  J 

EL  .« .  sine      -1    «  «  DH,^, ,  sine 

Cli 
c  c  3 


CH . . .  co-sme  } 


SM  PLANE  tBiGOKOSfBTRT.  pAAt^IX. 

BecaiM  NL  is  pandM  ta  CO,  and  JPQ  to  LO  md  the 

angles  at  6>  O,  and  N  rigbt  angles,  tke  triangles  CFG,  CLO, 

and  ELN  are  equiangular  (29  and  32. 1.),  consequently  (4. 6.) 

^.^    w,^      ^»     »^       »^      PO.CL    .    anil,  cos  ^ 
CF:  FGi.CLi  LO,  •.•  10=  (  =r) 

CFi  CG  : :  EL  :  E-N,  •.*  £2V=( — — ;— =) 


CF        '  R 


CF       '  R 

CFi  FG::EL:  LN,  •/  X^=-(— >-— -  =±) 

CK  R 


But  Eitf  (=itfi?+  EN^LO-^EN),  or  sin  ^+ J5=: 
sin  utf.  cos  JJ-fcos  ^.  sin  if 


CM  {=zCO'-MO=:CO'-LN),  or  eos  ^-ir£: 
toA  J.  COS  jB— sin  ^.  sin  £ 

B 


Dff  (:=:SO=:L0^L8^L0^E^),  or  sia ^--5: 
sin  u^.  cos  B— cos  A.  sin^  £    • 


CH (z=CO+OH^CO+LN)y  or  cos  ^-.JRar 
eos  A.  cos  5+sin  -rf.  sin  S 

^ ■  ^ 

44.  These  formula  for  the  sines  aod  co>sines  of  the  arcs 


A-JtB  which  are,  it  is  plain,  adapted  to  any  radius  B>  may  be 
simplified  and  rendered  more  convenient  ^putting  B=l  >  they 
will  then  become 
Formula   1.  Sin  ^+if=8in  A,  cos  j8+cob  A.  sin  B. 
%,  Cos  ^+ J3=cos  A.  cos  J?-^6in  A.  sin  B. 

3.  Sin  ^— ^sssiin  A,  cos  J3«-cos  A,  sin  J?. 

4.  Cos  A'^B^CQ»  A.  eos  ^-f-sin  A.  sin  5. 

45.  To  find  the  sine  and  cocaine  of  multiple  arcs,  that  is^  if  i^ 
foe  any  arc^  to  find  the  sine  and  co-sine  of  nA. 

Add  the^r^^  and  third  of  the  aboye  formul^B  togetJ^er«  and 
in  the  sum  let  ^  be  substituted  for  B,  and  B  for  A,  and  w^ 

shall  have  

sin  £4--^+sin  B'-Asz^  cos  A.  sin  B,  that  is^  • 

sin  £ -I- -4=2  cos  A.  aia  J3— sin  B— ^.  {Y), 


FiHT  IX.      INyESTIOATIOW  OF  tOBMVhM.  Ml 

Add  the  second  wadfwirlh  together^  and  substitute  S  for  A, 
and  ^for  B  as  before  :  then, 
cos  jg 4-^4. cos  B— ^5s2  cos  J.  cos  g;  that  i?, 
cos  ^-f^=2co8  ^:  cos  5- cos  JB— -rf  (Z) 

Let  n--i.A=B;  this  value  beings  substituted  for  B  in  the 
expressions  Y  and  Z,  we  have  the  two  following  theorems  for 
the  sines  and  co-sines  of  multiple  arcs,  viz. 
Theor,  1.  Sin  nif=2  cos  J.  sin  n-rl  -<^— sin  n— 2^. 

2.   Cos  n//2=2   cos  ^.  cos  n— 1  -^— cos  n— 2-^. 

Ifi  which  general  theorems,  if  n  be  expounded  by  1, 2,  3, 
^»  ^,  &c.  we  have  the  formulae  for  all  particular  mukiple  orcif 
viz.  if 

„--2.  /  *•  ^"^    3if  =c2  cos  A.  sin  ^  (from  theor.  1.) 

*  ^  6.  Cos  2A  =2  cos  A.  cos  /^— cos  0  (=  1)  (theor.  2.) 
fi=r3  /   '^'  ^^^  SJ=:2  cos  J,  Bin  2-<#— sin  J  (theor.  1.) 

'  ^   8.  Cos  S.<f  =2  cos  A.  cos  2^— cos  A  (theor.  2.) 
„_^  r   9.  Sin  4-^=2  cos  A,  sin  3-4— sin  2-4  (theor.  1.) 

^10.  Cos  44  =:2  <;o8  A.  cos  3-4— cos  2-4  (theor.  2.) 
„_5  f  1 1.  Sin   5-4—2  cos  A.  sin  4^— sin  3-4  (theor.  1.) 

*  1 12.  Cos  5^ =2  cos  -4.  cos  4-4— cos  3 A  (theor.  2.) 
&£.  &c.  &c. 

46.  These  formulae  may  be  continued  to  any  length,  and  by 
means  of  them*  the  sine  and  co-sine  of  evety  degree  and  minute 
of  the  quadrant,  may  be  computed,  as  will  be  shewn ;  but,  hav" 
log  found  the  sines  and  co-sines  to  the  end  of  the  first  30  de« 
grees  by  this  method,  those  from  30^  to  60^  may  be  obtained 
by  an  easier  process,  by  means  of  the  following  formula. 

Add  formula;  1  ami  3  (Art.  44.)  together,  and  sine  A^6 
+sin  -4—5=2  sin -4.  cos  JJ;  let-45=30P,  then  will -sin.  -4^2^ 
(cor.  Art.  34)  -,  substitute  these  values  of  A  and  sin.  A  in  thd 
above  expression,  and  it  will  become 
•in  300+JB-l-sin  30— Jg— (2xtXCOS  B^)  cosJg; 
•••  Formula  13.  sin  3G-iri3=co6  JB— sin  30— A 

47.  The  tangents  of  two  unequal  arcs  A  and  B  being  given, 
to  find  the  tangents  and  co-tangents  of  their  sum  and  difference^ 

It  has  been  shewn  (Art.  3^.)y  that  when  radius  =1,  the 

sine 

tangent  of  any  arc  =: r— :   wherefore,   bv  substituting  for 

°  ^  co-^me  '  ° 

c  c  4 


SK  PLANE  TRIGONOMETRY.  Part  IX 

the  sine  and  eo^ine  their  respective  values  as  given  in  the  for- 
mulae. Art.  44.  we  shall  have 


Formula  14.  Tan  ^+5=^--^±^= 

cos  A-jrB 

sin  A,  cos  jB+cos  A.  sin  B 


cos  J.  cos  B^&in  A,  sin  B  ' 


F.lS.Tan^— ^= 


sin  A-^B       sin  A.  cos  B—eosA.  sin  5 


cos  A-^B      ^^  -^-  c*^^  -B-f  sin  A,  sin  5' 

If  both  terms  of  the  right  hand  fractions  be  divided  by 
€ot  A,  cos  By  they  wiU  become 

sin  A       sin  B 

4 


F.  16.  Tan  -i+jB= 


cos  A       cos  J?        tacn  ^+tan  B 


sin  -4.  sin  B      1— tan  ^.  tan  B 

1— •  ^'  ■       ■  •  ■ 

cos  -4.  cos  B  (Art.  35.) 

sin  ^.       sin  B 


F,  17.  Tan  A^B^ 


cos  ^.       cos  JB         tan  -^— tan  B 


sin  ^.  sin  B  ~  l-|-tari-4.  tan  B 
l-f  ■  — 

cos  ^.  COS  J?  (Art.  35^.) 

■ 

.^      cos  A-^-B  , .^  ^^  ^      1— tan-4.tan^ 

F  18.  Cotan^+B=-;r*5^  (Art.35.>=  ^^^^^^.  . 

cos  -4— JS      1 H- tan  A.  tan  B 


F.  19.  Cotan^— J3=-T 


sm 


A—B       t^^  -^— tan  JB 


48.  To  find  the  tangents  and  co-tangents  of  multiple  arcs; 
that  is,  if  A  be  any  arct  to  find  the  tangent  and  co-tangent 
of  nA. 

.        tan  -<^-f-tan  B       ^ 

Since  tan  ^+B^  ^_,^j^^  j^^  (Art.  47.)   First,  let 

B=Aj  then 
F.  20.  Taii  2^=  (tan  2TB=)     ^  *^^  ^ 


1— tan2]« 


F.  21.  Co-tan  2^=  ( —-  Art.  35.=) 


L_Arf    «f.^Nl-^!^*_         1 


tmm^m^> 


tan  2^  *    '^    2  tan  ^        2  tan  i< 


tan  ^1* 

c^^P^-  (Art.  35.  analogy  5.)  4.  co-tan  ^-4.  tan  A. 

Secondly.  Let  J9=s2-dr,  then  will 


Pabt  IX.     INVESTIGATION  OF  R>RMUL.£.  89S 

^     2  tan  -4 
tan  A+ 


«^     «         .       tan -^+ tan  2^  1— tan^ 

f.  28.  Tan  3^=    utu^-rtau^^ 


1— tan -rf.  tan 2-4     ,_2tanr^* 


1— tan  ^1« 
3  tan  -rf— taiT^l^ 

1—3  taiH?)* 
P.  23.  Co-tan  3^=  (— Vr  Art.  35.=)     ^  ""^  ^^^  ^  * 


tan  3^  3  tan  ^-tan  A]^ 

In  like  manner,  

1— 6Tan  A*+Uin~3\* 
F.  25.  Co-taa  4^=±ll^^-!+^3l. 


4  tan -4-4  tan  ^1* 
&c.  &c. 

49.  These  formulae  may  be  extended  to  every  minute  of  the 
quadrant  j  but  although  it  seemed  necessary  to  shew  how  the 
tangents  and  co-tangents  of  multiple  arcs  are  expressed  in  tcrms^ 
of  the  tangents  of  the  component  arcs  themselves,  yet  we  have 
shewn  how  to  compute  the  tangents  and  co-tangents  for  the 
first  45°  by  means  of  the  sines  and  co-sines,  which  is  in  many 
respects  preferable  to  the  above  method.  The  tangents  and 
co-tangents  of  arcs  above  45°,  may  be  found  by  a  very  easy 
process,  the  formula  for  which  is  deduced  as  follows : 
It  appears  from  formulae  16  and  17>  Art.  47.  that 

Tan  A+B  -== J «  >  let  ^=45°,  then  (Art.  16.  cor.) 

—     l+tan-<f.tan  B  '  ^ 

tan.  As=l, 

•— —      14- tan  B  

Uence^  tan  45°+J?=-r^^ =,  and  Un  45«--B= 

l—tan  B 

l--tattB 
l+tanB' 

Subtract  the  latter  from  the  former,  and 

Tan  45^TI-tan  i^B=I±^_J-±:ii= 
1— tanB      1-ftanB 

i+tan  JBt«-l  -tan  B)«        4  tan  fi         ^  ,   .       ,      ^  ^ 

■ : = — ==r— ;  but  smce  tan  2JB= 

1— tiOl*  l-taniil« 

2.  tanB      ^        ,  ,  ,  ^^      4.  tanB 

—         ■  -  (formula  20.  Art.  4S) ;  •/  2  tan  2B= — =r , 


SM  J^LANB  TKIGOKOMETRT*  Pavt  tX. 

for  thif  fraction  substitute  iU  equal  (2  tan  9B)  in  the  lastegoa- 
tion  but  one,  and  we  shall  have  tan  45"+B— tan  45®— 5= 

5  tan.  2B;  hence  arises  

Formula  26.  Tan  45M^=tan  45'*— B+2  tan  2B  -. 

THE    METHOD   OF   CONSTRUCTING   A   TABLE   O? 
SINES,  TANGENTS,  SECANTS,  AND  VERSED  SINES. 

50.  In  the  preceding  articles  the  methods  of  deriviqg  ex- 
pressions for  the  sines,  co-sines,  tangents,  &c.  of  the  sum, 
difference,  and  multiples  of  arcs  in  terms  of  the  sines,  co-sines, 
&c.  of  the  arcs  themselves,  have  been  shewn ;  but  before  we  can 
employ  these  formulae  in  the  actual  eoBsthiction  of  the  trigono- 
metrical canon,  in  which  the  numerical  values  of  the  sine,  tan- 
gent, &c.  of  arcs  for  every  minute  of  the  quadrant  are  usually 
exhibited,  it  will  be  necessary  to  compute  the  sine  and  co-sine 
of  1  minute,  and  from  these  we  shall  be  able^  by  means  of  what 
has  already  been  proved,  to  determine  not  only  the  numerical 
values  of  the  rest  of  the  sines  and  co-sines^  but  likewise  those 
'of  the  tangents,  co-tangents,  secants,  co-secants,  versed  sines^ 
and  CO- versed  sines,  which  constitute  the  entire  canon. 

51.  To  find  the  sine  and  cosine  of  an  arc  of  1',  the  radm 
being  unity. 

It  has  been  shewn   (part  8.  p.  231,  232.)   that   if  the 

Iradius  of  a  circle  be  unity,  the  semi-circumference  will  be 

3.1415926535898  nearly  -,  this  semi-circumference  consists  of  ISO 

degrees,  each  degree  being  60  minutes  j  that  is,  of  (180x60=) 

3.1415926535898 

10800  minutes ;   •.• -— ar. 0008906882086= the 

10800 

length  of  an  ate  of  1',  the  radius  being  unity. 

But  in  a  very  small  arc,  as  that  of  V,  the  sine  coincides 

indefinitely  near  with  the  arc  ",*  wherefore  the  above  nombei 

■^  The  trigonometrical  formuls,  iatroducecl  iato  this  work,  4re  those  odIj 
Which  are  necessary  for  the  construction  of  a  table  of  sines,  tangents,  &c. 
Several  of  tb«  French  and  G^nuaa  matbepiaticiaos  hare  excelled  in  this  spedcs 
of  investigation,  and  produced  a  great  variety  of  theorems  suited  to  eveiy 
ease  in  Trigonometry.  The  English  reader  will  find  a  collection  of  fonnul8, 
applicable  to  the  most  delicate  investigations  in  Mechanics,  Astronomy, 
&c.  in  Mr.  Boqinycflstle's  Treatise  on  Plane  and  Spherical  Trigonometrift 
London,  1806. 

B  In  SfaBptoqft  Doctrine  and  application  iff  Fluxions,  part  3.  p.  SOl*  io^ 


Fa*t1X.        CONStRUCnON  OP  S1N£S,  &c.  99B 

XHM90B$SI%,  &e.  may  be  tftken  Ibr.tke  length  of  the  sioc  of  l^ 

Wherefore  also  (Art.  S5.)  the  co-sine  of  V^^l-^sin  1')*= 
( V-9^999991538405,  &c.=)  .99999996. 

52.  Construetum  nf  the  mneg  and  ea-tmtsfram  O  U  S€P. 
Since  (Art.  51.)  the  sine  of  r» (.0008906888086,  &c.s> 
X)0029O9,  whieh  is  its  nearest  Tslue  to  seven  places  of  decimals, 
and  co-sine  of  l'=s  .99999996.  Let  ^=an  arc  of  1',  tlten  the 
above  numeral  values  being  substituted  respectively  for  sine  and 
co-sine  of  1'  in  formula  5.  Art.  45.  we  shall  have 

By  Fmnula  6.  sip  ^'=2  cos   1'.   sin  V =2  x  .99^9996  x 
.00O29O9=.OOO581S,  here  the  sine  cf  3'  is  found 

F.  e.  Cos  2'=2  cos  Kcos  r -*1  ^2 X. 99999996 X. 99999996 
-.1=:  ,9999998^  here  the  co^sine  of  2'  isfdund- 

F.  7.    Sin  3'=2   cos    T.  sin  2'— sin   1' =?  2  X  .99999996  x 
.0005818— .0002909=:  .0008727*  here  the  sine  of  3'  is  found. 

F.  8.   Cos  3'=2  cos    1'.  cos  2"— cos   1' =  2  x  .99999996  x 
.9999998— .99999996=.  9999996,  here  the  eo^ineofS'  is  found. 

F  9.  Sin  4=2  cos  1'.  sin  3'-sin  2'= 2  x  .99999996  x 
.0008727— . 0005818=  .001 1 636 . 

F.  \0.  Cos  4'=9*C08  V.  cos  3'— cos  2's2  x  .99999996  x 
.9999996— .9999998= .9999993. 

F.  11.  Sin  5'=«  cos  1'.  sin  4'— sin  3'=  .0014544. 

F.  12.  Cos  5'=2  cos  1'.  cos  4'— cos  3'=.99999«9.  And  m 
this  manner  proceed  to  find  the  sine  and  co*sine  of  every 
nunute  as  fiir  as  30**. 

52.  B.  To  find  the  sims  and  co^mes  from  30*  iQ  60* 


By  formula  13.  Art.  46.  sin  30°-|-5=coe  B— sin  30—^. 


i  "f 


in  Wrkce\  Pluxitnu,  p.  ««0.  »  w  shewn  tbat  (radiat  Uin^  1,)  the  siae  of  aojT 

.b00290S88i086)^     .0008908889086)^     .0002908893086)^ 

.oowms«208ir-.-r-— -^ + jXi T^:^ 

-I-  he.  a:.oeD3906Sfil676»&«.  «tb9  WM  f4  I'y  wbifih  4if«r8  from  ti»«  al>oT« 
tjcpvctfiioo  for  tiie  length  of  thf»  cirv  9f  l'  hy  w\f  .(^000.009141  i  tM  !«« 
th»  ar^  of  1'  ABd  its  tine,  ooiwtide  W  »  dmrn^l  plMSM  indwsiv^f  thMreisr^  thft 
liiieof  I'to^plaMsof  d«c»Mi9  (Hw  wvidNr  t<»  vbtf^ln  the  t»h)««  w  wwOly 
coKput«4)  tXMit^  coiMidit  wA  its  IW^ 


396  PLANE  TRIGONOMETRY.  Part  IX. 

°  Let  J5=sl,  then    sin   30^    1'=C08    I'-sin   29^  59'= 
.99999990 -.4997481  =  .5002519. 
jB=2'  .Sin  30''  2'=cos  2— sin  29«  68' c=. 9999998— .4994961 

==.5005037. 

J?=3'  .Sin  80«»  3'=  cos  3'— sin  29°  57'=:  .5007556. 

5=4'  .Sin  30'>  4'=:cos  4— sin  29°  56=  .6010073. 

5=5'  .Sin  30°  5  =cos  5'— sin  29°  55'=.501259l. 

&c.#  &c.  .  .&c. 

53.  Having  computed  the  sines  in  this  manner  as  fas  as  60"} 
the  co-sines  from  30^  to  60"  will  likewise  be  known  3  the  co-sine 
of  any  arc  above  30*^  being  the  same  as  the  sine  of  an  arc  as 
much  fceZoti;  60**. 

Thus,  cos  30**  l'=sin  59^'  59'=.8658799. 
cos  30°  2'=-sin  59°  58'=.S657S44. 
cos  30°  3'=sin  59°  57'=. 8655887- 
cos  30°  4'=:8in  59°  56'=  .8654430. 

&c.  &c.  &c. 

COS  60°    =sin  30°       =.5000000. 

54.  To  find  ^e  sines  and  co- sines  from  60°  to  90^. 

The  sine  of  any  arc  above  60^  is  the  same  as  the  co-sine  of 
an  arc  at  the  same  distance  below  30°  -,  and  in  like  manner,  the 
co-sine  of  an  arc  above  60°  is  the  same  as  the  sine  of  an  arc 
equally  below  30° :  thus. 

Sin  60°  l'=cos  29"  59'=  .8661708.  cos  60°  l'=sin  29° 
59' =.4997481. 


Sin   60*  2'=cos  29o  58' 
Sin  /  6(y  3'= cos  22'  5/ 


cos  60°  2'=sin  29°  58' 
cos  60°  3=  sin  29°  57' 


&c.  &c. 

55.  To  find  the  versed  sines  and  co-versed  sines  of  the  quadrant. 
Jn  any  arc  less  than  90*^  tlie  versed  sine  is  found  by  sub- 
tracting the  co-sine  from  radius  (cor.  Art.  14.);   and  in  ares 
greater  than  90°,  it  is  found  by  adding  the  co-sine  to  radius :  thus. 


'  •  The  learner  is  supposed  (in  this  and  the  following  articles,)  to  bare  com- 
pated  all  the  preceding  sines,  co^sines,  tangents,  &c. ;  if  he  has  not,  he  mast, 
in  order  to  work  the  examples,  take  them  from  a  table.  By  means  of  th« 
fdrmnlse  here  given,  any  natural  sine,  tangent,  secant.  Sec.  in  the  table,  yfbifih. 
is  supected  to  be  wrong,  may  be  examined,  and  if  necessary,  corrected. 


Part  IX.        CONSTRUCTION  OF  SINES,  &c.  397 

ver.sin  r=l— cos  l'=(l  — .99999996=:)  .00000004 
ver.  sin  2  =  1  —cos  ^'=  (1  —  .9999998= )  .0000002 
ver.  sin  ;V=rl— cos  3' =(1  — .9999996=)  .0000004* 
ver.  sin  4'= 1  —cos  4'=0000007 
ver.  sin  5'=il-.cos  5=.000001l 

&c.  &c. 

▼er.  sin  90°  l'=l  +cos  89**  69'=1.0002909 
ver.  sin  90^  2'rsl+cos  89^  58'=1.0005818 

&c.  &c. 

Versed  sines  for  arcs  greater  than  90,  do  not  occur  in  the  com*, 
mon  tables. 

56.  The  co-versed  sine  is  found  by  subtracting  the  sine  from 
the  i-adius  (cor.  Art.  15,) ;  thus, 

co-versed  sin  r=asl— sin  l'a=(l — 0002909=)  .9997091 
CO- versed  sin  2'=1— sin  2'=  (1-. 0005818=)  .9994182 
co-versed  sin  3  =  1— -sin  3=  (1— .0008727=)  9991273 
&c.»  &c. 

57.  To  find  the  tangents  and  co-tangents  from  0'  to  45". 

By  Art.  35.  anal.  1.  it  appears  that  the  tangent  of  any  arc 

^=  (radius  being  1.)  = 

vtanr  I  =illLL;=C^22!2£e-=)  .0002909 

co-tan   89°  59'   J       cos  1        \99999996 

*->  2'  .  I  =ii!?i;=  (:^22551i  =)  .0005818 

co-tan   89°   58    J       cos  2        \9999998 

tans'  ,_sin3l^   0008727        ^3^ 

)o  57    J       co-sin  3'     \9999996     ^ 

tan  4'  1         sin  4'  .0011636 


sine 
co-sine 


co-tan   89^  57'    ^       co-sin  3'     \9999996 

,l=—^=(^^^-=)  0011636 
co-tan   89°  56    J       co-sin  4'      \9999993 

&c.  &c. 

And  proceed  in  this  manner  to  45**. 

58.   To  find  the  tangents  and  co-tangents  from  45^  to  90°. 
Because  (formula  26.  Art.  49.)  the  t  ngents  of  45*^  +  i?  = 
(an.  45°— -B+2  tan.  2-B;  therefore  if 

5=1',  then  1*^^  ^^7.0  ^^  }  =tan   44°   59+2  tan  2=5 
t  co-tan  44°  59  -^ 

(.9994184  +  2  X. 0005818=)  1.0C058^0. 


368  PLANE  TRIGOIfOllRRT.  PitT  IX. 

1.0011642. 

n— Q'  /tan  450  3'        1  ,  .    , 

1.0017469. 

£=4  ....  I^"'/^''*'  ,^   }=tan  440   56+2  tan  S^^ 
*.  co-tan  44. 56.  -^ 

1.0033298.  &c.  &c. 

And  in  this  manner  the  tangent  <if  ev«rf  suoceedii^  minute  of 

ihe  remainder  of  the  quadrant^  must  be  found. 

59.  To  find  the  seeants  and  c6'$ecaittB  ^  the  ^uadroftt. 

By  the  second  analogy  Art.  35.  fire  have  seie  /i:x r-  ths 

•^  ^'^  cos  Jt 

fadius  being  tmity;  whence  if 

_  fsec  1'  1         1  1       _ 

/T-r,  then^^  g^  g^  59' J "^  cos  1'  "^  .99999996""^ 
1.00000004.  • 

if^3   .  .  •  .  <  ^^  ^    .  Vac ^^r  ■*  -  ss) 

(^co-sec  89®  ST'J      cos  3      \9999996    ' 

^0000004. 

r6ec5'  1      J ,       1        _, 

"^-"^  •  •  '  •  jco-sec  89®  55'  |-  cos  5'""\9999989'"^ 
1.0000011. 

{sec  T                1         1 
orvo  £.0'   >= -^=1.0000021 
so-sec  89®  63  J     cos  7^ 

&c«  &c. 

60.  By  this  method  the  sejcants  and  co-se<sants  of  every  minute 
pf  the  quadnii^t  may  be  computed^  but  it  is  necessary  to  employ 
it  only  for  the  odd  minutes  -,  the  secants  and  co-secants  of  the 
even  minutes  may  be  obtained  by  a  process  which  is  somewhat 
more  easy ;  a^  follows 

By  art.  41.  tan  4+sec  ^ssco-tan  -^  90—^. 

•.•  sec  ^=co-tan  4. 90— <^— tan  4. 

("see  2'  *)  HQ  ^fi' 

Let  4=b2',  theni  ^  ^  ^^  .^/  J.=^(co-tan  -— taa  «' 

'         1  co-sec  89®  58  J      ^  2 

=)  co-tan  44»  59'-tan2'=s(|  .0005819— .0005818=5)1.0000001. 


p«>T  m     cofiflniiucnoN  or  sikes,  &e.         p6» 

a(1.601l642— .0011636=)  1.0000006. 

=(l.p017469-. 0017455=)  1.0000016. 

(^co-sec  89°  52  J 
=  1XHX)0027. 

&c.  &c. 

61.  The  numbers  thus  computed  are  called  natural  sines, 
tangents>  &c.  they  are  computed  for  every  degree  and  mimite  of 
the  quadrant,  and  arranged  ia  eight  columns^  titled  at  the  top 
and  bottom  3  these  together  constitute  the  table  of  natural  sines^ 
tangents^  &c.  directiooB  for  the  use  of  which  are  given  in  the 
introduction  to  every  system  of  trigoaosietric^il  tables  >*. 

OF   THE  TABLE  OF   LOGARITHMIC    SINES, 

TANGENTS,  &c. 

69.  The  logarithmic  or  artificial  sines,  tangents,  &c.  are  the 
l^rkhms  of  the  sines,  tangents,  &c.  computed  to  the  radius 
io)  '®=ieO0000000Oj  for  since  the  sines,  co-sines,  and  many  of 
the  versed  stnes  and  tangents  c<»Bputed  to  the  radius  1  are 
proper  fractions,  their  logarithms  will  have  a  negative  indexj 
(v(4. 1.  page  287.)  but  by  assuming  the  above  number  for  radius, 
these  fractions  become  Whole  numbers,  their  logarithms  affir- 
Hoative,  and  the  figures  expressing  any  sine,  tangent,  &c.  will  be 
the  same  in  both  cases,  as  likewise  their  logarithms,  excepting 
the  indices^  which  (as  we  have  observed)  will  he  frequently  nega* 
tive  in  the  former  case,  but  always  affirmative  in  the  lattCTj 
therefore,  in  order  to  find  the  logarithm  of  the  sine  of  an  arc, 
ejaculated  to  the  radius  10)'^,  we  most  add  10  to  the  index  of  the 
logafrithm  of  the  same  sine  to  the  radius  1 :  for,  let  r=  the  radius, 
«=<fee  sine  of  any  arc  to  rad.  r^  Ri=^a  different  radius,  S=the 
sine  of  an  arc  (to  rad.  R)  simitar  to  the  former,  then  {Art.  38.) 


P  For  an  accoont  of  the  tables  of  sines,  tangents,  &c  with  ample  directions 
to  assist  the  learner  in  their  use,  see  Dtt  Hutton's  Math^ftwticai  Tables, 
iedit.  p.  151,152. 


400  PLANE  TRIGONOMETRY.  Fart  IX. 

r:R::8zSi  which  if  r=cl  and  Hs=10^**,  becomes  1  :  lo!'®  : : 

s :  S,  V  5=io^'Ox5,  •/  hg.  S^lOxlog.  lO+log.  s:={8rnce  log. 

10=1)  10+ Zog.«.     Q.  E.  D. 
Examples.— 1.  To  find  the  logarithmic  sine  of  l'. 
To  log.  of  .000^909  (s=«i«  1')  =—4.46374^7 

jidd 10 

The  sum  is 6A6S7437=:thelog.sine 

of  X  to  radius  10000000000. « 
^.  To  find  the  logarithmic  tangent  of  2*.  35'=s 
To  log.  of  .0451183  (=fa»  2^35') =—2.6543527 
Add 10 

The  sum 8.6543527  m  Ihelog. 

tangent  of  2%  35'. 

3.  To  find  the  logarithmic  secant  of  7*.  5'; 

The  log.  of  1.0076908  (=5cc  7*  5')=0.0033273 
Add 10 

The  log.  secant  of  7*  5'.=  10.0033273 

.    4.  To  find  the  logarithmic  versed  sine  of  20"  12'. 

To  log.  of  .0615070  (=»er.  s.  of  20'  12')  ==-2.7889245 

Add 10 

The  log.  versed  sine  of  20«»  12'=  8.7889245 

In  this  manner  the  logarithmic  sines>  co-sines^  tangents,  &c. 
are  computed  -,  viz.  by  adding  10  to  the  index  of  the  logarithm 
of  the  nat-ural  sine^  co-eine^  tangent,  &c.  respectively  comespond- 
ing  to  the  radius  1  '. 

Having  shewn  the  method  of  computing  the  trigonometrical 
canon,  both  in  natural  numbers  and  fogarithajs,  the  next  thing 
to  be  done  is  to  demonstrate  the  propositions  on  which  the 
practical  part  of  trigonometry  is  founded. 

•the  fundamental  theorems  of  plane 

trigonometry. 

.63.  In  a  right  angled  triangle  the  hypothenuse  :  is  to  either  of 
the  sides  : :  as  radius  :  to  the  sine  of  the  angle  opposite  to  that 
side. 

4  By  the  preceding  rules  any  logarithmic  sine,  tangent,  secant,  &c.  in  the 
table,  suspected  to  be  inaccurate,  may  be  examined,  and  the  error  (if  aoy 
should  be  found)  corrected. 

'  The  log.  sine  of  1'  (as  here  given)  exceeds  the  truth  by  .0000176  becaasc 
,tbe  sine  of  T  is  only  .000390888  and  not  .0002909.  See  Art.  51. 


PaztIX.         tONDAMENTAL  theorems. 


401 


Let  JOB  be  a  triangle,  rigiit  angled  at  Ai  frOm  €  as  a 

centre  with  any  radius  CD  describe  a  circle  I>£,  and  draw  DF 

perpendicular  to  CA. 

Because    DF  is  parallel  to  BJ  (^8.1.) 

CBiBAtiCD:  DF  and!  .^  ^v 

CB,CA.,CDx  CF         S^      '^ 

But  1>F  ia 
the  sine  of  the 
angle  C  (Art.  12.), 

Md  CF  is  the  co-  D 

sine  of  the  angle 
C  (Art.  13),  or 
the  sine  of  the 
angle  (CDJPs)  jB; 
••  hyp.  CB  :  side 
Bil::  radius  (CD)  ^ 
:  sin  ang.  C  (DF)  oppoaite  to  BA:  in  like  manner  hyp.  CB  :  side 
CA::  radius  {CD) :  sin.  ang.  B  (CF)  opposite  to  CA,    Q.  £.  D. 

64.  If  CD  be  the  radius  to  which  the  trigonometrical  canon 
18  computed,  then  will  DF  be  the  sine  of  C,  and  CF  the  sine  of 
B,2iS  actually  exhibited  in  the  eanon;  and  therefore,  having 
the  hypothenuae  CB,  and  one  side  BA^  of  a  right  angled  tri- 
angle given,  the  angle  C  (opposite  BA)  may  be  found,  for  CB  : 
BA  : :  tabular  radius  :  tabular  sine  of  C,  which  sine  being  found 
in  the  table,  the  angle  of  which  it  ia  the  sine,  will  be  known. 

Hence>  the  angle  C  being  known,  the  angle  £=90»—  C  is 
likewise  known. 

65.  In  a  right  ftngled  trifloogle,  one  of  the  sides  about  the 
light  angle  :  ia  to  the  other  : :  as  radius  :  to  the  tangent  of  the 
angle  opposite  the  latter  side. 

About  the  angular  point  C»  of  the  triangle  ABC,  with  any 
radius  CE,  describe  the  arc  DE  aa  before,  and  draw  £G  at  right 
angles  to  C^  (II.  1.)  meeting  CB  in  G,  EG  will  be  the  tangent 
of  the  angle  C  (Art.  16.)  •.•  CA:AB::  CE:  EG  (4. 6.) ;  that 
K  side  CA  :  side  AB  : :  radius  :  tan.  ang.  C. 

In  like  manner,  if  from  B  as  a  centre  with  the  radius  BA 
a  circle  be  described,  AC  will  be  the  tangent  of  the  angle  B; 
and  it  may  in  like  manner  be  ahewn,  that  BA  :  AC  : :  radius  : 
taxu  ang  B.    Q.  £.  t>. 

TOL.  11.  p  d 


4M 


PLAN£  TRXGONaMETRT. 


Fait  IX. 


6(1.  If  C£  bft  the  radioa  to  which  the  canon  la  computed, 
£d  will  be  the  tabular  tangent  of  C;  wherefore^  shoe  €A : 
AB  ::  CE:  EG,  we  have  only  to  find  EG  in  the  tangents,  and 
its  corresponding  tngle  C  will  be  known }  wharefore  the  two 
sides  about  the  right  angle  of  any  right  angled  triangle  being 
given,  the  angle  C,  and  likewise  the  angle  B  (=:90*^€^)  in»f 
be  found. 

€7*  The  sides  of  any  plane  triangle  are  to  each  other  as  the 
tines  6f  their  opposite  angles. 

Let  ABC  be  a  triangle,  from  B  draw  BD 'perpendicular  to 
AC  produced  if  necessary ;  and  CE  perpendicular  to  AB, 

If  a  circle  b^  described  from  B  as  a  centre,  with  the  radius 
BC,  then  it  is  evident  that  CE  will  be  the  sine  of  the  angle 
ABC}  and  if  from  the  centre  C,  with  the  same  radius,  a  circle  be 
described,  BD  wfll  be  the  Mne  of  the  angle  BCA  (Art.  12.)  f 


wherefore,  since  the  angle  A  is  common  to  the  right  angled  trL 
angles  AEC,  ADB,  these  triangles  are  equiangular  (3^.  1.),  and 
AB:BD::AC:CE  {4.6)  .'  AB  :  AC  ::  BD  :  CE  (16.5.); 
that  is,  side  AJBisUie  AC  : :  sin.  ai^.  ACB  oppaii^Q  AM  :  sin. 
ang.  ABC  opposite  AC.    Q.  £.  D. 

In  the  case  in  which  the  perpend&eular  BD  fiills  without  tbe 
triangle  ABC,  BD  is  actucUl^  the  sine  oi  the  exterior  angle 
BCD  i  but  BCA  k  the  supplement  of  BCD  (13. 1.  asad  Art.  8.) 
and  since  the  sine  (^  an.  angle  J^likewiBe  tt^e  sine  of  ite  supple- 
ment (cor.  Art.  12.)  BD  is  therefore  the  sine  of  the  angle  BCA. 

68.  Hence,  if  we  have  two  sides  AB,  AC  oi  saiy  triangle 
ghen,  and  likewise  an  angle  ACB  opposite  (AB)  one  of  them  ; 
the  angle  ABC  opposite  the  other  given  side  (AC)   may  be 
found  i  and  thence  the  renmining  angle  A.    For  since  AB : 
AC  : :  sin.  ang.  ACB  :  sin.  ang.  ABC,  the  three  first  terms  beiog 


pAfet  IX.        FimDAMESTAh.  THEOREMS.  40) 

given,  the  fourth,  or  sine  of  ABCy  atid  consequently  the  atigle 
j^BC  is  known  3  whence  also  the  angle  ^=  180^*^^8^480 
is  known.  Lastly,  from  the  two  given  sides  AB,  AC,  and  thd 
three  angles  which  we  have  found,  the  third  side  BC  will  be 
obtained,  for  invcrtendo,  sin.  ang.  ABC :  sin.  ang.  BAC : :  side  AC 
:  side  BC, 

69.  If  half  the  difierence  of  two  quantities  be  added  to  half 
their  sum,  the  result  will  be  the  gfeater  of  the  two  proposed 
quantities  -,  but  if  half  the  did^rence  be  taken  from  half  their 
sum,  the  result  will  be  the  kss. 

Thus,  let  A  and  B  be  two  quantities,  of  which  A  is  the 
greater;  S:^ their  sum,  i>=± their  difference. 

And  A^B^dS''^''^^'^' 


Their  sum  ^A=:S+D,  •.•  A=z~—ts—-^—. 

Their  difference  2B=S-1>,  •.'  B=— — -  =-5^— -  Q.E. D. 

S      D  ,5 

Cor,  Hence,  if  from  {A=)  ■q-+-x-  we  take  •^,  the  remain* 

der  is  — -,  that  is,  *'  if  half  the  sum  be  subtracted  from  the 
2 

greater,  the  remainder  id  half  the  difference." 

70.  If  within  a  triangle,  a  perpendicular  be  drawn  from  the 
opposite  angle  to  the  base,  then  will  the  base  :  be  to  the  sum  of 
the  other  two  sides  : :  as  the  difference  of  these  sides  :  to  the. 
difference  of  the  segments  of  the  base. 

Liet  ABC  be  a  triangle,  having  the  straight  line  CD  drawa. 
from  the  angle  C  perpendicular  to  the  base  ABs  then  will  A  8 
:  AC-^  CB  : :  AC-CB  :  AD—DB. 

From  C  as  a  centre  with  the  distance  CB  the  least  of  the  t wa 
sides,  describe  the  circle  EBF,  cutting  CB  in  £,  and  AC  pro^ 
duced  in  G  and  JP;  then  because  CF:x:  CB  (15  def.  1.)  AFzzAQ 


]>d  2 


404 


PLANE  TRIGONOMETRY. 


Part  IX. 


+  CB=s:the  suip  of  the  sides  j 
and  because  CG=±€B,  AC^ 
€B=x  (AC-^CGzt^)  ^G±=the 
di£fereDce  of  the  sides.  M&o, 
since  DE=:DB  (3.3.),  AD--^ 
DB^iAD-^-DEz^)  ^E=the 
differenqe  of  the  segments  -A..  E 
(AD  and  DB)  of  the  base. 

Because  from  the  point  A  without  the  circle,  AB  and  AF 
are  drawn  cutting  the  circle,  AB.AE^AF.AG  (cor.  36.3,),-.* 
AB  :  AF::  AG  :  AE  (16.6.) ;  that  is,  the  base  :  sum  of  the 
sides  : :  difference  of  the  sides  :  difference  of  the  segments  of 
the  base.    Q.  E.  D. 

When  the  three  sides  of  a  triangle  are  given,  the  angles  are 
found  by  this  proposition. 

71.  In  a  plane' triangle,  twice  the  rectangle  contained  by  any 
two  sides,  is  to  the  diffefrence  of  the  sum  of  the  squares  of  these 
two  sides  and  the  square  of  the  base,  as  radius  to  the  co-sine  of 
the  angle  contained  by  the  two  sides. 

Let  ABC  be  a  triangle  2^B.BC:  31i?^4-Sc|«-:33« :: 
radius  ;  co-sine  of  ABC  Draw  AD  per- 
^ndicular  to  BC  (produced  if  neces- 
sary), then  52) «  ^-'icl*  =^2  ^  2C5.  BI> 
(13.2  ),  vZ5)«+Sc|*— 3C|^=2CB.BD; 
but  ^CS.BA  :  ^CB.BD  :  :  AB  :  BD 
(1. 6.)  5  that  is,  twice  the  rectangle  con- 
tained by  the  sides  :  is  to  the  difference 
of  the  sum  of  the  squares  of  the  sides, 
and  the  square  of  the  base  : :  as  AB  :  to 
BD;  but  B  being  the  centre,  and  AB 
radius,  BD  will  be  the  co-sine  of  the  angle  ABC  (Art.  13.),  •.• 
twice  the  rectangle  contained  by  the  sides,  is  to  the  difference  of 
the  sum  of  the  squares  of  these  two  sides  and  the  square  of  the 
base,  as  radius,  to  the  co-sine  of  the  angle  contained  by  the  two 
sides)  and  the  same  may  in  like  manner  be  proved  when  the 
angle  at  B  is  obtuse,  by  using  the  I2th  proposition  of  the  second 
book  of  Euclid,  instead  of  the  13th.    Q.  E.  D. 

When  the  three  sides  only  of  a  plane  triangle  are  given, 


S 


Past  IX.        HJNDAMENTAL  THEOREMS.  40S 

the  angles  may  be  found  by  means  of  this  proposition^  withput 
letting  fall  a  perpendicular^  as  In  the  preceding  article* 

7^«  In  a  plaice  triangle^  tfj^^um  of  any  two  sides  :  is  to  their 
difference  ; ;  as  the  tangent  of  half  the  sum  of  the  angles  at  the 
jbase  s  to  the  tangent  of  half  the  difference. 

Jjet  ABC  be  a  triangle^  from  C  as  a  centre  with  the  .least 
side  CB  as  radius^  describe  the  circle  EBF-^  produce  AC  to  F, 
join  BE,  BF,  and  draw  ED  perpendicular  to  EB. 

Because  CE^CF^CB,  AF=i{AC-\-CF^)  AC+CBzsthe 
sum  of  the  sides,  and  AE 
s  (AC^CE=)  AC-CB=z 
difference  of  the  sides.   Also  C, 

tCB=iCB4+  CAB  j(32. 1.)  ^ 

s;sthe  sum  of  the  angles  at 
the  bwe,  •.•  FEB=i{^FCB  ^ 
by  30. 3.=)  half  the  sum  of 
the  angles  at  the  base.  And 
since  CEzszCB,  the  angle  CEBszCBE  (6.  i.) }  but  CEB:=zCAB 
-^EBA  (3«.  1.)  5  •/  CBE=iCAB-^EBA;  to  each  of  these  equals 
add  EBA,  %•  {CBE-\'EBA=i)  CBA^CAB-i-^EBA  or  CBA-^ 
CABz=:^EBA;  that  is,  ^EBA^^the  difference  of  the  angles 
(CBA,  CAB)  at  the  base,  •.•  EBA^half  the  difference  of  the  angles 
at  the  base.  Now  since  EBFis  a  right  angle  (31. 3.)»  and  BED 
a  right  angle  by  construction,  if  from  £  as  a  cenire  with  the 
radius  EB  a  circle  be  described,  it  is  evident  that  FB  is  the  tan^ 
gent  of  FEB  (Art.  16.)  j  that  is,  FB  is  tlie  tangent  of  half  the 
sum  of  the  angles  {CAB,  CBA)  at  the  base;  and  if  from  ^  as  a 
centime  with  the  same  radius  (EB)  a  circle  be  described,  it  will 
be  equally  plain  that  ED  is  the  tangent  of  EBA;  that  is,  ED 
is  the  tangent  of  half  the  difference  of  the  angles  {CAB,  CBA) 
at  the  base.  Again,  becaui^e  ED  is  parallel  to  FB  (27*  1  -),  and 
the  angle  A  common,  the  two  triangles  AFB,  AED  are  equi- 
angular (29. 1.),  •.•  AF:  FB  iiAEiED  (4.  6.)  and  AF.AE: : 
FB :  ED  (16. 5.)  ^  that  is,  the  sum  of  the  sides  :  is  to  their 
difference  : :  as  the  tangent  of  half  the  sum  of  the  angles  at  the 
base  :  to  the  tangent  of  half  their  difference.    Q.  E.  D. 

When  two  sides  and  the  included  angle  are  given,  the  re- 
maining angles  may  be  fo^nd  by  this  proposition  with  the  help 
of  Art.  69. 

Dd  3 


4W  PLANE  TKIGOMOMETRT*  Fast  IL 

SOLUTION  OF  THE  CASES  OF  PLANE  TRJANGLB$. 

73.  There  agre  three  ways  of  solving  trigonometrical  problems, 
V17.  hy  geometrical  conMtruction,  h^rithmetical  computation,  and 
hutrumentally,  or  bj  the'  scale  and  compasses.  The  first  of  these 
methods  has  been  already  explained  in  part  8.  under  the  head 
of  Practical  Geometry ;  the  second  consists  in  the  application  of 
the  principles  laid  down  in  the  foregoing  theorems,  by  the  help 
of  either  natural  numbers,  or  logarithms  3  and  by  the  third, 
the  proportions  are  worked  with  a  pair  of  compasses  on  the 
Ganters*  scale  'j  the  method  of  doing  which  will  be  explained  in 
the  foHowing  examples,  where  the  conditions  are  exhibited  in 
th^  form  of  a  Rule  of  Three  stating,  having  either  thefirti 
and  second  terms^  or  the^r^^  and  third,  always  of  the  same  Idnd. 

74.  iVhen  the  first  axd  second  terms  are  of  the  same  kind. 
Extend  the  compasses  from  the  first  term  to  the  s^o^iid,  on 

that  line  of  the  Gunter  which  is  of  the  same  name  with  Ihfise 
terms ;  this  extent  will  reach  from  the  third  term  to  the  fourth, 
on  the  line  which  is  of  the  same  name  with  the  third  and  fourtli. 

75.  When  the  first  and  third  terms  are  of  the  seme  hmd. 
Extend  the  compasses  (on  the  proper  line)  fircHn  the  first 

to  the  third  ;  that  extent  will  reach  (en  the  proper  line)  from 
the  second  to  the  fourth  -,  observing  in  all  eases,  that  when  the 
proportion  is  increasing,  the  extent  must  be  taken  forwards  oa 

*  Tbk  scale  was  inrented  by  the  fUr.  Edmimd  Guattr,  B.D.  professor  of 
Afttffoaomy  at  Grctbam  College,  probably  about  the  year  1$$4 ;  it  it  a  bioad 
4(1  f  o)er  tWQ  feet  in  length,  on  which  are  laid  down  (besides  all  the  lines  com- 
mofl  to  the  plape  scale)  logaritluuic  lines  of  nvmbers,  sines,  versed  sines,  tan* 
ctntf,  meridional  parts,  eqaal  parts,  sine  rhumbs,  and  tangent  rhombs ;  that  is, 
t^e  actual  lengths  (taken  on  a  scale  of  equal  parts)  are  expressed  by  the  figures 
constituting  the  Ic^arithms  of  the  quantities  in  question.  With  these  logar- 
Kbmic  scales,  all  questions  relating  to  proportion  in  numbers  may  be  solved, 
fb»>tlie  compasses  being  extended  fmm  the  first  term  to  the  second  or  third, 
t(at  extent  will  reach  from  the  second,  or  from  the  third  to  the  fourth,  aocordiog 
as  t^e  ^rst  and  second,  or  first  and  third  terms  are  of  the  same  kind.  For  aii 
ample  description  of  this  scale,  see  Robertson's  EUmentt  of  Navigation, 
vol.  1.  p.  114.  4th.  edit*  likewise  Mr.  Donne*s  directions  usually  sold  with  his 
improved  scale ;  and  for  an  account  of  the  improrements  by  Mr.  Robertson, 
see  a  tract  on  the  subject,  published  in  1778,  by  William  Monntaine,  Esq. 
F.  R.  S. 


FamtUC:     of  BKfHT  AM0LI2>  TBUNOLES.  Mf 

the  sqOfi^  but  VirlMti  the  ptoptije^oA  kdecreaib^Atiumi  be  tiken 


SOLUTION  OF  RIOHT  ANGLED  TRIANGLES. 

76.  Case  1.  Given  the  bjpothenuse  AB,  and  one  side  AC,  of* 
i7gbt  angled  triangk  -,  to  find  tbe  j^waining  side  BC,  wd  tlie 
angles  A  and  £  ".  ^^^^ 

Because^^5cl«4^«==351«  (47. 1.)  /  BCJi «=r45;*-^Cl«> 

and  SC=;  v'^^'— ^**  whmce  BC  is  found . 
likewise  (Art.  63.)  hyp.  AB  :  tade  AC  :i 
radius  :  sin.  angle   B;   that  is,  ain  B^ 

^Cx  radius  ,     ,        ..•  i         .     « 

^p 5   or  by  logarithms  <,  log.  sin  B 

=log.  AC+log.  rod.-— log. .AB i  whence  the 
angle  B  is  found,  both  by  natural  numbers 
and  logarithms. 

Lastly^  since  the  three  angles  of  any  tri- 
aagje  aire  equal  to  two  rig^t  angles  (32. 1.)  ^ 
=  180»,  and  the  angle  C  (a  right  angle)=90%  •.•  B+A:=^ 
(180«>-.C=lScr-9(y=)  90^  but  the  angle  B  has  been  found, 
•/  .4=90— B  is  likewise  known  •. 

By  a  similar  process  AB  and  BC  being  given>  AC  and  the 
angles  B  and  A  may  be  found. 


*  Before  yon  begin  to  work  any  qne3tion  in  Trigonometry,  you  mast  draw  » 
sketch  resembling,  as  nearly  as  you  can  guess,  the  figure  intended  ;  pladny 
letters  at  the  angles,  and  eacb  number  given  in  the  question  opposite  the  tide 
or  angle  to  which  it  belongs ;  some  authors  mark  the  given  sides  and  angles  by 
a  small  stroke,  drawn  across  the  given  side,  or  issuing  from  the  given  angle  ; 
t^e  unknown  paits  they  mark  with  a  dphtr  (o). 

*  It  must  be  remembered,  that  multiplication  of  natural  numbers  is  per- 
fomed  hy  the  addiiiM  of  their  k>garithm8,  division  by  subtraction,  involution 
l»y  wutHpHcaiiony  and  evolution  by  divvrioHf  if  these  particulars  be  kept  in 
mind,  there  will  be  no  difficalty  in  solving  tri^nometrical  problems  by  logar- 
itlmis,  see  vol.  l.  part.  8. 

«-  The  angle  A  may  be  found  in  the  same  page  of  the  table  in  which  B  is 
fomnd ;  thus,  if  the  degrees  and  minutes  contained  in  B  be  foand  at  the  top 
and  on  the  iefi  hand  respectively,  of  the  page,  those  contained  in  A  will  be 
fvoo^jkt  theftoMom  and  on  tint  right;  viz.  the  degrees  at  the  bottom  of  the  page, 
sutd  the  minutes  on  therrighPhand,  in  a  fine  with  tlie  minutes  in  B» 

n  d  4 


4M  HAKE.TBlGONOMSrRY.  Pa&t  IX-^ 

•  BzAKFtBft."-!.  Qhtm  the  liypotlieimse  ABssl9CK  and  tb6 
perpendicular  ^Cs95>  to  find  the  base  BC  and  the  angk&  4 
andJB. 

B9  caiutruetion. 
Draw  any  straight  line  £C,  at  C  draw  Ci^  perpendicular  tq 
BC,  and  make  it  eqiul  to  95  taken  from  any  convenient  scale 
of  equal  partS}  from  ^  as  a  centre  with  the  radius  190  takeiv 
from  the  same  scale^  cross  CB  in  J3,  and  join  4B.  Take  the 
length  of  CB  in  the  compasses,  and  apply  it  to  the  abo?e« 
mentioned  scale>  and  it  wiU  be  found  to  measure  7B  nearly; 
next  measure  the  angles  A  and  B  by  the  scale  of  chords  or  the 
protractor,  and  they  will  be  known,  viz.  ^ss38*  and  ^s=5f2*« 
nearly  *. 

By  cakulaiion. 

First,  to  find  BC.    We  have  BCs=  v'SS)*— ^«= 


(^i5o)«-96>=^5S76=)  73.3143,  &c.  f 

Seeondhf,    to  find   the   angle   B.     We  have   sin    £=; 

^^2?=(^^^=).7916666  the  natural  sine  of  B,  and  the 

nearest  angle  in  the  table  corresponding  with  this  sine  is  52^ 
30'  •)  wherefore  the  angle  B=52*  20',  and^=(90"— J5=90'-» 
62'  20'=)  37*  40'. 


>  The  sides  and  angles  of  triangles  are  yery  ezpeditioosly  determined  both 
by  the  plane  scale  and  the  Ganter,  but  these  methods  are  not  to  be  depend^ 
on  fn  cases  where  accnracy  is  required ;  they  are  neyertbeless  nsefal  where 
great  exactness  is  no  object,  and  as  convenient  checks  on  the  method  of 
calculation. 

y  The  side  £C  may  likewise  be  ibund  trigonometricallyi  after  the  angle  A 

AB.%m  A 

has  been  found ;  thus  (Art.  63.)  AB  :  BC : :  rad  :  sin  A^  •.•  BC^ ;; — > 

rad 

this  solution  may  be  performed  by  the  Gunter ;  thus,  extend  on  the  sines  from 
900  to  37<>4,  this  extent  will  reach  on  the  numbers  from  120  to  7d-^sJSC 
nearly. 

>  This,  although  it  is  the  angle  which  has  the  nearest  sine  in  the  table  to 
the  above,  is  not  perfectly  exact ;  the  natural  sine  of  58*  SO'  being  only 
.7916792.  which  is  less  than  .7916666  by  .0000874}  now  the  sine  of  52«8l' 
exceeds  that  of  52(>  20'  by  1 777 >  therefore  our  angle  52<>  20'  is  too  small  by 
-rrrr  of  a  minute  ;  that  is,  by  29"  -tttt  '  whence,  in  strict  exactness,  aogl« 
5=52«  20'  29"  iVrV^J  and  angle-<rf=37*>  39'  SO"  -rf'^fj" 


FiBt  IX.      OP  MGHT  ANGLED  TRIANGLES.  4W 

The  same  by  loganthms.    Since  log.  sin  B^log,  ifC+log. 
jfad.-log.  ^iJ,  V  to  log.  ^C=log.  95=  .......   1.9777236 

Add  log.  radius  =log.  10000000000=  10.0000000 

And  from  the  sum  = 11.9777336 

Subtract  log.  JB—log.  120  =  ...  .    2.0791812 
■  Remains  log.  sin  J?=52»20'=  ....     9.89854^4 
Whence  angle  -4=  (90^—5=:)  ST  40'  as  before. 

Jnstrumentally,  by  the  Gunter, 
Extend  the  compasses  from  120  to  95  on  the  line  (of  num- 
bers) marked  Num.  that  extent  will  reach  from  (radius)  90*  on 
the  line  (of  sines)  marked  sin,  to  52'*4.=52*  20'=the  angle  P. 
We  cannot  find  the  side  BC  by  this  method^  without  anticipating 
case  4. 

2.'  In  the  right  angled  triangle  JBC,  given  the  hypothenuse 
i*B=I35,  and  the  perpendicular  ^C=  108,  required  the  ba^e 


*  An  observation  similar  to  that  in  the  preceding  note  occurs  iiere :  the  log. 
tine  in  the  table  which  is  the  nearest  to  the  above,  is  that  of  52<*  20',  vis. 
9.8984944,  bat  this  is  less  than  the  above,  being  too  small  by  480,  wherefore 
520  ir  ig  too  ii^Q  foff  the  an^le  B;  now  the  difference  between  the  log.  sine 
of  52<>  20',  and  that  of  52«  21'  is  975,  whence  the  above  value  of  B  is  -Stt-  of  a 
minute,  or  29"-^  too  small  j  that  is,  the  angle  J3=62«  20'  29"tt-,  and  -.^«37^ 
S9'  ao'^T^  by  this  mode  of  calculation. 

It  is  worth  .while  to  observe,  that  the  difference  of  about  -x-w  of  a  second 
between  this  result,  and  that  in  the  foregoing  note,  arises  from  the  circum- 
stance of  the  logarithms,  as  well  as  the  sines,  being  approximations,  and  not 
absolutely  exact. 

When  the  sine,  tangent,  &c.  found  by  operation  is  not  in  the  table, 
1.  take  the  nearest  from  the  table,  and  find  the  difference  between  that  and  the 
one  found  by  operation;  call  this  difference  the  numerator.  2.  Find  the 
difference  of  the  next  greater  and  next  less  than  that  found  by  operation,  and 
call  this  difference  the  deneminator,  3.  Multiply  the  numerator  by  60  and 
divide  the  product  by  the  denominator,  the  quotient  will  be  seconds,  which  must 
be  added  to,  or  subtracted  from  the  degrees  and  minutes  corresponding  to  the 
nearest  tabular  number,  according  as  that  number  is  less  or  greater  than  the 
namber  found  by  operation. 

This  rale  will  serve  both  for  natural  and  logarithmic  sines,  tangents,  Sec.  and 
Tikewise  for  the  logarithms  of  numbers,  observing  in  the  latter  case  (instead  of 
multiplying  by  60)  to  subjoin  a  cipher  to  the  numerator,  and  having  divided 
by  the  denominator,  the  first  quotient  figure  must  occupy  one  place  to  the  right 
ol  the  right  hand  figare  in  the  nearest  tabular  number,  and  be  added,  or  sub* 
tracted,  according  as  that  namber  is  too  little,  or  too  great. 


\ 


*4 

410  Pl4/^£  14EUOOM0a|BQntY.  Pakt  IX. 

BQ»  vA  tb« fliaglcs  JkvfAB^  An$.  BC^^h  ang.  A^=99^ hi, 

5.  Givi^  AB^Q9l,  BCatl6,  required  th^  remaining  side 
and  angfef  ?    Am.  AC^19^  ang.  ^as43*  5'^  cii^,  J3=47'  55'. 

77,  Case  2.  Giuen  the  two  sides  AC  and  CB,  to  find  the 
hypothenuse  AB  and  the  angles  ^  and  B, 

first,  (47. 1.)  ^-6==  ^^/ACJl^+VSi"^}  whence  AB  is  found. 
Secondly,  (Art.  65.)  AC  :  CB  : :  radius  :  tangent  ang.  A:  or 

tan  A:sz  """•'••  '-•"  }  and  by  logarithms,  log.  tan.  -^=log.  BC 

+log.  rad.— log.  AC,  •/  the  angle  A  is  found,  both  by  natural 
numbers  and  logarithms,  and  the  angle  B^^^Cf—A  is  likewise 
found. 

ExAMPLjEs. — 1.  Given  the  side  ^C=123,  and  the  side  CJ5s= 
132,  to  find  the  hypothenuse  AB  and  the  angles  A  and  B. 

By  calculation  \ 

F*r#<,  ^5=  V^*  +  CB)«=;r  V  123l«+  132l«=  ^32553= 
180.424. 

o       J,     t        .      ,        T  ^     CBxrad.     132 

Secondly,  by  natural  munbers,  tan  ^o^ — >r    "'^123^ 

1.0T31707=natural  tangent  of  47*  l'=ang.  A,  \'  ang.  -B= 

(9(r— ^^)  {W-.47'  1  '=42«  59'. 

Thirdly,  by  logarithm,  log.  tan.  ulalqg.  CB-J-log.  yad.- 

log.  ^J?  •.•  to  log.  CB  13a= 2.1»5739 

Add    log.    radius    10000000000=10.0000000 

And  from  the  sum  = 12.1205739 

Subtract  log.  ^B  123  = 2.0899051 

Remains  lag.  tan.  ang.  ^==47^  l'=  10.0306^88 
And  ang.  B^W^^A:=i4aPbtf  as  before. 

Instrumenlally, 
Extend  the  compasses  from  123  to  132  on  the  line  (of 
numbers)  marked  Num,  this  extent  will  reach  from  (radius  =) 
45^  on  the  line  (of  tangents)  marked  Tan.  to  47^  I'scthe  angle  A. 


T 


^  In  this  and  the  foUowiaf  e^Mc  of  rigU  aa|^  triasf  let,  tibc  ^onstracilian 
n  purposely  emitted,  it  beiog  perfectly-  «a«y  and-  obvioss,  frani  wittt  kas  liaea 
given  on  the  subject  in  the  PractUi^.  G^ameti^,.wmu  the  end  of  part  9* 


fARt  IX      OF  SIGHT  ANEOJD  TRIANGLES. 


411 


The  side  JB  is  not  foaifd  vutrumeaially  for  a  reason  simi- 
lar to  that  before  given. 

2.  The  perpendicular  AC^^^tOO,  and  the  base  BC=110  of 
a  right  angled  triangle  ABC  being  given,  required  the  hypothe* 
nuse  AB,  and  the  ^ngles  A  and  B  ?  An9»  i#B=;^8.254^  ang. 
^=28°  49',  ang.  B=6V  11'. 

3.  Given  AC=^4,  and  BC=S,  to  find  AB,  and  the  angles  A 
and  B.    Ans.  AB=zS,  ang.A=zSe^  52',  ang.  5=53°  8'. 

78.  Ceue  3.  The  hypothenuse  AB  and  the  angle  B  being 
given^  to  find  the  sides  AC,  CB*  and  the  angle  A. 

First,  since  the  angle  at  JS  is  given, 
the  angle  A=z90P-^B. 

Secondly ^  (Art.  63.)  AB  :  ^C  : :  radius 

: sin  ang. B •.•  AC^^  ^d  "  ' ^  •'^^ Ipg.^C 

=:log.  sin  B+log.  AB—log.  radiqs ;  whence 
AC  is  found  both  by  natural  numbers  and 
logarithms. 

Thirdly,  5F«=J?C)«  +  CffI«  (47. 1.)  '/ 
VSi^^ABi^-^-lT)^  and  C5= 

^AB'^AC.AB-'AC  (cor.  5.2.);  also  log. 

CW^'^^^^^^^^-  ^^-^C,  . .  ^j  ^  ^^„„^^  boti^  by 

2 
natural  numbers  and  logarithms. 

ExAMPi^Es.-^!.  Given  the  hypothenuse  AB^=^\6o,  and  the 
angle  5=35"  30',  to  find  the  sides  AC,  CB,  and  the  angle  A. 

By  calculation. 
First,  ang.  ^=9Q»'-J?=«^  (90^-35^  30'=)  54^30'. 

X     Ar^     ^^^  B.AB       ,  . 

Secondly,  (by  naitural  numbers)  AC=^— — ^ — =(smce 

raa=l,  sin  35°  30' x ^-8=)  .580703x16.^=5=95.815996  5  but 
the  same  may  be  done  more  r^dily  by  logarithms  -,  thus,  be- 
cause Jog.  -^C'=log.  sin  5+ log.  AB'-^oQ.  rad. 

•.•  To  log.  sin  B.  or  35°  30'=^  ....    9.7639540 


Add  log.  AB.  or  165=   .  . 

And  from  their  suin=  •  • 
l^ubtract  log.  radius^^  .  .  . 
Remains  log.  AC  95.816= 


2.2174839 
U. 98 14379 
10.0000000 

1.9814379 


T 


\ 


f 


419  PLANB  TEUGOMOM BTRT;    ^     Fait  IX 


TUrdlxj,   CB^  ss/AB^-ACAB—AC^ 


<V:16&+95.816x  165-95.816S  ^860.816x69.184= 
V18044.?94144=)  134^29. 

The  same  by  logarithms,  log.   C^^^ 
\i^:AB-\-A€^\og.  AB—AC 


that  is,  to  log.  AJ^-^AC,  or  260.816= 2.4163348 

Add  log.  AB-^AC,  or    69.184=:  .  . . .  1.8400057 

The  sum  divided  by  2 2)4.2568399 

Gives  the  log.  of  CB=:  134.329  » .  . .  .^2.1281699 

Instrumentally. 

1.  Extend  from  (radius  or)  90*  to  35*  30'  (sang.  B)  on  tbe 
}ine  of  sines;  this  extent  will  reach  from  165  (backwards)  to 
^bout  95  ^  on  the  line  of  numbers^  for  the  side  AC  (opposite 
the  ang.  B.) 

2.  £xtend  on  the  Une  of  sines^  from  90"  to  54*  30'  (comp. 
B.)  'y  this  extent  will  reach  on  the  lines  of  numbers  from  165  to 
fkbout  134  -iV  for  the  side  CB, 

Ex.— 2.  Given  the  hypothenuse  4B=25,  and  the  aogh 
£=49%  to  find  the  sides  AC,  CB,  and  the  angle  A}  Ans. 
4C=  18.893,  CB=  16.4017,  ang.  A=z4l\ 

3.  Given  ^BslOO,  and  the  angle  ^=45^  to  find  the  lest? 
Ans.  BC=-iC=70.7108,  ang.  B=45^ 

79.  Case  4.  Ope  side  AC,  and  its  adjacent  angle  A  hemg 
given,  to  find  the  other  sides  AB^  BC,  and  the  remaining 
angle  B, 

Ftrsti  angle  5=90**—^. 

Secondly,  because  (Art.  67.)  AC :  CB 

^     .       ^       ^»    sin  A.AC 
: :  sin   B  :  sm   A,  *.•  CJ5=:  — : — =-—  3  and 

log.  CB=log.  sin  -4-l-log.  .4C— log.  sin  B. 
Thirdly,  because  (Art.  63.)  AB  :  AC:i 

radius  :  sin  B, '.'  AB^ — /   ^  :  also  log. 

sm  B 

AB=log.  AC +log.  rad.  —log.  sin  B.  j^ 

Examples. — 1.  Given  the  perpendicular 
^C=  1023,  and  the  angle  ^=12*»  45' 5  to  find  the  angle  B,  and 
the  remaining  sides  AB,  BC. 


PiiBT  IX.     OF  BIGHT  At<6LBD  TBIAMGLES.  4i^ 

B^  calculation. 

First,  aog.  B=i90»-^=:(90»-12r4b'=)  77^  15'. 

c        ^1     ^^    HnJ,AC     .2206974X1083     ^^,  .^lo 

Secondly,  CB=z — .     _     = ^.o.qq =231.4812; 

^  sin  jB  .9753423 

and  by  logarithms^  log.  CBs=log.  sin  -^+log.  -^C— log.  sin  B; 

that  is,  to  log.  sin  J.  12^  45'= 9.3437973 

Add  log.  AC  1023=  . 3.0098756 

From  the  sum  = 12.3536729 

Subtract  log.  8in,B  77**  15'=  ....    99891571 

Gives  log.  CB.  231.4812= 2.3645158 

«,L.   «      ^„    -^C.rad.      1023x1     ,^,««^^ 

Thirdly,  AB=:  —. — -- = =  1048.862. 

^  sin  5       .9753423 

And  by  logarithms^  log.  .-^J5=log.  -^C+log.  rad— log. 

sin  B;  that  is,  to  log.  AC  1023= 3.0098756 

Add  log.  radius= 10.0000000 

And  frorn'the  sum= 13.0098756 

Subtww't  log.  sin  J5  77^  15'=  . . .    9.9891571 

Gives  1<^.  AB  1048.862= 3.0207185 

Instrumentally, 

1.  To  find  CB,  extend  from  (sin  B,  to  sin  A,  that  is,  from) 
sin  77^7  to  sin  12K  3  this  extent  will  reach  on  the  line  of  num*« 
hers  from  (AC)  1023  to  2314-. 

2.  To  find  AB,  extend  from  (sin  B  to  radius,  that  is,  from) 
77^-i^  to  90^  on  the  sines;  this  extent  will  reach  from  1023  to 
about  1049  on  the  numbers^ 

Ex. — 2.  Given  the  perpendicular  ifC=400,  and  the  angle 
A=^4T^  S(f,  to  find  the  hypothenuse  AB,  the  base  BC,  and 
the  angle  jB?  Ans,  \^B=592.072^  BC=436.52^  ang,  J?= 
42«  Stf . 

3.  Given  ^tf C=82,  ang.  ^1=33^  13'^  to  fikid  the  rest  ?  Ans. 
ABssi979^  CB=63.69,  ang.  B=^%69  Alf. 

SOLUTION  OF  THE  CASES  OF  OBLIQUE  ANGLED 

TRIANGLES. 

The  foregoing  calculations  are  efiected  both  by  natural 
numbers  and  logarithms,  serving  as  a  useful  exercise  for  the 
learner;  but  principally  to  shew,  that  both  methods  termiimte 
in  the  same  result. 


• 


\ 


414  PLAN&  tRfCKlNOMBIllT.  PaAyIK. 

Trigonometrical  operartidtis  are  liowever  seldom  performed 
by  the  natitml  aumherft,  abd  tkeneffere^  in  the  fottMvitag  cases^ 
we  ghall  employ  only  the  logarithmic  ptoOMs. 

80.  Case.  I.  Let  there  be  given  the  two  angles  B  and  C,  and 
the  side  AC  opposite  to  one  of  them  j  to  find  the  angle  J,  and 
the  sides  JJ5  and  JSC. 

First,  the  aisles  ^  and  C  A 

being    given,    and    ^  =  ISO^— 
B+C,  the  angle  A  wUl  be  inown. 

Secondly,    (Art.  6f.)'  AC  : 

AB  : :  sin  i?  :  sin  C  *•'  AB:^ 

ACAnC        ^,        .^    ,       »' ~' 'C 

■  M  ,„  ■  ;-or  by  lo^itopms,  l^g. 

AB=:\og,  -4C+log.  sin  C— log.  sin  B;  '.'  ABisinown. 

Thirdly,  (Art.  67.)  AC  i  CB  ::  sin  5  :  sin  ^  •/  GB= 

AC, sin  A     ^    ,        .  *     *  ^^  ^  i^-' 

— .    p    .    By  logArithms,  log.  CBaftlog.^+log.  sin  4- log. 

sin  B;  '.'  CB  is  known. 

Examples — 1.  Given  the  angle  5=46®,  the  ai^Ie  Car59^ 
and  the  side  AC  (opposite  JB)r=i^O;  to  find  the  angjte-4  and  the 
sides  y^jS,  BC.  "^ 

.    By  construction.  ... 

From  any  scakof  eqnai  pavta.  take  ACsslQO,  at  C  ixttfae 
the  angle  ACB=i59y  and  at  A  make  the  angle  GrfBsat(l8d'»- 
B+C=1800-46o  +  59<>:?:)  75^5  then  take  the  length  of  J^, 
and  of  BC  respectively  in  the  compasses^  and  apply  them  to  the 
above-mentioned  scale,  and  AB  will=143,  £C=161^ 

By  computation. 
1.  Log.  ^B=slog.  ifC-hlog,  sin  C'^log.  tin  B 

•/  To  log.  AC  IWic 34>7P1813 

Add  log.  sin  C.59 9.9330656 

And  from  the  ^m:t: lB.0l3246a 

Subtract  log.  sin  B  4€P=st 9.8569341 

Remains  log.  AB  14^.9845=  . ,  .  2.1653127 


PaAt  IX.    OF  OBLIQUE  AN6L£D  TRIANGLES.        41$ 

2.  Log.  CBisilo^:  A€+\og,  sin  .<— log.  sin  B. 

\'  To  log.  AC  190= 2.079181« 

Add  log.  sin  J  75o= ...    9.9849438 

And  from  the  snni= 18.0641250 

Subtract  log.  sin  B  46<>= 9.8569341 

Ronaaina  leg.  CB  161.1354 . 4 .  .    2.2071909 

Instrumentallj^. 

1.  Extend  on  the  sines  from  46*"  (ang.  B),  to  59^  (ang,  C) ; 
this  extent  will  reach  on  the  numbers  from  120  {AC),  to  about 
143  (AB). 

2.  Extend  from  46°  to  75°  on  the  sines  -,  this  extent  wilt  reach 
froDd  120  {AC),  to  about  161  {CB),  on  the  numbers. 

Bx.  2.  Giveft  the  angle  AstBSP  43',  the  angle  Css7#  Y, 
and  the  side  ABta^eiO',  to  §nd  the  angle  B,  and  the  sides  AC, 
CB  }     Am.  ang.  B^Af  10^^  jlCss46&.08»  C£s542. 

3.  Given  the  side  ^^=1075,  the  angle  ^=34'^  46^,  and  the 
angle  0=22"*  5' ;  to  find  the  r«st  ?  Am.  BC=2394,  ^C=  1630.5, 
any. -rf=123»9'. 

81.  Case  2.  Let  there  be  given  the  two  sides  AB,  AC,  and 
the  angle  B,  opposite  AC:  to  find  the  angle  B^Cand  C>  and 
the  remaining  side  BC, 

Mrst,  {Art  67)  AC  : 
AB :  :  sin  £ :  sm  C;  -.*  sin 

C= ■;■■■'  5  which  by 

logarithms  is,  log.  sin  C^ 
log.  AB-^log,  sin  J&— log. 

AC;  •••  angle  C  is  known,     ^  X  /a    x\ 

Secondly,  angle   l?.^C 
=180— jB+C,  •/  angie  B-iC  w  ifc«oio». 


•^ 


<  This  case  will  be  always  ambiguous  when  the  given  angle  B  is  acute,  &fld 
AB  greater  than  AC,  (a;i  in  the  first  example) ;  for  the  above  expression  \^ 
the  sine  of  both  AsB^Axa,  or  of  its  supplement  AzB  (for  the  sine  of  an 
angle  and  the  sine  of  its  supplement  are  the  same,  by  cor.  Art.  1 S.) ;  conse- 
quently the  angle  A  will  be  either  BAx  or  BAts,  according  as  the  angle  AsB, 
6r  its  stipplement  AzB  be  taken ;  and  the  correspondiqg  value  oi  BC  will 
be  either  Bx  or  Bz,    But  if  the  given  angle  be  either  obtuse,  or  a  right ' 


416  PLANE  TRIGONOliETRT.  Part  £| 

Thirdbf,  (Art. 67-)  JC iBC-^nn  Bisin  BAC,   .•  BCz 

jiCsin  BAC 

: — = —  j  that  is,  by  logaritfams,  log.  5C=Iog.  -4C4-lo| 

sill  Xy 

sin  J?ufC— log.  sin  B:  */  JSC  if  known. 

Examples. — 1.  Given  AB=204,  ^C=145,  and  the  angle  J 
=35®;  to  find  the  side  BC  and  the  angles  BJC  and  C. 

jBy  coMtruciion. 

Draw  .<^jB  and  make  it =204  by  any  scale  of  equal  parts 
and  make  the  angle  J?=35®  5  from  .^  as  a  centre  with  the  radio 
(AC=)  145  taken  from  the  same  scale,  cross  jBCin  z  and  jf, 
join  Az,  Ax,  either  of  which  will  be  AC  3  then  will  Bz  or  Bx  bi 
the  value  of  BC,  these  being  measured  by  the  above  scale>  ivill 
be  BzzsSl^  and  Bx=252j.  for  the  values  of  AC;  also  by  the 
scale  of  chords,  or  protractor,  BAx=z9V,  BAz^l^^  for  the 
corresponding  values  of  BAG;  likewise  ^J5=:54^  AzB=s  I26*i 
for  those  of  C. 

By  calculation. 

To  find  the  angle  C. 

Because  log.  sin  Cslog.  AB+lc^,  sin  J?— log.  AC; 

V  To  log.  AB  204= 2.3096302 

Add  log.  sin  B  35«= 9.7585913 

From  this  sum= • . . . .  12.0682215 

Subtract  log.  AC  145= 2.1613680 

Remains  log.  sin  C-l  or  its  supp.  >  =9.9068535 

I  viz.  1260  12' J 

Next,  to  find  the  angle  BAC. 

^350+53048'  ^        f   Q»>^ 

First,  B+C:=}         or  S=J       or 

I350  + 1260  n'J       1 16l«  12' 


angle,  each  of  the  remaining  angles  will  be  acnte  (32. 1.) ;  therefore  when  the 
angle  B  is  either  obtuse,  or  a  right  angle,  C  muH  be  acute ;  consequentijr  when 
B  it  not  less  than  a  right  angle,  no  ambiguity  can  possibly  take  place 

If  the  angle  B  (in  any  proposed  example  nnder  this  case)  be  either  acute, 
obtuse,  or  a  right  angle,  and  AC  greater  than»^B,  there  is  no  ambiguity ;  but 
it  must  be  remarked,  that  if  JiChe  less  than  j^B  X  nat.  sin  B  (or  the  peiptB* 
dicnlar  drawn  from  A  to  the  base  BCt)  the  question  is  impossible. 


^ 


eaBgfcf 


Ai^i»iiaTlX.    OF  OBLIQUE  ANGLED  TBIANGLfiS.        417 

^..^ /angleS-4C=180— £4-C=J  or  >=-J      or 

Lastly,  to  find  tfie  side  BC 
Since  log.  BC=log.  ^C+log.  sin  B^C-log.  sin  B. 

If  BACt=i9l^  12' 

To  log.  AC  145= 2.1613680 

/•  Of  12'     ^ 
nl  pr*  Add  log.  sin  BAC'l  or  its  sup.  >  9*9999047 

thenar  t  88*48'      J 

luiii  And  from  tbe  sums 12.1612727 

orir'  Subtract  log.  sin  B  SS^rs 9.7585913 

de;H  Reoiaiiis  kg.  ^Css262.744a  . . .  2.4026814 

^^'  If  i9i#Ca  18* 48' 

,  fcf  To  log.  irfC  145SS 2.1613680 

B=l^  Add  kig.sio^i#€iy4tfg    9.5082141 

And  from  the  sums  ....  11.6695821 
Subtract  log.  sin  B  35**=  . . .  9.7585913 
Remains  log.  £0=81.4687=:  1.9109906 

ifMrumentallif, 

To  find  the  angle  C,  Extend  the  compasses  from  204  to 
145  on  the  line  of  numbers^  that  extent  will  reach,  on  the  sines 
from  35*  to  53'  48',  the  supplement  of  which  is  126"  12',  either 
of  these  is  the  angle  C. 

To  find  the  side  BC.  Extend  on  the  sines  from  35^  to  88* 
48'>.that  extent  will  reach  on  the  numbers  from  145  to  253 ;  or 
extend  on  the  sines  from  35°  to  18°  48',  this  will  reach  from  145 
to  Sli  on  the  line  of  numbers. 

Ex.— 2.  Given  the  side  ^£=266,  BC^  179,  and  the  angle 
C=107°40'$  to  find  ^C,  and  the  angles  A  and  B?  Ans, 
i<C=  149. 8.  ang.  A=:S9^  53',  ang.  B=32°  27'. 

3.  Giwn  -rfC=236,  ^C=350,  and  the  angle  B=38°40'j 
required  the  rest?  Ans.  AS==IS4A7,  or  S62.04,  ang,  -4= 
67<>  54',  or  112°  6',  ang.  C=73°  26'  or  29°  14'. 

82.  Case  3.  Let  the  two  sides  BA^  AC,  and  the  included  angle 
A,  be  given  5  to  find  the  side  BC  and  tlie  angles  B  and  C 


VOL.  II.  K  e 


\ 


418  PLANE  TRIGONOMETRY.  FftHi-IX. 

Let  AB  y  AC,  then  (18. 1,)  ^ 

the  ang.  C^B',  and  since  B^ C 
=  180°-^  (32. 1.)  4.  C-fB=:ix 
18Cy>— -4=90P— i^j   V  ^C-k-B 
it  known. 

But  (Art.  73.)  ABj^AC  :  'B^ ^C 

AB-^AC  (::  tan  ^C+B  ;  tan 

4^   C-B)   : :   tan   90-4-  A  :  tan  i  C-B,    /  tan  4.  C^ 

AB-^AatangO-^^  A     .     ,        .  

IfB+^C '   ^  loganthms,  log.  tan  4.  C—B  log. 

-^^— -4C+log.  tan  90-4-  ^— log.  ^iB+^C  •.•  4-  C^B  » 
JmotD». 

Whence  (Art.  69.)  the  greater  angle  C=4^  C+Jg-fj.  C— B^ 

and  the  leas^vix.  5=4-  C-|-B— f  C— B> 
'.'  ^  ang2e«  C  and  R  are  known. 

Lastly,   (Art.  67.)  AB  :  BC  ::  sin  C  :  sin  -^,  •••   BC^ 

AB.sin  A 

— ^j^-^  >  oy  logarithms,  log.  J5C=log.  -^B+ Jog.  sin  -rf— log. 

sin  C  •.•  BC  is  known. 

Examples — 1.  Given  AB:=:90,  .rfC=30,  and  the  angle  A 
szSOPy  t^find  lihe  aide  BC  and  the  angles  JB  and  C. 

Bif  construction. 
Make  AB^20  by  any  scale  of  equal  parts,  at  A  (with  the 
scale  of  chords  or  protractor)  make  the  angle  BACz=i8GP,  and 
make  AC^SO,  by  the  above  scale  of  equal  parts,  join  BC;  then, 
the  angles  B  and  C,  and  the  side  BC  being  measured,  will  be 
as  foUowsi  viz.  ang.  B=63o  24',  ang.  C=z3GP  36',  side  BC=33, 
nearly. 

By  carculation. 

4-  B4.C=:9O<^-4.^:=(9O0-4(y>=:)  50°;  this  being  knoway 
in  ^rder  to  find  4  B~C,  we  have  log,  tan  4  B— C=(log. 
-rfC— -4B+log.  tan  W-^^A-^log.AC-^ABzs)  log;  10+ log. 
tan  5(y>— log.  50. 

•.•  To  log.  10= ,    :i.ooooooo 

Add  log.  tan  SO^rs 10.0761865 

From  the  sums  ........  1 1.0761865 

Subti-act  log.  50a±   .  ......     1.6989700 

Remwns  tott4  B-^C  13®  24'4.      9.377216& 


PjibtIX.    of  <»LfQtJ£  AKG1.BB  triangles.       4i9 


Also  ^  ^+C;++  JB--C=i£60» + 13«  34'i=sdSo  24'4-=aiJgle  B. 
And  4.  B+  C— i  B— 1'=»60«- 13*»  24'4^s=»6*  35'i.=cat)gle  C, 
Lastly^  log.  BCalog.  ^B+log.  sin  ^— log.  sin  C; 
V  To  log.  ^B  20=  ........     1.3010300 

Add  log.  sin  A  80«=s {».99335l5 

From  the  sums 11.2943815 

Subtract  log.  sin  C  36^  35^4-  .  .    9.7753250 
Remains  log.  BC  33.0412stt  .  .     1.5190565 

Irutrumentally, 

For  the  first  proportion^  extend  from  50  to  10  on  the 
numbers)  this  extent  wiU  reach  on  the  tangents  from  50^  (the 
contrary  way,  because  the  tangents  above  45^  are  set  back 
again  f)  to  about  S^'i,  that  is^  from  45<>  to  13''4-. 

Extend,  for  the  second  proportion,  from  36*^  36'  to  80^  on 
the  sines ;  tliis  extent  will  reach  from  20  to  about  33  on  the 
numbers. 

Ex.-^2.  Given  the' side  ABsz^lB,  the  side  ^C=478.d, 
and  the  included  angle  AzsSi9  AQ'i  to  find  EC,  and  the 
Angles  B  and  C?  Am.  BC=s:326.1,  ang.  Bsl23«  9',  ang,  Csa 
220  6'. 

3.  Given  ifB=116,  AC=zB7,  and  the  angle  Jr=115^  37' > 
required  the  rest  ?  Ansi  BCs  172.5>  ang.  B^^V  ^\  ang.  C» 
37«  20'. 

G3.  Case  4.  Let  the  three  sides  ABy  BC,  and  CA,  of  the  tri-^ 
angle  ABC  be  given  5  to  find  the  three  angles  A^  B,  and  C. 


^»" 


i^iar 


^  When  the  ratio  to  be  niemiared  is  in  the  tangents,  and  one  of  the  term» 
below,  and  the  other  above  45* ;  ba¥in$  talwn  the  extent  of  the.  two  fbruer 
tenne  on  the  nombers,  &c.  as  the  case  may  be,  Kppiy  this  distance  00  the  tan- 
gents, from  45»  downwards  (to  the  left)  and  let  the  foot  of  (be  compasses 
rest  on  this  point,  which  for  distinction  we  will  call  s;  with  00*6  foot  on  o^ 
bring  the  other  foot  from  45%  to  the  given  term  of  the  ratio;  apply  the 
distance  (of  z  from  the  given  term)  from  45^  downwards,  then,  one  foot  ol 
the  compasses  being  on  45,  the  other  will  (with  this  extent)  exactly  readk  the 
term  re^^uired  to  be  founds 


£e2r 


4^0 


l^LANE  TRIGOTJOMEtlnr. 


Past  % 


Bnt,  By  ieiUngfaU  o  perpendicvlar  AD. 
Let  BJ  be thegreater  side,  AC  j^ 

the  less^  and  BC  the  base;  then 
(Art.  70.)  BC  :  BA+AC  : :  BA^ 
AC  :   BD-^DC,  -.-  BD-DC:s 


BAJfACBA^AC 
BC 


and  log. 


i^D-Dc'=Iog.  5-4+^C+log. 

BA^AC—log.  BC '.'  BD'-DC   u    Awwic».     But  BD+DC 

{szBC)  M  5riw»,  ".'  *A«  AaZtre*  of  <^efc  arc  likewise  known. 


Bin  (Art.  69.) 


BD+DC .  BD^DC       , 
. ^ ^ and 

2  2' 

JBD+DC    BD-DC 


2  2 

•.'  l/jc  segments  BDy  DC  are  known. 

Now  in  the  right  angled  triangle  ABD  we  have  AS,  BD 
and  the  right  angle  ADB  given. 
•/   (Art.  63.)  AB  :  BD  : :  rad.  :  sin  -BJfD,  or  sin  BADss 

^^.    In  logarithms,  log.  sin  ^^D=log.  ^JD+lo-log. 
AB  • 

AB;  •/  J?-4I>  is  known,  •.•  also  its  complement}  viz.  the  mgle 
ABC  is  known. 

And  in  the  right  angled  tntakgU  ADC  we  h^ve  AC,  CD 
and  the  right  angle  -41>C  given,  •.•  as  above,  CA  :  CD::  rad. : 

nn  CAD,  or  sin  CAD=s      '^  ' .    By  logaritha»,  log.  sin  CAD 

=log.  CD+10— log.  CA\-  CAh,  and  consequeHilly  fts  com- 
ptetnent,  viz.  the  angle  Cis  known. 

Also  BACz=^BAD+DAC  is  known. 

The  solution  without  a  perpendkulmr. 

By  Art.  71-  2  BA.AC ;  b1)^+A(\*'^Sc^  : :  mditts :  cos^T 

^    rad.53l*+^Cl«-5r}» 
,,cos^= -^ZaC 

s  (since  rad=:  1,  see  also  cor.  5. 2.) 

'  2"B^  ^C  •      ^ 

logarithms,  log,  co^  A=lO+]og' 

Sa"*  +  AC -^BC.ACSC -log,  ^BA-^lo^.^AC  '.'  the  angle  A 
is  known  f  and  B+  CslSO^--^,  to  find  the  angles  B  and  C. 


f^Kf  DC.    OF  qi||4tfJS  AN^1|D  TBIA{4GLES.        4SI 


(Art.  S%.)  Log,  tan,  i-  B^CzsJog.  AC-^AB -^-log.  tan.  M'-^^A 
-log.  AC^ABj  then  4- ^4- 0+4- ^- Cs=ang.  Bi  .     ^^^  ^^ 

4.B-HC-i5-C=ang.  c'    ^ 
-^ence  ^&e  three  {mgle$.A,  B,  and  C,  are  known*. 

Examples.— 1.  Given  the  aMe  -rfB=12,  AC=^U,  and  JBC= 
10^  to  find  the  angles  A,  B,  and  C. 

£y  comtructum, 
1.  Draw  the  straight  line  JSC=:10>  taken  from  any  convenient 
.scale  of  eqjual  parts^  from  £  as  a  centre  with  the  radius  12  de- 
scribe an  arc,  and  froijoi  C  with  the  radius  11  cross  the  above  arc 
in  A,  (both  the  latter  distances  being  taken  from  the  same 
scale  with  JBC,)  and  join  4B,  AC' 

9.  Measure  the  angles  by  means  of  the  scale  of  diords^  or 
protractor^  and  they  will  be  nearly  as  follows)  viz.  A^hl^, 
JB==  59°^,  and  C=  6d°i. 

jPy  ealcnilaHon. 
First,  let  Ajy  he  perpendiculqar  to  BC;  .see  the  last  figure 
Jjutpne.    Jgf)+Z)C=JgC=10 

BD-DC^ ^- ^  =  -^^^2.3. 

^^     BD+DC    BD-'DC 
V  jBP==        ^^    r{-^  ;^5+.1.15==^.1.5   the  greater 

figment  3 

and  DC— ^ ?-- — — r-=^-^1.15=3.85  the  less  seg- 

meat ; 

Then  log.  sin.  54I>=log.  -BD+  lO^log.  ^5=0.7888751 
+  10—1.0791812=9.7096939  5  v  ang.  BiiD=30' 50',  and  v 
its  complement  5=f59*'  1(/. 

Jn  li)£e  manner^  log.  sip.  CAD^sslog.  CD+lO—log.  C-4=p 
0.5854607+ 10—1.0413927=9.5440680=  /  ang.  CAJQ^^O^ 
29^  >  the  complement  of  which  is  69*  3I'==the  angle  C. 

Also  the  ang.  BJC=BJD+C4D=30  50'+?0»  29'« 
61*  W. 

«  On  haviog  found  the  angle  A,  the  remainiDg  angles  B  and  Cmay  be  fonnd 

(perhii|»8  more  omyeiiieatly)  by  Art. 67.  thvts  BC:  CA ; :  tin  ^ :  sin  B^ 

CA,%vbl  a  ■ 

■     ^^      ;  V  B  w  ifcnott7»;  whence  also  C^X^O'-^A^k- B \  '•'  C  is  likewUe 

hnotvn, 

z  e  S 


4»  PLANE  TRICSONOMETRY.  Paet  I£. 

ThetobiiummMoui  aperpendictilari  see  tbe  kst  figure. 
Natural  cos  ^=5.__ 


2 1^^.^C  264 

.6250000  •/  angle  AszbV   Wy  •/  C+B««18a»— 51«  IS': 

128»  41',  and  -.-±5ik64*  20'4. 

2  • 


Ixjg.  tan.  =log.  ^B— itfC+log.  tan.  64»  2<y4^-log. 


-rfB+^C=0+10.31S4222— 1.3617279==«.9566944   / 

C-t-  B     C—  B 
•.  angle  C«s-~--  +-— -  «s64*  20'^ +5*  l(f^=^S^  31'. 

2  2 

angle  B—  -^ o"""^*"  20>-5'  l(/4.=59*  lO'. 

2  2  ' 

Instrumenially,  first  method. 

1.  Extend  from  10  to  23  on  the  line  of  numbers ;  this  extent 
will  reach^  on  the  same  line^  from  1  to  2iV>  the  difference  of  the 
segments  of  the  base. 

2.  Extend  from  12  to  6.15  on  the  numbers  -,  this  extent  will 
reach  on  the  sines  from  90"  (radius)  to  SO*  &0'=BJD,  the 
complement  of  which  is  59*  10'=ang.  B. 

3.  Extend  from  11  to  3.85  on  the  numbers  >  that  extent  will 
reach  from  90*  to  20"^  on  the  sines,  the  complement  of  which 
is  69i=^C, 

Second  method.  1.  Extend  from  264  («a=2B^.-^C)  to 
165(=SZ|*+^C*— *B9*)  on  the  numbers  j  that  extent  will 
reach  from  90°  to  384-  on  the  sines^  the  complement  of  which 
}s  51-;-==angle  A. 

2.  Extend  on  the  numbers  from  23  to  1 3  that  extent  will 
reach  from  64* j.  to  45*" }  and  back  again  to  54.  on  the  tangents,  for 
half  the  difference  of  the  angles  B  and  C. 

Ex.  2.  Given  the  three  sides,  viz.  -^B=100,  AC=s»40,  and 
BC=s70.25  'y  to  find  the  three  angles  ?  Ans.  ang,  A=33''  35', 
ang.  B=l&'  22^  ang.  C=128«  3'. 

3.  Given  ^B»:^68.95,  JC=^7^,  and  BC^^OO,  to  find  the 
Jingles?  Ans.  ang.  -^=112**  6\  ang.  B=3S^  40'>  ang.  0=» 
.^9^  J4'. 


I^BTIX.  1NA0CB»IBL£  ISUQfiTS  &  DISTANCES.  4Si3 

THE  APPLICATION  OF  PLANE  TRIGONOMETRY  TO 
THE  FINDING  OF  THE  HEIGHTS  AND  DISTANCES 
OF  INACCESSIBLE  OBJECTS. 

The  uses  to  whick  Hane  Trigonometry  may  lie  applied  are 
50  various  and  extensive,  that  merely  to  point  them  out  would 
require  a  very  large  vc^ume  3  and  to  understand  them>  the  stu- 
dent must  be  well  acquainted  with  Geography,  Astronomy^  and 
the  numerous  branches  of  Natural  Philosophy^  of  which  this 
science  fnrms  a  necessary  part.  At  present  we  shall  confine  our* 
selves  to  one  of  its  immediate  and  obvious  applications^  namely^ 
that  of  determining  the  hdghts  and  distances  4)f  inaccessible 
objects. 

The  following  instruments  are  used  in  this  branch  of  men« 
suration>  namely,  a  quadrant,  a  theodolite,  a  mariner's. compass, 
a  perambulator,  Gunter's  chain,  measuring  tapes,  a  measuring 
rod,  station  staves,  and  arrows  ^  the  description  and  uses  of 
which  are  as  follow : 

84.  The  uuadrant  'is  an  instrument  for  measuring  angles 
in  a  vertical  position  9  that  is,  to  determine<the  angular  altitude 

'  Besides  the  common  surveying  qimdrant,  of  Which  that  described  abore  is 
the  simplest  form,  there  are  yarioas  other- kinds,  as  the  astronomical  quadrant, 
the  sinical  quadrant,  the  herodictical  quadrant,  i>ayis's,  Gunter's,  Hadlej'Sy 
Oole's,  CoUins's,  Adao^s's,  9fid  some  others.  Quadrants  may  be  bad  at  any 
f  rice  from  one  to  twelTe  guineas. 

The  height  of  an  object  may  be  taken  in  two  senses,  viz.  1.  its  perpendicu- 
lar distance  (in  fathoms,  yards,  feet,  &c.)  from  the  ground ;  3.  its  angular 
height,  or  the  number  of  degrees  contained  iti  the  angle  St  the  eye  of  the  ob* 
«enrer,  ^hicb  the  perpendicular  height  subtends ;  the  former  we  have,  for 
wfiitinction,  denexnioated  hHgkl,  the  latter  tUtUude. 


I  1fiC4 


4M 


HMfi  fnoiO»<mMXKr. 


IPawIX. 


♦♦ 


d  BBCf  pfOpOMQ  OCQCCt. 

jtBC  18  a  quadiant,  to 
the  centre  C  of  which 
the  weight  IFisfredy 
Mttpended,  b^  meaot  of 
the  string  CW$  <«  are 
two  sights^  through 
which  the  eye  of  an 
observer  at  Jl  sees  the 
object  O* 

The  arc  AB  of 
the  quadrant  is  divided 
Into  degrees,  which  are 
subdivided  into  halves, 
quarters,  or  single 
minutes.  In  using  this 

instrument,  the  obser-  Wk^'-^X^^^t^-^.^^A'Z^''''^^^^^^^  D 

ver  turns  it  about  the 

centre  C,  until  the  oh* 

ject  O  is  visible  through  the  sights  «# ;  and  as  he  turns  it,  the 

line  CW,  revolving  freely  about  the  centre  C,  moves  along  the 
circumference  AB^  when  he  sees  the  object  0  through  the 
sights,  the  arc  BWvinXL  be  the  measure  of  its  angular  altitude, 
that  is,  of  the  angle  OAD, 

Draw  OD  perpendicular,  and  AD  parallel  to  the  plane  of 
the  horizon  \  then  because  the  angles  at  E  and  D  are  rigM 
angles  and  the  angle  A  common,  the  triangles  CAE,  OAD  are 
equiangular  (33.  1.),  •/  the  angle  ACE=iAOD;  but  DOA-\- 
DAO»  (a  right  angles)  ACB,  from  these  equals  take  the 
equals  DOA^ECA,  aad  the  remainder  DA€hssECB.  And 
since  the  arc  BfV  is  the  measure  of  the  angle  ECB  (Part  8. 
Art.  237.)  it  is  likewise  the  measure  of  DAO,  or  of  the  angular 
altitude  of  the  object  0  above  the  plane  of  the  horizon. 

85.  The  theodolite  S  in  its  simplest  form,  consists  of  a  brass 


I  Some  of  the  best  theodolites  are  adapted  to  measuring  vertical  as  well  as 
hwizonttJ  aSgles  to  a  single  minute ;  being  fitted  with  vertical  arch,  lerel, 
telescopic  sights,  and  rack-work  motions.  The  prices  of  theodolites  are  from 
two  to  ibrty  guineas.  The  circumferentor  is  an  instrument  for  measuring 
horizontal  angles,  chiefly  used  in  wood  lands,  and  its  price  is  from  two  to  fire 


B 


Taut  a.  INACClBSIBIiB  BBteSTS  &  DUCTANCES.  48( 

€irc^e  of  about  a  foot  in  diameter,  having  it«  circumference 
divided  into  360  degrees,  and  these  subdivided  into  halves, 
quarters,  or  minutes;  the  index  sCs  turns  About  the  centre  C# 
and  has  fixed  on  it  two  sights  s  s;  there  are  likewise  fixed  oo 
the  cir^mierence  two*sights  »  n ;  this  cirde  i«  fixed  in  {^  hori* 
zbntal  position  on  three  legs  of  a  convenient  height  for  making 
observations. 

The  theodolite  is  used  ^  ^ 

for  measuring  the  angular 
distanciM  of  objects  situated 
on  the  plane  of  the  hori- 
zon j  thus. 

Let  A  and  B  be  two 
objects,  place  the  instru- 
ment in  such  a  position 
that  one  of  them,  as  A,  may 
be  aeen  through  the  fixied 
fights  n  and  n  by  an  eye 
atF. 

Turn  the  index  9  s 
about  the  centre  C,  until 
Che  other  ot^ect  B  appears 
trough  the  sights  $  s  to 
an  eye  situated  at  E ;  then  will  the  angle  ACB^  which  is  meli«ared 
by  the  arc  nr^  be  the  angular  distance  of  the  given  objects  A 
and^. 

86.  Thb  mariner's  compass  ^  is  an  instrument  used  for  find- 
ing the  position  or  bearings  of  objects  with  respect  to  the  meri- 
dian, and  for  determinii^  the  counie  of  a  ship :  wliat  principaHy 
requires  explanation  is  the  eard ;  it  is  a  round  piece  of  stiff 
pasteboard,  having  its  circumference  divided  into  thirty-two 


c 


gnineas.  The  semicircle  is  a  macb  simpler  and  cheaper  iustrument  than  the 
theodolite,  and  serves  very  well  for  measuring  angles  on  the  plane  of  tbe 
horizon  where  very  great  accuracy  is  not  required. 

^  The  invention  of  the  mariner's  compass  is  usually  afcribed  to  Flavio 
Gioia,  an  Italian,  A.D.  1302;  but  it  is  stated  by  some  authors  that  the 
Chinese  had  a  knowledge  of  it  as  early  as  the  year  1 1 30  beibre  Christ.  The 
price  of  this  usefcrl  instrument  is  from  half- a- crown  to  twelve  guineas. 


4s<  MANE  TuieosraiiEniY.         put  is. 

eqml  pMts,  cdled 
poMb ;  K  Bteel  wire, 
called  (he  needle, 
vrbMi  hu  beea 
rubbed  with  «  kttd- 
ttoDe,  it  fixed  acroM 
the  und^  ude  of 
the  card  from  N  to 
S,  by  which  means 
(when  the  card  is 
ezactl;  balanced  on 
id  centre)  the  pMot 
N  is  directed  to  the 
north,  and  conse- 
quently the  point  S 
to    the  south,  and 

each  (rf  the  remaining  pcnnts  to  its  .reapective  paaition  in  dw 
horizon  j  in  the  centre  of  the  card  URdemeath,  b  fixed  a  finely 
polished  conical  brass  socket,  aboat  one  third  of  an  inch  deep. 
The  compass  box  is  a  basin  of  brass  or  wood,  having  a  fine 
pointed  steel  needle  fixed  perpendicularly  in  its  bottom :  on  the 
point  of  this,  the  above-meationed  socket  in  the  bottom  <^  the, 
card  being  placed,  the  card  is  bidanced  and  turns  freely  as  im- 
plied by  the  attractive  force  of  the  magnet.  The  box  is  sus- 
pended within  a  brats  hoop  or  ring,  by  means  oi  two  gimbdi 
placed  on  opposite  sides,  which  serve  as  an  axis,  and  admit  &«e 
motion  i  and  this  hoop  is  in  like  manner  suspended  on  the  oppo- 
site sides  of  a  square  wooden  box  by  gimbols,  at  90"  distance 
fixtm  the  former,  a  contrivance  intended  to  secure  the  horizontil 
position  of  the  inner  box  and  card,  wliatever  may  he  the  motion 
of  the  ship  in  which  the  compass  is  placed  '. 


*  Tboie  wbo  crou  (oreita,  deurti,  and  aDinbabited  coantiiag,  find  thi> 
|D>tniiiient  a  nectunj  compsDioa  to  direct  themi  tbej  kerp  tbe  oompan 
alwBjt  before  tbtm,  and  Ibllaw  tbe  direction  of  tbat  poiDt  which  indicate]  tbe 
■itnUion  of  tbe  place  tbe;  wisb  to  arrive  at.  Tbe  like  metbod  U  emplojel 
id  heerios  a  ibip,  wbicb  i(  kept  in  •och  «  pmitioo,  tbat  the  propoeed  pm^ 
majp,  of  ill  o«H  accord,  etand  in  a  direction  towanU  tbe  bead  irf  tbe  ibip. 
Note,NbElDeB]UHrfiliy«Jl;  tiJli E,  itortknordMiuti HEhK.  mrrlk^u* 
in  iMrdt  j  N  £,  northeait,  &c  &c.  Y^icb  will  be  eaillj  uaderahrad. 


NORTil^ 

Pts 

Degrees 

SOUTH        1 

Nb£ 

Nb  W 

1 

no  16' 

SbE 

Sb  W 

Kne 

NNW 

3 

es  30 

SSE 

SSW 

N£bN 

NWbN 

3 

33    45 

SEbS 

SWbS 

NE 

NW 

4 

45     0 

S£ 

SW 

NEbE 

NWbW 

5 

56    15 

SEbE 

SWbW 

£N£ 

WN  W 

6 

67   80 

ESE 

WSW 

BbN 

WhN 
WEST 

7 

78   45 

EbS 

WbS 

EAST 

8 

90     0 

EAST 

WEST 

^abtI^.  inaccessible  heights  &  distances.  4«r 

87.  A  table  shewing  the  degrees  and  minutes  that  every  point  of 
tlie  compass  makes  with  the  meridian  ^ 

Exjdawttion. 

In  the  preceding  figure 
tbe  line  N  S  is  called  the 
'mei^idia$iHne  s  the  two  first 
colamns  of  the  table  ex- 
tend from  nxurth  both  ways 
to  east  and  west,  as  tb« 
two  last  do  from  south  ^ 
the  two  first  points  in  the 
first  and  second  columns 
make  the  same  angle  with 
the  meridian  line  N  S  (ll« 
15')  reckoning  frvm  the  north  point,  that  the  two  first  in  the  5th  and  6th 
columns  do,  reckoning  from  the  south,  and  the  like  is  CTidently  true  of  the 
points  in  any  horizontal  line  of  the  table.  The  angles  made  by  the  points  in 
tbe  first  and  second  columns  witii  the  meridian  are  therefore  measured  by  the 
arc9  intercepted  between  them  and  the  north  point,  viz.  tbe  first  column,  on 
the  east  side  of  north  ;  and  the  second  on  the  W€9t :  in  like  manner  the  angles 
made  by  the  points  in  the  5th  and  6th  columns  with  the  meridian  are  measured 
by  the  respective  arc*  intercepted  between  them  and  the  touth  point,  those  in 
the  5th  column  being  on  the  east  of  south,  and  those  in  the  sixth  on  the  west: 
for  example,  N  N  E  is  92<»  30'  to  the  east  ef  north,  N  N  W  is  the  same  distance 
W€9t  of  north  ;  SS  E  is  the  same  distance  east  of  smithy  and  S  S  W  is  tbe  same 
distance  west  of  south.  In  the  third  column  each  number  denotes  the  distance 
from  north  or  soUtb  of  the  points  agdinst  which  it  stands ;  and  tbe  numbers 
in  the  fourth  column  shew  the  degrees  ai^d  minutes  of  ihe  arc  intercepted 
between  the  north  or  souths  and  the  points  against  which  they  stand. 

88.  The  use  of  the  above  Table. 
When  a  question  is  proposed  in  which  the  conditions 
require  that  lines  should  be  drawn  in  given  positions  with 
the  meridian  expressed  in  points  of  the  compass,  the  construc- 
tion may  be  made  with  the  greatest  fiaicility,  by  means  of  this 
table  5  to  eflfect  which  this  is  the 

KuLE. — 1.  Describe  a  circle  and  draw  the  diameter  NS  for 
the  meridian,  N  being  the  north  point,  S  the  south. 

2.  Take  the  degrees  and  minutes  from  the  table  which  cor- 
respond with  the  points  mentioned  in  the  question,  and  mea- 
sure arcs  from  the  meridian  equal  ta  them. 


^  The  table  is  thus  constructed :  divide  360  (»  the  number  of  degrees  in 
the  circumference  pf  a  circle)  by  32  (=»  the  number  of  points  in  the  compass,} 
and  the  quotient  is  ^  part  of  the  circumference  — 11<*  15',  or  1  point  of  the 
compass  ;  this  doubled  is  23<*  30'  for  two  points  ;  its  triple  is  3d<*45'  for  three 
]H>ints,  and  so  on. 


4S8 


fUNK  TBiaO»rQMinEf . 


»iwIX 


8.  Ikiw  liQei  Uiraigli  4ie  ecvtre  |p  t^  pmote  Hups 
sured,  and  construd  your  figure  by  drawing  its  aides  retpectiT^f 
parallel  to  these,  and  each  of  its  proper  length  taken  fipom  a 
scale  of  equal  parts. 

4.  If  the  position  of  one  of  the  lines  be  required^  draw  f 
line  parallel  to  it  through  the  centre  of  the  circle^  measure  tht 
angle  this  line  malces  with  the  meridian,  then  the  point  of  the 
compass  which  stands  opposite  this  measure  will  give  the  besyr* 
ings  or  position  required  ';  and  its  length,  taken  in  the  eom* 
passes,  and  applied  to  a  sqale  of  eqpal  parts>  will  give  it| 
measure. 

Examples. — 1.  A  man  intends  to  travel  from  C  to  Z  which 
lies  N  N  W  from  C  6  miles,  but  he  must  arst  call  ^t  D»  whi4 
lies  N  £  3  miles,  then  at  A  N  b  W  frt>m  D  5  miles,  and  lastly 
at  £',  which  is  S  Wfrom  H  41-  miles;  at  Hhowfar  is  he  distant 
from  Z,  and  what  course  must  he  travel  to  arrive  there  > 

Here  I  first  draw  cCZ  through 
the  point  d,  distant  2t®  SO^  from 
N  (answering  to  N  N  W) ;  next 
I  draw  itb  at  45^  distance  from  N 
(answering  to  N  E)  ;  next  1  draw 
rn  at  II®  15'  distance  on  the  left 
of  N  (answering  to  N  b  W) }  and 
since  a8zsNbzat4&^,  it  is  plain 
that  ah  will  be  the  S  W  as  well 
as  the  N£  line.  I  then  take 
CDs3,  draw  DH  parallel  to  rn 
and  make  it=5,  whence  I  draw 
HK  parallel  to  ab  and  make  it= 
4,,  1  then  join  KZ  and  find  its 
measure  to  be  2^  miles  nearly, 
and  its  bearings  (shewn  by  the 
paraUel  xv,  the  position  of  which 
is.  measured    by   the    arc    Nv) 


'  Ttie  pQtitiop,  or  heariji|pi  ctf  »  Une  nipy  likewise  he  known  by  simply 
4fAwiqg  .a  meridian  from  the  g^Tep  point,  and  measnrii^  the  angle  which  that 
iiae  mak^  witli  it  $  the  d^reet  cont^pfd  in  it  .beiq|  fo^nd  in  the  table  wiU 
shew  the  point  of  the  compass  required. 


PiRT  IX.  INACG^^lBLlft  itfiiaairS  k  DISTANCES.   489 

N  Ifl^  fi"*,  ^ilkt  tt> )lbJ& f>^B>  of  74^egraM  to  tbe eastward 
of  north  by  east 

2.  B  is  8  ihileB  NW  From  C,  and  il  4  miles  N  from  B-, 
requirad  th«  course  uid  distance  frwn  AtoCP  An».  €ourse  S 
3P4E.    Distance  11  mites. 

3.  A  ship  sailed  S  E  12  leagues^  N  N  E  20  leagues,  and 
NN  W  SO  leagues  $  required  her  distance  from  the  point  sailed 
from^  and  hter  course  back  ? 

89.  The  perambulator  »,  called  also  a  pedometer,  waywiser, 
and  surveying  wheely  is  an  instrument  for  measuring  large  dis- 
tances on  ground  nearly  level ;  it  consists  of  a  wheel  8^  feet  in 
circumference,  which  the  noeasurer  drives  before  him,  by  means 
of  two  handles,  fixed  at  the  end  oi  a  hollow  shaft,  terminating 
in  two  cheeks  to  receive  the  wheel,  and  in  which  its  axis  turns. 
The  wheel  goes  over  one  pole  of  ground  in  every  two  revolu- 
tions, and  its  motion  is  communicated  by  the  intervention  of 
various  clock-work  movements  within  the  shaft,  to  a  dial,  fixed 
near  the  handles,  the  index  of  which  points  out  the  distance 
passed  over.    * 

The  Gunter's  chain  "*  is  used  to  measure  smaller  distances 
tban  those  to  which  the  perambulator  is  applied  j  its  length  is 
66  feet=:22  yardsrs4  poles,  and  is  divided  into  100  links,  each 
7,92  inches  in  length.  This  is  the  most  convenient  instrument 
of  any  that  has  been  contrived  for  measuring  land,  because  10 


•*■  ttt!  bearings  «f  two  tAjectB  from  vackotber  may  be  estimated  otther  in 
<fc^ee»,  or  po^t ;  degrees  may  be  tarned  into  fiomts,  or  poiats  into  degve^ 
l»y  referring  to  the  table ;  thtts,  if  an  object  bear  8«»  46'  to  the  east  of  sooth* 
bytaming  tethfefeblc  I  ftttd  that  the  exact  pof  nit  of  bearing  is  SEbS;  if 
it  bear  25*»to  the  ^est  of  north,  the  bearing  'm  ptim*  n  NNW«»a«'W; 
that  !i,'j|«aO''Wt8tof  NNW.  Or  the  Teckonlng  mf^bemadetotiie  neareet 
^rterpoiki,  tfciis  N14»4'W  is  N  bW^W;  S««»7<{-£  U  SS£i£; 
ikVXe  manner  Nier4« 41' £  is  N£ b£4  E,  fto.  fte. 

■  Ttie  prtce  of  Ifliis  instrtiment  Is  Mit  «v«  to  ten  gnioeas.  l^ie  name 
hdm^&r  is  likewise  appHed  to  «h  instraifleiit  of  a  watch  tiae  ^  the  ipockct, 
tot  Kscertidning  distances,  either  walking  or  ridiag,  and  eoels  from  tfafiee  to 
tfteen  guftteas.  Tbe  ptra'ftrbiilator,  CWmten^s  «iate,  and  ta^es,wiU  measofe 
iWth  »uiB<irent  txactn^s  for  most  fmrfM^s  whete  <he  gtomid  is  level,  but 
where  it  is  not,  distances  should  be  TofBd  by  trigonott^tikal  ealevkttien. 

•  theOnritfcr«sc1iain  Will  cdl*  IHMn  iws  to  foOTteen  thlUiDgB,  aooov^to^  to 
its  strength,  apdihe  pie(le«ftidli  ^ 


'430  PLANE  TRIGONOMETRY.  P«t  IX, 

diains  in  lengthy  aod  oaeia  breadlii^  (sslOOOOO  squave  links) 
make  just  an  acre. 

91.  The  measuring  tapes  '  are  of  cftxe,  two^  thfee,  or  !b«r 
poles  in  length;  they  are  applied  to  the  same  purposes  as  tbe 
chain,  and,  if  kept  dry,  will  measure  with  tolerable  exactness. 

92.  The  measuring  rod  may  be  of  six>  eight,  or  ten  feet  in 
length ;  it  is  divided  into  single  feet,  which  are  subdivided  inta 
halves  and  quarters,  or  into  tenths  of  a  foot,  for  the  convenience 
of  measuring  small  distances. 

93.  Station  staves  or  prickets,  are  staves  of  about  five  or  six 
feet  in  length,  having  a  small  flag  fixed  at  one  end,  the  other 
end  being  sharpened  to  a  point  for  fixing  in  the  ground; 
these  staves  are  used  in  measuring,  for  marking  stations,  which 
are  required  to  be  seen  and  distinguished  at  a  distance. 

94.  The  arrows  arc  of  wood  or  iron,  pointed  at  one  ehd, 
and  their  use  is  to  stick  in  the  ground  as  a  mark,  at  the  end  of 
every  chain  or  other  measure. 

95.  Fboblems% 

Prob,  1.  An  observer  at  113  feet  distance  from  the  foot  of  an 
obelisk,  finds  its  angular  altitude  to  be  40^ ;  required  its  height, 
that  of  the  observer's  eye  above  the  plane  of  the  horizon  beic^ 
5  feet? 

p  These  tapes  are  sold  at  the  sbops  of  the  niathttniatica}  instrament  makert, 
and  cost  from  five  to  twelve  shUHngs>  according  to  their  length. 

The  above  instruoients^  at  the  prices  we  have  mentioned^  will  perhi^  be 
found  too  expensive  for  the  student's  pocket ;  in  that  case  his  own  ingeouitjE 
may  supply  him  with  all  that  is  necessary  for  measuring  vertical  and  horizon- 
tal angles  and  distances.  A  theodolite  may  be  made  with  a  circular  piece  of 
stiff  pasteboard,  gradnated  and  nailed  (through  its  centre)  on  the  top  of  a 
piece  of  mop^stick,  the  other  end  of  the  etiek  being  sharpened  to  a  point  for 
fixing  it  in  the  ground.  A  qoadraat  likewise  may  be  made  of  pasteboard,.  ' 
in  like  manner  graduated,  and  having  a  piece  of  lead,  or  a  stone,  hung  frobi  itt 
centre  by  a  strnig.  The  chain  or  tapes  may  have  their  place  supplied  by  a 
string  previously  measored,  divided,  and  subdivided,  according  to  the  mind  of 
the  operator.  The  measuring  rod  may  be  made  of  any  stick,  of  a  proper  length 
and  thickness.  The  station  staves  may  be  made  of  sticks  having  one  end 
pointed  and  the  other  split,  for  the  purpose  of  holding  a  piece  of  white  papcr^ 
and  the  arrows  may  be  cut  ou^  of  any  hedge. 

With  apparaitus  of  this  kind,  I  have  frequently  known  altitudes  and  difitanff^ 
determined,  with  sufficient  ezactii^  for  any  commoo  purpose. 


PaatIX.   inaccessible  heights  &  distances.   431 


Npte,  In  ftftdiag  the  height  of  ofatiects^  to  the  observed  hngtit 

must  be  added,  that  of  the  obeerver*9  eye  above  the  ylaoe  of 
the  horizon. 

Let  AB 

be  the  obelisk, 
CB  the  dis- 
tance of  the 
observer,  and 
J?£  the  height 
of  his  eye ; 
then  JIE  is 
the  part  re- 
quired to  be 
found. 

In  the  tri- 
angle ACE,  we  have  given  C£=11S,  the  angle  -rfC£=40%' 
consequently  C.il£s:(90— 40ss)   60^,  and  the  angle   CEA  a 
right  angle;  to  find  ^£. 

Now  (Art.  67)  CE:  EA  ::  sin  A  :  sin  ACE,  •.•  EAss 

CE.  Bin  ACE 

"    ^,     2 ,  and  log.  E^srlog.  C£+log.  sin  ^C£— log.  sin  A 

3s8.05SO784+9.a0S0675-9.884254O=:1.9768919>  the  natural 
Hittiber,  corresponding  to  which  is  94.8182s=il£,  *.*  ^£+££=a 
94.81884- 5=:99.8182  feetss99  feet  9  inches  iff4==the  height 
lequired. 

Pro6. 2.  The  angular  altitude  of  a  spire,  known  to  be  137 
feet  high^  is  51^  \  now  supposing  the  height  of  the  observer's  eye 
to  be  5-i-  feet,  how  &r  is^he  distant  from  the  foot  of  the  spire  ? 

l^oit.  In  questions  of  this  kind,  the  height  of  the  eye  must 
be  subtracted  from  the  given 
height,  previous  to  the  operation. 
Here  are  given  ^£sl37y 
£B=S^,  •.•  ^£=137^5^=s 
131.5,  AEiy  a  right  angle,  and 
angle  ^JD£=:51^  •••  ang.  DAE-zs: 
(900— 5P=)  390. .  (Art.  67.)  I>£ 
:  E-df  : :  sin  BAE  :  sin  ADE  •.• 
EA.  sin  BAE 


I>£s 


sin  ADE 

131.5  xsm  39^         ,  .._    ,  |^ 

r-zTT — >  ••'  IPB  i>£=log       n 

sm  51*       ^       ^  ^^ 


433  PLANS  TRIGONOHKTRY.  Fast  IX. 

ISl.S+Iop.  tin  Sd-log.  Oa  51"=«.ll»258+9.7»B«7lfl- 
9.6906(n6=S.0Cr%95O  .-  i)£  so  106.487  feet=IOS  §e^  fi 
inches  -rfr- 

Prob.  3.  Wanting  to  calculate  the  perpendicular  height  of  & 
cliff,  I  took  its  angular  altitude  IS"  3<y,  but  after  measuring 
950  yards  in  a  direct  tine  towards  its  base,  I  was  unexpectedly 
Slopped  by  a  river;  here  however  T  ag;ain  took  its  altitude  69° 
SO'i  required  the  height  of  the  cliff,  and  my  distance  from  the 
centre  of  its  base  P 
Let  ^  be  the 
first  station,  B  the 
second,  C  the  sum- 
mit of  the  cliff,  and 
D  its  base;  then 
^6=950,  the  an- 
gle    ^=ir     30',  -A- 
angle /iBC=  (1800 

—690  3tf=)  110"  

30'  ■.■  ang^CB=(18O-12''3O'  +  ll0',30'=18O'-123'=)  S7»i 
■/  in  the  triang^  ABC  we  have  Ote  side  AB  and  the  three 
angles  given,  to  find    BC.    Now  (Art.  67.)  AB  :  BC  ::  ua 


sin  ACB ' 

sin  .rf— log.  sin  ACB=  (log  950+log.  sin  12»30'— log.  sin 
57°=)  2.9777236+9.3353368-9.9235914=2.3894690,  ■.■  BC 
=24S,17I;  having  ftinnd  BC,  there  is  given  in  the  triangle 
.  BCD  the  right  angle  BBC,  the  angle  CBI>=69"  Stf,  the  angle 
BCI>=(90*-69"  30'=)  80P3tf  and  the  side  BC=iM5.m,-.* 

(Art.  63.)  BC :  BD  ::  rad  :  sin  BCD,  ;■  BD=f^'^^"= 

85.8608  yards.     Also  (Art.  63.)  BC  :  CD  ::  rad  :  rin  CBDj 

BC  sin  CUT} 
:■  CD= — '-—^ =249.645  yiuds. 

Prob.  4.  Two  persons,  situated  at  jt  and  B,  distant  ^  miles, 
observed  a  bright  spot  in  a  thunder  doud  at  the  same  instanlj 
its  altitude  at  A  was  46°,  and  at  B  63° 30';  required  its  perpe»- 
dicular  hei^t  Irom  the  earth  ? 

BtsI.  Angle  .<iCB={180"— 46«+6S''  SB's)  70* ,30",  Aen 

(Art.  67.)  AB  -.BC::  sia  ACB :  sin  BAC,  w  SC=^^^^^  '< 


1 


P^HT  K. .  INACCESSIKLB  HSfOHTS  ft  DISTANCES.    433 


=2.1361^  miles.    Wherefore  in  the, rj|ght  apgled  triapgle  BCD, 

BC  :  CD  : :  rad  :  sin  CBD  (Art. 63),    /  CD^^^' ^'^^^^^ 

/-03/f/  rad 

1.9117  mile=th6  height  required. 

Prob.  5.  Two  towns,  A  and  B,  are  invisible  and  inaccessible 
to  each  other,  by  reason  of  an  impassible  mountain,  situated 
between  them;  but  both  of  them  are  visible  and  accessible 
from  the  point  C,  viz.  A  bears  N  E  from  C  distance  3  miles,  and 
B  bears  N  b  W  from  C  distance  B-^  miles  3  required  the  bearings 
and  distance  of  A  and  B  from  each  other  ? 

First,  Since  CJ  lies  N  E,  or  45® 
on  the  east  of  the  meridian,  and  CB 
lies  N  b  W  or  11^  15'  on  the  west, 

V  angle  C=  (45°+ 11°  15'=)  56o 
15';  •.•  (Art.  72.)  CB-\-CA  :  CB— 

tA  : :  tan  — - — :  tan  — - — :  or 

8.95  :  2.25  : :  tan  61°  52'i  :  tan  27** 

V  57''  3  then  (Art.  69.)  angle  4= 
(6F  52' 30''+ 27^  1'  57''=)  880  54' 
27",  and  angle  B={61^  52'  3(/'- 
27°  r57"=)  34®  50'  33";  next, 
(Art.  67.)  C^  ;  ^B  : :  sin  B  :  sin  C, 

...  ^B-:£:!:i!^=  4.36606    miles, 
sm  B 

Lastly,  through  the  centre  C  draw 

ab  parallel  to  AB,  and  measure  the 

circumference  Net,  and  it  will  be  found  to  contain  46®  6',  which, 

by  refi^rring  to  the  table  (Art.  87.)»  will  be  found  to  answer  to 

the  N  W  point  nearly;  that  is,  B  bears  from  A  N  W  1®  6'  W 

distance  (4.36606  miles=)  4  miles  3  furlongs  nearly. 

VOL.  1|.  F  f 


4ft4  FLAKE  TBIGONOHEniT.  Part  IX. 

Prob.  C.  A  general  wriring  wHh  his  army  on  the  b2nk  of  a 
river  is  deairoiu  irf  crawiDg  it,  but  there  are  two  of  the  enetaft 
fortresses,  jI  and  B,  on  the  opposite  shore,  and  he  wishes  to 
know  their  bearings  and  distance  from  each  other;  for  this  pur- 
pose two  stations  C  and  D  are  chosen  close  to  the  river  side,  C 
being  directly  east,  from  D  at-i  mile  disljmce ;  at  C  the  angles 
are  as  follow,  viz.  ACB=6eP,  BCD=3Vi  at  D  the  angles  are 
jlDB=e2',  jtDC=Si*.  Now  suppose  be  crosses  directly  froni 
the  point  D,  required  the  bearings  and  distance  of  ^  and  5 
from  each,  other  i  the  width  of  the  rirer  at  the  point  of  croesiDB;, 


Part  IX.   INACCESSIfiLE  HEIGHTS  &  DISTANCES.  435 

and  thp  distance  of  the  point  wbere  he  proposes  to  land  from 
AarndB} 

First  In  the  triangle  DACy  there  are  given  DC=^  mile=r 
.75,  the  angle  ^DC=64%  DC^=  (32'*+ 68°=)  100,  and  DJC 
=  (180—164=)  16*^5  to  find  DA.    By  Art.  67.  DC :  DA  : ; 

'     ^.r.     .     T^ry.        « ^     DCxsiu  DCA     .75  X  sitt  100° 

»in  DAC  :  sin  DCA,  •/  DA^ _-__= r— r^^—  = 

sin  DAC  sm  16° 

3.67963  miles. 

Secondly,  In  the  triangle  BDC,  there  are  given  DCzsjB, 

^JDC=(62°+64°=)  ,126°,   i>C£=32°,  wad    I>J5C=  ( 180°— 

126°-f  32°=)  22°,  to  find  BD.     By  Art.  67.  DC.BD::  sin 

T^Di-      •     r^^D         i>n     I>C X sin  DC5     .75  X sin  32° 

DBC  :  sm  DCS,  •.•  5D= : — ~-^,     =  — .    ^^^ — = 

sinDJ^C  8in.22° 

1.06095  miles. 

Thirdly.    In  the  triangle  BDA  there  are  given   DA=s 
2.67963^  j!?D=  1.06095,  and  the  included  angle  AD£=:eQ'* ;  to 

find  the  angles  DBA,  BAD,  and  the  side  BA.  Now r 

180°— 62° 


=  59°=half  the  sum  of  the  angles  DBA,  BAD  at 

the  base}  also  ^D+I^^— 2. 67963 +1.0609$  =s 3. 74058= sum 
of  the  sidfes,  and  ^D— D J? =2. 67963— 1.06093 =1.61 868as 
diff.  of  the  sides.     But  (Art.  72.)  AD^DB  :  AD-^BD  : :  tan 

DBA+BAD  DBA-^BAD      , 

-=^-^^ :  tan -,  that  is,  3.74058  :  1.61868  : : 

2  2 

,^    1.6 i 868  X  tan  59°     ,      «„«  .^  ^„     ,.  ,^  .^     ^.«. 

tan  59°  :   — - — =tan  35<»  42^  5"=half  the  difference 

3.74058 

of  the  angles  DBA,  BAD  at  the  base. 

•  ^A  t  150  ^  /59°+35°  42'  5''= 94°  42'  5''  =the  angle  DBA. 

•/  (Art.  09.)  1 5^p_35.  42,  5//-230  17/  55'/=,the  angle  BAD. 

Also  (Art.  67)    BD  :   B^  : :  sin  BAD  :  sin  B£>^,  •.•  jB^= 

.BD  X  sin  BD^     1 .06095  x  sin  62°     ^  „    •^^      ., 

:^ — K-7T^ — =    .    ^00  ,^/ .>// =^'36842  miles. 

8in:J9.iJ>  sin  23°  17' 55'" 

Fourt^y.  In  the  triangle  DBE  there  are  given  the  angle  E 
a  right  angle  D-8£=(180°— i)B^=180°-94°  42  5''=)  85« 
IT  55",  the  angle  BD£=(9Q°— DB£=90°-85°  I7'  55"=) 
4°  42'  5",  and  the  side  BD=  1.06095  -,  to  find  the  sides  BE  and 
DE. 

By  Art,  63.   DB  :   BJE  :  :  rad  :  sin  BDE,  '.'  BE  = 

F  f  2 


43«  PLANE  TRIGONOMETRY.  Part  K. 

DBx&lnBDE     1 .06095  x  sin  4M^' 5''      ^«^^,«      .,       ' 

; = ; =  .086958  mile  = 

rad.  rad. 

somewhat  more  than  150  yards. 

^  I)B  X  sin  DBE 

Also  DB  :  DE  ::  rad  :  sin  DBE,  •.'  D£= —z 

rad. 

1.06095  X  sin  85^  17'  55     ,     .  ^^^     „ 

= —  1 .05738  Diile. 

rad. 

Lastly.  Since  the  line  CD  lies  directly  east  sind  west,  any 

line  CN  drawn  perpendicular  to  it  wiH  represent  the  meridian^ 

and  the  acute  angle  BNC,  which  AB  makes  with  CN,  will  be  the 

bearings  of  B  from  A  ;  this  angle  may  be  very  readily  determined 

in  the  present  instance ;  for  since  the  two  opposite  angles  DCN 

and  DEN  of  the  quadrilateral  DENC  are  two  right  angles,  the 

two  remaining  angles  EDC-^ENCsz^   right  angles  (cor.  1* 

3«.  \.)',  but  EZ>C=(4«  4«'  5''4-«2<'  +  64o=5)   130^  42'  5'',  v 

EiVC=(l800-J5Z>C=rlSO°-130'*  42'  5''=)  49^  17'  55'^  which 

in  the  table  (Art.  87.)  answers  to  S  W  4*»  17'  55''  W  or  S  W  i  W 

nearly  j  for  the  bearings  of  B  from  A, 

Prob.  7.  Required  the  perpendicular  height  of  the  spire  of  a 
church,  the  angular  altitude  of  which  is  40^ ;  the  observer  being 
187  feet  distant,  and  his-eye  54^  feet  from  the  ground  ?  Answer 
K0AS7feet, 

6.  The  angular  altitude  of  an  observatory  is  53**,  its  perpen- 
dicular height  129  feet,  and  the  height  of  the  eye  5  feet ;  re- 
quired the  distance  of  the  observer?     Ans.  93.4407 /eef. 

9.  A  ladder  30  feet  long  reaches  23  feet  up  a  bdiiding ;  re- 
quired the  angle  of  inclination  at  the  foot,  and  its' distance  from 
the  wall?     Ans.  inclination  50>  3'  SO'';  distance  19.261 3 /ciif. 

10.  A  shore  1 1  feet  long,  in  order  to  support  a  wall,  is  placed 
so  that  the  angle  at  bottom  is  double  the  angle  at  tc^,  how  high 
tip  the  v^all  does  it  reach,  atid  how  fhr  distant  from  the  wall  is 
its  foot  ?     Ans.  heigHt  9.52628  feet ;  distance  %rf^eL 

11.  Required  the  altitude  of  the  sun,  when  the 'length  of  a 
iDan*8  shadow  is  double  its  height,  and  likewise  when  it  is^ODly 
half  its  height?  Ans.  26°  34'  5''  in  the  first  case,  and  63^ 
25'  55'''  in  the  second. 

12.  A  maypole  being  broken  by  a  sudden  gust  of  \Vind,  the 
Upper  par*  (which  still  adhered  by  some  splinters  to  the  stumps 
inade  with  the  ground  at  15  feet  distance  from  the  stunip,  an 


PARxlXi   INACCESSIBLE  HEIGHTS  &  DISTANCES.   437 

AOgle  of  7^  30';  required  the  height  oi  the  maypole  and  the 
leqgth  of  each  of  the  pieces  ?  Ans.  stump  29.2072  feet,  upper 
€nd  80,46^6  feet,  whole  length  ^9-6696  feet. 

13.  A  ship  having  sailed  234  miles  between  the  south  and 
WjBst^  finds  herself  96  miles  distant  from  the  meridian  she  sailed 
from  i  required  her  course  and  difference  of  latitude  ^  ?  Ans. 
course  SSW  2*  13'  15'^  west;  diff,  of  latitude  213.401  miles 
9outh. 

.  14.  There  are  three  towns  A,  B,  and  C;  from  5  to  C  the 
distance  is  7.625  miles  3  at  B  the  towns  A  and  C  subtend  an 
angle  of  51°  ^5',  and  at  C  the  towns  A  and  B  make  an  angle 
of  37°  21^5  required  the  distance  from  A  to  each  of  the  other 
two  ?  Ans.  from  A  to  B  4.6275  miles,  from  A  to  C  5.9482$ 
miles.  *' 

15.  Within  sight  of  my  house  there  is  a  church  and  a  mil], 
^e  former  is  distant  2.875  miles^  the  latter  4.24625  miles,  and 
they  subtend  ^n  angle  of  47°  23' 3  required  the  distance  from 
the  mill  to  the  church  ?     Ans,  3.125  miles. 

16.  A  &rmer  has  a  triangular  field,  the  sides  of  which  are  as 
follow,  viz.  AB:^7S0  yards,  -4C=690,  and  JBC=8505  he  is 
desirous  of  dividing  it  into  two  pails  by  a  l^dge  from  A,  per- 
pendicular to  BC;  required  its  length,  and  likewise  whereabouts 
it  will  meet  the  hedge  BC  ?  Ans.  length  585.31  yards;  distance 
from  C  365.2942  yards. 

17.  "^  A  man  travels  from  ^  to  jB  5^  miles,  then  bending  a 
little  to  the  right  hand  of  the  direct  road,  he  arrives  at  C  distant 
from  B  3  miles  -,  from  C  both  A  and  B  are  visible  under  an 
angle  of  25»4-  -,  what  is  his  distance  from  home  by  the  shortest 
cut  ?     Ans.  7.796  miles. 

18.  A  man  having  ti-avelled  from  ^  to  ^  5-4-  miles,  attempts 


p  The  angle  wbich  the  directum  m  sk^  soils  nm  makes  with  the  meridiaD,  n 
called  her  course,  whence  in  the  present  case,  constract  a  right  angled  triangle, 
the  bypothenuse  of  wbich  is=2d4,  this  will  be  her  distance,  the  ba8^»S6  will 
be  her  departure,  and  the  perpendicular  will  be  her  difference  of  latitude ;  and 
the  same  in  all  cases  of  plain  sailing. 

4  Problems  similar  to  this  and  the  following  one,  are  given  by  Ludlam,  to 
shew  how  the  apparent  ambi|^ity  of  a  problem  is  sometimes  corrected  by  the 
wording  ;  particular  attention  mast  be  paid  to  '  bending  a  little  to  the  right" 
in  prob.  n .  2in\  *  attempt*  to  return*  in  prob.  18.  and  the  solution  will  be 
attended  with  no  difficulty. 

Ff3 


k 


.  438  PLANE  TRIGONOMETRY.  Pakt  IX, 

to  return,  but  a  thick  fog  coming  on,  he  roistakefii  bis  way,  and 
takes  a  road  which  tends  a  little  to  the  right  hand  of  bis  proposed 
rout  5  arriving  at  C,  3  miles  from  B,  he  discovers  his  mistake, 
and  the  fug  clearing  up,  he  sees  both  A  and  B  under  an  angle 
of  154% )  how  far  is  he  distant  from  home  ?     Ans.  2.38  miles. 

19.  In  order  to  measure  the  breadth  of  a  harbour's  mouth,  a 
station  was  taken  at  its  inner  extremity,  where  the  angle  made 
by  the  two  projecting  points  which  form  the  harbour  was  ob« 
served,  viz.  33®  40'  -,  the  line  bisecting  this  angle  being  pro- 
duced 1900  yards  backward  and  another  observation  made,  the 
fore-mentioned  points  were  found  to  subtend  an  angle  of  17* 
SO';  required  the  breadth  of  the  said  entrance,  and  how  for  the 
harbour  extends  inltoid?  Ans.  breadth  751.904  yards,  perp. 
extent  inland  124*2.6  j^ards. 

'SO.  Three  trees  are  planted  in  such  a  manner  that  the  angle 
at  A  is  double  the  angle  at  B,  and  the  angle  at  B  double  that  at 
C,  and  a  line  of  234  yards  wiU  just  reach  round  them ;  required 
their  respective  distances  ?  '  Ans.  ABss46,346B  yards,  ACsz 
83.5135  yards,  BC=  104.14  yards, 

21.  in  order  to  determine  the  distance  between  two  inaccessi- 
ble batteries  A  and  B,  two  stations  X  and  Z  were  chosen,  distant 
from  each  other  4541.8  yards ;  at  AT  the  following  angles  were 
taken,  viz.  AXDszW  34'-,  BXZ=i46»  16' 5  at  Z  the  angles 
were  XZA=^96<>  44',  XZB:szmo  23';  required  the  distance  of 
the  batteries  from  each  other?     Ans.  3373.1  yards. 

22.  Two  ships  leave  a  port  together^  A  steers  S  W; 
6SE,  and  sails  twice  as  fast  as  A:  at  the  end  of 
they  arrive  at  ports  55S  miles  apart ;  now,  supposing 
to  have  blown  equally  from  one  point  during  the  wh^Kflime; 
at  what  rate  per  hour  did  the  ships  run  ?  '  Ans.  A  3.l^k  miles 
per  hour,  B  6.243.  • 


»  If  *5stbc  least  angle,  viz  C;  then  2x=»  B,  and  As^A,  whence  7*=' 180, 
and  jp:fe-"-5^^=  25«  42'^.  Assume  either  of  the  sides  of  any  convenient  length, 
and  find  (by  Art.  ^7.)  the  two  remaining  sides ;  then  say,  as  the  sam  of  these 
three  sides  :  to  the  given  snm  234  : :  either  of  the  sides  :  the  corresponding 
side  of  the  proposed  triangle. 

*  From  any  point  draw  two  indefinite  lines  in  the  proposed  directions,  from 
the  table  (  \rt.  87.)  Assame  any  length  in  the  S  W  line  for  A*%  distance,  and 
take  double  that  length  in  the  other  line  for  ^s ;  join  these  points  by  a 
straight  line,  and  fad  its  length  (Art.  72y  69,  and  67.)  ;  then  say^  as  this  line  : 


P&BTIX.    INACgfiSSIBLE  HEIGHTS  &  DISTANCES.  430 

:,-9t.  From  one  of  the  aoglea  of  a  rsctangular  met 
.are  two  straight  foot  paths,  ooe  leading  to  the  oppc 
and  the  other  to  a  stile  1 10  jaiila  distant  from  it }  thi 
with  the  two  patiis,  forms  a  triangle,  of  which  the 
as  the  numbers  9, 3,  and  10  j  what  sum  will  pay  fbrth 
making;,  and  carting  of  tiie  said  meadow  at  37«.  Sd. 
J.n$.  7L  Si.  ^d. 

24.  There  are  three  seaport  towns  J,  B,  xdA  C 
£  S  £,  atid  Cj  £  by  N  from  J :  a  telegraph  is  erected,  for  the 
purpose  of  speedy  communication  with  the  metrtqxdis,  at  5 
miles  distance  from  each  of  the  towns,  and  in  the  line  4Ci 
required  the  distance  of  B  from  J  and  C,  and  its  bearings  from 
the  lel^raph  ?  Ant.  from  B  to  A  8  J147  mifc»,  Jirom  B  to  C 
5.55S7  milet}  and  B  heart  S  £  b  S/rom  the  lekgr^ph. 

35.  Aflag-staffisplacedon  acaetlewalll63 feet  long, in  sm^ 
A  situation  ib^t  a  line  of  100  feet  in  length  will  reach  fh>m  its 
4op  to  one  end  of  the  wall,  and  a  line  of  89  feet  from  iu  top 
to  the  other  j  required  the  height  of  the  flag-staff,  and  its  dis- 
tance from  the  extremities  of  the  wall }  Ant.  height  47.7344 
jtel;  dittance  from  une  extremity  87.8773  feet,  front  the  oilier 
75. 1237 /ee*. 

-  36.  la  the  hedge  of  ^a  drctilar  inclosure  500  yards  in  diame- 
ter three  tixes  A,  B,  and  C  vere  planted  'in  such  a  roauter, 
that  if  straight, lines  be  drawn  from  each  to  the  other  two,  the 
Angle  at  A  will  be  double  the  angle  at  B,  and  the  angle  at  C 
douUe  of  A  >ad  B  together  j  required  the  distance  between 
<»ery  two  of  the  ti«es  '?  Am.  from  A  to  B  433.013  y(ird*,_/rMn 
BtoC  321.394  yardt,  and  from  A  to  C  171  01  yarrff. 

Jti  atamed  ilistaoce  : ;  SS8  :  jft  real  dittantc  ;  wheoce  alu  B't  diMance  wiU 
be  fuaod ;  uul  the  iJislBncc  dirided  by  the  nainbei  of  boun,  will  give  tbc  rate 
o(  lailiog  per  hanr.  9 

'  To  find  tbc  aagles,  sf  tbe  Dotc  an  prob.  SO.  Ta  find  the  t\6.tt ;  Firit,, 
nUh  the  ruliuiSSO  d«Kribe  a  cirels,  and  frum  it  cut  off  >  setimeat  canUiBing 
SB  ai^  equid  to  the  grealett  angls  of  tbe  proposod  trimf  le  (34. 3.),  draw 
■troight  lines  rrnio  the  extremities  of  thU  chord  to  the  ceotie,  and  an  Uoyelei 
triangle  will  he  formed  by  the)e  three  lines,  of  Khich  the  vertical  an^e  {M  tl^ 
centre)  vKI  lie  duubk  the  lupplemeM  of  (he  laid  greatut  angle  (SO  and  84.  a.), 
and  the  three  angles  of  this  isosceles  triangle  will  be  known  (39.  I.). 
Secosily,  find  tbe  b  ise  (Art.  67.)  which  will  be  tbe  greatest  side  of  the  pro- 
pmcd  triaugle  (19. 1  ),  whence  the  two  remaining  >idei  irill  likewise  be  found 
by  Art.  67. 

Ff4 


440  PLANB  TBICK>NOMBTRT.  Paut  IX. 

27.  An  £ogl]^  sloCip  of  war  having  orders  to  survey  an 
enemy's  port,  placed  two  boats  A  and  B  at  1100  &thoms  dis^ 
tance  apart^  A  being  directly  east  from  B :  at  the  inner  ex* 
tremity  of  the  harbour  there  is  a  spire  visible  from  the  boats^ 
likewise  a  castle  on  one  point  of  the  entrance^  and  a  light-house 
on  the  others  at  J  the  castle  bore  SSW,  the  spire  S  W  by  S, 
and  the  light-house  W  S  W.  At  J9  the  castle  bore  S  B,  the  spire 
south>  and  the  light-house  S  by  W  ^  required  the  kmgth  and 
breadth  of  the  harbour  ?  Ans.  length  from  middle  of  entrance 
loss  futhoms;  breadth  of  Entrance  9iO.S9  fathoms. 

2S.  On  the  c^posite  sides  of  an  impaasil^k  wood,  two  citisB 
A  and  B  are  situated  ^  C  is  a  town  visible  from  A  and  B,  dis- 
tant from  the  former  3  miles,  and  from  the  latter  2,  and  they 
make  at  C  an  angle  of  'iSP  5  now,  it  is  desirable  to  cut  a  passage 
lh>m  A  to  B,  and  an  engineer  undertakes  to  make  one,  19  feet 
wide,  at  7«-  6<i.  per  square  yard;  the  inhabitants  of  A  agree  te 
furnish  4  of  the  expense,  which  th^  can  accomplish,  by  ev^ 
7  persons  paying  31  shillings  5  those  of  B  can  make  up  the 
remainder,  by  every  six  persons  subscribing  33  shillings ;  re- 
quired the  number  of  inhabitants  in  A  and  B  ?  Ans.  A  43626, 
B  8839,  to  the  nearest  unit. 

S9^  An  isosceles  triangle  has  each  of  the  angles  at  the  base 
double  that  at  the  vertex ;  now,  if  the  vertical  ai^le  be  bisected, 
and  either  of  the  angles  at  the  base  trisected,  the  segment  of 
the  trisecting  line,  intercepted  between  the  opposite  side  and 
the  bisecting  line,  will  be  three  inches ;  required  the  sides  of 
the  triangle?  Ans.  each  of  the  equal  &ides  13.8314  inches;  the 
base  8.35371  inches. 

30.  In  a  circle,  whose  radius  is  5,  a  triangle  is  inscribed,  and 
the  perpendiculars  from  the  centre  of  the  circle  to  the  sides  of 
the  triangle  are  as  1,  3,  an^4  -,  required  the  sides  and  angles  of 
the  triangle  ? 

31.  The  altitude  of  a  balloon  as  seen  from  A  was  47°,  and  its 
bearings  SE;  from  B,  which  is  ^4-  miles  south  of  A,  it  bore 
NE  b  N'j  required  the  perpendicular  height  of  the  balloon, 
and  its  distance  from  B  ? 


/ 


<_  J 


PART  X. 


THE   CONIC  SECTIONS. 


HISTORICAL  INTRODUCTION. 

If  a  solid  be  cut  into  two  parts  by  a  plane  passing 
through  it,  the  surface  oiade  jn  the  solid  by  the  cutti^og 
plane,  is  called  A. SECTION. 

If  a  fixed  point  be  takep  above  a  plape,  and  one  of 
fhe  extremities  of  a  atraigbt  line  parsing  through  it  b^ 
made  to  describe  a  circle  <>n  the  plane,  then  will  the  seg* 
ments  of  this  line  by  their  revolution,  describe  two  solids 
(one  on  each  side  of  the  fixed  poipt)  which  are  called 

OPPOSITB  CONES  '• 

A  plaDe  may  be  mad(&  to  cat  a  cone  five  ways;^rs/t, 
by  passin  g  through  the  vertex  and  the  base ;  secondly, 
by  passing  through  the  cone  parallel  to  the  base ;  thirdly, 
by  passing  through  it  parallel  to  its  sides;  fourthly^ 
by  passing  through  the  side  of  the  cone  and  the  base, 
so  as  likewise  to  cut  the  opposite  cone;  and^thly,  so  as 
to  cut  its  opposite  sides  in  unequal  angles  *^,  or  in  a  posi- 
tion not  parallel  to  the  base. 
~    •  - — • ■ — —*: -  . 

*  If  the  segment  of  the  geDerating  line  between  the  fixed  point  and  the 
base  be  o!P>^>givea  length,  the  cone  described  by  ita  motion  will  be  A  right 
COKE,  ha^iog.  Hs  m»  peipendicuUr  to  the  bate ;  but  if  the  Ungth  of  i\yt 
segment  be  variable  in  any  given  ratio,  so  as  to  become  in  one  revolution  a 
fnaximum  and  a  minimum,  the  Cone  produced  will  be  an  oblique  coke,  and 
Hs  axis  will  make  an  oblique  angle  with  the  base. 

**  Of  course  a  right  oone  is  hare  understood ;  for  if  the  cone  be  oblique, 
the  base,  which  is  a  circle,  will  <ut  the  opposite  sides  in  unequal  anglrs,  and 
the  segment  made  by  cutting  them  in  eqtial  angle*  will  evidently  be  an  ellipse. 


44«  qONIC  SECTIONS.  Pabt  X. 

If  the  plane  pass  through  the  vertex  and  the  base,  the 
section  is  a  triangle ;  if  it  be  parallel  to  the  base,  the 
section  is  a  circle ;  if-  parallel  to  the  side  of  the  cone, 
the  section  is  called  a  pababola;  if  the  plane  pass 
through  the  side  and  cut  the  opposite  cone,  the  section 
is  called  an  hyperbola;  and  if  it  cut  the  opposite 
sides  of  the  cone  at  unequal  angles,  the  section  h  called 

AN  ELLIPSE. 

The  triangle  and  circle  pertain  to  common  elementary 
Geometry,  and  are  treated  of  in  the  Elements  of  Euclid; 
the  parabola,  the  ellipse,  and  the  hyperbola,  are  the  three 
figures  which  are  denominated  the  conic  sections. 

There  are  three  ways  in  which  these  curves  may  be 
conceived  to  arise,  from  each  of  which  their  properties 
may  be  satisfactorily  determined  ;^r9f,  by  the  section  of  a 
cone  by  a  plane,  as  above  described,  which  is  the  genuine 
method  of  the  ancients ;  secondly y  by  algebraic  equations, 
wherein  their  chief  properties  are  exhibited,  and  frooi 
whence  their  other  properties  are  easily  deduced,  accord- 
ing to  the  methods  of  Fermat,  Des  Cartes,  Roberval, 
Schooten,  Sir  Isaac  Newton,  and  others  of  the  moderns; 
thirdly y  these  curves  may  be  described  on  a  plane  by 
local  motion,  and  their  properties  determined  as  in  other 
plane  figures  from  their  definition,  and  the  principles  of 
their  construction.  This  method  is  employed  in  the 
following  pages. 

»  _ 

W  H  E  N,  or  f rom  whom  the  ancient  Greek  geometricians 
first  acquired  a  knowledge  of  the  nature  aqd  properties 
of  the  cone  and  its  sections,  we  are  not  fully  informed,  al* 
though  there  is  every  reason  to  suppose  that  the  discovery 
owes  its  origin  to  that  inventive  genius,  and  indefatigable 
application  to  science,  which  distinguished  that  learned 
people  above  all  the  other  nations  of  antiquity.    Some 


PartX.       ^       INTRODUCTION.  443 

of  the  most  remarkable  properties  of  these  curves  were 
in  all  probability  known  to  the  Greeks  as  early  as  the 
fifth  century  before  Christ,  as  the  study  of  them  appears 
to  have  been  cultivated  (perhaps  not  as  a  new  subject) 
in  the  time  of  Plato,  A.  C.  SQO.  We  are  indeed  told,  that 
until  his  time  the  conic  sections  were  not  introduced 
into  Geometry,  and  to  him  the  honour  of  incorporating 
them  with  that  science  is  usually  ascribed.  We  have 
nothing  remaining  of  his  expressly  on  the  subject,  the 
early  history  of  which,  in  common  with  that  of  almost 
every  other  branch  of  science,  is  involved  in  impene- 
trable obscurity. 

The  first  writer  on  this  branch  of  Geometry,  of  whom 
we  have  any  certain  account,  was  Aristaeus,  the  disciple 
and  friend  of  Plato,  A.  C.  380.  He  wrote,  a  treatise  con- 
sisting of  five  books,  on  the  Conic  Sections ;  but  unfor- 
tunately this  work,  which  is  said  to  have  been  much 
valued  by  the  ancients,  has  not  descended  to  us.  Me- 
nechmus,  by  means  of  the  intersections  of  these  curves 
(which  appears  to  have  been  the  earliest  instance  of  the 
kind)  shewed  the  method  of  finding  two  mean  proper- 
tionals,  and  thence  the  duplication  of  the  cube;  others 
applied  the  same  theory,  with  equal  success,  to  the  tri- 
section  of  an  angle;  these  curious  and  difficult  problems 
were  attempted' by  almost  every  geometrician  of  this 
period,  but  the  solution  (as  we  have  remarked  in  another 
place)  has  never  yet  been  effected  by  pure  elementary 
Geometry.  Archytas,  Eudoxus,  Philolaus,  Denostratus, 
and  many  others,  chiefly  of  the  Platonic  school,  pene- 
trated deeply  into  this  branch,  and  carried  it  to  an 
amazing  extent;  succeeding  geometers  enriched  it  by 
the  addition  of  several  oiher  Curves  as  the  cycloid, 
cissoid,  couchoicl,  quadratrix,  spiral,  Seethe  whole  form- 
ing a  branch  of  science  justly  considered  by  the  ancients 


444  CONIC  SECTIONg.  Part  X. 

AS  possessing  a  more  elevated  nature  ihan- common  Geo- 
metry,  and  on  this  account  they  distinguished  it  by  the 
name  of  TH£  moHBR  or  sublime  geometry. 

Euclid  of  Alexandria^  the  celebrated  author  of  the 
BlementSji  A.  C.  280;  wrote  four  books  on  the  Conic 
Sections,  as  we  learn  from  Pappus  and  Proclus  ;  but  the 
work  has  not  descended  to  modern  times.  Archimedes 
was  profoundly  skilled  in  every  part  of  science,  es- 
pecially Geometry,  which  he  valued  above  every  othet 
pursuit ;  it  appears  that  he  wrote  a  work  which  is  lost^ 
expressly  on  the  subject  we  are  considering,  and  his 
writings  which  remain  respecting  spiral  lines,  conoids, 
and  spheroids,  the  quadrature  of  the  parabola,  &c.  are 
sufficient  proofs  thai  he  was  deeply  skilled  in  the  theory 
of  the  Conic  Sections.  In  his  tract  on  the  parabola  he 
has  proved  by  two  ingenious  methods,  that  the  area  of 
the  parabola  is  two-thirds  that  of  its  circumscribing 
rectangle ;  which  is  said  to  be  the  earliest  instance  on 
record  of  the  absolute  and  rigorous  quadrature  of  a  space 
included  between  right  lines  and  a  curve.  But  the  most 
perfect  work  of  the  kind  among  the  ancients  is  a  trfsatise 
originally  consis-ting  of  eight  books  by  ApoUonius  Per- 
gaeus  of  the  Alexandrian  School,  A.  C.  (230.  The  first 
four  only  of  these,  have  descended  to  us  in  their  original 
Greek,  the  fifth,  sixth,  and  seventh,  in  an  Arabic  version ; 
the  eighth  has  not  been  found,  but  Dr.  Halley  has  sup- 
plied an  eighth  book  in  his  edition,  printed  at  Oxford, 

in  1710. 

This  excellent  treatise  is  the  most  ancient  work  in  our 
possession,  on  the  subject;  it  supplied  a  model  for  the 
earliest  writers  among  the  moderns,  and  still  maintains' 
its  classical  authority :  the  improvements  on  the  system 
of  ApoUonius  by  modern  geometricians  are  comparatively 
few,  except  such  as  depend  on  the  application  of  Algebra 


Part  X.  INTRODUGtION-  445 

and  the  Newtonian  Analysis.  Hitherto  the  ancients  had 
admitted  the  right  cone  only  (of  which  the  axis  is  per- 
pendicular to  the  base)  into  their  Geometry ;  they  sup- 
posed all  the  three  sections  to  be  made  by  a  plane  cutting 
the  cone  at  right  angles  to  its  side.  According  to  this' 
method,  if  the  cone  be  right  angled  (dcf.  18. 11.),  the 
section  will  be  a  parabola;  if  acute  angled,  the  section 
will  be  an  ellipse;  and  if  obtuse  angled,  an  hyperbola; 
hence  they  named  the  parabola.  The  section  of  a  right 
angled  cone;  the  ellipse,  The  section  of  an  acute  angled 
cone;  and  the  hyperbola.  The  section  of  an  obtuse  angled 
cone.  But  Apollonius  first  shewed  that  the  three  sections 
depend  only  on  the  diiSerent  inclinations  of  the  cutting 
plane,  and  may  all  be  obtained  from  the  same  coiie, 
whether  it  be  right  or  oblique,  and  whether  the  angle  of 
its  vertex  be  right,  acute,  or  obtuse.  Pappus -of  Alex* 
andria,  who  flourished  in  the  fourth  century  after  Christ, 
wrote  valuable  lemmata  and  observatrons  on  the  writings 
of  Apollonius,  particularly  on  the  conies,  which  ^re  to  be 
found  in  the  seventh  book  of  his  Mathematical  Collet'^ 
tiotts:  and  Eutocius,  who  lived  about  a  ocndtury  later^ 
composed  an  elaborate  commentary  on  sevitm\  of  the 
propositions. 

In  I  J£e  John  Werner  published,  at  Nofreitiberg,  0ome 
tracts  on  the  subject;  and  drboiit  thresame time  Frtmcia-* 
cus  Maurolycns,  Abbotof  St.  Maria  del  Porta^  id  Sicily/ 
published  a  treatise  on  the  Conic  Sections^  which  has 
been  highly  spoken  of  by  somre  oSf  oor  be^t  geometers 
for  its  perspicuity  and  eleganoe.  The  applicacioA  of 
Algebra  to  Geometry,  first  generally  intrbdnced  by  Vieta^ 
and  afterwards  improved  and  extended  by  Dest^Cartesy 
Fermat,  Torricellius,  and  others,  furnished  means  for  the 
further  developement  of  the  nature  and  properties  of 
Curves.     The  indivisibles  of  Roberval  and  Cavalerius; 


446  CONIC  SECTIONS.  Part  X, 

the  AriUmitic  of  L^iiet,  by  Dr.  Wallis;  die  Theory  of 
Evoiuies,  by  Huygens;  the  Method  of  Tangents,  by  Dr. 
Barrow,  &c.  were  discoveries  which  supplied  additional 
means  for  extending  the  theory  or  facilitating  the  several 
applications  of  the  doctrine ;  bat  that  which  rendered  the 
most  complete  and  essential  service  to  this  department 
of  science,  was  the  discovery  of  the  method  of  Flaxions 
by  Sir  Isaac  Newton,  which  took  place  about  the  year 
\66S. 

The  principal  modern  writers  on  the  Conic  Sections 
are,  Mydorgtus,  Trevigar,  Gregory  St.  Vincent,  De 
Witte,  De  la  Hire,  De  1'  Hopilal,  Dr.  Wallis,  Milne,. 
Dr.  Simson,  Emerson,  Muller,  Steel,  Jack,  Dr.  Robertson, 
&c*  The  ProperticM  of  the  Conic  Sections,  by  Williain 
Jones,  Esq,  F.  R  S.  published  by  Mr.  John  Robertson, 
in  1774,  is  a  tract  in  which  is  coflnprised  a  very  great 
number  of  properties  deduced  in  a  most  compendioos 
and  general  manner,  within  the  narrow  compass  of  24 
pages.  Dr.  Hamilton's  Conic  Sections  is  a  very  elegant 
and  ample  work ;  Dr.  Hutton's  treatise  on  the  subject 
will  be  found  easy  and  useful.  The  introductory  tracts 
of  the  Rev.  Messrs.  Vince  and  Peacock  are  the  shortest 
and  plainest  elementary  pieces  which  have  been  put  into 
the  hands  of  students ;  on  the  plun  of  these  (especiaUj 
the  latter)  the  following  compendium  was  drawn  up,  in 
wbich  it  is  hoped  there  wiU  be  found  some  improve* 
ments.  A  coarse  of  Lectures  on  the  Conic  Sections  has 
lately  been  published  by  the  Rev.  Mr.  Bridge,  of  the 
East  India  College,  I  have  not  seen  the  work,  and 
therefore  cannot  speak  of  it,  but  the  talents  of  the 
author  are  well  known. ' 


PaxtX. 


THE  PARABOLA. 


447 


I 


THE  PARABOLA. 


BBFINITIONS. 


strai^t 


moving  parallel  to  itself  at  right  angles  to  xy ;  and  if  another 
straight  line  FP  revolve  about  F,  so  that  FP  be  always  equal  to 
MP  J  the  point  P  will  trace  out  the  curve  DVPb,  which  is  called 

A  PABABOLA. 

2.  The  straight  line  xy  is  called  the  dirsctbix^  and  the  poiAt 
Fthbvocus. 


3.  If  through  the  focus  F,  a  straight  line  BZ  be  drawn  per- 
pendicular to  the  directrix  xy,  cutting  the  parabola  in  V,  VZ  is 
called  THE  AXIS  of  the  parabola,  and  V,  the  vertex. 


U^  CONIC  SECnOlSB;  PIky  X.: 

Car.  Hence,  because  jFP  is  alwayssPJIf  (Art.  1,)^  when  P  by 
its  motion  arrives  at  V,  FP  becomes  FF,  and  PM  beccmies  VH, 
\'Fr=zVH. 

4.  A  straight  line  drawn  through  the  focus  F,  perpen^cular 
to  the  axis  VZ,  and  meetings  the  cunre  both  ways,  is  called 

THE    LATUS    RECTUM,    Or    PRINCIPAL    PARAMETER.       ThU8    DB 

is  tfie  latus  rectum.    In  some  of  the  following  articler^  the 
latus  rectum  is  denoted  by  the  letter  L, 

5.  Any  straight  line  perpendicular  to  the  axis  TZ,  meeting 
the  curve,  is  called  an  ordinate  to  the  axis  3  dnd  the  part 
of  the  2LXh  intercepted  between  the  vertex  Fand  any  ordinate, 
is  called  the  abscissa.  Thtis  NP  is  an  ordinate  to  the  axis,  and 
NV  its  corresponding  abscissa. 

6.  A  straight  line  meeting  the  curve  in  any  point,  and  which 
being  produced  does  not  cut  it»  is  called  a  tangent  to  the 
parabola  at  that  point.     Thus  FT  is  a  tangent  at  the  point  P. 

7*  A  tangent  drawn  from  the  eixtremity  of  the  latus  rectum^ 
is  called  the  focal  tan«ent»     Thus  DHis  the  focal  tangent. 

8.  If  an  ordinate  and  a  tangent  be  drawn  from  the  same 
point  in  the  curve^  that  part  of  the  axis  produced^  which  is 
intercepted  between  their  extremities^  is  called  the  .sub-tan- 
gent. Thus  P  being  any  point  from  whence  the  tangent  FT  and 
the  ordinate  FN  are  drawn,  NT  is  the  sub-tangent  to  the  point  P. 

9.  A  straight  line  drawn  perpendicular  to  the  tangent  from 
the  point  of  contact^  and  meeting  the  axis^  is  called  th& 
NORMAL.     Thus  PG  is  the  normal  to  the  point  P. 

10.  If  a  normal  and  an  ordinate  be  drawn  to  the  same  point 
in  the  curve^  that  part  of  the  axis  intercepted  between  them,  is 
called  THE  SUB-NORMAL.  Thus  NG  is  the  sub-normal  to  the 
point  P.     ^ 

11.  A  straight  linei  drawn  from  any  point  in  the  curve, 
parallel  to  the  axis>  is  called  a  diameter  to  that  point  -,  and 
the  point  in  which  iit  meets  the  curve,  is  called  the  vertex  to 
THAT  DIAMETER.  Thus  PX  is  a  diameter  to  the  point  P,  mid  P 
is  its  vertex. 

12.  A  straight  line  drawn  through  the  focus  F,  parallel  to  the 
tangent  at  any  point,  and  terminated  both  ways  by  the  curve,  is 

called    THE  PARAMETER  TO  THE  DIAMETER  of  which  that   point 

is  the  vertex.     Tlius  db  b  the  parameter  to  the  diameter  PX. 


Paw  X;  TB£  PAVABOLA,  44^ 

13.  A  atraigfat  line  ^brawn  from  any  diamet^>  parallel  to  a 
tangent  at  its  vertex,  and  meeting  the  enrre,  U  called  an 
OBOiNATB  tff  that  diai^ter.  Thus  vn  U  an  ordinate  to  ike 
diamttir  PX, 


PROPERTIBS  OF  THE  PARABOLA  \ 

14.  The  straight  line  FP,  drawn  from  the  focus  F,  to  any 

point  P  in  the  curve,  is  equal  to  tbe  sum  of  the  s^noents  FF 

and  FiVof  the  axis  inteit:epted  between  the  vertex  and  the  {bcus» 

and  between  the  vertex  and  the  ordinate  -,  that  is,  JRP=s  FJV+ 
FR 

For  FPz=iPM  (Art.  1.)  =:HN  (34.  1.)  sFiyT+FHa  (cor. 
Art.  3.)  VN^  VF.    Q.  E.  D. 

Cor.  1.  Hence,  when  ^  cdneides  with  3,  N  will  coincide 
>«^ith  F,  Fi\r  will  become  VF,  and  FP  wiU  become  FB;  -.'  FB^  • 
^^F,  and  D£^4FF,  or  the  latus  rectum  is  equal  to  four  times 
ths  distance  of  the  focus  from  the  vertex. 

Cor.  2.  Hence  FP—Fi^=  FSsshalf  the  latus  rectum,  for  FP 
-  (FF+FiV=)  %VF-^FNi  V  FP-^FNzsL^VF^^FB. 

15.  The  straight  line  PT,  which  bisects  the  angle  FPM,  is  a 
ttn^eiit  to  the  parabola  at  P.    See  the  following  figure. 

For  if  not,  let  it  cut  the  curve  in  P  and  p,  join  Fp,  FM, 
pM;  draw  jwit  perpendicular  to  HM,  and  join  FM  cutting  PT 
m  Y.  Then  in  the  triangles  FPY,  MPY,  tPz=iMP  (Art.  1.), 
'^Fis  oomaion,  and  the  angle  FPFaaHPF  (by  hypothecs),  •/ 

It  will  be  proper  to  iafona  the  student  before  he  begiat  to  study  the  Cooi« 
S«ctioQs,  that  he  ought  to  be  thoroughly  nuutor  of  the  first  six  books  of  Euclid^ 
^d  to  know  tomething  of  the  elerenth  and  twelfth  }  the  doctrine  of  propor- 
tion,  as  delivered  in  part  4.  paf •  49  U  8f  of  this  vofmt  tntMt  liktwise  be 
^ell  understood,  as  its  apflicatton  cootinnally  occurs  i«  tha  foUowimg  yaigea. 


'OL,  H.  G  g 


t> 


460 


CONIC  SBCTIOI®. 


Pam  X. 


FKss  JIfK  Mid  the  angle  jyp=  If  FP 

(4. 1.)  */  io  the  triaDgles  FVp,  MYp, 
the  two  sides  FV,  YpTs^MY,  Yp,  and 
the  included  angles  fTpsMFp,  -.* 
Fjl>=spM(4. 1)5  but  fp=pm  (Art.  1.), 
*.'  pM^pm:  '.*  the  angle  pmM^ 
pMm  (5.  l.)>  hut  pmM  b  a  right 
angle  (Art.  1.),  '.*  pMm  is  also  a  right 
angle,  which  is  impossible  (17*  l.)? 
\'  PT  does  not  cut  the  parabola, 
consequently  it  is  a  tangent  (Art.  6.) 
Q.  £.  D. 

16.  The  tangent  FF  at  the  vertex  F,  is  perpendicular  to  the 
as^s  FZ. 

For  since  the  tsuigent  PTcuts  Fift  at  right  angles  in  what- 
ever point  of  the  curve  P  be  taken  (Art.  15.),  •,•  when  P  coin- 
cides  with  F,  FP  will  coincide  with  FF,  Pilf  with  Fft  and  FM 
with  FH;  •/  the  tangent  FY  is  perpendicular  to  (FJIf,  that  is, 
to)JFfl:    Q,E.D. 

Cor,  I.  Hence,  because  TPand  ilfP  are  parallel  FTP^TPM 
(29.  l.)=PPr  (Art.  16.),  /  FT:izFP. 

Cor.  2.  Hence  FM,  FY,  and  PT  intersect  each  other  in  the 
point  Y.  For  fy=F^,and  (cor.  Art.  3.)  FF^^FH,  /  (9.6.) 
FY  is  parallel  to  HM^  and  consequently  perpendicular  to  TZ; 
':  FY  is  a  tangent  at  F. 

17.  The  focal  tangent  DH,  the  dir^trix  xy,  and  the  axis  TZ, 
intersect  each  other  in  the  point  H.  (See  the  figure  to  Art.  3.) 

For  FC=FH  (Art.  14.),  •.•  by  the  preceding  corollary,  the 
tangent  meets  the  axis  at  the  point  H,  where  the  axis  and 
directrix  intersect.    Q.  E.  D. 

18.  Jf  »r  be  an  ordinate  to  the  diameter  PJST  cutting  FP  in  r, 
(see  the  figure  to  Art.  3.)  Pr^Pv :  for  Prv^^rPT  (29.  l.)aa 
TPM  (Art.  15.)=  P»r  (39.  1.)  •.•  Pr::>>.Pv  (6. 1.).    Q.  E.  J>. 

19.  The  straight  line  PFis  a  mean  proportional  between  FP 
and  FF.     See  the  figure  to  Art  16. 

For  since  FYT  is  a  right  angle  (Art  16.)  and  YF  perpea- 
dicular  to  FT,  l^F:  FY ::  FY  :  FF  (cor.  8.  6.),  but  rP=PP 
(cor.  Art.  16.),  .' FP  :  FY ::  FY :  FF.    Q.  E.  D. 


Fjlkt  X, 


PARABOLA. 


«1 


Cor.  Hence  FP :FF::FI^:f¥^ (coir.  1, da 6.) > consequently 
FY^^FP.FF{16. 6.),  and  4FF»«b4FF.FP;  but  4l^=:the  latus 
rectum  (Art.  14.  cor.)  whidi  beiog  denoted  by  L,  we  have  4FF* 


20.  The  line  fP  varies  as  FY^. 
For,  let  P  and  p  be  two  points 
in  the  curve,  from  whence  the  tangents 
PT,  pt,  are  drawn,  and  let  FY  and  Fy 
Be  perpendiciilar  to  the  tangents  re- 
spectively.     Then,    because     FY^ssa 
TP,FV,  and  Fi/^—Fp.FV  (by  the  pre- 
ceding cor.)  •••  FY^xFy^:-,  (FP.FF: 
fy.Fr  : :  by  1. 6.)  FT :  Fp^,  v  FP  ec 
FT*.    Q.  E.D. 


iVWf.  Tbe  figure  tothls  Art.  is  inaccinately      n 
cot;  ^^moit  be  understood  as  a  straig^ht  lino 
at  right  angles  to  TZ, 


21.  If  PP  be  produced  through  F  and  njeet  the  curve 
again  in  p,  then  will  4PP.  JFp 

=»I.PP-hl^. 

For  FP^FB—PM^ 
FH=:NH'^FH=FN.    And 

JFB— fp=JFH— pw=Pff— 
H»==jRi,    /  FP-FB  :  FJ? 
-l^E)  ::  Py:P/*  ::   (4.6.) 
FPxFp,   /    (16.6.)  PP.Fp 
-FJ.i^= FP.FB  -  FP.Fp; 

or  2FP.Fp=FP.Fg+ 
FB.Fp^FBFP-itPp  \' 
(since  2FB^L  by  Art.- 4.) 
4FPFp=L.FP+f3[>. 
Cor.  Hence,  if  4a=: L,  A'=  FP,  and  j?2=  Fp,  the  last  expression 


will  become  4J&=4a.A'+*,  or  J&=a-X'+ax,  •.•  — ^ — +-v^. 

a       X      .A 

22.  If  c  be  the  co-sine  of  the  angle  FFP  to  radius  1,  then 

2FF 

1  — c  >  \        ' 

For  -PP»fW+FF(Art.l4.)=iW^+FiV^+FF==2rFiFy 

og2 


4» 


CONIC  88CI1QHB. 


Pa&tX 


BQt±FN :  IvP  : :  (sia  FTN:  miam  i :  cos  PFN  i  niditiB  : :  oos 
VFP  the  8upp.  of  PiV:  xaditti : :  )+€  :  1  b^r  Art.  63.  |wrt  0. 
•••  (l«. «.)  ±FN:=^c.FF ;  .•  fF^jVJS^  VF^^FF^FV (Axt.  14.) 
=)  SFF+cfP,-  ••  (fP— c.FP>  or)  1— c.PP=:9FF,  or  «P= 

.    Q.  E.  D. 

1— c  '    - 

23.  The  sub-tangent  NT=:2VN.     See  the  figure  to  ^rf .  20. 
Let  rr  be  a  tangent  at  F meeting  P Tin  T,  then  FF being 

perpendicular  to  PT  (Art.  15.),  an4  FP=Fr  (cor.  Art.  16.); 
also  FF  common,  to  the  two  triangles  FPK,  FTY,  theae  trian- 
gles are  similar  and  equal  (47  and  4. 1),  •.•  PY=yT.  But  F^ 
is  perpendicular  to  the  axis  VZ  (Art.  16.),  •/  it  is  parallel  to  the 
ordinate  NP ;  \' PY :  YT : :  ^iNT :  VT  (2.  6.) ;  byt  PY=rT/ 
VN=  FT  (prop.  -4.5.)  -,  '.•  Ae  sub-tangent  Ntr=zivN,    Q.  ]E,D: 

24.  If  P»r  be  parallel  to  the  tangent  PT,  and  vM  perpen- 
dicular to  the  axis  VZ,  (tee  the  figure  to  Art.  30.),  then  RM=^ 
2FN;  for  the  triangles  TNP,  RMv  being  equiangular  (29. 1.) 
TN  iNP'.iRM:  Mv  (4. 6.).  But  NP=Mv  (34. 1.)  /  JR3f= 
TN  ( 14. 5.)  =2 FJY  (Art.  23.)     Q.  E.  D. 

25.  If  two  parabolas  VR  and  VK  be  described  on  the  same 
axis    FZ,  and  the  ordinate  NQ 

meet  Ffl,  FK  in  P  and  Q,  then 
will  the  tangents  at  P  and  Q 
intersect  the  axis  FZ  produced 
in  the  same  point  T;  for  FN  is 
the  common  abscissa  to  the  or- 
dinates  NP,  NQ  of  both  parabo- 
las, and  NT—2FN  in  both 
(Art.  23  )     Q.  B.  D. 

26.  The  square  of  the  ordinate 
is  equal  to  the  rectangle  contained 
by  the  latus  rectum  and  abscissa^ 
or  PN'^^L.FN 

For  FP=:  FN+  FF(Art.  14.) 
•••  FP^^FN^^FF^+^FEFN 
(4.2.).  But  FN''+FF'-z=^PT.FN 
4-FA«  (7.2.),  V  FP^^<jtVF.VN 
+  FZV» + 2  FF.  FiVr=4  FF  riV-f- 

FN^,    But  FP^^PN^+Fm  (47.  1.),    /  Pm+FN^:^4FRFN 
^EN' 5  •.•  PN^^4FF.FN^(cot.  1.  Ait  140  L.FN.    Q.  B.  D. 


Pa»t  X  THE  PARABOLA.  458 

Cor.  HtiMe^  41  my  cardinate  i'N»y>  it&  abeetea  Or^Ar,  and 
the  latus  rectum=4a,  the  expi-ession  P^=L.  FN  will  beeom^ 
y'«4a4r;  whifih  &»  the  equation  of  the  parabQla»  conaideced  as  a 
geooietrical  curve. 

27.  llie  abscissa  varies  as  the  square  of  the  ordinate. 
Let  PN  and  pn  be  any  two  ordinates  to  the  axis  VZ;  then 
because  PN^=zL.FN,  and  pn^—LTn  (Art.  36.),  PiV»  :  p»«  : : 
L.FN :  I..F«  : :  (15. 5.)  FN :  Tw,  •/  (Art.  97.  part  4.)    FN  cc 
PA*.    Q.  E.D. 

9S.  If  two  parabolas  FR  and  FK  be  described  on  the  same 
a3ds  rz,  and  the  ordinate  NQ  meets  FP  in  P ;  then  will  PN 
and  QA  have  to  one  another  a  given  ratia 

Produce  np  to  q,  then  (Art.  27.)  PiV»  :pn*::  FN :  Fn:: 
QN^iqn^i  '-:  (22.  6,)  PN :  pn  ::  QN :  qn,  and  (16.  5.)  PN : 
QiV  : :  p»  «  qu.     Q.  E.  D, 

29.  The  area  FATP  :  the  area  FNQ  iiPNi  QN. 
For>  let  the  abscissa  FZ  be  divided  into  the  equal  parts  Nn 
nrf'rm,  &c.  and  qomplete  the  parallelograms  Pn,  Qt^,  pr,  qt,  <m> 
tfiiy  &c.  these  ha^ng  equal  altitudes  (Nn,  nr,  rm,  &€.)  are  to  Otte 
'  another  aa  their  basea  (1. 6.)» 
•.•  PmQni:  NP'.NQ 

pr  :  qr  -.:  np  :  nq  ::  (Art.  28.)  NP  :  NQ 
itnibmi:  rs  :  rh::  (Art.  ^8.)  NP  :  NQ 
V  (12.5,)  P»4-|ir+«n^+&c. :  Qw+^r-fftm-f  &C.  ::  iVP  :  NQ 
(15. 5.).  Wherefore^  if  the  magnitude  of  the  parts  An,  nr,  rm, 
&C.  be  diiaifiishjBdt  aod  their  number  increased  indefinitelyi  the 
61^  of  ^  tb?  parallelograms  between  Faiid  mx  will  approxi-* 
mat^  iade^ni^ely  near  to  the  ar^e^  of  the  ourvitineal  space  Fxm ; 
as  ihe  sum  of  the  parallelograms  between  F  and  ytn  willa  to  the 
(nirvil^neal  $]^e  Fjfmi  '.:  the  area  FPxm  ;  the  area  FQym  : ; 
NP  :  AQ.     Q.  E.  D. 

Cpr.  Hen^^e,  if  from  my  P^int  P  iu  the  ws,  straight  lines 
FP^  FQ  be  drawn,  the  curvilio^  fu^  FFP  i  the  curvilineal 
area  FFQ  : :  itf^  5  NQ. 

yor  tbetrijuigle  P^^:  PCA::  NP  :  AQ  (I.  6.) 

And  FP A :  FQA : :  AP  :  NQ  (as  shewn  above.) 
aW FPAiPPA;:  FOA:PQA(U.  5) 
...  Fpjff^PPN  :  FQN--FQN :  AP  :  NQ  (19. 6.)  . 
TJiat  i*,  the  area  FFP  :  the  area  FPQ  ; :  AP  :  NQ. 

Gg3 


454 


CONIC  SECTIONS. 


PartX. 


SO.  The  sub-normal  is  equal  to  lialf  the  latus  Tectum,  that  is. 

For  TPG  18  a  right  angle  (Art.  9. 10.),  from  which  NP  U 
drawn  perpendicular  to  the  base  TG  (Art.  5.),  '.•  TN :  NP  :•" 
NP  :  NG  (cor.  8.  6.) ;  v  T 
TNNG^Nf'^  (17.  6.)== 
L.VN (Art.  ^6.) y\'  Tff:  FN  r 
II  L  :  NG  (16.  6.).  But 
TN=^VN  (Art.23.).  •.*£=: 
QNG  (prop.  D.5.),  and  their 
lialves  are  ecjuaj,  or  ffG^ 
-^  L.    Q.  E.  D. 

Qyr.  X,  If  from  F  as  a 
centre  with  the  distance  FT 
=FP  a  circle  be  described^ 
it  will  pass  through  G;  for  M 
T  and  P  being  in  the  cir-  ^ 
ouniference,  and  JP  G  a  right 
angle,  the  point  G  will  like- 
wise be  in  the  circumference 
<31.  S.)i  •/  FP=FG,  and 
the  angle  FPG=zFGP(5. 1.). 

Cor.  2.    Hence-  also  the 

angle  rPP=FGP+FPG  (32.  I.)  =:2FGP. 

31.  If  GA*  be  drawn  perpendicular  to  FP,  then  will  PK 

For  the  triangles  PGJT,  PGN  having  PJTG.  PNG  right 
angles,  GPK=PGN  (cor.  1.  Art.  SO.),  and  PG  common,  are 
wmilar  and  equal   (26.  1.);    •/  PK=NG=z^L  (Art.  30.) 
Q.  E.  1>. 

32.  If  nv  be  an  ordinate  to  the  diameter  PX,  then  wiU  «©«= 
4FP.Po. 

Because  the  triangles  RAn,  RMv  are  similar,  jRJtf»  :  RJ* 
: :  (M»«  =)  NP'  :  An^  (4.  6.  and  22.  6.)  : :  FiV :  F^  (Art.  27.)  j 
but  iiM»  =  ieXMiW)«=/J^»4.^M»+2fi^.^i«f  (4.2,),  v 
(prop.  E.  5.)  RM'  :  JM'  +^SJ.AM  : :  FN:  AN  : :  (2FN^) 
RMiStAN;:  RM'  :  ^RM.AN  (15.5);  v  AM'+^RA.AM^ 
9,RM.AN  (9.6.),  or  AM' =±<iRM,AN^^RA.AM.  But  R,i=^ 
RM^AM,  and  AN=s=AM^MN\et  these  values  of  RA  and  AN 
be  substituted  in  the  foregoing  expression,  and  it  will  be- 


Pj.«T  Xv  THE  PARABOLA.  4S5 

come  ^iV'-='iRM.AM-MN—^RM~AM.JM^<iRM.Aia- 
^R3LMN-'2RM  JM+2.1M')=-^RM MN+^AM' ,  or  AM' 
=B=^RM.MS.  But  since  TB=:zPv=^MN  (34.  \.),  ■.•  RM^^TN 
^fW(Art.  23.);  ■-■  tbe  above  expression  AM'^IRM.Hti^ 
4VN:MN=4rN.Pv. 

Now  m*  :  Xb£*=)  am*  : :  Re*  :  BJIf  (4.  6.  and  22. 6.)  : : 
iIM*+Jlfp'  (47.  I.)  :  BM»  ::  (siDce  RM=^VN,  and  Jtft>»= 
yP^=4rN.VF.  by  Art.  26.)  4rJV*+4FW.rF  :  4rAf«  ;: 
4FN+4rF.FN  :  AVN*  : :  ^Art.  14.)  4fP  :  4rN  : :  4FP.Pv  : 
4VN.Pvi  that  ia,  no*  :  AM*  ::  4FP.i'o  :  AVN.Pv,  but  it 
has  been  proved  ubove  that  AM'=4FN.Pv,  ■.■  (14.  5.)  no»=;= 
4FP.FV.     Q.  E.  D.  . 

And  in  like  manner,  if  RI—IM  be  substituted  for  RM, 
and  compoiition  be  used  instead  of  conoeraion,  it  may  be  shewn 
that  fcp*=4fy./*i' ;  con^quenlly  ni'=if;  that  is,  any  djameter 
jpX  bisects  its  ordinates. 

Cor.  I.  Because  ■kFP.Pii=nv*.  and  FP  is  constant,  ■.■  Pv  «e 
nt-'.  •.■  also  On  «  OP*. 

Cot.  "2.  If  from  any  point  p  in  the  diameter  PA",  ordinates  itfi 
b6  drawn  cutting  P.ir  in  a  given  angle,  and  having  a  givea  ratio 
to  vb  I  the  curve  passing  through  all  the  points  B  will  be  a  para- 
bola. For  rb  :  vB  being  by  hypothesis  a  given  ratio,  vb"  :  vS* 
ia  likewise  given  i  but  (cor.  1,)  c6«  (^nC)  «  Po,  ■.■  pB»  «  Pv. 
Cor.  3.  Since  AM:'=iVN.Po,  as  shewn  above,  and  Pv=MN 
(34.  1.),  '.■  Am=4yN.NM. 

Cor.  4.  Let  Pbe  the  parameter  to  the  diameter  PX  then  when 
n&pa3sesthroughthefbcuBF,itbecumes  the  parameter  (Art.  12.), 
uid  the  point  r  coincides  with  F:  ■.■  Pr^Pv=FP  (Art.  17.), 
and  because  nr*=4FP.Pr,  ■.■  n6'=4»r''  (4.4.)=4x4fP.Po= 
16  FP»  (since  Pu=fr),  that  is  P*==16PP'. -.■  P=4F/'. 

33.  If  no  he  an  ordinate  to  the  diameter  PX,  and  nTn  tan- 
gent at  »,  the  sub-tangent  tt  7"  will  ^..,-4— _  T  n 
be  bisected  by  the  vertex  P.                  ^'"^ 

Produce  nv  to  meet  the  carve 
in  b,  produce  nT  to  E,  and  draw  Eb 
parallel    to   TX.      Then    (cor.  1. 
Art.  38.)  PTibE  ::  nT>tnE*  ;■. 
•    (4.  6.  and  22.  6.)    oT*  :  bE';  :■  , 

(l6.6.)Pr.6JS"=6E.tir',orPr.&£  I 

=vT^,  V   (ir.6.)  PPirr::  cT:  ^ 

Gg4 


^ 


CONIC  SBCTION& 


FAmxi 


bE  : :  (4.  6.  and  16.  B,)w:nb::  (Art.  32.)  1:3;  that  is^  the 
sub^tangent  vT  is  bisected  in  the  point  P.    Q.  E.  D. 

Cor.  Hence,  if  67*  be  a  tangent  at  b,  the  two  tangents  nTt  5T 
and  the  diameter  TXwiW  intersect  each  other  in  the  same  point 
T;  and  in  like  manner^  if  other  parabolas  be  described  upoi^ 
the  diameter  PX,  by  either  increasing  or  decreasing  the  ordinate 
nv,  or  its  inclination  to  the  diameter,  the  tangents  will  all  pw 
through  the  point  T,  as  appears  from  the  precedioip  demon* 
strati  on. 

34.  If  several  circles  be  described  upon  as  many  diametext  of 
different  lengths,  these  circles  will  have  different  d^rees  of 
curvatmre,  as  is  plain ;  and  if  the  diameter  be  increased  and 
decreased  indefinitely,  and  circlea  be  described  from  the  same 
centre  through  every  point  of  the  increased  or  diminished  dia-* 
meter,  these  circles  will  possess  all  possible  degrees  of  curvature, 
Hence  it  follows,  that  if  a  point  be  assumed  in  ^y  curvc^^ 
circle  may  be  found  which  will  coincide  with  an  indefinitely 
small  portion  of  that  curve  at  the  assumed  point,  so  that  the 
curve  and  the  circle  will  have  the  same  tangent^i  and  the  sane 
djsfiection  from  the  tangent  at  that  point ;  this  circle  is  called 
Tus  ciKCL£  OF  cuEVATURa  to  the  proposed  point. 


PabtX; 


TUB  PASABGHLA. 


4&t 


36*  If  P  be  tba  loew  of  a  paralx^  and  P  any  point  m  the 
curva,  the  chord  of  curvature  to  the  point  P  which  pasees 
through  f  h  equal  to  4/P. 

^  Let  Fr  be  an  indefinitely  imall  «r«of  the  paraMn^  coin- 
ciding with  the  circle  of  «u#vsiture  FHK  (Art.  S4.)|  then  the 
Hoe  nR  may  be  considered  as  common  to  both  5  join  iiP»  nH, 
produce  the  latter  to  M,  and  draw  ae  parallel  to  the  tan^nt 
PY.  Then  ainoe  the  angle  RPnz^^RHP  (99.  S.)>  and  nP  k 
indefinitely  near  a  coincidence  with  RP,  the  triangles  PHt^, 
PnR  may  be  considered  as  equiangular,  *.*  PH  i  Pn::  Pni  nR 
(4.6.)  and  (27.6.)  Pn'h^PHjnRi  bqt  mce  the  arc  is  in  its 
nascent  state  (or  indefinitely  small)  Pn^s^nv,  '.*  (ae^asby  cor. 
Art.  19.)  4W.P»=Fn«=Plf  aHj  but  nfi=Prs«  (Art.  18.)  Pv, 
*.' 4FP.Pv:^PH.Pv,oxPB^^^FP.    Q.£.Di. 

C^r.  1.  Hence,  because  4fP=:^he  panuneter  (cor.  4*  Artf  ^0,>» 
•.'  tbe.i^ord  of  c«qrtat\ire  passing  t}iro«!gh  the  focua  ia  equal  lo 
the  parameter. 

Cor.  9.  If  the  diameter  PK  be  drawn,  HK  jQined,  and  fY 
drawn  perpendicular  to  PY,  the  triangles  PHK,  PKF  will  he 
equiangular,  since  YFP^szHPK  (99. 1.)  end  the  angles  at  H 
and  Fright  angles  (31. 3.  and  by  construction)  •.•  FYi  FP  : :  PH, 
:PK::  (because  4FF=PH)  AFP  :  PK.  Hence,  if  a  tangent  be 
drawn  to  any  poipt  in  the  parabola,  and  a  perpendicular  to  the 
tangent,  he  drawn  from  the  fecus,  the  <Kameter  of  the  circle  of 
Gunratme  to  that  point,  will  be  readily  determined. 

36.  If  a  cone  be  cut  by  a  plane  parallel  to  its  side^  the  sectioA- 
will  be  a  parabola. 

Let  ABO  be  a  cone,  and  let  the  plane  VHK  pass  through 
it;  parallel  to  the  side  AB,  the  section 
HPVQK  will  be  a  parabola. 

tiet  the  plane  HVK  be  perpendi- 
cular to  the  plane  BAG,  the  common 
section  being  VS:  PDQE  a  section  of 
the  cone  parallel  to  the  base,  conse- 
quently a  circle,  PQ  and  DE  its  com- 
mon sections  with  the  fbre«mentioned 
planes,  and  draw  FF  parallel  to  DE, 
•/  since  the  planes  BVK,  PDQE  are 
perpendicular  to  BAC,  their  comnoift 
section  PQ  will  be  peffwipdjcmlar  !• 


458 


CONIC  sEcrroNs. 


Part  X.' 


BAO  (19. 1 1 .)  and  coaseqOBatly  to  the  lines  DE,  V$  (def;3. 11.)  ^ 
and  because  DE  the  diameter  of  tliejarcle  FDQE  cuts  FQ  at 
right  angles,  FC=CQ  (3.  S.),  v  DC.CEsiFO  (14.  3.)  Now 
the  triangles  VCE,  AFF  being  nmilar  FC :  C£  : :  AFi  FV 
(4-  6.)  Let  AFiFVtiFF'.L  (11.6.)  v  FC:  C£  ::  FFiL 
(11.  5.)}  V  FC.LsiCE.FF (16.6.)  ^DC.CE  (34.  1.)  =PC»,  •.• 
(Art.  26.)  HFJi:  is  a  iiambohi  of  wUdi  PC  is  an  ordinate  to  the 
axis,  FC  the  correspondent  abscissa,  and.  L  the  latus  rectum. 
Q.  £.  a 


THE  ELLIPSE. 


BBFINITIONS. 

37'  If  two  straight  lines  PP,  SF  intersecting  each  other  in 
P,  revolre  about  the  fixed  points  Pand  S,  so  that  PP+5P  be 
always  the  same,  the  point  P  will  trace  out  the  curve  PFKU, 
which  is  called  an  ellipse. 

38  The  points  P  and  5  about  which  FP  and  SP  revolve,  arc 
called  THE  FOCI. 

39.  The  straight  line  which  joins  the  Ibci  being  produced 
both  ways  to  the  curve,  is  called  the  major  axis  *".     Thus  VU. 
is  the  major  axis. 

40.  If  the  major  axis  VU  be  bisected  in  C,  C  is  c^led  the 

CENTRE  of  the  ellipse. 

41 .  The  straight  line  drawn  through  the  centre  perpendicular 
to  the  m^or  axis,  and  terminated  both  ways  by  the  curve,  is 
called  THE  MINOR  AXIS  \     Thus  EK  is  the  minor  axis. 

42.  Any  straight 
line  passing  through 
the  centre,  and  ter- 
minated both  ways 
by  the  curve,  is  called 

A     DIAMETER.       ThuS 

BX  is  a  diameter  of 
the  ellipse. 


j:  d 


c  It  b  also  named  7%e  irmuverte  axU. 

4  It  is  likewise  frequently  named  Z%0  etmJvigmU  a«i». 


Part  X.  TttE  BLLIKS.  460^ 

43.  The  eiftremity  of  any  ittametar  is  ctUed  its  viunxJ 
Thus  V  and  U  are  the  vertices  of  the  major  axis,  E  and  K  of  the 
minor  axiSy  and  B  and  X  of  the  diameter  BX, 

44.  A  straight  line  drawn  throii^  the  focus^  i^rpendicQlar 
to  the  nisgor  axis,  and  terminated  both  ways  by  the  curve,  is 
called  TBS  latus  rectum  or  principal  parameter.  Thus 
3I>  is  the  latus  rectum. 

45.  A  straight  line  meeting  the  ellipse-  in  any  point,  and 
which  being  produced  does  not  ait  it,  is  called  a  tangent  to 
that  point.     Tims  BT  is  a  tangent  at  the  point  B. 

46«  The  tangent  to  the  point  B  or  D,  the  extreipity  of  the 
latus  rectum,  is  called  the  focal  tangent.  Thus  BTis  the 
focal  tangent, 

47*  The  atraiight  line  drawn  perpendieular  to  the  major  ajus 
produced,  through  the  point  in  which  the  focal  tangent  meets 
it>  is  called  the  directrix.     Thus  xy  is  the  directrix, 

48.  Any  strai  ht  line  drawn  from  the  curve,  perpendicular  to 
tiie  major  axis,  is  called  an  ordinate  to  the  axis.  Thus  FN  is 
an  ordinate  to  the  ajiis, 

49.  The  parts  of  the  axis  intercepted  between  its  vertices  and 
tiie  ordinate,  are  called  abscissas.  Thus  VN  and  NU  are  ah' 
mssas  to  the  ordinate  PN, 

50.  If  from  the  vertex  of  any  diameter  a  tangent  be  drawn/ 
suay  sttaigfot  line  paraUel  to  the  tangent  terminated  by  the  dia- 
ineter  and  the  curve,  is  called  an  ordinate  to  that  diameter } 
and  the  intercepted  parts  of  the  diameter  are  called  abscissas. 
Thus  dv  is  an  ordinate  to  the  ^meter  BX,  and  Bv,  vX  abscissas* 

51.  If  the  ordinate  pass  through^  the  centre*  and  meet  the 
curve. botli  ways,  it  is  called  the  conjugate  d.iam£ter  *;  and 
if  it  pass  through  the  focus,  it  is  called  the  parameter  to  that 
diameter.  Thus  DG  is  the  conjugate  diameter^  and  db  thepara* 
meter,  both  to  the  diameter  BXr 

PROPERTIES  OF  THE  ELLIPSE. 

52.  The  sum  of  the  two  straight  lines  drawn  from  the  foci  of 
an  ellipse  to  any  point  in  the  curve,  is  equal  to  the  major  axis. 


•  And  ID  general,  if  each  of  two  diameters  be  parallel  to  the  tangent  at 
the  vertex  of  the  other,  these  diameters  are  called  conjugate*  to  each  other, 
^^c  8ub.taDf.enty  normal^  and  sab-Dormal,  are  the  same  as  in  the  parabola. 


460  COOnC  SKTidllS.  Fakt  Y. 

TIhv^  VPUmKy  poiol  in  the €«¥•,  Ibea  FF-i-  PS^  VUt=i%VC. 

For  (Art.  37.)  ^^4^  ra«p/'l7+ I7«i  that  ii»  2i?r+JPS» 
atrS4*iJ^&  V  ^FVmWS,  aad  fF^USf  and  (Art. 37)  i^F+ 
J^«»i^i»+«Fw,  F«^r  US:nfVU^SlVa    Q.  E,  J>. 

Csr.  1.  Hoiee,  beGMiK  FV^  VS^^WC}  bf  adding  VT  to 
both>  57+  TF:=2CT;  and  by  taking  27^  frun  thia^  ST^TF 
9s%C7-^%TFmStCF. 

Car.^,  Hanee,  bacmiaB  <Art.40«)  CWsb^CU,  and  FF^sUS 
(as  proved  abov#)  v  CV^FV^CU^  US^  or  CF^CS. 

Car.  3.  Hence,  SF^FU-^FFm^FC^FF^  a^d  in  like  oMn- 
Mf  it  wppem  tbat  JPasd  FC«^  &P. 

Cor.  4.  Hence,  because  jFP+5P=9rC,  by  taking  %SF  fh>» 
bcxth  FF^SB^2FC^2SF,  or  (ttnca  ^Pisrji  FC-^i^p,  by 
0OT.a,)»«FP-.9FC 

53.  Tbe  latua  rectum  is  less  tban  4Fi^i  £air  BF-^BS=^VU 
(Art.  37,)7xt^FF^FSXATU  52.) ;  and  since  BS  i&  greater  than 
FS,  BF  must  be  less  than  S^VF,  and  {^BF^)  BD 1^  than  4Ff • 
Q.B.D. 

M,  A  straight  Una  dnMivn  frooi  tkefQct»»^ili«  v^rtw  of  tli# 
minor  axis  is  equal  to  half  the  msgor  a^,.or  FB^ F€%  Sw  ikf 
folhwiHgJtgwre. 

For  since  fCs^CS  (ear.  1.  Art.  ^%)  and  Cfi  ^to^imm  te  the 
two  lri«»glea  FCE^  SCS  (and  tbe  anglea  at  C  rigbt  a«gi«i 
(Art.  41.)  •••  FE^ES  (4.  i.)i  U»t  (Art.  »7.)  I'fi^JE*  t*at  is 
2irjS5»Fi;fe»«FC,  /  FZ^FC.    a  S.  D< 

Cor.  1.  And  in  like  manner  it  may  be  shews  that  FK^tzKSts 
ti8:izEF^FC,\'^^  tbe  triangles  FEC,  FKC,  FK^FE,  tbe 
angles  trt  C  «Fe  right  anglss,  and  the  side  FC  is  common, 
whence  (««.  1.)  EC^CK. 

Cor.  2.  Hence  £0  =:  FjE*  .-. flC  (47.  1)  aFCT^-^FC*™ 
(cor.  5.2.)  FC-  FC.  FC-k-  FC;=  VF.  FU. 

55.  If  on  the  major  axis  as  a  diameter  a  ciftJe  be  d^cribed, 
and  the  latus  rectum  be  produced  to  meet  the  circymferenoe  in 
k,  then  will  Ffc=EC.  For  (14.2.)  P%*=FF.Fr=  (cof.  2. 
Art.  54.)  EC*,  •••  Fk=EC. 

5$.  Jim  btus  rectum  k  ^  Uurd  proportional  to  the  ms^or  apd 
minor  eves,  er  FU :  JSJT: :  EK:  BD. 


Part  X. 


THE  SLLIPSZL 


461 


^VC'^BF  (cop.i. 
Art.  59.)  •.•  BS" 
35: 4  FC8  -f.  JfF«  -* 
4rCJiF.  Bat  BS^ 
=  BF«  +  FS* 
(47.  1.)  •/  4F0«+ 
#F»^4FC.aF=tt 
JBF*-fF5t/4FC« 

— 4rc.BF=js«= 
(4.  «.)  4inc«  •.* 

(Vc«—FC«  by  cor. 

3.  Art.  54.«)  EO«rC.JBP  (Art,  56;)  |  •.'  (IT- 6.)  FC 1  EC  : : 

EC  :  BF;  whence  (15. 6.)  VU :  EK  i :  EK :  BD.    Q.  E.  D. 

Cor.  1.  If  L  (=BD)  be  the  Utui  recttiiii»  theo  (lioce  Fl/jft 
2  FC)  L.2KC=E^  (17.  6.) 

Cor.  2.  Hence,  of  the  major  and  minor  aKBA  and  ktus  rectum* ; 
any  two  being  given,  the  third  may  be  found. 

57.  If  FP  and  SP  be  drawn  from  the  foti,  to  any  point  P  in 
the  carve,  and  FP  be  produced  to  M,  the  straight  line  PT 
which  bisects  the  exterior  angle  FPM  is  a  tangent  to  the  dfipse. 

Make  PM:=  PF,  join  MF,  let  P  T  if 
possible,  intersect  the  curve  in  p,  and  join 
Mp,  Fp.  Then  because  MP^FP,  the 
angle  PJkrF=Pf3»f  (5. 1.)  MPr:=^FPr  by 
hypothesis,  and  Pr  common,  /  (4.  1.) 
ilfrssFr,  and  the  angle  MrP^FrP;  >r 
then  in  the  triangles  Mpr,  Fpr,  Mr=^Fr, 
pr  common,  and  the  angles  at  r  are 
equal,  •.•  (4. 1.)  Mp—Fp.  But  (20. 1.) 
5p+pi)f  y  SM,  that  is  >  SP+PM,  that 
is  >  SP^PF  (because  PF^PM)  that  is 

).  Sp-hpF  (because  Sp4"pF=  SP-fPP 
by  Art.37.)5;.*  since  .Sp+pM  J>  *^+pF, 
if  5^  be  taken  from  both  pM  )>  pF;  but 
it  has  been  shewn  that  pM=pF;  V  Mp 
and  pFare  both  equal  and  unequal  to  each 
other,  which  is  Absurd?  •.'  PT  does  not 
intersect  the  curve  in  any  other  point  p;  PT  is  therefore  a 
tangent  at  P.    Q.E.  D.  ^ 


4Gi 


CONIC  SECTIONS. 


Past  X. 


Cor.  1.  It  18  plain  that  the  nearer  the  point  p  be  to  F,  the 
greater  will  be  the  angle  FpM;  and  therefore  when  p  coin- 
cides with  V,  the  lines  Fjp,  pM  will  coincide  with  fP^,  FT,  and 
the  angle  FpM  will  become = two  right  angles  >  but  the  tangent 
at  (p  which  now  coincides  with)  F  bisects  this  angle,  */  the 
tangent  at  T  is  at  right  angles  to  the  axis  FU, 

Car:^.  Hence  (prop.  A.  6.)  STiTF::  SP  :  PP. 

'  Car.  3.  Hence,  straight  lines  drawn  from  the  ibd  to  any  point 
jn  the  curve>  make  equal  angles  with  the  tangent  at  that  pdnt, 
for  the  angle  iPS^^MPT  (15.  l.)=FPT. 

Cor.  4.  Hence  the  triangles  FPY,  SPi  will  be  dimilar,  and 
(4.6.)  SP'.Stii  FP'.FY. 

68.  Let  P  be  any  point  in  the  ellipse;  join  FP,  SP,  then  if 
SG  and  FG  be  drawn  parallel  to  these  respectively,  the  point  G 
where  they  meet  will  be  in  the  curve. 

For  since  FPSG 
b  a  parallelogram*  FG  ^ 

^GSz=^SP  +  FP 
(34.1.)-.'  G  is  a  point 
in  the  ellipse  by  Art. 
37.   Q.E.D. 

Cor.  Since  PG  and 
FS  bisect  each  other 
in  C  (part  8.  Art.  241. 
cor.)>  C  is  the  centre 
of  the  ellipse  (cor.  1. 
Art.  59.),  and  PG  a 
diameter  (Art.  42.)> 
*.'  all  the  diameters 
of  the  ellipse  are  bi- 
sected by  the  centre. 

59.  if /2r  be  a  tan- 
gent at  G,  it  will  be 
parallel  to  Tt. 

For  since  SGr+SGF+FGRzsz^  right  angles  (13.  and 
cor.  1. 15.1.),  =5P^+5PP4-PPr,  and  SGF^SPF  (34.1.), 
by  taking  the  latter  equals  from  the  former,  the  remainders  SGr 
+PGR=5Pe+PPr,  that  is,  (cor.  3.  Art.  57.)  ^FGR=:^2SPt, 
or  FGR^SPts  but  PGF^GPS  (29. 1.)}  add  these  equali  to 
the  preceding,  and  FGR+PGF^zSPt-^^  GPS^  that  is,  PGRsi 
GPt,  •.•  (27. 1.)  Rr  is  parallel  to  Tt.    Q.  B.  D. 


Pa>tX- 


THE  EliLIFSE. 


Cor.  Hence,  if  HD  be  a  ccmjtigate  diameter  to  PO,  taagents 
at  D  and  H  will  be  parallel^  and  the  four  tan^ntt  r/>  tr,  rR, 
and  RT  will  form  a  parailelflgram  circumMvibed  about  the 
ellipse. 

60.  If  HD  be  drawn  through  the  centra,  parallel  to  Tl  a 
tangent  at  P>  cuUiag  SP  in  the  point  E,  then  will  P£^  UC. 

Draw  FN  parallel,  and  Pa  perpendicular  to  HD,  Because 
NF  ia  parallel. to  tT  (30. 1.),.  and  the  angles  at  o  right  angles, 
'.'  the  angles  oPT,  oPt  are  right  angles  (99.  l.)»  or  oPTssoPt, 
but  FPT=:iSPt  (cor.  3.  Art.  57.),  •.*  by.  taking  the  latter  fmn 
the  former  FPq^NPo,  ':  PNz^PFz  (33. 1.),  the  aisles  at  z 
(=sthe  angles  at  o  by,  99.  1 )  right  angles,  and  Pz  is  com- 
mon to  the  triangles  PzN,  PzF,  •/  (26.  1.)  PN:=^PF.  And 
shice  EC  is  parallel  to  NF  a  side  of  the  triangle  SNF,  and  SC^ 

:=CF  (cor.  1.  Art.  53.),  v  5£=£i\r  (2. 60  i  •.*  SP+PF  (=:^ 
SiV^-|-iVP-f.pjF=2£i\r+2  NP)  ==  2PE.    But   «^P  +  Pf =2  l/C 

(Art. 52.), •.•  2 PJS=  {SP + PP=) 2 l^C, and PE^UC.   Q. fi D. 

61.  If  perpendiculars  be  drawn  from  the  foci  to  any  tangent, 
axikd  a  circle  be  described  on  the  major  axis  as  a  diameter,  the 
points  in  which  the  perpendiculars  intei^ct  the  tangent  shall 
be  in  the  circumference  of  the  circle. 

Let  P^,<Sr  be  drawn 
perpendicular  to  er  a  ^ 
tangent  at  P,  join  SP 
and  produce,  it  to  meet 
Ft  produced  in  F,  and 
join  Ct  Then  in  the 
triangles  PtF.PtY,  the 
angle  tPFssztPY  (Art. 
57.)»  the  angles  at  t  are 
right  angles,  and  Ft  is^ 
common,  V  (26.1.)  FP 
=3  PF  and  P*=^F;  also 
PCszCS  (cor.  1.  Art.  62.)  •••  Ct  is  parallel  to  S^  (2. 6),  and  the 
triangles  FCt,  FSY  are  similar,  •.•  PC;  CtziFS  i  SY  (4.6,). 
But  PC  =  ^PS,  V  a  =  i5Fx=4.5P+PF=4SP+PP  = 
(Art.  52.)  4-  VUszFCj  /  since  Ct^CF,  the  points  ^  and  Fare 
in  the  circumference  of  the  cirde  whose  centre  is  C,  and  in  like 
manner  it  may  be  proveil  that  T  is  in  the  circumference. 
Q.  E.  D. 


COMK  flBcTIOKS. 


FAstJL 


01.&  Hie  i«6tttigte  TtAT^EC^.  Vrtiifaoil  »  to  JT  uid 
join  CA,  tben  beoauBe  tTR  U  a  right  ukgle,  the  segmetit  iTM 
h  4  umUAn^  (SL.3.),  v  <C  and  CJ2  tnoetinj^  at  thecMitM, 
will  constitute  the  diameter,  and  be  ia  the  same  strai^t  Um, 
V  the  aa^  tCF^SCR  (15. 1.)  and  IC  CJPccAa  CS  respec 
tivdy,  V  (4. 1.)  f]t«&B|  •/  Fi.Srfc35JtS2te(a5.3.)  rS.8Um 
(Art.  54.  cor.  3.)  £C*.    Q.  fi.  D. 

Car.  1.  Hedce  Fi  .EC  .t  JBC:  ST  (17.  d.),  •.•  IV  :  EC»  i: 
J?# :  5r  (eor.«.  20.«.)  ! :  FP  :  SP  (4. 6.  because  the  triaiigiei 
KP,  STP  are  similar) : :  FP  r  %VK:^tP  (because  FP+^P;* 
«I^C,  Alt.  59)  Whetefoie  pntting  FCiiso,  ECisA,  FP^jf, 
and  Ft^^y  the  analogy  Fl«  :  EC»  ::  fp  t  ^rC^FP  becomdi 


y*  :  I*  : :  « :  2a— x,  •/  ^^ 


which  «i|uatioa  expreasts  Um 


2a-- J? 

imtfire  of  the  ellipse  considered  as  a  ^piM,  described  by  the 
retohition  of  FF  about  the  centre  F. 

Cbr.2.  Because  Ft*  :EO::FF:  SP  {car.  1.)  •.•  4Fh  :  4£0 
(=Fi:»=I.2rC,  cor.  1.  Art.  56.)  ::  l.FP  :  LSP,  v  (16.5.) 
4Ft^  :  LSP  : :  U^fC  i  l.SP  : :  2FC  :  SP  : :  ^VC  :  gFC—FP. 

62.  If  BT  be  the  focal  tangcttt,  thu  tdU  CF.CT^VO. 
See  the  following  Jigure. 

Because  (cor. 2.  Art. 57)  ST.tFz.SB:  BF»  •/  by  eoifr» 
position  and  division  (18.  and  17. 5.)  STf  TF :  JST*^TF  i :  iKH 
+BF  :  SB'-BF,  or  (cor.  1.  Art.  52.)  2Cr-  2CF: :  SB-¥^St  i 
Sg^BF.v  (15.5.)  2Cr.2CF:  4CF«  : j  (SA^AP.SB^BF : 
SB-^BF.SB--BP  : : )  SB+bt]^  :  5B«-BP»  (^.  5.2.).  Bwt 
BFS  is  a  right  ang^k,  /  (47.  1.)  SB^^hP^^PS^^  (4.2.) 
4CF*,  •.•  (prop.  A.  & )  2Cr.2CF=S^4-jgrFp«=  F&»  (Art:  62.)* 

4ro^4.2.),  V  cr.CF=:rc*. 

Cor.  Hence,  because  Cr=CF+Fr,;-.-  (CT-CF=)  f:F^+ 
CF.FTz^VL\  V  CF.Fr=FC«-CF»=£C^  (w.2.  Art. 54.)' 


t 


.J 


>  t 


.  -i      .  - 


fkUT  X; 


THB  £LUMB. 


4€6 


^ 

T 

M 

y- 

\ 

Ti 

y^ 

V 

/«\\ 

• 

J. 

\ 

/ 

F 

N 

^ 

• 

/       "" 

K 

C 

/ 

/r 

\ 

r 

/ 

tt.  If  m  be  Amm  perpendicvkr  t6  the  dinetrix  yr^  tken 
wMl  PP  :  i»lf  : :  i?C  :  VC. 

Let  PiV^  be  perpen* 
dicuhur    to    V\J,    then 

SP-i-FF.SP—PP*,  V 
(16.  6.)  SP^PF  :  SN:- 
i^P  ::  SN+NF  :  SP+ 
PP.  Bui  (Art.  52.  cor.  4.) 
5P—  PP  =  «  PC -«  PP  ; 
and    SN^NF^SC'^CN 

-iVPss  CP-2^P+  Ci^=: 
^CN;  likewise  SN-i^NF 
s^^CF;  and  SP+PP= 
2FC  (Art.  52.)  j  by  sub- 
stituting these  values  for 
tbeir  equals  in  the  above 
analogy^  it  becomes  2FC 
^^FP  :  2CiV  : :  2cP  :  ^ 

2FC  : :  (Art.  62.)  2PC  : 

2Cr;  .  (15.  5 )  FC'^FP  zCNi:  VC:  Cr, subtract  the  former 
antecedent  from  the  latter,  and  the  former  consequent  from  the 
latter,  then  (yC—VC+FP:  CT-^CN  ::  VCiCT-,  that  is,) 
FP  :  (Nr=)  PMiiVCiCT::  (Art.  62.)  CF  :  VC.     Q.  E.  D. 

Cor.  Hence,  if  the  centre  C  be  supposed  at  an  infinite  distance 
from  Vj  CF  may  be  considered  as  equal  to  FC,  •.*  FPssPM,  and 
the  curve  in  this  case  at  every  finite  distance,  becomes  a  para- 
bola. See  Art.  1. 

64.  If  PF  be  produced  to  meet  the  curve  in  p,  then  will 
SFP.Fp=FB.FF-i^Fp. 

Because  FFiNT::  CF:  CV  (Art.  63.),  if  P  be  supposed 
to  coincide  with  B,  the  point  N  will  coincide  with  P,  and  the 
straight  line  FP  will  become  P6 ;  *.*  the  above  proportion  will 

become FB  :FT::CF:  CP;  •••  since ^^  \^^j  i:CF  :  CF, 


^^ 


*  Por  (47. 1 .)  SP* -5iV«  +  ATPg  an4  PF^^lfF^  ■¥  XPu'-  SP*— 
PFi'  ^SIV^^JVF* or  (cof . S. «.)  SP-i- PF,SP^jHF»SN+  /VF.dN-^NF 
as  above. 

VOL.  u.  ah 


M*  CONIC  SECnOMS.  Paut  X 

rPzNT::FB:Ft  (11. 5.) ;  K«t  NTtsPM  (S4.  l.>,  •>  F^ : 
PM  ::  FB:  FT,  •/  FBiFT::  FP^FB  :  (PM-Fr=)  FIT. 
In  like  maBDCT  it  may  be  sbewn  tliat'FB  :  FTi :  Fp  s  fm,  v 
FBzFT: :  PB-^Fp  :  (FT-^pm^J  Fn;  v  (H.  5  )  FP-^FB : 
FN::FB^Fp<Fn.  Bat  tlie  triangleft  FPN,  Fpii  are  simikr, 
•••  (4.6.)  FN :  FP  ::  Fn  :  Fp,  and  ex  ^uo  («.  5.)  FP^FB : 
FP  : :  FB^Fp :  Fp,  •.'  (16. 6.)  FP.Fp-'FB.Fp==JPBJPP'^ 
FP^Fp,  /  by  transposition  «FP.Fp=(FB.FP+FS.Fp3:) 
FB.FP^Pp.    Q.KD. 

Cor.  Hence,  if  Ffi=/,  FP=sX^  and  Fpas3%  the  atove  con- 


clusion expressed  algebraically  will  be  2Xx^lJC+x,  or  —=: 
1       1 


Jir"*"x' 

65.  If  c  be  tbe  co-sine  of  the  angle  UFP  to  radius  1,  then 
will  FP  :EC::EC:  VC-c.CF. 

Because  (Art.  63.)  FP  :  PM  :  :  FC :  VC,  /  (16.6.)  FPJ€ 
zsFaPMz=iFC.FT±FN=zFaFT±FC.F^=  (bec2Lme  FC.FT= 
EO  Art.  62.  cor.)  EC*±FC.FN.  But  FY  :  FP  : :  -j- c :  i; 
/  (16. 6.)  TFN=c.FP,  and  ±FC.FN=c.FaFP,  /  by  sub- 
stituting  this  latter  quantity  for  its  equal  in  the  above  equa- 
tion, it  becomes  FP.FC=EC*+c.FC.FP;  v(?P.n:— c.JTFF 
=)  FP.  FC— c.fr=£C,  V  (17. 6.)  FP:EC::EC:  VC-^c.FC. 
Q.  E.  D. 

Cor.  If  VC  be  infinite,  FC  and  VC  mav  be  considered  as 
equal,  and  Aie  above  analogy  becomes  FP :  EC  : :  EC:  1— c.FC 
But  (Art.  56.)  EC  :^L  : :  FC  :  EC,  '.'  ex  <gyw  (2vg.  5.)  FP  : 
^L  ::  (FC:  1-c.FC  ::  )  1  :  1-c,  or  (16.6.)  1— c.]RP=r4.I, 

and  FP=- as  in  the  parabola,  see  Art.  ^S.  * 

1— c 

€6.  If  on  the  major  axis  as  a  diameter,  a  circle  be  described, 
and  P^T  an  ordinate  to  the'm^jor  aoiis  be  produced* to  meet  the 
circumference  in  Q,  and  if  .c  be  the  co-6ine  of  the  angle  FCQ 
to  radius  1 ;  then  will  FP=:  VC^cFC. 


P#»»X. 


THB  £LLTfSB. 


4fr 


a*^  «*«»Mpa» 


Beoftwe  (Art.  63.)  S?r?W.5N-2VjFs==SP+i?P.SP-JPP, 

(16.6.)5V(seeArt.  63.) 
SCF:2rC^2PP::2FC 
=  SCAT,  or  CF  :  FC^FP 
::  VC  :  CN  ::  (because 
QCzzVQQCiCN.  Butu 
(Art.  es.  part  9.)  QC  : 

Ci^::l:c,vCF:  TC- 
W  i :  1  :  c,  •.    (16.  «.) 

c.CF=Fc~FP,  and  FP 

==FC-c.CF.    Q.E.  D. 

67.  If  PN  be  an  ordinate  to  the  axis,  then  will  UN.NV :  PN^ 

iiFC^iEC^.  

For  (Art.  63.)  SN+NF .  5>^-2^F=5P+  FP .  SP--FF, 
'-'SN^NFz  SP-^PF  ::  SP-^PF  :  SN+NF  (16.6.);  but 
8N^NF=z  (5C+CArr-]^^F=:CF#-NF+CAr=)  2Ci^;  Ukcwise 
5P+PF=^arC  (Art  52.)  j  also  SP-PF=:2SP— 2 FC  (Art.  52. 
cor.  4.);  and  lastly,  SN-^NPss^SCf  •.•  substituting  these  four 
values  for  their  equals  in  the  above  analogy,  it  becomes  ^CN : 
^VC  : :  2S|>-.2rC  :  25C;  /  (15.  5.)  CNiVC::  SP-^FC ;  SC. 
. .  f  (18. 5.)  UN  :VC.:  SC-^SP--  VC  ;  SC. 
*    «-  {17.  6.)  FNiFC::  SC-SP-i-  FC  :  SC. 

From  the  former  of  these  (12. 5.)  UN  :FC::  IW4-5C+ 
5P-FC:  FC'^'SC::  SP  +  SN:  UF } 

And  from  the  latter  (19.5.)  FN i  PC::  FN-SC+SP^ 

rC:  FC^SC::8PSNi  US;  V  (prop.  G. 5.)  UN.NF:  FC^ 

::  SPi-SN.SP^SN  :   UF.US  ::    (cor.  5.2.  SP^-SN^^sz 

by  47. 1.)   PN^  :  JSC*  (because  UF.USz=zFF.FU  see  cor.  2. 

Art.  54.)  •.•  (16.  5.)  UN.  NF  :  PiST*  : :  FC*  :  ECK    Q.  E.  D. 

Cor.  y.  Hence,  because  UN=VC^CN,  fs^d  NF=^  (FC-r-CN 
^)  UC-CN,'.'  UN.NV=^VC^CN,UC'-CN-^{coT.^.%.) 
UC^^CNi  '.'  also  FC*--CN^  :  PiV*  : :  FC*  :  ECK 

fjor.  2.  Hence,  if  FQ;ssa,  ECssb,  CNssx,  and  PiV^jf,  the 
expression  FC^^CN*  :PN^::FC*  :  JEC*  becomes  aJ»-«*  :  y* 

:  i«  •/  ifisz— .  a*—i*  which  is  the  equation  of  the  ellipse. 

Hh  2 


: :  02 


4m  come  sECTioi^.  part  x. 

Ow.  S.  Hence  VN.'NV^  PN^i  that  is,  the  rectangle  con- 
tained by  the  abscissae  varies  as  the  square  of  the  ordinate. 

68.  If  Pn  be  an  ordinate  to  the  minor  axis  EK,  then  in  like 
manner  En.nK  :  /V  :  £C*  :  TC*. 

For  Pn=CN,  and  PN^Cn-,  VC-Pn^  .  Cn^  :  FC*  :  EC* 
(Art.  67.  cor.  1.),  •.•  (16.  5.)  CV*  ^Pn"^  :  FC«  ; :  Cn«  :  ECS  .- 
(17.  5.)  P»«  :  FC«  : :  EC*-Cfi«  :  £C«  : :  (cor.  5.2.)  EC+Cn, 
EC^Cn  :EO  ::  En,nK :  EC*-,  •.*  (16.  5.)  P«»  :  En.nJST :  :  VC* 
:  EC^  and  (prop.  B.5.)  EumK  :  P»«  : :  £C«  :  VC\     Q.  E.  D. 

69.  If  on  the  major  axis  UK  as  a  diameter,  a  circle  UQVhe 
described  and  NQ  an  ordinate  to  the  axis  be  drawn  cutting  the 
ellipse  in  P,  and  the  circle  in  Q ;  then  will  PN :  QN : :.  EC 
:  VC. 

For  OAr«=  UN,NV  (14. 2.)  •.*  (Art  67.)  QN^  :  PiV^*  :  :  FC* 
:  ECS  •••  (22. 6)QN:PN::VCi  EC,  -,'  (prop.  B.5.)  PiV  ;  <?iyr 
: :  EC:  VC.     Q.  E.  D. 

Cor,  1.  In  like  manner,  if  on  the  minor  axis  EIIl  as  a  diame- 
ter the  circle  EqK  be  described,  it  mayi)e  shewn  that  Pn  :  qn 
::VC:  EC. 

Cor.  2.  Hence  the  area  VPN :  UQN  : :  (UC=^)  EC  :  VC  as 
in  the  parabola,  (Art.  29.)}  in  like  manner  VPN :  VQN  : :  EC 
:  VC,  •/  UPV :  UQV : :  (2EC  :  2FC  : : )  EC  :  VC.  Also,  if  any 
point  S  be  taken  in  the  axis,  and  SP,  SQ  be  joined,  the  area 
UFS  :  area  UQS  ::  EC  :  {UC=^)  VC  as  in  the  parabola, 
cor.  Art.  29. 

70.  If  a  mean  proportional  R  be  found  between  VC  and  EC, 
and  with  it  as  radius,  a  circle  be  described  >  the  area  of  thit, 
circle  will  be  equal  to  the  area  of  the  ellipse. 

For  the  area  UPV :  area  UQV  ::EC:  VC  (cor.  2.  Art.  69.) 
and  since  VC  :  R  ::  R:  EC,  •.•  (2.  12,  and  cor.  2,  20.  6.)  area 
of  circle  UQV  "whose  radius  is  VC  :  area  of  circle  who^e  radius 
is  -S  : :  VC :  EC;  this  proportion  being  compounded  with  the 
first,  we  have  UPV.UQV :  UQVx2Lrea,  of  circ.  whose  rad.  is  R 
: :  EC.VC  :,  VC.EC;  that  is,  (15.5.)  elliptical  area  UPV:  circu- 
lar area  whose  rad.  \s  R::  {EC.VC  :  ECVC : : )  1  :  I ;  or  the 
area  of  the  circle  is  equal  to  the  area  of  the  ellipse.  *  Q.  E.  D. 
Cor.  1.  Since  (cor.  2.  Art.  69.)  UPV-,  UQV  : :  EC:  VC  :: 
(15.  5.)  EC.  VC  :  VC^  •••  (16.  5  )  UPV ;  EICVC  : ;  UQV :  VC^; 


Pam  X. 


TH£  ELLIPBB. 


4M 


-.'  <15. 5.)  atfea  of  ellipse  :  ECFC  : :  area  of  oirc.  whose  diam.  Is 
UF :  FC^.  But  the  area  of  the  circle  varies  as  FC*  («,  12.)  j 
•/  the  are^  of  the  ellipse  varies  as  ECFC. 

Car.  2.  Because  FC :  EC  : :  EC :  ^L  (Art.  56),  •/  FCi^Li: 
FC^  :  EC*  (cor.  2,  20.  6.)  j  but  VN.NF :  PN*  : :  FC*  :  £C* 
<Art.  67.),  •/  UN.NF  (cm-  FC^^CN^,  Art.  67.  cor.  1.)  :  PJV^«  :  t 
/^C :  4-L;  -/  since  f'C  and  ^L  are  constant  quantities  VN.NF 

Cor.  3.  Hence,  if  the  major  axis  UF  become  infinite,  the  curve 
at  all  finite  distances  fkom  the  vertex  U  vdll  be  n  parsd>o]a3  for 
NF  being  infinite  will  be  constant^  and  *.*  UN  oe  PN^  which 
(Art.  27-)  is  the  distinguishing  property  of  the  parabola. 

Cor.  4.  The  curve  UPF  which  arises  by  diminishing  the  ordi- 
nates  NQ  oi  the  circle  in  a  given  ratio^  is  an  ellipse. 

For,  let  EC:  UC::  PN:  QN,  then  if  an  eUipee  be  de^ 
scribed  on  UFaa  the  major  axis*  having  EK  for  its  minor  axis, 
we  shall  have  (Art.  69.)  UC :  EC  : :  QN :  ordinate  of  the  ellipse ; 
and  from  the  preceding  analogy  (prop.  B.  5.)  UC :  EC  : :  QN  i 
PN  •••  PiV^=an  ordinate  of  the  ellipse  (9.  5.),  or  the  curve 
passing  through  P  is  an  ellipse.  In  like  manner  it  may  be 
shewn,  that  if  the  ordinates  QN  of  the  circle  be  increased  in 
any  given  ratio,  the  curve  described  upon  UF  as  a  minor  aju8> 
and  passing  through  the  extremities  of  the  increased  ordinates^ 
will  be  an  ellipse. 

71.  If  a  plane  be  inclined  in  any  angle  to  the  plane  of  a  cir- 
cle, and  if  straight  lines  be  drawn  from  every  poiut  in  the  cir- 
cumference, perpendicular  to  the  inclined  plane,  the  curve  which 
passes  through  the  extremities  of  all  the  perpendicalars  will  be 
an  ellipse. 

Let  C7£r«rbeacir-  u^ >^^ 

cle,  and  the  perpendi- 
culars Uu,  Ee,  Fd,  Kk, 
&c.  meeting  the  inclined 
plane  GuvO  in  the  points 
tt,  e,  r,  k ;  the  figure  ueok 
will  be  an  ellipse. 

Let  UF  be  a  diame- 
ter of  the  circle  parallel 
U>  QO  the  common  sec- 
tion of  the  planes^  and 


470 


OOMIC  SICTfQN& 


MP  St  right  moglet  to  UV;  draw  GU,  OFevh  pcndU  to  JtfP« 
join  Ov  and  dnmr  Mp,  Qu  panAel  to  it,  join  Nn,  Fp,  IBecasxtt 
MFisa,  paralkkgnm  Jro is innIM to  1^ (94.  l.)«  tat  ITF is 
perpendicular  to  the  plane  iiifn  hj  eonstradion  (4. 11.)  '•'  MO 
u  abo  perpendicnlar  to  the  plana  MMn  (8. 11.)  *•'  w  is  perpen- 
iiicniar  to  MNn  (19.  il.)  .-  no  is  panUel  to  UFifi.  11.)  $  aai 
dnce  the  planes  MpP,  VuFv  aie  hath  at  right  aisles  to  the 
plane  GF,  their  common  section  Nn  is  at  right  ang^  to  it 
(19.  lU),  V  Nn  18  parallel  to  Pp  (6.  11.)$  v  jm  :  nJf : :  PNi 
NM  («.6.)and  fmiPN.i  nMz  JVJf  (1«.  5.)  : :  radial  2  cMiae 
FMp  (part  9.  Art.  63.)  the  angle  of  inclination  of  the  planes,  or 
pn  :  PN  in  a  given  ratio,  *•*  (cor.  4.  Art.  70.)  a^nlp  is  an  ellipse. 
Q.  £.  D. 

Cor.  Hence  the  oblique  section  of  a  cinder  is  an  fSlipse,  ef 
winch  the  minor  axis  is  the  diameter  of  the  cylinder. 

79.  If  a  circle  be  described  on  the  major  axis  as  a  diameter, 
and  any  ordinate  ^TP  be  drawn  meeting  the  cirde  in  Q»  tangeate 
at  P  and  Q  will  meet  the  axis  prodoeed  in  the  same  point  T. 

For  if  possible,  let  Qr  be  a  tangent  to  the  drde  in  Q,  and 
PT  not  a  tangent  to  the  ellipse,  but  cut  it 
in  P  and  p;  draw  np  and  produce  it  to 
meet  TQ  produced  in  in;* then  since  the 
triangles  TNPy  Tup,  as  also  TNQ,  Tnm  are 
similar  (32.  \)  PN  :pn:i  NT  inT  11  QN  x 
vm(4. 6.).  But  PiV :  QAT: :  pa :  qn{hxi.  69.), 
-.-  PN :pn  ::QN:qm  (16. 5.).  But  by  the 
first  analogy  PN :  pa  : :  QN :  ma,  -.*  QNi 
qn  : :  QN:  ma,  v  (9.5.)  qnszmu,  the  less 
to  the  greater,  which  is  impossible ;  *.*  TP  m 
which  meets  the  ellipse  in  P  does  not  cut 
it,  it  must  therefore  be  a  tangent  to  the 
ellipse.  In  like  manner  (s^  the  figure  to 
Art.  ee) ;  since  Pn  :  qn  (^nC)  : :  FC  :  EC 
(cor.  1.  Art.  69.),  it  may  be  shewn  that 
tangents  at  P  and  q  cut  the  minor  as^  in 
the  same  point  t    Q.  £.  D. 

Cor.  1.  Because  CQris  a  right  angle  (IS.  a.  see  the  iguie 
to  Art.  72  ),  CN:CQ::CQ:CT  (cor.  8. 6.)  5  but  CFarOQ,  / 
CN.CF.iCF:  CT.  In  like  manner  it  is  shewn  that  fsce  the 
figure  to  Art.  66.)  Cn  iCEiiCE:  Ct. 


PA&rX. 


fFHB  ELLIfSE. 


4n 


Cor.  2.  TN.NCszQN^  (cor.S.e,  and  iy.6.)«CQ*— CW» 
07. 1.)=  FC«-CiV«=  (cor.  1.  Art.  670  VN.NU. 

Cer,  3,  The  sub-tang«nt  NT  i«  greater  than  ^FNg  for  sinc^ 
<ly  the  precedHig)  TN.NCszFNMUr-'  {16.6.)  NTiFN  i: 
NUiNCi  bat  CU  >  2^^C,  •/  (JVC+Cl^ss)  i^I^  >  ai^C,  %'  2Vr  > 

Cor.  4.  If  PG  be  the  normal^  then  <cor.  8. 6.  and  17-  6.) 
TNMQ^PNK  and  TJV.l^C  :  TN.NG  ::  FC  :  ^L  (cor.l. 
Art,  67.  and  eor.  1.  Art. 7«.)  •/  NC:  NG;:FUiL  (15. 5.). 

73.  mrP  be  a  diamettr  and  JTO  its  conjugate,  then  PM 
being  drawn  perpendicular  to  KO  catting  the  aw  FU  in  G>  thfi 
sect^ngle  PM.PG^EC\ 

For  if  Cy  be  drawn  parallel  to  PM,  the  angle  PGNsttyCG 
(89.  L)»  bnt  yCG+yC<=(CC/=) 
a  right  'iM^gle,  and  ytC-\-yCt=:A 
right  angle  (32. 1.),  /  yCG+yCt 
s=yiC-\-yCt;  take  away  the  com- 
mon angle  yCtf  and  the  remainder 
yCG=:yiC,  /  PGNsx(yCG^)  ytC, 
and  PNG:ss  Cyt  being  right  angles ; 
•/  the  triangles  PGN,  Cyt  are  equi- 
angular (32. 1.)  J  and  PG  :  (PN:sl 
by  34. 1.)  C»  : :  Ct  i  (Cyzs)  PM 
(4.  6.)  5  •••  PM.PGz=Cn.Ct  (16. 6.) 
iss  JSC*  by  cor.  1.  Art.  72.   Q.  E.  D. 

74.  Join  PS,  th^n  if  PO  be 
drawn  perpendicular  to  T^^and  Gk 
perpendicular  to  PS,  Pkss^L. 

For  the  angles  at  k  and  M  being  right  angles,  and  the 
angle  kPM  common,  the  triangles  PMR,  PkG  are  equiangular 
(32. 1.)  •.•  PRiPM  ::  PG:  Pk  (4.6.),  and  PR.Pkz=zPM.PG 
(16. 6.)=EC^  (Art.  73),  •.•  (PR-hy  Art.  60.)  FC :  EC : :  EC : 
Pk  (16. 6.).  But  FCzEC  : :  EC  :  4.L  (Art.  56.),  •.'  Pk^iL 
(9. 5.).    O.  E.  D. 

75.  If  PC;  CO  be  semi'Conjugate  diameters,  and  PN,  Om  be 
perpendicular  to  the  axis,  then  will  CN*+  Cm<ss  FCK 

For  FC^-^Cm^  :  Om*  : :  FC*  :  EC*  (cor.  1.  Art.  67.)  : : 
FC^'-CN^  :  Piyr*  (Art.  67.)  But  OC  being  parallel  to  tT,  and 
the  angles  at  m  and  A^ right  angles,  -.•  (29. 1.)  the  triangles  COm, 

nh  4 


4fe  CONIC  8BCn(»«.  Pamt  X. 

PNT  are  similar,  and  (4. 6.)  Om  :  Cm  : :  PN :  NT;  /  («. «.) 
Oni<  :  Cm*  : :  PN*  :  NT\  '.'  from  this  and  the  Brst  analqgy 
(2^.  5.)  VC*-Cm*  :  Cm*  ::  VC^-^CN*  :  NT*.  But  CN.NT  i 
lVr«  ::CN:NT  (1.6.)  /  hy  iiiTenion  Cm* :  FC*— Cm*  : :  NT 
:  CN,  and  by  companUon  FC*  :  FC*^Cn^  ziCTzCNiz  (1. 6.) 
CA^^CT :  CiV*  But  FC*=CN.CT  (cor,  I.  Art  73.)*  *•'  ''^'- 
Cm«s=  CN«  (14. 5.),  •.-  FC*=:  C»r»+  Cm*     Q.  £.  D. 

Car.  1.  Hence  FC»— Cy=:0»»»,  v  Cm*  :  FN*  ::  VC*  I 
EC*  by  the  first  anak^  in  the  proposition,  and  Cm  :  PN:: 
FC :  EC  (23.  6.).  In  like  manner,  because  FC^^Cm^szCN*, 
V  CN*  :  Om*  ::  FC«  :  £CSand  CN^t  Om::  FC i  EC. 

Cor.  3.  Henoe  also  Cm  :  FN  ::CN:  Om,  •/  (16. 6.)  Cm.Oni 
zsPN.CN. 

76.  If  PN,  Om  be  perpendicular  to  the  axis  FU,  and  PC,  CO 
semi-conjugate  diameters*  then  will  PN'  -{^Om*  szEC* . 

For  CN'  :  Om*  : :  FC*  :  EC*  (cor.  1.  Art.  75.),  : :  FC«- 
CN'  :  PN*  (cor.  1 .  Art.  67.)  *•*  summing  the  antecedents  and 
consequents  (13. 6.)  FC*  :  Om*'\'PN*  n  FC*^CN*  :  PN*  :: 
(Art  67)  FC*  :  EC*,  /  Om*  +  PN*=zEC*  by  14. 5.  Q.  E. D. 
Cor.  1.  Because  CP  and  CO  are  semi-conjugate  diameters  to. 
each  other,  '.*  CP  will  be  parallel  to  a  tangent  at  0;  and  Cn*-^ 
Cr'=  (Om*  -^PN'  34. 1.=)  EC'-,  and  hence  the  same  relation 
subsists  between  the  ordinates  and  abscissas  to  the  minor  axis^ 
that  does  between  those  to  the  msyor  axis. 

77.  CP*'^CO*z=:FC*-}-EC*. 

For  FC*=iCN*  +  Cm*  (Art.  75.),  and  JBC«=PJV*  +  Om» 
(Art.  76.)  3  V  FC  +EC*=z{CN*  +  PN*  +  Cm*  +  0m*=:)  CP* 
-^  CO' (47.1.).    Q.  E.  D. 

78.  If  Fe  a  tangent  to  the  major  axis,  be  made  e^pal  to  the 
semi-minor  axis,  and  eC  be  joined  cutting  PN,  any  ordinate  to 
the  msgor  axis  in  Jf ;  then  will  MN'  -k-PN*  =  Fe*. 


pART'i. 


THE  ELLIPSE. 


473 


For  the  triangles  eFC  and  JUNC  being 
rimilar  (2  6.)  Ve  :  MN : :  CF :  CN,  and  Fe' 
:  MN'  : :  CF*  :  CN^  (92.  6.),  /  Ve'  :  Fe'- 
Jtfiyr*  :  :  CF»  :  Cr»-CiV'  (prop.  E.  5.)  :: 
Fe»  :  P2V*  (cor.  1.  Art.  67.) }  '.•  Ve'  -^MN* 
as  /  2\r»(l4.  5),  and  consequently  MN*  +  PN' 
sFc    Q.  ED. 

Cor.  Because  ArN'  +  PJV»=:(Fe»=)  EC', 
i£e  afeo  <Ae  fgure  to  Art,  73.  and  Oi»'  +  iW 
5=£C'  (Art.  76);  ••  MN=Om,  and  JIfO 
being  joined^  it  will  be  parallel  to  the  axis 
vil  (33.  I.).  Hence,  if  a  straight  line  OC  be 
drawn  from  the  extremity  0  of  the  parallel 
JIfO,  through  the  centre  C,  it  will  be  the  conjugate  diameter  to 
PC;  and  henoe  by  this  proposition,  having  any  diameter  of  an 
ellipse  given^  the  position  of  its  conjugate  may  be  readily 
determined* 

78.  If  PC,  CO  be  semi-conjugate  diameters,  and  PM  be 
drawn  perpendicular  to  CO  (see  the  figure  to  Art,  73.)  then  will 
CCPM^FCEC. 

Because  PNt  Om  are  perpendicular  to  the  axis,  and  Cy 
perpendicular  to  the  tangent,  *.*  (cor.  1.  Art.  75.)  CN :  Om  : : 
FC :  EC,  and  (16.  5.)  CN :  FC : :  Om  i  EC;  and  the  jtriaogles 
TCy,  OCm,  being  similar  CT :Cy  i :  CO  i  (ha  (4.  6),  the  two 
latter  analogies  being  compounded  (prop.  F.5.)'CW.Cr:  FCCy 
iiCO:  EC;  but  (because  CN.CT^FC\  cor.  1.  Art.  72.)  TCV 
:  FCCy  ::FC:Cy  ::C0  :  EC;  V  (16. 6.)  FCEC^OCCy^ 
OC.PM  {S4. 1.)    Q.  £.  D. 

Car.  1.  Let  FC:=^a.  EC^h,  PCzzx,  Cyszy,  then  (Art.  77) 
CO»a:(FC*+ JSC'-PC'ss)  a»4.6»-x%  v  y«  =  (Cy»=: 
FC»,EC'  aH* 


CO' 


+  6*— «•* 


Cor,  2.  Hence,  if  at  the  vertices  of  two  diameters  which  are 
coiQugates  to  each  other,  tangents  be  drawn,  a  parallelogram 
will  be  circamsei'ib(*d  about  the  ellipse,  the  area  of  which  is 
4C0.PM  a  constant  quantity.    See  the  figure  to  Art.  58. 

79.  If  CP,  CO  be  senii^oo^jugate  diameters,  then  will  FP,SP, 
szCO». 


474 


CXnUG  8BCTK>NS. 


PastX 


For  the  trianglM  SJH,  PRM,  FPTvre  uoukr,  because  TF, 
PM,  and  iS  are  parallel^  the  apgl^  at 
r^  ilf,  and  I  right  angles,  and  TPF:sztPS 

(cor. 3.  Art.57.)  =  PBM  (^  l.)|    /  SP 
:Si::PR:  PM,  and  FPiFT  .:  PR: 
PM  (4. 6.)^  these  analogies  being  com- 
pcmnded  (prop.  F.  5.)  SP.FP  :  St.FT  : : 
PR*  :  PM*.    But  (Art.  78.)   rC.J5C= 
OC.PM, '.'  (rC==by  Art.  eo.)  PR  :  PJf 
: :  OC  :  JBC  (16. 6.) ;  and  FB«  :  PM*  : : 
0C« :  EC*  (23. 6.)  j  /  from  above  SP.FP 
:  8t,FT : :  0C\:  JSC* ;  but  StFT^EC* 
(Art.  61.  B.)  V  SP.FP=:zOC'  (14.5.) 
Q.  £.  D. 

80.  l^et  OX  be  the  eoGJugate  aad  <2o  an  ordinate  to  the  dia« 

meter  PG,  then  wUl  Pv.vG  :  Qv*  : :  PC*  :  CO*. 

Draw  PA'^  tTn,  QH,  and  Om  perpendicular  to  the  axis  FIT/ 

and  or  parallel  to  it.    Then  because  PN  is  par^lel  to  Qr,  or  to 

TN,  and  <?o  to  PT,  the  triangles  PTiV;  Qvr  are  equiangular, 

and  (4.  6.)  Qr  :  (rj?=by  34. 1,)  Hn::     ' 

CN 
PN  :  J^r,  V  Qr  :  j^Bn  ::  PN. 

CM 

(~.i^:5)  CN  (part  4.  Art.  75.)  }• 

bat  vhi  Cn  ::  PNiCN  («.  6.)  j  •.• 
by  adding  the  antecedents  together, 
and  tlie  consequents  together  (12.  5.)  q 
in  the  two  last  analogies,  Qr-f  on : 

CN  * 

^.  Hn+  Cn  : :  ^PN :  2CN,  or  QH 
NT 

CN 


NT 
andQH 


.iJbi+Cn  ::  PiV^ :  CJV (15. 5.), 


'NT 


.Hn+Cn]*  ::  PiV  : 


CN*  (92. 6.).    But  (cor.  1.  Art.  6?.)  FC»-CH« :  QH  •  : :  FC» 
^CiV' :  PJVr*  (being  each  as  VC* :  EC)  /  ex  aqw  («.  5.) 

CN  . 

VC-^CfI^:—.Hn^i^^::FC'^CN':CN*  ::    (cor.9. 

ArtW.)  CNNTiCN':-.  (15.50  f^TiCNs  v  (since.  C^: 


JPaktX. 


THXBLLIPSB. 


-♦W 


rc  3 :  FC i  CT  bf  eot.l.  Axt.nst  iribnice,  by  oor.  9, 90.  d, 
Cy :  CT::  CN' :  rC'  =  ^.CiV^*)  FC'-  CH*  or  its  equal 


*  cr  ^_      ^  .     _       NT CN 


'  (16. 6j  and  {wt  4. 


cr  CT 

actually  squaring  and  muUiplyuig  j)  •/  ^k^  .  C-AT'  —  ^^  .  Cn'  s9 


cr 


C2y 


CN 


~.Hn'  (by  reduction,  and  from  the  figure);  •/  CN*'^Cn'=:i 


CN 


CT 


j^.Hn*  (by  dividing  by  ^),  or  NT.CJS'-^Cn^szCNJSn*  -,  \' 

(16.  6.)  CN'-^Cn'  :Hh*  ::  CN :  NT  :  i  (by  inversion  in  th* 
7th  analogy^  above)  CN»  :  FC'^CN^i  :•  (16.5.)  CN'^Cn*  : 
CiSr*  : :  H»»  :  VC'-^CN'-,  but  (^.6.)  CJV :  Cn  : :  CP  :  C»,  V 
CN'^Cn*  :  CA^«  : :  CP«-C»»  :  CP'  (part  4.  Art.  69.).  Also, 
(by  similar  triang.  and  22.  6.)  rv^szHn*  :  (Cw'ssby  Art.  75.) 
VC'-^CN'  ::  $»•  :  CG'j  •.•  (CP«  — Oi7«=cor.  5.2.)  Pv.vG  : 
CP*  ::  Q©«  :  C0«,  and  (16.5.)  Pto.cG  :  Q©*  : :  PC*  :  CO*, 
Q.  £.  D. 

Cor.  Hence  it  may  likewise  be  shewn  by  similar  reasoning^ 
that  if  Q«  be  produced  to  meet  the  curve  again  in  9»  Pv.vG  : 
qv  ::  PC*  :  CJT',  -.'  Qv  :  qo  :  :  CO  :  CX.  But  CO^CX 
(car.  Art.  58.),  •-•  Qv^qv. 

81.  The  parameter  P  to  any  diameter  PG  is  a  third  propor-* 
tional  to  the  major  axis  and  conjugate  diameter;  that  is,  FU : 
OX::  OX:  P. 

Let  the  ordinate  Qv  passing 
tfaroi^fa  the  fbcus  F  meet  the 
curve  9gain  In  9;  thea  will  Qq 
be  the  parameter  to  the  dimw*  q 
ter  PG,  and  (cor.  Art.  80)  Qvx^P. 
Because  (Pv.vGx)  PC'-^Cv^:  Qv* 
: :  PC*  :  CO*  (Art.  80.)  /  Qv*  : 
PC'^CV  : :  CO*  :  PC«(iHX>p.  B.5.) 
But  because  Ce  is  parallel  to  vP  ^^ 
(Art.  60.)  Pe^FC,  v  PC*^Cv*  : 
(Pe^  wi8e««=)  ?€•—€♦"  J :  P^  :  Pr» 
t ;  PC*  :  Pe*  -.*  «r  ^equto  (08. 5.*] 


47« 


COHK  aBCTMNS. 


PAttX. 


Fe*^er*  ;;  CO*  :  (Pe'»)  FC>. .  But  JV»— (Se'ae)  er»: 


Pe+er .  Pe— er  (cor.  5.*2.)=(Ait.  60.)  CP.  5P=(Art.  79.)  OO'*, 
•/  Qo-  :  CO*  : :  C0»  :  VC*  and  (««.  6.)  0©  :  CO  : :  CO :  VC, 
v(l5.5.)2CP:«CO::«CO:SrC,]lhati8P:OJir::  OJT:  FU 
orFUiOXiiOXiP.    Q.B.D. 

82.  If  two  ellipses  RPZ,  RQZ  bave  a  common  diameter  RZ, 
from  any  point  N  in  which  iVP  and  NQ  an  ordinate  to  each  of 
tliem  be  drawn,  then  will  the  tangents  at  F  and  Q  meet  tbe 
diameter  RZ  produced  in  the  same  point  T, 

Draw  TP  a  tangent  to  the  ellipse  RPZ 
and  join  TQ;  TQ  shall  be  a  tangent  to  the 
eQipse  RQZ.  For  if  not^  let  7X2  meet  the 
curve  again  in  g  and  draw  the  ordipates  nq, 
np  and  produce  np,  TP  to  meet  in  r.  Then 
PN'  :pn'  ::  RKNZ :  Rn^Z  : :  QN'  :  qn' 
(cor.  S.  Art.70.)a  •.'  P^  i  pn  ::  QN  ;  qn 
(92.6.).  But  the  triangles  PNT,  mT  are 
siBiilar,  as  are  also  QNT,  qnT;  -.'  PN  i  m 
: :  NT :  uT  (4. 6.)  i:  QNi  qn,  •/  PAT  :pn:: 
PN  :  rn  (11. 5.),  '.'  pn^rn  (14.5.)«  the  less 
equal  to  tiie  greater^  which  is  absurd  ^  -.*  TQ 
meets  the  curve  no  where  but  in  Q,  conse- 
quently touches  it  in  Q.    Q.  £.  D. 

Cor.  Heni»,  if  RZ  be  bisected  in  C,  the 
point  C  will  be  the  centre  of  both  ellipses^ 
and  (cor.  1.  Art.  72)  CN  :CR  ::CR:  CT. 

83.  If  RPZ  be  an  ellipse,  of  which  RZ  is  a  diameter,  atiid  if 
from  every  point  in  RZ,  straight  lines  QN  be  drawn,  having  aoy 
given  ratio  to  the  ordinates  PN,  and  cutting  the  diameter  RZ 
in  any  given  angle^  then  shall  the  curve  passing  throtj^  It,  2, 
and  all  the  points  Q  be  an  ellipse. 

For  since  by  hypothesis  PN  iQNiiOC:  oC,  (22. 6.)  PN* 
;  QN'  ::  OC*  :  oC\  But  (Art.  80.)  RN.NZ :  PN'  : :  CR'  : 
0C\  /  ex  aquo  (22.5.)  RNSZ  :  QN'  ::  CR'  :  Co'  which 
(by  Art.  80.)  is  the  property  of  the  ellipse  5  '.-  the  curve  RQoZ 
is  an  ellipse.    Q.  E.  D. 

84.  If  PQJIf  0  be  the  drcfe  of  cnrvatuie  .at  the  point  P  in 
the  eUipse  PFU,  PG  the  diameter  of  curvature^  and  PH,  Pv 


Part  X. 


THB  ELLIPSE. 


477 


tJie  chords  of  curvfttore  paasiog  thretigh  the  centre  C>  and  focus 
F  respectively  i  then  wiU 

CP:CO::CO:  \PH. 

PK:CO:  :  CO  :  ^O. 

VCiCOxiCO  :  \Pv. 
Join  PC  and  produce  it  to  M,  and  join  Gt),  HQ  and  QP;  draw 
the   tangent   TP,  and  through  I?  and  C  draw  Qr,  OCK  each 
parallel  to  TP,  then  will  OC  he  the  semi-conjugate  diameter 
and  Qr  an  ordinate  to  PH^  and 
let  QP  be  the  arc  in  its  nascent 
state,  which  may  therefore  be 
considered  as  common  to  the 
circle  and  ellipse.     Then   be- 
cause the  angle   TPQ=zPHQ 
(32. 3.)=PQr  (29. 1.)  and  QPr 
is  common4o  the  two  triangles 
QPr,  QPH,  these  triangles  are 

equiangular  (a^.l.)^/  Pr  :  PQ 
::  PQ:  PH  (4,6.)  r^' Pr.PH 
saPO'sr  (since  the  arc  QP  is 
indefinitely  small,  see  Art.  35.)  Qr';  ••  Pr.rM  :  Pr.rH :  :  PC*  : 
CO'  (Art.  80.),  •/  (rM :  rH,  that  is  since  r  and  P  are  indefi- 
nitely near  coinciding)  2PC  :  PH  :  :  PC  :  CO'-,  •.•  (15. 5.)  PC 
:  ^PH :  :  PC  :  C0\  •.•  (cor.  3, 20.  6.)  PC  :  CO  :  :  CO  :  ^PH. 
Since  CK  is  parallel  to  TP,  and  TP  perpendicular  to  PG 
(cor,  16. 3.),  CKP  is  a  right  angle  (29. 1.),  also  PHG  is  a  right 
angle  (31.3.),  and  the  angle  ffPG  common  to  the  triangkt 
PKC,  PHG  •.•  these  triangles  are  equiangular,  and  (4. 6.)  PK  : 
PC  ::  PH  :  PG  ::  ^PH  :  •i.PG.  But  PC  :  CO  :  :  CO  : 
^PH,  '.'  ex  aquo  (22. 5.)  PK  :  PC  :  :  CO  ;  \PH,  and  PC 
:  CO  ::  ^PH :  ^PG,  •.'  PK  :  C(f  :  :  CO:  ^PG.  Again,  the 
triangles  PnK,  PvG  having  the  angles  at  K  and  r  right  angles, 
and  the  angle  at  P  common^  are  similar  (32. 1.)  j  •/  (P«=by 
Art.  60.)  VC  :  PK  ::  PG  :  Pv  :  :  ^PG  :  ^Pv,  and  PK  :  CO  :  t: 
CO  :  ^PG  '.'  ex  aquo  FC  :  CO  :  :  CO  :  ^Pv.     Q.  E.  D. 

Cor.  Hence  VU :  ^CO  :  :  ^CO  :  Pv,  that  is,  the  chord  of 
cnrvature  Pv  which  passes  through  the  focus  F,  is  a  third 
proportional  to  the  major  axis,  and  the  conjugate  diameter,  and 
is  consequently  equsd  to  the  parameter  of  the  diameter  PM.^ 
(Art.  81.) 


478  COme^  SBCntmS.  Past  X. 


*> 

A' 

^  £^ 

' — >A7 

%\j 

^/^yv^,. 

••/5Cr^ 

%- 

-^ 

^^^ 

-A 

u  t 

^ 

66«  If  n  plmiB  out  a  oone  oo  u  ndtber  i^OMot  the  base  nor 
be  parallel  to  it^  the  section  wiU  be  an  dUpse. 

Let  ABD  be  a  ooae»  and  let  tbe  section  VEUK  be  perpen- 
dicular to  ABC  the  plane  of  the  geneiBtiiig  triangle,  VU  being 
their-common  section,  and  the  section  FiXid  be  parallel  to  the 
base  and  therefore  a  circle,  and  let  its 
common    sections    with    ABD    and 
VEUK  be  cd  and  PQ-,   let  oEKb 
be  a  section  likewise  parallel  to  the 
base,  bisecting  FU  in  C,  having  EK 
and  ah  for  its  common  sections  with 
the  planes  F£27J^and  ABD.  Because 
ABD  and  FcQd  are  both  p^^ndicu- 
lar  to  VEUK,  their  common  section 

FQ  is  perpendicidar  to  ABD  (19.  J 1.)  b^^^ ^d 

and  therefore  perpendicular  to  VU 
and  cd  (conv.  4. 11.),  in  like  manner 

it  may  be  shewn  that  EK  is  perpendicular  to  VU  and  ah,  '.*  EK. 
and  PQ  are  bisected  in  C  and  iV  (3. 3.)  j  and  since  cd  and  ah  are 
parallel  (16. 11.),  '.*  the  triangles  UNt,  UCa  are  ecpiianguhur, 
and  UNiNci.  UC:  Ca,  also  AF :  IVd  : :  (CF=)  UC :  C6,  / 
by  compounding  the  terms  of  these  aoalqgiesT/M^r:  Ncfid 
t:  17C»  :CtLC6.  But  Nc.NdsiFN'  and  Ca.CbszEO  (14.3,), 
V  i/iV.NK  :  PiV  ::  l/C*  :  £C'  which  (by  Art.  67.)  is  the 
propwty  of  tbe  ellipse ;  therefore  VEUK  is  an  elKpse,  come- 
quently  if  acone  be  cut  by  a  plane  which  neither  meets  the  base 
nor  is^paraUel  to  it^  the  section  will  be  an  elHpse.    Q.  £.  D. 

THE  HYPERBOLA. 

DEFINITIONS. 

86.  If  two  straight  lines  JPP,  SP  revolve  about  the  fixed 
points  F  and  S,  and  intersect  each  other  in  P,  so  that  SF-^-FF 

.  may  alwi^  e«[ual  any  given  straight  line  Z,  the  point  P  witf 
describe  the  figuiie  PVR  which  is  called  am  HYPEaaoLii. 

87.  H  two  straight  lines  Fp,  Sp  revcdve  in  liloe  manner  about 
F  and  S,  so  that  Fp^Sp  may  always  equal  the  given  stra^ht 
line  Z,  the  point  p  will  likewise  describe  an  hyperbola  pUr^ 

«this  figure  and  the  former,  with  respect  to  each  otber^  are  called 

OFPOSITB  HTrBKBOLAS. 


Pabt  X« 


THE  HYPBBBOLA. 


479 


86.  Tte  fisBd  poinu  Fund  5ftboat  whtab  tb0  itaiigiit 
FP  and  5P^  jF)»  and  Sp  revolve,  are  called  thb  foci. 

m.  \i  F,  S  km  joined,  the  itnogbt  line  l^K  intercepted 
between  the  oj^poiite  hyperbolae  ie  allied  Tsa  major  axis,  and 
the  pointB  {/,  FarecaUedTHs  ratircirAi.  vbbtxoss. 

90.  If  UV  be  biBected  in  C,  the  point  C  is  called  thb 

CENTEB. 

IT. 


91.  If  through  the  centre  C  the  straight  line  £1^^  be  drawn 
perpendicular  to  the  major  axis  UV,  and  if  from  F  as  a  centre, 
with  the  distance  CF  a  circle  be  described,  cutting  EK  in  the 
points  £  «9d  K»  the  straight  line  ££  is  called  the  mi^ob  axis. 

Cor.  Hence  £C=Ci^  (3.3.). 

9^.  If  JBC=€F,  that  is,  EjP=  UV  the  hyperbola  is  called 

EaUII.ATBBAL. 

98.  H  *wtth  EK  as  a  major  axis,  «nd  UVt&  a  minor  axis  two 
•pforite  hyfierboks  GEH,  gKh  be  desoiibed,  these  are  called 

CO«nEr«AT£  HYPEBBOLAS. 

94.  Any  straigl^  line  passing  throu^  the  centre  C,  and 
teiw^ated    by   liie    two    opposite    hyperboks,    Ib   called    a 

9»A1IBTE«. 

Thtt9  Pp  u  a  diameter  to  the  point  P^  or  p, 

95.  A  straight  iisie  soeetlng  the  ewrve  at  any  p^nt,  and  which 
being  produced  does  not  cut  it,  is  called  a  tangent  to  that 
poivit. 

Thus  PT  \Ba  tangent  at  the  point  P. 

9CL  If  JP|p  be  a4iameter,  and  PTa  tangent  at  thepoint  P> 
and  tl^ough  the  centre  C  a  straight  line  Hg  be  drawn  paralM 


480 


CCfMIC  SBCnONS. 


PAMtiL. 


to  the  tangoit  FT,  Uie  Due  Bg  is  Gdkd  tu  cmjugatb 

PIAMBTER  to  Fp. 

97.  If  through  the  focus  F  a  straight  line  DB  be  drawn, 
perpendkmlar  to  the  axis  i%  meeting  the  curve  in  B  and  D, 
DB  is  called  thb  latds  rbctum  or  principal  parameter. 

98.  A  tangent  at  th^  extremity 
of  the  latus  rectum  produced  to 
meet  the  axis>  is  called  the  focal 

TANGENT. 

Thus  BT  is  the  focal  tangent. 

99.  A  straight  line  drawn 
through  the  point  where  the  focal 
tangent  meets  the  axis,  and  parallel 
to  the  latus  rectum^  is  called  thb 

DIBBCTKIX. 

*      TT^iw  xy  is  the  directrix. 

100.  A  straight  line  drawn  from 
any  pcHnt  in  the  curve,  perpendicu- 
lar to  the  axis,  is  called  an  ohdi- 

NATB  TO  THB  AXIS  at  that  pOtUt. 

Thus  FN  is  an  ordinate  to  the  axis  at  the  point  P. 

100  B.  The  segments  of  the  axis,  ii^rcepted  between  the 
ordinate  and  the  vertices  of  the  opposite  hyperbolas,  are  called 

ABSCISSAS. 

Thus  V  and  V  being  the  vertices,  and  FN  the  mr^^naie,  VN 
and  NU  are  the  abscissas. 

101.  if  PG  be  a  diameter  and  Pr  the  tangent  at  the  point 
P»  a  straight  line  drawn  from  any  point  Q  in  the  curve,  puaUel 
to  FT,  and  meeting  FG  produced  in  v,  is  called  an  obdinatb 
to  the  diameter  PG;  see  the  figure  to  Art.  141. 

103.  If  the  ordinate  to  any  diameter  pass  through  the  focus, 
and  meet  the  curve  on  the  opposite  side,  the  ordinate  thus 
produced  is  called  thb  parambtbb  to  that  diameter. 

Thus  bd  is  the  parameter  to  the  diameter  FG,    See  the 
figure  to  Art.  141. 

103.  An  asymplote  is  a  straight  line  passing  through  the 
centre,  which  continually  approaches  the  curve,  but  does  not 
meet  it,  except  at  an  Infinite  distance  from  the  vertex;  or,  it  is 
a  tangent  to  the  curve  at  im  infinite  distance. 


TAid   (tte  tM  Sguni   tp.  Act,.  134)   CX,  Cs   arc   <Ae 

PBOFERTIES  €&  TH£  HYPERBOLA. 

104.  Tlie  diflerence  of  the  two  strw^t  lines  drawn  from  (he 
Ibd  to  any  point  in  the  curve,  is  equal  to  the  ms^or  axisj  that; 
is,  SP^FFss  UV^SLFC.  (See  the  figure  to  Art.  89.) 

For  since  SP-^FP  is  aconstant  quamRy  in  whatever  point 
of  the  curve  P  be  talcen  (Art.  86.)«  let  the  points  P,  p  be  sup- 
posed to  arrive  at  F  and  U  respectively,  then  SP  will  become 
SF,  and  FP  wiU  become  FF,  •••  5P— FP  will  become  SF^FF; 
in  like  manner  fy^Sp  will  (by  the  arrival  of  the  point  p  at  C7) 
become  FUSU,  v  SF^FF^FU^SU  (Art.  87.)  5  but  5F= 
FU-^-SUand  FU^FU-^FFr*  FU-^-SU-FF^zFU-k^FF^SU 
'.'^SV=i^FFwad  SU::^FFi  v  SP^BP^SF-^FF^SF^SUz^i 
VFz=i(hn.  90.)  9^FC.    Q.  B.D, 

Cor,  1.  Hence  the  foci  are  equally  distant  from  the  centre 
and  likewise  ttom  the  vertices,  that  b,  SC^FC,  SV^FF,  and 
SF^FU. 

Car,  2.  Hence  SC^UF'\'FPt=9V€-\^FP :  and  5P+FP=s 
2FC-f-2FP. 

Car.  3.  Because  BS-^iTsxVF  (see  the  figure  to  Art.  97.) 
=sfS-2FF,  and  BS^  >*  F9  •/  JIP  >.«rP  and  («BFsr>  BD  ^ 
4FF '.*  the  latus  rectum  is  greater  than  four  times  the  distance 
of  the  focus  F  from  the  vertex  F. 

105.  The  rectangle  FF.FUssEC*  (see  the  figure  to  Art.  89.) 
For  EC'^FE'--  FC*  (47. 1.)  =rC'-.  FC»  (Art.  91.)  = 


FC-^FC.  FC-'FC  (cor.  5. «.).     But  FC  +  ^'C  =  FU  (cor.  1. 
Art.  104.)  and  FC'-FCz:^  FF,  .•  FF.  FUz^EC'.    Q.  E.  D. 
For  the  same  reason  C75.SF=EC». 

106.  The  latus  rectum  is  a  third  proportional  to  the  m^or. 
and  minor  asus;  or  FU:  EK  ::  EKi  BD  (see  the  figure  to 
Art.  97.). 

Because  B5»=2FC+FB)«  (cor.  2.  Art.  104.)=a4rC»-f  FB* 
+4FC.FB(4.«.).  And  BS«=?FS»+F£»  (47.  l.)=4FC'  + 
FB»(4.2.),  •/  4FC»+4rC.FB=s4FC»j  and  FO-^FC.FB^ 
FC*i  \'  Fe.FB=sFC«  —  FC» »(Art.  105.)  £ <>,  •••  FC  E 
EC  :  FB  (17. «.),  ':FU  .EK: :  £«  :  BjD  (lo.  6.)    Q.  E.  D 

VOL.  It.  1  i 


«  •  • 


4M 


€DNICi^CTIOK& 


BkwmXi 


Cor  I .  Henoe  J2C»  ss^L.TC,  vtiA  HT*  «1»>K  - 
Car.  2.  Hence,  in  the  equilateial  hypefbdia,  because  fHtTatt  JUT 
(Art.  99.)  •••  BDszEK  (prop.  A.5.)i  that  is,  the  nugor  axis, 
minor  axisi  and  latys  lectam,  are  etfotl  to  eskth  iflkldr. 

lor.  If  FP,  8P  be  drawii  from  the  im  to  any  fioCnt  P  in 
the  curre,  the  itraight  liaa  Pr  whkh  Useda  the  angle  J^M 
will  be  a  tafcigent  at  P. 

For  if  not,  let  Prmeet  the  hyper- 
bola again  in  p,  drale  FF  perpendicular 
to  Pr  meeting  it  in  Y,  prodooe  FV  to 
m,  and  join  pS,  pm,  afid  pF.  • 

In  the  tHengles  FPF,  mPT,  the 
angle  mPY^FPT  by  hjrpothdeli,  the 
angles  at  Frigfat  angles  by  coa6tir0di6n^ 
and  PFcOnnnon,  •••  {9SA.)  FFasihFj 
'.'  in  the  triangles  Ff^Y,  mpY,  thfe  sides 
FY»  YpctmY,  Vp  each  tOr  eMh,  and 
the  ineliMkd  aogies  at  Fright  aagtasi, 
V  (4.1.)  Fp^mp;  \'  5p--pF=:5|p— 
pm.  Bat  5p— pFae5P-*PF<Aft  Sa) 
s=5F— Pm=5i»,  •••  Sp-^pmssSm,  and 
<8fp:a:i$«f+jMn  which  (30*  1.)  i^  afannd^, 
-/  TP  cannot  possibly  iwet  the  hyp** 
b<4a  aginn  In  any  point  ;p^  \*  JP  tonchai 
the  curve.    Q.  E.  D. 

Cor.  I.  Hence  the  tangent  at  the  vertex  Vis  perpendicular  to 
ihe  axis  SF,     See  cor,  1.  Art  5j'. 

Cor.  2.  Hence  (3.  6.)  ST.  TP  .:  kP  :  PP.' 

lOS.  All  the  diameters  of  the  brperbola  ana  bi00Bffadi  bf  dr 
centre  C.    (See  the  figiu*e  to  AiC  W.) 

Complete  the  paralldegiam  PSpF,  then  (34. 1.)  SpssPP 
and  SPsxpF,  •.•  Fp-'-^BdbSP^PF,  /  (Alt.  87)  the  point  p  is 
in  the  oppo^te  hypeibok;  join  Pp,  \*  (pai*t  8.  AR.  84i.  cer.) 
SC^CFand  pCsstCP,  and  the  like  may  be  shewn  of  any  othe^ 
diameter.    Q.  E.  D. 

'  Cor^  1.  Hence  the  tangents  PTy  pi  at  the  points  P  ami  p  are 
]pmllel«  fcir  since  (84. 1.)  SFF±zSpF  and  these  an^  are 
bfsecCkl  by  PT  and  pi  (Art.  167)  their  halves  will  bee^lfal; 
that'is^rPpss  l>P,  /<^,  1.)  Prfelparallel  iopi. 


?^mmXi 


THS  PYf BRSQ^ 


m 


Cor.  «.  PeipDe,  if  taffgenits  U  dirawn  at  the  ejttieiuUiw  of 
tW9  GQi^u^^  dis^meti^  ff ,  ^*  the  four  Isw^g^Mts  will  fwii 
9.ffkr»lle}pgca^« 

109.  If  CR  be  paraUel  to  a  tangent  PT,  cut^iig  FB  prodimd 
in  H,  then  will  PHas  FC. 

Ihw.^lpaimWtoCiB.aniiwtt;'*.    XJw  tiWW  th» 

angle  PSi^SFV 
(99.  1.)  *  yPF 
(Art.  107  )  «  *f5 


•  • 


(29. 1.). 

P/(«;i.)=rJf*«+ 

it/;    the»   (€.«.) 

HL  But  (XXM*.  1. 
Ajt.  104.)  K?=fc 
CS,  '.*  (prop.  A.5.) 

fP4-PK=2P«+ 
PP.  But  (cor.  3. 
Art.  104)  P55= 
«r04-FP,v«P« 
+  FP^^FC-i^FP,  V  PRtaiVa    Q.  B,  D. 

1 10.  If  the  tangent  PT  he  produced,  and  «traig!it  lines  SZ, 
FY  be  drawn  from  the  foci  parpendieukr  talt,  the  points  Y 
and  Z  will  be  in  the  circumference  of  the  circle  described  ea 
the  major  axis  UF  as  a  diameter. 

Join  CFand  produ^  FY  to  meet  SP  In  ^m,  then  since  the 
triangles,  mPY,  FPY  are  equal  and  similar  (Art.  40f .),  PF* 
mY  and  FPrzMPi  v  Sin  =  <SP— Pwi=)  SP-^BP^sz^VG 
(Art.  109.).  Next,  because  FC=CS  and  i>r=r«,  •.•  (2.  6.) 
CF  is  pairallel  to  Sm,  and  the  triangles  PCF,  FSm  at«  equi- 
angular (29. 1.),  s-  PC  :  CF  : :  FS  :  ««  (4.  6  ) )  but  ^^Crcj-IV, 
•/  CYs:^Sm  (16.  and  prop.  J>.^.)x^FC  by  what  has  been  shewn 
above,  •.•  Fand  F  are  in  the  circumference  «f  the  circle  of  wbieh 
C  is  the  centre.  Produce  I^  to  n,  then  since  CY^'^m^  •.' 
nYssSm,  and  they  are  paralteli  •••  (33. 1.)  Sn,  mY  are  equal 
and  parallel,  and  if  Sn  be  produced  to  meet  YZ  in  Z,  then 
SZYJtmYZi;^%  right  w^les  (29. 1.)  5  buj  mYZ  is  a  rijjjit  angle 

ii2  ' 


484 


CONIC  SBCTIOKS. 


PrwitX. 


'.'  SZV  is  a  right  angle;  that  is,  the  straight  line  passing 
through  S  and  n  Is  pefpendleolar  to  YZ;  and  since  nY  is  a 
diameter  of  the  circle,  and  nZY  a  right  angle,  Z  is  in  the 
dicun^renoe  (31. 3.)    Q.  K.  D. 

HI.  The  rectangle  FY, SZ=  EC'. 

'  For  since  Z  is  alight  angle  (Art.  IIQ.)*  and  nC,  CY  meet 
at  the  centre  C,  they  are  both  in  the  same  straight  line  (31. 3.) 
•.'  FCYzsSCtt  (15. 1.).  also  SC^CF  (cor.  1.  Art.  104.),  and 
nC=zCY>  ••  (4. 1.)  FY=zSn.  But  (cor.  3a.3.)  S^&iznVS.SU; 
that  is,  Pr.5Zr=F5.Sl7=(Art.  105.)  EC*.    Q.  E.  D. 

Cor.  1.  Because  the  triangles  fPF,  SPZ  have  the  angles  at 
P  equal  (Art.  107.)  and  the  angles  at  Y  and  Z  right  angles, 
ihey  are  equiangular  (3^.  1.),  and  FY :  FP  : :  8Z  :  SP  (4. 6.), 
•/  FY :  SZ::  FP  :  SP  (16.  5.).  But  FYJ8Z=EC^i  '.-(17. 6) 
FY:  EC::  EC  :  5Z,  and  (cor.  2, 20. 6.)  FY^  :  KC^  ::  FY :  SZ^ 
But  since  PF :  SZ  : :  FP  :  SP  •/  FF*  :  EC*  ::  FP  i  (SP^) 
'2VC-^FP  (cor.  2.  Art.  104.)  If  VC=:ia,  EC:=zb,  FP=zx.  ajid 
FY==y  the  last  pn^KH-tion  becomes  y*  :  A*  : :  a? :  2a +x,  •/  y*=: 

6*3: 
2a+j:* 

Cor.  2.  Hence  4Jpy«  ;  4EC*  : :  FP  :  SP. : :  L.FP  ;  X.5P 
(15.5).  V4fy«:  L.FP::  4EC«  :  L.5P  (16.  5,)  ::Lx2KC'  : 
L.SP  : :  %VC :  (SP=)  ^FC-^FP  (car.  Art.  104.) 

112.  If  £D  be  the  focal  tan- 
gent,   then    wEl    the    rectangle 
CECr=  VC*. 

For  since  (cor.  2.  Art.  107.) 
STiFT  ::  SB:  BF,  \'  (18.  and 
17.6.)  5r+JFTor2CF:  5r— Er 
or2Cr  ::  SB-^BF:  SJSSF,  v 
(15.6.)  4CF«  :  4CECr  :; 
SB+  BF .  SB  "  BF:  (SB  —  BF. 
SJB-^BFsz)  SB'-^BF]^  ::  5B«- 
BF^:  (I7F«=)  4rC«,  see  Art.  86. 
But  since  SFB  h  a  right  angle, 
4Ci^  (=SE«)=6'JB*— J5E*  (47. 1.). 
V  4CF..Cr=x4rC*  (14.  5.)  and 
CF.Cr^FC*.    Q.E.  D. 


-•  For  «/^C:  2EC  : :  sEC:  /*  (Art.  106.)  v  L.^FC^aEC^ , 


PAtt  X.  THE  HYPERBOLA.  *m 

Cor.  1.  Hence  (IT.C.)  CF.VC::  VOi  CT. 
Car.  9.  Because  CT^CF-FT,  v  €F.  CF-^CF.IT^CRCf 
«FC«5  •/  CF.FTraCF'-FC^^ (Art.  105.)  JSC*  v  CFi  EC 
ECiFT. 


113.  If  from  any  point  P  in  the  carve,  PM  be  drawn  per- 
pendicular to  the  directrix  xy,  then  wttl  FP  i  PM  ::  CF:  CV.  • 

Join  SP  and  draw  PN  peri)endicular  to  the  aitis  UV,  pro- 
duced, then  because  (47. 1)  SP*=SN'  +NP'  and  FP«=JW« 
+^^P^  by  taking  the  latter  from  the  former  .SP*— JFP«=^jy« 
-IW*,  that  is  (cor.  5. 2.)  SP-^FP.  SP.--FP=:8N^FN.SN>^FN; 
V  (16.6.)  SP^FP  :  SN-^-FN  ::  SN-^FN  :  SP^FP.  But 
(cor.  2.  Art.  104.)  5P+JFP=3FC+2PP/  also  SN+FN=:^Se 
+  CN  +  FN=:CF+CN  +  FN=2CN,  and  SN'-FN=:SF=t 
2CF,  likewise  (Art.  104.)  SP^FP=z^VC ;  '.•  if  instead  of  the 
terms  of  the  above  analogy,  their  equals  be  substituted,  we 
shall  have  ^VC-\'^FP  :  ^CN. : :  ^CF  :  2FC,  or  FC^FP  i  CN 
'i  CF:  VC  :i  (cor.  1.  Art.  112.)  VC :  CT,  •.•  (cor.  19. 5.)  FP  i 
{NT=:)PM::  VCiCTiiCFiVC.    Q.  E.  D. 

Cor.  Hence,  if  P  be  supposed  to  coincide  with  B,  FP  will 
become  FB  and  PM  wiU=:Fr;  *.*  the  above  analogy  becomes 
(FP  :  PM::)FB:FT::FC:  VC. 

114.  If  PF  be  produced  to  meet  the  curve  again  in  p,  then 
will  2PP.  Pp=FB.FP-\-Fjf>. 

Because  (cor.  Art.  113.)  FP  :  PM  : ;  FB  .  FT,  •.•  (16  and 
cor.  19. 5.)  FP  -FB  :  {PM--FT^)  FN :  i  FB  :  FT.  3ut  (con 
Art.  113.)  FB:  FT  ::  Fjp  :  pm,  ':  FB-^Fp  :  (FT— pni=)  Fh  :  : 
FB  :  JFT}  /  FP'-FB  :  FN:-.  FB--Fp  :  Fn.  But  the  triangle^ 
FPN.Fjm  are  similar,  r  FN :  FP  ::  Fn.  Fp,  •/  (22. 5.)  FP-^FB : 
FP  : ;  FB-^Fp  :  Fp, '.'  (16. 6.)  FP.Fp^FB.Fp^^FB.FP'-FP.Fp; 
or  2FP,Fp^FB.FP+Fp.    Q.  E.  D.  . 

Cor.    Hence,  if  PJ5=t  FP^X,  and  Pp=sx,  we  shaU  have 

—        2    jsr+«    11 

2Xj;=?.A'+x,  and— =--^=~+jr.  , 

115.  If  c  be  the  co-sine  of  the  angle  PFU  to  the  radius  l^ 
then  wiU  FP  :  EC  : :  EC  :  VC-^cCF. 

For  (Art.  113.)  FP  :  PM  :  :  CF :  VC,  \'  (16.  eyppyc^^ 
CF.PM^  (34.  l.y CF.TN=CF.TF+FN=^CFTF^^CF.FN.  But 
(cor.  2.  Art.  112.)  CF.TF=EC\  and  (Art.  63.  Pai^  9.)  PA^^ 

I  1  3 


4M 


cONtc  st/(yti6m. 


t^«#£. 


FF  ::  ±c  :  Ij'v  itTz^  ^e.Pf(l6.6.),  and  ¥CF.1^:= 
^c.FP,Cf,  •/  frdfii  the  flrst'cqaa«6ii  by  sabstittitidn  FP.VC=i 
tC^-^cFF/t,  or  /!P.  r(?+(f.K>.CF=lSC*,  that  is  /P  FC+d.CF 
=EC«5  .•  (16. 6.)  FF:  EC::  EC:  FC-^c.CF.    Q.  E.  1>. 

116.  If  PN  Ife  aa  ontinatt  to  tke  nu^r  aods  FC7,  thm&  t?iU 

VN.NUi  PiV«  :  :  FC*  :  £(  ». 

P<*  (Aft.  113)  iF7?>.  5P-  PFtti  ^^i5^  IVF.  N8^ NF, 
V  (15. 6.)  m-^NFi  BP^PF: :  SF^PF:  NS^NFi^hkh  by 
I^Utltution  (as  in  tbe  htttf  part  Of  Art.  il5.)  becomes  ^CN  : 
itrC  : :  ^FC+fi/P  :  ^CF,  •.•  CJV :  FC  : :  FC-^FP  :  CF;  whence 
by  coiApoaition  (ir.  5.);6iid  divbion  (17. 5.) 
^«  obiyn  the  iblldwio^  analogieli,  viz. 
15f«#  CJT-  FC :  FC : :  VC+FP^CF:  CF. 
SMkdIlf  CN^ VC .VCii  VC-^FF^ CF: CF 
By  adding^  tbe  anteo^cfepts  Udd  cotts^qu^tltd 
Uk  the  flwt,  abd  siibtnictin^  in  tba  s^cobd 
(IS.  and  19. 5.)  we  have 
CN^VC:  VC  ::  CN^FP^CF  :<:F+FC 
::  FP^-rt^iCF-k^VC 
CN-^FC  :FC  ::  FP^CF-^CN :  CF-FC 
::  FP-^FN:  CF-^VC 
'.'  coropounding  the  ratios  (23. 6.) 
gJg-- FC .  C?y+FC- :  FC^  : yTF^TpN. 
FP^FN  :  CF+FC.CF^IfC,  or  m. 
m:  VC^t:  iFP-^FN^^)PIf^:  (CF«^F^7*«.byArt.  106,) 
BO  •/  alteirftalely  FiV.  JVi7 :  Pm  iiVC^i  ECK    Q.  E.  D. 

Cor.  Heaoe»  because  VN.  NU=  CN^^VC .  CN-k-  FCa*  CJV«r- 
^C*  (oor.  6.8.)  •/  by  substitution  C!iy«— FC«  :  FiV*  : :  FC*  : 
iC«i   wherefore,  if  VC^a,  ECs^h  CN^:^,  wd  Pi^^ry^  ^ 

shall  have  a*— a«  :  y»  : :  a«  :  I^,  whence  y*;±:^ .  JrZ^. 
«  •  . 

117.  If  two  hyperbolas  F/\  VQ  b«  described  m  the  same 
mioor  axis,  having  eC  Mi,  EC  respectively  for  the  temi-roinor 
axes ;  and  if  NP  be  produced  to  Q,  then  will  QN  •  FN  •  • 
afc:£C 

For  (Art.  116.)  (  ^'f  •  ^^  ^  ^^'  : :  ^'C»  :  JEC* 
V  kar  li^iio  CJV*  :  FJV»  : ;  gC»  :  EC*,  &nd  r«6.  t5. j  QN :  FN ; : 


Cor.  L  Hence  it  may  be  shewn,  as  in  lxUf%.  Ihil  tangents 
at  P  and  Q  will  meet  the  axis  produced  in  the samepoint  T; 
that  the  area  VQN :  am^  FPN : :  eC :  EC,  and  that  if  i^  be  aqy 
point  in  the  axis,  the  area  VQF  :  area  FPF  : :  «C :  EC. 

Car.  2.  Hence,  if  VQ  be  an  equilaterai  hyperbola,  or  VCss^eQ 
(Art.  92.)  J  then  since  VN.  Ntl »  QN'  : :  TC  r  #C*  (Art.  11«.) 
FN.NU=z  QN'  (prop.  A.6.) 

118.  In  the  equilaterai  hyperbola,  the  latus  rectum  is  equal 
to  the  minor  axis,  that  is  ^FbssSteC. 

For  since  (Art.  105.)  Vt.FU^eC*,  if  the  point  N  be 
supposed  to  coincide  with  JP,  the  expression  (cor.  9.  Art  \\T^ 
VN.NU^QN*  will  become  VF.FU^Fb\  %•  F6'=cC',>t= 
eC,  and  gPiasaeC.    Q.  £.D. 

Cor.  1.  Hence  it  again  appears  that  the  miyqr  axis,  minor 
axis,  and  latus  rectum  of  an  equilateral  hyperbola,  are  e<^ual  to 
each  other. 

Cor.  «.  Hence,  because  (Art.  106.)  VC.ECiiEC  :  BF,  %• 
(cor.  2,  20. 6.)  VC  xBF'.i  VC*  :  EC'.  But  (Art.  116.)  VC  : 
EC*  ::  VN.NU  or  CN'-^CV  :  P^^  v  Fli.NU  or  CA^«- 
CF'.PN'::VC:BF.. 

119.  If  Fit  be  an  ordinate  to  the  minor  axis  £C,  then  wili 
Cn*  +  EC'  '<  FBP  : :  £0  :  FC'  (see  the  fisBowuig  figure.) 

fkir  (34. 1.)  Fm=xNCBod  Ckm^NP  \'  (eor.  Art  116.)  P*f 
f!f  FC  :  0»'  : :  FC^  :  £C,  *.*  by  addh^  anteccdenU  and  CQHr 
sequentB  Pn'  :  Ca'.-fCC'  : :  FC^  ;  £C^'  And  by  invei]aiQn  Cn^ 

120.  If  PN  any  ordinate  to  the  majm*  axis  be  produced  to 

meet  the  conjugate  hyperbofta  In  Bf,  then  wiH  ii^^*— PjW*=» 

2EC».  .  . 

•  > 

•  •   •    •  ;  V        ^ 

•■•         .^.-  •     .  .  •  .  •  •  '       .      •      ^ 

•    -  '  '.  • »    1 

.  »  '  *  '  • 

.     ■  ■    \ 


ii4 


4tt 


CONK  SBGTIOBli. 


Fait  1, 


B0CM»«  (cor*  Alt. 

116.)a«-JBC»:n^:: 
JBO  :  VO  •/  (16. 5.) 
a* -EC*  :  EO  i: 
(n6«=)  CJV»  :  FC*,  ittul 
(17.5.)  C6*-.2EC«  : 
CJB«  ::  CN*^CF*  : 
CV^  : :  (by  alternation 
and  inversion  in  cor.  2. 
Art.,  lib.)  FN^:  EC*, 

V  (9.5.)  a«-2i;c«= 

2£C«,  but  (34.1.)  a- 

niyr«— pjvr«=:  2  EC«  5 

and  in  like  manner  it 
maybe  shewn,  that  if  , 

hn  be  produced  to  meet  the  hyperbola  VP  in  (he  point  w^  fr6«— 
n6»=2FC'.    Q.E.D. 

121.  If  PT  be  a  tangent  at  the  point  P,  tben  wfll  OJV^cr 
zsiFO.  ^  ' 

Because  (cor.  2.  Art.  loy.)  STi  TF  ::  5P;  PF,  v  divi- 
dendo  et  componendo)  Sr—  TF :  574-  TF  : ;  SP'^PF  :  SP4- 
PF\  that  is  (sec  Art.  liS.)  %CT  r  gjF  ; ;  2FC :  5P+  PF  But 
(Art.  113.)  SN^NF.SN^NF^SF:Iff.sFTpF,  -.•  since 
SN^NF=.SF,  SP^PF^^rC  (Art.  104.)>  and  SN^NF^ 
9CH,  by  substitution  5E.2CJV^=2rC.SP+PF  /  (16.6,)  5E  ; 
2VC::  SP-hPF  :  ^CN;  but  it  has  been  shewn  that  ^CTiSP 
: :  2FC  ;  SP^PF  '.•  fjr  dr^uo  2Cr  :  2rC  : :  2FC  ;  2CW  that  is 
CT',VC:,VC:  CN,  •.•  (17. 6.)  CW.CTs FC».     Q.  E.  D. 

Cor.  1.  Because  NT^^rCN-^CT,  •.•  €N.NTs:CN.CN^ CT« 
tiV»  -  CN.CTsz  CN'  -  FC . 

Cor,  2.  Because  in  the  equilateral  hyperbola  CN^-^FC'ss 
PN*  (because  FCs^EC,  see  the  cor.  to  Art.  116.)  •  •  CN  NT= 
(fN'-^-FC'^z)  PN\ 

Ccr.  3.  Hence  also^  in  the  conjugate  hyperbola  En,  if  ps  be 
an  ordinate  to  the  axis  Eg,  and  pT  a  tangent  at  p,  then  will 
C«.Cr=EC'. 


Past  X.  THE  UYFES»9Lh.  48B 

1^.  If  Pit  be  an  ordinate  to  the  minor  v&%  BC,9Bd  the 
tangent  Pt  meet  EC  in  t,  then  will  Cn.C^s£C'. 

Be'cause  (Art.  121.)  CN.CT=FC',  v  (17.  6.)  €N :  FC  : : 
FC  :  Cr,  •/  (cor.  2,  20  6.)  Ci\r :  Cr  : :  CN'  '•  VC\  /  (17.  5.) 
NT:  CTi:  CN'  —  FC«  :VC'::  (because  bf  cor.  Art.  116.  CN' 
"-VC^  :  PiV*  : :  FC  :  i:c»,  by  alternation)  PJV*  :  EC.  But 
the  triangles  TPN,  TtC  are  similar,  •.•  (4. 6.)  NT :  CT::  PN: 
Ct;  '.'  (from  above)  PN :  Ct::  PN»  :  EC%  '/  (16. 6.)  PN.EC^ 
rrO.PiV,  or  EC'=^Ct.PN;  But  (34. 1.)  P.VssC»,  v  Oi.C/= 
-EC.    Q.  E.  D. 

Tor.  Hence,  because  Cn.CfssEC'icor,  3.  Art.  121.)  •.•  0».C« 
=  C«.Cr  and  Ct^Cty  that  is.  if  the  perpendicular  Pn  cut  the 
conjugate  hyperbola  in  p,  and  tangents  be  drawn  at  P  and  p, 
the  points  i  and  T  where  they  meet  the  minor  axis^  will  be 
equally  distant  from  the  centre.  C;  and  conversely,  if  Ct=CTy 
the  perpendicular  Pn  will  pass  through  the  point  p, 

123.  The  same  things  remaining  nt  :nT::  nP'  :  np». 

For  by  the  preceding  corollary  Cn.Ct:=iEC\  '.•  (17. 6.)  Cn  : 
EC  ::  EC:  Ct,  \-  (cor.  2,  20.  6. )  Cn:Cf  ::  Cn'  :  EC,  v 
(componendo et dividendo)  Cfi+CTornf :  Cn-^-CfoTnt::  Cn» 
+EC'  :  Cn'  —  ECK  But  (Art.  119.)  Cn'-^-EC'  ;  Pn*  :  EC  r 
FC»  and  (cor.  Art.  116.)  Cil?-r-C£'  :  up'  ::  EC  :  FC'  •.• 
(11.5.)  Cn*+EC'  :  nP'  ::  Oi»-«£C»  :  «pS  •.'  (alternando) 
Cn«  +  £C*  :  Cn'  ---EC'  ; :  nP'  :  np»  5  that  is,  nt :  nt  : :  nP'  : 
up'.    Q.  £.  D. 

124.  The  normals  at  P^and  p  will  meet  the  minor  axis  in  the 
same  point  g.  • 

For  the  angles  gpT,  gPt  being  right  angles  nP^=znt.ng and 
np'2s:nT,ng  (14.2.)  \-  «P«  :  «p*  ;  :  nt.ng  :  nt.ng,^y  {Art,  123.) 
nt :  nT  ::  nt.ng  :  nTng  : :  ng  :  ng  ;  that  is,  the  normals  at  P  and 
p  cut  the  minor* axis  at  equal  distances  from  rt  or  in  the  satiie 
point  g,    Q.  E.  D. 

Cor.  In  like  manner  it  is  shewn,  that  if  NP  be  produced  to 
meet  the  conjugate  hyperbola  in  n^  the  normals  from  these 
points  will  meet  the  major  axis  in  the*  same  point  G. 

125-  If  CR  be  parallel  to  a  tangent  at  P,  and  MPG  perpen- 
dicuiar  to  it*  then  will  the  rectangle  PMPG^EC'. 

Let  PN  be  the  ordinate^  and  di*aw  Cm  perpendicular  to  the 
tangent  Pt*    Because  in  the  triangles  PTG^  CTt,  the  angles  at 


496 


CONIC  8BCT10NS. 


Pakt  X. 


7d. 


* 

/ 

1 

/ 

1 

X— 

/  > 

<? 

K 

S 

/: 

/ 

^ 

^ 

\ 

/^ 

v\ 

^ 

Ni 

a 

P  and  C  are  right  angle»^  and  the 
vertical  angles  at  Teqaal,  \*Ctm 
ssPGi^t  and  the  angles  at  m  and 
N  being  right  angles^  the*  remam- 
ing  angle  tCm^sNPG,  \'  CnU, 
PNG  are  equiangtdar^  and  (4.  6.) 
Cm  :  C*  : :  P^  :  PG,  •.•  (16.  6.) 
Cm.PG^Ct,PN,  but  Cmz=:PM 
(34.1.),  ••  PM.PG^Ct.PNx=:EC' 
(Art.  122.)    Q.  la.  D. 

126.  If  from  the  point  P  the 
normal  PG  be  drawn,  PF  joined, 
and  OH  drawn  perpendiccdar  to  ^ 
PP,  then  will  PH=^L. 

Produce  GP,  FP  to  M  and  It,  then  because  the  angles  at 
H  and  M  are  right  angles  and  those  at  P  vertical,  the  triangles 
PHG,  PUR  ape  equiangular,  and  (4. 6.)  i^PG  :  FH  : :  PH  : 
PJf ,  V  (16.6.)  Pfl.P«:='PG,Pilf=^  (Art.  1^50  JEC'=;?{'»or.l; 
Art  106,)  +L.FC.  But  (Art  109.)  Pfi=  TC,  v  PHJ^R=xi 
iZ.PR,  ov  rH=^.    Q.£.P. 

127.  If  CR  be  paraUel  totliBtangisnt  at  P,  and  PN,  RH 
perpendicular  to  the  im^or  axis,  then  vm.  CN^ ^CE* v^VC* . 

Draw  tR  an  ordinate  ta  the  unnor  aais,  and  produce  it  ijb 
Q,  and  draw  the  ordiBale  Qui,  Then  (43or.  Art.  116.)  Cn^^CF^ 
:  Qn*  ::  CN'-^CV*  :  PN  and  Qf-^VC  :  RM»  : :  CN't^ 
CF'  I  PN'.  But  (Art.  120.)  Qr»— 2^»=2CF%  •.•  Qr'^VC^ 
=5  FC  +  flr«= (34. 1  .)^rC» 
+  Cif*,  •/  by  substitution 
yC'JtCW  :RH'  ::CN' 

—  rC»  ;  PN'.  But  the 
triangles  CitH,  TPN  are 
similar,  •.•  (4. 6.)  EH :  CfT 
: :  PiV  :  TN,  and  (22. 6.) 
JR/P:  CW  :;  PN".  TN\ 
•••  eo?  €equo  VO  +  CH^  s 
CH'  ; :  CN'-'FC  :  TiST^  ? :  (cor.  1.  Art.  121.)  CN.NT :  TN* 
itCNi  TN,  •••  by  conversion  (prop.  E.  5.)  FC'^CH*  :  F€»  : : 
CN :  (CN-  r2yr=)  cr  -. :  (l.  6.)  C2^»^ :  ^iV^.CT.  But  (AitrWl.) 
rC»  =  CN.CT.  •/  (14.6.)  FC'  +  Cff'±=:CN',  r  CN\-€H'=^ 
B.  D. 


«. 

• 

H 

^ 

X 

y 

Tm  _^f^f' 

Q 

V 

s 

wt^^*^*      -^^^ 

*          (^^^      f^Mr" 

.^.. 

\ 

c 

^ 

•  • 

• 

• 

. 

NvX 

-^T. 

"K     «2r      ^ 

i 

y^ 

J 

% 

• 

Pah*  It.  THfi  trft^tOlA.  491 

C(ff.  Hence  CH^ {^tClf* *^r&)  t  PN'  tr  FC*  :  «C»  (cor. 
Art,  1 16.)  and  fff :  PIf  ::f^€:  EC  (24.  (?.) 

1^8.  The  same  f  hixigs  remaifting  CN  -.  KH : :  VC :  EC, 
For  (Art.  127)  JP^C'  +  CJ?'  f  RW  ::  CN'-^P^C'  :  JPiV'  :: 
(cor.  Art.  127.)   ^C*  :  £Cv  and   rC»  +  C^'  =  CiSrs  v  C2^»  : 
RW  : :  rC  :  EC*  and  (22. 6?.)  CNiRlt: :  VC  :  EC.    Q.  E.  D. 

12a  If  CR  be  parfllkl  to  the  tangent  PTand  PN,  RH  ordi- 
nates  to  the  major  axis,  then  will  RH'-^PN's^EC'. 

Because  (Art.  128)  CN'  :  RH*  ::  FC'  :  EC  : :  CiV^»— 
VC'  :  PiV'  by  subtracting  antecedents  and  consequents  VC*  : 
RH'^PN'  ::  CiV^»  — FC  :  PN'  : :  FC«  :  EG*,  V  (14.5.)  JIH« 
^-Pm^zECK    Q.  E.  D. 

Cor.  Because  rv*-Ct>«=liH«— P^"  (34.  l.)s±JEC«>  and  CiV* 
^CH^=zyC*  (Art.  127.)»  ••  i^  ^i*  be  conjugate  to  CR.  CR  k 
also  conjugate  to  CP,  ' 

130.  If  CP  and  Cil  be  semi-conjugate  diameters^  then  will 

CP«— CB«=FC«— J5:c«. 

Because  (Art.  127.)  CN^—CH^^FC^,  and  (Art.  129.) 
RB^-^PN'^EC*,  •••  by  subtracting  the  latter  from  the  former 
CN*  +  PN»-'CH*'-RH'==:FC*'--BCi.  But  (47. 1)  CP'=: 
CfPj^^PN*,  and  Cfi«=  CH«  +  RH\  •.•  (Ci^"  +  PiST*  - 
C£P4-JI£/«=)CP«— C«»=KC*— EC*.    Q.  B.  D. 

131.  The  same  things  remaining,  if  PL  be  drawn  perpendi- 
cular to  CR,  then  will  CR,PL=FC.EC 

Draw  Cm  parallel  to  PL,  then  because  (Art  128.)  CAT :  RH 
::FC:  EC,  \'  (16  5,)  CNiVCiiRH:  EC.  Bat  the  triangles 
CTin,  RCH  (having  the  alternate  angles  RCH,  CTm  equal 
t29. 1.),  and  the  angles  at  H  and  m  right  anglers)  are  similar, 
and  (4. 6.)  CT:  Cm  : :  CR  :  RH,  '.'  (compounding  the  two 
latter  proportions,)  CiVT.CT  (=by  Art.  121.)  VC  :  VCXm  :: 
RtfCR  :  RH.EC  :  :  CR  :  EC^  \'  (15.  5.)  VC  :  Cm  . ,  CR  i  EC, 
'.'  {\6.6.)=:CR.Cm=^VCEC i  but  Cm= PL  (34. 1.),  •.•  CRPL 
^VCEC.    Q.  E.l>. 

Cor.  1.  Hence  (16.6.)  VC :  PL  ::  CR  :  £C,  and  (22.6.) 
rC«  :Pi*::Cft*  :£C^ 

Cor.  2.  Let  VC—a,  EC=::b,  CP^x,  and  PL=y;  then  because 

ah^CR4f,  '•*  V*==^^-    But  <Art.  130.)  ;!P»-.C£«ara»-i»,  .• 


4» 


CONIC  SSCFIONS. 


Part  X. 


■ 

Cor.  3.  Heooe^  If  Umgento  be  dravm  «t  tbe  ^tremities  of  any 
two  conjugate  diaoieten  (cor.  %  Art  108.)  a  paraUelogram  wOi 
be  formed,  and  all  the  panillelogramB  that  can  be  formed  by  the 
tangents  in  thb  manner  are  equal  to  each  other,  as  appean  from 
the  foregoing  demonstration,  being  each  equal  to  2FC2£C= 
VU.EK:  see  the  figure  to  Art.  133? 

133.  If  C^  be  a  semi-conjugate  to  Cl\  then  wiU  FP.FS 

Let  FP  and 
CA  be    produced 
to  meet  in  R,  and 
draw  FY,  SZ  per- 
pendicular  to  the 
tangent     at     P. 
Then  the  triangles 
FPY,  PRL,  and 
SPZ  being  equi- 
angular,-    (4.  6.) 
FP  :FY::PR: 
PL  and  SP  :  SZ 
::    PR  :    PL,   '.' 
compounding    ' 
these  proportions 
FP.SP  :  FYSZ 
: :   PR^  I  PL'  :: 

(Art.  109.)   VC^  :  PL^  : :  (cor.  1.  Art.  131.)  CJ*  :  ECK    But 
(Art.  111.)  FYSZ:=EC\  v  (14.5.)  FP,SP^CJ^.    Q.E.D. 

•  133.  If  through  the  vertex  V  the  straight  line  €k  be  drawn 
equal  and  parallel  to  the  minor  axis  EK,  and  from  the  centre  C 
straight  lines  GM,  Cm  be  drawn  through  e  and  k  meeting  any 
ordinate  {PN)  to  the  major  axis,  produced  in  M  and  m-,  theq 
willPM.Pw=rc».     See  the  following  figure. 

Because  (cor.  Art.  116.)   CN'-^VC'  :  PN'  ::  FC»  :  EC 
and   (4.  aid  22.6.)    CiV*  ;  iVilf»  ::    TC/  :  (FcV=:)  EC,  v 
(19.5.)    FC  :  NM'-^PN'  ::  FC='  :  EC',  •/   (14.5.)   W'- 
Pjy^  =  EC*  =  Fe*.      But  (cor.  5.2.)  iVM«  -  PiV*  = 
NM+PN.  NM-PN^PMPmi  •.•  PM.Pm=z  Ve^    Q.  E.  D. 

Cor.  1.  Hence,  in  like  manner  pfn.pM  may  be  shewn  to  be 
equal  to  Vk^=::Ve^,  •••  PM,Pm=ipm.pM ;  and  if  any  other  line 


Paht  X . 


THE  HYPERBOLA. 


493 


J^  be  drawn  parallel  tm^Mm  cut- 
ting the  curve  in  Qq,  then  by 
similar  reasoning  it  is  shewn  tibat 
FM.Pm^QX,Qx=qx.qX. 

134.  The  straight  lines  CM,  Cm 
continually  approach  the  curve 
but  do  not  meet  it  at  any  finite 
distance  from  the  centre  C,  and 
therefore  (Art.  103.)  CM  and  Cm 
are  asymptotes  to  the  hyperbola. 
Because  PM.Pm^iFe'^  (Art. 

133.), PH «  4-  (Art.  Ill  Part 4.) 

that  is  PM  and  Pm  are  inversely 
as  each  other,  or  as  Pm  increases, 
Pilf  decreases ;  and  when  Pm  be- 
comes infinitely  great,  PM  be- 
comes infinitely  small  3  that  is,  at 
any  finite  distance  it  does  not  entirely  vanish.  For  the  same 
reason  as  pM  increases,  pm  decreases ;  and  at  an  infinite  distance 
^XHn  C  becomes  infinitely  small, .  but  does  not  vanish  >.  '.'  CM 
and  Cm  continually  approach  the  curve,  but  do  not  meet  it  at 
any  finite  distance,  they  are  therefore  asymptotes. 

Cor,  1.  Hence  it  appears  that  CM.  Cm  are  likewise  asympto- 
tes to  the  conjugate  hyperbolas  >  for  Te,  Vk  being  respectively 
equal  and  parallel  to  EC,  CK,  %•  (33. 1.)  Ee,  Kk  will  each  be 
equal  and  parallel  to  VC;  and  by  the  same  reasoning  it  is  plain 
that  CMt  Cm  continually  approach  the  conjugate  hyperbolas, 
but  do  not  meet  them  at  any  finite  distance  from  the  centre. 

Cor.^.  If  VE  be  joined,  the  right  angled  triangles  FfiC, 
FeC  having  CE=  Fe  and  VC  common,  are  equal  in  all  respects 
(4. 1.)  •/  VE^eC,  and  the  angle  CVE^FCe.  In  like  manner 
it  foUows  that  VKzs:Ck,  and  since  £C=  CIT  (Art.  108.)  /  the 
right  angled  parallelograms  CEeF,  CKkF  are  equal  (36. 1.) 
and  consequently  similar,  and  the  four  diameters  Ce,  BF,  Ck, 
KF  are  equal,  •.•  (cor.  Art.  241.  Part  8.)  CD,  De,  ED,  DF, 
CZ,  Zk,  KZ,  ZFnre  equal  to  each  other  5  and  because  FkzsCK 
iszEC  \'  (33. 1.)  Brand  Ck  are  parallel 5  in  like  manner  it  is 
plain  that  JlTrand  Ce  are  paralkL 


494  CONIC  sscmom.  P4&t:s. 

IW.  The  pasitkm  of  anjr  dMuD^lor  ^^  nsftBCt  to  the  «9(i9 
li^iiig:  given,  that  of  its  conjuigaie  inajr  ^  ^etermiiiedi  for 
(Art.  133.)  NM^--FN*^EC*,  md  (Art.  1^,)  RU^^PN*:^;^ 
EC^  -r  NM^RH,  \'  if  CP  be  a  semMliMiieter^  fX^  w  m^ 
nate  at  P  to  the  major  axis  produced  to  the  point  ilf  in  the 
asymptote,  and  MR  be  drawn  peraMel  to  I9ie  nugor  aaaa,  tlien  if 
RC  be  joined,  MC  win  be  oot^jtigale  toCPhj^  eat.  to  ^rt.  If9, 
And  in  the  same  manner  the  position  of  'tiie  oonjugatte  to  any 
other  diameter  is  known.    Q.  fi.  I>. 

136.  If  a  straight  line  Xx  be  drawn  in  any  position  cutting 
the  curve  in  Qq,  and  the  tangent  TPt  be  parallel  to  it,  then 
win  QX.Qxz=iPT.Pt    See  the  figure  to  Jrt.  141. 

Through  (^  and  P  draw  ¥f»,  Zt  fMrpeodkiihr  to  the  4ids| 
then  the  triangles  XQfV,  TPZ,  wQx,  and  zPi  being  similar  QW 
:  QXi:  PZ  :  Pr(4.6.)  and  Qm  :  Qx  n  Pz  :  Pt^  these  propor- 
tions being  compounded  QW.Qw  :  QX,Qx  t:  PZ.Pz  ;  PT^t. 
But  (cor.  Art  133.)  QfV.QwzsPZP^-  (14.5.)  QX.Qx.=PT.Pt 
Q.  £.  D. 

Cor.  By  simihu*  reasoniiig  gjr.^jr^P7JP<>//  QXQx=f4;i?.9X 

137.  The  same  eonstrutftion  miMMiing  QXs^x, 

For  QX  Qx=  QXQ9  -h  qx^QX.Qq + QX.qx.  And  ^x.g3r= 
qx.qQ+QX±zqx.qQ'^qx.QX;  •/  (since  Qiir.<?j?=:^x.gJr  by  the 
preceding  corollary)  QXQq + QX.qxss  qx,qQ + qx,QXj  from  these 
equals  take  away  QX.qx,  and  the  remainders  are  equal,  viz. 
QX.Qq:s:qx.Qqy  divide  both  sides  by  Qq,  and  QJTs^x.  Q.  E.  D. 
Cor.  Hence,  if  ^  move  parallel  to  itself  so  as  to  coincide 
with  Tty  the  points  Q  and  q  will  each  coincide  with  P,  and  Q^ 
will  vanish  -,  also  QXand  qx  will  coincide  with,  and  be  equal  to 
TP  and  <P respectively ;  •.•  (since  QX=zqx)  T/>=<P,  •.•  QX.Qx 
:=zTP*. 

■ 

138.  The  same  construction  remainiiig  if  through  P,  the  clia- 
meter  Gv  he  drawni  Qvssqv. 

Becanae  the  trimi^ei  XvC,  TPC  are  dimilar^  and  ako  xoC, 

tF<^;    /  (4.^.aiwi  W.  5.,)  fJT:  Pr  : :  i?C:  PC  : :  we:  PL    BMt 

PTszPi  by  the  pneoeding  eor.  *.-  (14.  &.)  o2r=s«a;.    But  (Art. 

1370  CJ^=^*i  ••  ivX-QX:^»x^qx  or  Q»5»«v.  'O;  £.  D. 

Cor.  Hence  cJT'  -t?jQ» =Pr».    For  (eor.  6. 2.)  nX^-^^vQ^zs; 

vX^vQ .  »JK^.f©Q=QXQa?=(cor.  Art.  137.)  TP*. 


FUtX.  THS  13TPBSBDU.  49» 

IS9.  If  PB.  VD  bo  .pmlW  to  u  Mymiitolt  Cs,  tbea  nlB 
PB.Cltssf'D.CD    Sae4iuJigwnto,.M.l»i. 

TlAough  the  pomta  F  and  F  dnw  the  suaigbt  lines  ek, 
Umtmtix  perpendlcalitr  M  theftxli  CN,  and  fd,  Vo  psnUel  to 
CX  DacauH  the  triantlea  Plffi;  PeD,  i^dm,  and  Ttufc  u« 
liorHar, -■-(4.«.)i'a:l'M::  PJ)  :  FAaad  (Prf»)  CHi  Pn  u 
(»'«>=)  CD  ^  n  aad  bf  oonponnding  i>if.C«  :  PJtf.Pm  :: 
VD.CD  :  fe.yk.  But  (Art.  1S3.)  P«.P«»:(r«'s«)re*ffc,  -., 
(14.5.)  PB.CH=yD.CD.     Q.  E,  D. 

Cor.   X.    Hence,    became   (cor.  S.  Art.  134.)  CD^FD,  v 

Cor.  t.  Hence  aho,  if  PSbe  produced  to  meet  the  conjugate 
hyperbola  in  R,  RH,Ca=^ED.CD==FD.CD=CJy*  or  riJ«. 

Cor.  3.  Hence,  because  PH.Ca={CD's=)  RB-CH,  hj  dU 
vidiog  these  equab  by  CB,  PB=fRB. 

I40.  If  PT  be  a  tsogent  at  P  mestinp  Ibe  asymptotes  is  T 
and  a;  andCRbejiMDed,tikaaniltCAaitd  TX*  be  paiaJlel  and 
CJt=TP-PX. 

For  P^^bdngperallcl  tnCToMwdeof  thetmngle  CXT. 
:•  (8.  6)  PX :  PTi-.XB:  «C.  But  (cor.  Art.  i37.)  PXi=PT, 
V  (prop.  A.6.)  ABr«HCr.Inthe  triai^lesPJfff,  flCHthare 
are  the  two  sUea  XB,  BP=  (;B,  BR  respeotivelyj  and  the  vei^ 
tical  angles  at  B  equnl  (15. 1,)  ■-■  PX^iPTif)  CR;  also  tl»e 
angle  HRC^BPX  (4. 1.)  ■.  CR  and  fPX  are  pandlel  (27. 1.) 
«.  E.  D. 

.Ml.  If  PG 
«ni  DO  be  con- 
JBgate  dime- 
teib,  and  Qt>  as 
orainatetoPG, 
then  will  Pv.vO 
;'<ie»::  CP'  : 

At  the  point 
P  draw  the 
tangent  Pr, 
and  f»T)dure 
the  ordinate  vQ 
to  oteet  the 
asymptote  in  X. 


4M 


CXmiC  SECTIOK& 


PaktX. 


tbetf^  mace  CD,  PT,  aad  vlTaieptfiifel  (Art.  96>  ibl.),  TP  is 
therefore  parallel  to  Aa  a  akle  of  the  triangle  XO^  */  (3.6.) 
r»  :  »X^: :  CP  :  PT,  afid  <«.  6.)  Co*  :  eJI?  : :  CP*  :  FT^  •/ 
(19.6.)  rp*-»CP*  :  Pjr^— Pr»  :;  CP*  :  PST*.  But  l.Cb^— 
CP«as:  (cor.  6. «.)  Cb— C*P .  Op + CP=  P».t>0.  «.  (cor.  Art.  1S8.) 
vJf« ^  Oi>«=t: PT*  or  «;if»-  PT»a:  ©1^.  S.  (Art.  140.)  P  T*  CD  ; 
*.'  subetHuting  theee  results^  for  their  equals  in  the  above  aaa* 
logy,  it  becomed  Pv.vG  :  <?»«  : :  CP^  :  CD^.  Q.  E.  D. 
Cor.  Hence  Pv.vG  «  Qtr*. 

14^.  The  parjimeter  P  to  any  diameter  PG  is  a  third  propor- 
tional to  the  major  axis  VU,  and  .the  conjogate  DO  to  the  dia- 
meter PG;  that  isrP  :  DO  ::D0:  VU. 

Let  ilfiii  be  the  ordinate  to  the  diameter  PG  which  passes 
through  the  focus  F,  which  19  therefore  the  parameter  P 
(Art.  10^.)  5  then  will  Mv^^P  (Art.  138.).  Then  because  CD, 
PJIf  are  parallel,  Cr  :  CPi:  Fe  :  Pe  (9. 6.),  and  Cr^  :  CP^  : : 
F^  :  Pe«  (««.6.),  ♦.'  dividendo  C^^^CP^  :  CP«  : :  F^^P^  : 
Pe\  But  (Art.  141.)  Pr.rO  :  Mr^::  CP' :  CD»  j  \-  alternando 
{PrrGzs:)  Cr*-CP»  :  CP«  : :  Mr^  :  CD\'/  Mr^  :  CD*  : :  Fc« 
— Pe*  :  Pie'.  But  /c*  — Pe» a  Fe^Pe .  Pe-fPc  (cor.  5.  2.)  =: 
PP. PS  (Art.  109.)  ::=:CD'  (Art.  13^2.);  /  3fr«  :  CD'  :  •  CD'  ; 
Pe»  and  (22. 6.)  Mr  :  CD  : :  CD  :  (Pesby  Art.  169.)  PC;  '.• 
(15.  5.)  2Mr  or  P  :  DO  ::  DO  :  Ptf.    Q.  B.  D. 

143.  If  two  hyperbolas  PQq,  PW^  be 
described  on  the  same  diameter  GP  and 
from  any  point  N  in  it  the  ordinates  .A^Q^ 
A7F  be  drawn,  A'Q  shall  have  a  given  ratio 
to  NW. 

In  GP  produced  take  any  other  point 
n,  and  from  it  draw  the  ordi  nates  nq,  nw ; 
then  (cor.  Art.  141.).  PiV:iV<^  :  Pn.nG  :i 
NQ*  :  nq'  : «  NH^^  :  nw^-,  \'  NQ :  nq  :  iNfF 
:  nw  (22. 6.),  and  A^Q  :  NfV  ::  nq  :  nw" 
(16. 5.).    O.  E.  D. 

Cor.  1.  Hence,  as  in  the  parabola 
(Art.  29,  and  cor.)  and  the  ellipse  (Art.  69. 
cor.  2.)  the  area  NQP:  area  NWP  in  a  given 
ratio.  Abo,  if  any  point  v  be  taken  in  the 
axis  and  vQ,  vW  be  joined,  the  area  PQt> : 
thcarea  PWv  in  a  given  ratio. 


J 


^AKT  X.  THE  HYPERBOLA.  4d7 

Cbr.  2.  Hcnce^  if  FQq  be  an  hyperbola,  and  ham  erery  point 
N,  n,  &€.  in  the  diameter,  ordinatee  NQ,  nq,  &c.  be  drawn,  and 
if  fitiaJgbt  lines  NW,  nw,  &c.  be  drawn  irom  the  points  N^  n,  &e. 
making  a  given  angle  with  NQ,  nq^  &c.  and  having  a  given  ratio 
to  each  other,  the  curve  FWio  passing  through  P,  and  the  ex* 
treoiitiea  of  those  line$,  will  be  an  hyperbda,  iiaving  FG  for  its 
diameter. 

For  NQ*  :  NW*  : :  nq*  :  nw«  : :  PNNO  :  Pn.nO,  that  is, 
nq^^PN.NG  (cor.  Art.  141.)  which  is  the  property  of  the 
hyperbola. 

144.  If  two  hyperbolas  PQq,  PWw  be  described  on  the  same 
diameter  PG,  and  NQ,  NWan  ordinate  to  each  be  drawn  from 
the  same  point  N,  tangents  at  Q  and  fV  will  intersect  the  dia* 
meter  PG  in  the  same  point  T. 

Let  QTbe  a  tangent  at  Q,  and  join  TW;  TW]&  a  tangent ; 
for  if  not,  let  it  meet  the  hyperbola  again  in  to,  draw  the  ordi- 
nates  nw,  nq,  and  produce  nq  to  meet  the  tangent  TQ  produced 
in  t.  Then  because  the  triangles  QTN,  sTNare  similar,  as  also 
TNfF,  Tnw,  v  (4.6.)  NQim  (::  TN :  Tn)  ::  ISWinw,  But 
(Art.  143.)  NQinq::  NWx  nw,  \'  NQ:n$::  NQ:nq  •.•  (9.  5.) 
ns^nq,  the  greater  equal  to  the  less,  which  is  absurd;  *.*  T9V 
which  noeets  the  hyp^i)ola,  cannot  cut  it ;  T9F  is  therefore  a 
tangent.    Q.E.D. 

Car.  Hence,  if  GP  be  the  major  axis  of  the  hyperbola  PQp, 
since  (cor.  1.  Art.  117.)  tangents  at  Q  and  FT  will  in  like  man- 
ner meet  the  axis  6P in  the  same  point  T,  -.*  (Art.  ISl.)  CN.CT 
szCP*,  '.'  (17.  6.)  CN  :CP::CP:  CT. 

145.  If  PM  be  the  diameter  of  curvature  at  the  point  P,  and 
PL,  PR  chords  of  curvature,  the  former  passing  through  the 
centre  C,  and  the  latter  throogfa  the  focus  F,  then  wiU  AfP  pro« 
duced  be  perpendicular  to  the  semi-conjugate  diameter  EC,  and 

PCiCE::CE:^PL 

PH'.CBiiCEi^PM 

FC:CE:iCE:^PR 

FirMt  Let  FQ  be  a  nascent  arc  common  to  the  hyperbola  and 

circle  of  curvature,  draw  Qv  parallel  to  the  tangent  PT,  join 

VOL.  II.  K  k 


496 


CONIC  SECTIONS. 


Part  X. 


CF,9nA  draw  the  chords  PQ,  QL,  LM,  MR.  Then  the  triangles 
QPv»  QPL  having  the  angle  QPv  commoo,  and  (99. 1.)  PQv^ 
rP0=(32.3.)  QLP,  are  equiangular,  •/  (4.6.)  F»  :  jPQ  : :  PQ  : 
PL,  '.'  (l7.6.)  Pv.PL 
^siPQ^l  but  since  the 
arc  PQ  is  indefinitely 
small,  Qv  and  PQ  will 
be  indefinitely  near  a 
coincidence,  and  there- 
fore may  be  considered 
as  equal,  •.•  Pv,PL^ 
PQ'=zQv*,  also  for 
the  same  reason  oC=s 
PC. 

But    (Art.   141.) 
Pv.vG:{Qv*=)Pv,PL 

::  PC'  :  C£S  V 
(15.5.)  (rG=)  2PC: 
PL  ::  PC:  ^PLi: 
PC*  :  CE*,  /  (cor.  2, 
90.6.)FC:C£::C£ 

:  4^FL. 

Secondly.  The  tri- 
angles PCfl,  PML 
having  the  vertical  an- 
gles at  P  equal  (15. 1.)  and  likewise  the  angles  at  H  and  L  right 
angles  (31. 3.  and  construction),  are  equiangular,  and  PH :  PC 
::  PL  :  PM  ii^PLi^  PM ;  but  by  the  former  case  PC  :  CE 
::  CEz  ^PL,  /  ex  aquo  PH :  CE  : :  CE  i^PM. 

Thirdly.  The  triangles  PKH,  PMR  are  »milar  (15. 1,  31. 3. 
and  construction)  /  PK  :  PH  ::  PM :  PR  (4. 6.)  :  :  i^PM  1 1 
PR  (15. 5.).  But,  as  in  the  preceding  case  PHiCEzzCEi 
i  PM,  \'  c»  aquo  {PKsiby  Art.  109.)  FC  i  CE  : :  CE  :  ^PR. 
Q.  E.  D. 

Cor.  Hence,  because  2rC  :  2CE : :  'ZCE  :  PiJ  by  the  above, 

and  ^FC:^CE::^CE  :  the  parameter 
(Art.  142.)  '.*  the  chord  of  curvature  PR,  passing  through  the 
focus,  b  equal  to  the  parameter. 


Pakt  X. 


THE  HYPERBOLA. 


499 


146.  If  a  cone  ABD  be  out  by  a  plane  PFp  which  meets  the 
opposite  cone  Md  in  any  point  U  except  the  rertex,  the  section 
FFp  will  be  an  hyperbola. 

Let  dHhKA  be  the 
opposite  cone,  let  BD 
be  perpendicular  to  pP  ; 
bisect  UV  in  C,  draw 
VL,  CF,  US,  and  bd 
parallel  to  the  diameter 
BD  of  the  base,  then 
will  the  section  passing 
through  FL,  CF,  US, 
and  bd  ]}arallel  to  the 
base  be  circles  (13. 1^.) 
and  HK,  Pp  the  inter- 
sections of  the  cutting 
plane  with  the  planes  of 
the  circles  HbKd,  pBPD 
will  be  parallel  (16. 11.).  Draw  Cr  a  tangent  to  the  circle  TFs, 
then  (36.3.)  BN.ND=PN^  and  bn.nd=:Kn^,  also  8C.CF=zCT\ 
Now  the  triangles  FNB,  sCF  are  similar,  as  are  UND,  UCF,  •.. 
(4. 6.)  VN:  NB::  FC'.Qt  and  UN :  ND::  UC:  CF,  /  (com- 
pounding these  analogies)  FN. UN :  BN.ND  : :  FC.UC :  Cs.CF. 
that  is,  FNNU :  PA«  : :  FC»  :  CT^  '.•  (Art.  116.)  the  figure 
PFp  is  an  hyperbola,  Cthe  centre,  CFthe  semi-miyor  axis,  and 
CT  the  semi-minor  axis.    Q.  E.  D. 

Cor.  Hence  the  section  HUK  will  be  the  opposite  hyperbola 
to  PTp  and  similar  to  it  -,  for  Fn  :  nd  ::  FC  :  Cs  and  Un  :  nb  ii 
UC  :  OF,  •.•  (compounding)  Vn.nU  :  dn^nb  : :  UCVC  :  Cs.CF,  or 
(as  above)  Fn.nU  :nK^::  FC^  :  CT*. 


The  foregoing  are  the  principal  and  most  useful  properties 
of  the  Conic  Sections ;  a  branch  of  knowledge^  which  is  abso- 
lutely necessaiy  to  prepare  the  Student  for  the  Physico  Mathe- 
matical Sciences;  many  more  properties  of  these  celebrated 
curves  might  have  been  added,  if  our  prescribed  limits  had  per- 
mitted ',  but  it  would  require  a  large  volume,  to  treat  the  subject 
in  that  comprehensive  and  circumstantial  manner,  which  its  im- 
portance demands)  we  must  therefore  refer  the  reader^  for  a 


MO  COMIC  SECTIONS.  Pabt  X. 

more  ample  detail,  to  the  writingB  of  AfM>noDiai>  De  TH^pital, 
Hamilton,  £merBoii,  &c.  observing  in  conclusion,  that  what  is 
liere  given  wiU,  aa  for  as  relates  to  this  subject,  be  fully  suffieieBt 
to  enable  turn  to  understand  Sir  Isaac  Newtan*s  Frincipia,  or 
any  othor  work  usually  read  by  Students,  on  Mathematical 
Philosophy  and  Astronomy. 


THE   END. 


«    ■   I II 1 1 » ' 


Printed  by  Bartlett  and  Newnu,  Oxford. 


LIST  OF  SUBSCRIBERS. 


A. 

AcLAND>  Sir  Thomaa  Dyke,  Bart  Kilkrton,  Deron. 

Agnew,  John^  Esq.  Harefidd. 

Agnew^  Mrs. 

Agnew^  Henry,  Esq.  Wadh^un  CoH  Oxford. 

Allan,  Grant,  Esq.  B.  A.  Balham  HilL 

Allan;  Colin,  Esq.  Stock  Exchange,  London. 

Amos, ' '     ', Esq.  St.  John's  Coll.  Cambridge*    ^ 

Annesley,  Rev.'  C.  All  Souls,  Oxford. 

Antrobus,  Philip,  Esq.  Cheam  Park. 

Antrobus,  John,  Esq.  Cheam. 

Arbothnot,  Edmund,  Esq.  Long  Ditton. 

Austen,  John  Thomas,  Esq.  B.  A.  Swifts,  Kent    .    .  two  copies, 

B. 
BouTerie,  The  Hon.  B. 

Baker,  tlie  B[ev.  Ir.  P.  M.  A.  St.  John's  CoU.  Cambridge. 
Banrell,  C.  Esq.  London  Docks. 
Barton,  Nath.  Esq.  Baker  Street,  London. 
Bean,  The  Rev.  John,  B;  A.-  St.  Paul's  School,  London. 
Beniiet,  The  Hev,  Ws  6.  D.  I^ecior  of  Cheam. 
Bentley,  Bei^amin,  Esq.  Sutton. 

Berens,  Joseph,  Esq.  Kevington,  Kent two  copies. 

Berens,  Joseph,  Jun.  Esq two  copies, 

Berens,  Henry,  Esq.  Lincoln's  Inn,  London     .    .    ,   six  copies, 
Berens,  The  Rev.  Edw^d    •     .     .  .  .    ^.    .    .     .    .  two  copies. 
Berens,  Richard,  Esq.  All  Souls,  Oxford      .    ...    six  copies. 
Best,  WOliam  Baliol,  Esq.  Magdalen  Coll.  Oxford  .    two  copies. 
Bishop,  Rev.  W.  M,  A.  Oriel  Coll.  Oxford, 
Birkhead,  Bei^jamip,  Esq.  Reigate. 
Bland,  The  Rev.  M.  M.  A.  St.  John's  Coll.  CamWidge. 
Blackman,  H.  Hamage,  Esq.  St.  John's  Coll.  Cambridge. 
Boulderson,  Henry,  Esq.  India  House,  London; 

VOL.  IZ,  JL  1  ' 


1 


508  LIST  OF  SUBSCRIBEIISL 

BiWf,  Edward,  Esq.  Sbiie. 

Bny,  Bus* 

Bny,  Edwani,  Jim.  Eiq. 

Bnj,  Reginald,  Esq. 

Bridges,  Sir  Hcory,  Beddii^gtoii. 

Bffmriie,TlieBeT.G.A.M.A.TkiiiitfCoILCainb^^  teocopier. 

Bfown,  Robert,  Esq.  SCieatbam two  eofie§i. 

Brown,  lir.  Tbonias,'Jun.  EweH 

BaUer,  Thomas,  Esq.  Trinity  Square,  London      .    •  too  copes. 

Butler,  Mr.  Williain,  Clirist  Churdi,  Ozibtd. 

Butler,  lir.  John,  High  Ercall,  Salop. 

C. 

Cancellor,  John,  Esq.  Gower  Street,  London. 

Can,  T.  Esq.  B.  A.  St  John's  CoIL  Cambridge. 

Cassan,  Stephen  Hyde,  Esq.  Magdalen  Hall,  Oxford    two  capku 

Cator«  John  Barwell,  Esq.  Beckenham,  Kent   ,     .    .  six  copies^ 

Cholmle^  Lewin,  Esq.  M.  A.  Ewell two  copiet» 

Clarke,  Robert,  Esq.  Tooting. 

Clarice,  John,  Esq.  Wamford  Court,  London. 

Cocks,  James,  Esq.  B£  P.  Shortgrove  HalL 

Cook,  John,  Esq. 

Cook,  Robert,  Esq. 

Cook, ,  Esq.  B.  A.  Trinity  ColL  Cambridge. 

Cooke,  The  Rev.  G.  M.A.  Fkofessor  of  Natural  FUOosaptif* 

OsfonL 
Corne,  Rev.  W.  Christ  Church,  Oxford. 

Crisp, ,  Esq.  B.  A.  Catharine  Hall*  Cambridge, 

Crosthwaite,  The  Rev.  Josqih,  B.  A.  St  Jolm's  GoH.  Canibn^ge. 

D. 
Davis,  Alexander,  Esq. 
Davis,  John,  Esq.  Lajton,  Essex. 
Dawes,  John  Edwin,  Esq. 
Day,  William,  Esq.  M.  ^  Albion  SdxodL,  Kew. 
De  la  Fite,  The  Rev.  Heniy,  M.  A. 
Demierre,  Robert,  Esq.  Putney. 
Dottcaster,  The  Rev.  J.  M.  A.  Christ's  ColL  Cambridge. 
Drane,  Thomas,  Jun.  Esq.  18,  Church  RoWj  Limehouse. 
Dring,  John  Robert,  Esq.  Edinburgh. 
Dundas,  William,  Esq.  Richmond. 


LIST  OF  SUBSCRIBERS.  503 

£. 

£ddison>  J.  Esq.  Kentish  Town. 

Edwards^  William^  Esq.  York  Row,  London. 

Elmsley,TheRev.  Peter,M.A.   ' 

Erobry,  The  Rev.  Edward,  M.  A.  Rector  of  St.  Pauls,  Covent 

Garden,  London. 
Estridge,  Mrs.  Carshalton. 

F,         '. 
Farish,  James,  Esq.  Cambridge. 

Fanner,  Samuel,  Esq.  M.  P.  Nonsuch  Park      .    .    .  two  eopiei* 
Fawcett,  Henry,  Esq.  M.  P.  Portland  Place,  London   two  copies. 
Fenton,  — ,  Esq.  B.  A.  Trinity  Coll,  Cambridge. 
Bske,  — ^-,  Esq.  B.  A.  St.  John's  CoIL  Cs^mbri^. 
Fleet,  Charles  Hussey,'  Esq.  Mawley  Place. 
Fleet,  William,  Esq. 
Fleming,  Samuel,  Esq.  M.  A. 

Forbes,  John  Uopton,  Esq.  9Q,  Ely  Place,  London  .  two  copies, 
Foster,  The  Rev.         ,  M.  A.  Ciapham. 
JFranco,  Jacob,  Esq.  Adelphi,  London. 
Ffanco,  John  Heniy,  Esq. 

G. 

Gardiner,  The  Rev.  Charles,  D.  D.  Rector  of  Sutton. 
Gardineir,  Rawson  Boddam,  Esq.  Calcutta  «...  two  copies. 
Gilpin,  The  Rev.  William,  M.  A.  Rector  of  Church  Pulverbateh 

two  copies. 
Gilpin,  The  Rev.  B.  M.  A.  Fellow  of  Chrisf  s  Coll.  Cambridge. 
Gilpin,  ,  Esq.  Trinity  Coll.  Cambridge. 

Godfrey,  — ,  Esq.  B.  A;  St.  John's  Coll.  Cambridge. 
Godachall,  The  Rev.  Samuel  Man,  M.  A.  Rector  of  Ockham. 
Grant,  The  Rev.  John  Thomas,  B.  A.  St.  John's  Coll.  Cambridge 

six  copie9^ 
Green,  Alexander  Holmer,  Esq.  120,  Long  Acre,  London. 
Greenwood,  John,  Esq.  Adelphi,  London. 
Grififin,— — ,  Esq.  Northumberland  Street,  London. 

H. 

Hildyard,  W.  Esq.  M.  A.  Trinity  Coll.  Oxford. 
Hall,  Ambrose,  Esq.  Hermitage,  Walton  on  the  HiD* 
Hall,  Humphrey,  Esq.  ditto. 

Lis 


504  -  LIST  OF  SUB8CRIBSR8. 

Harman,  C.  B.  Eiq.  Croydon.     • 

Hannan,  Thomas^  £6q.  Queen's  Coll.  Oxford. 

Hannan,  John^  Esq. 

Harenc,  Benjamin,  Esq ,.     .     .    .    «^^  .  koo  copiett 

Harris^  Quarks,  Jun.  Esq.  Crutched  Friars^  London    two  copet, 

Harris,  William,  Esq twa  copiet. 

Harris,  James  Dawson,  Esq.  Oporto* 

Harris,  Thomas,  Esq.  Cheam. 

Haycock,  Edward,  Esq. 

Hoare,  Henry,  Esq.  Mitcham  Grove.  '«   - 

Hoare,  Tbe  Rev.  Charles,  M.  A.  Rector  of  BlancMtmL 

Holmer,  George,  Esq.  Borough* 

Holmer,  John,  Esq. 

HoUond,  Richard,  Esq. 

Holmes, ,  Esq.  B.  A.  Benet  Q^.  Cambridge. 

HuU, J  Esq.  B.  A.  St.  John's  Coll.  Cambridge. 

I,  J; 

Ilchester,  The  Right  Hon.  the  Earl  of  ^. 

James,  John,  Esq.  M.  A.  Christ  Church,  OxfcMrd. 

James,  The  Rev.  E.  C.  M.  A.  Epsom      .    •.    .    •    .  hoo.oopm. 

Idle,  Christopher,  Esq.  M.  P.  Adelphi,  London     .     •  two  copies. 

Jeremy,  H.  Esq.  B.  A.  Trinity  Coll.  Cambridge. 

Johnson,  F.  Esq.  Isleworth.  .     *  - 

Jones,  Thd  Rev.  Morgan,  M.  A.  St.  John's  Coll.  Cambridge. 

Jones,  T.  Esq.  Grove,  Highgate. 

Isham,  Rev.  Dr.  Warden  of  All  Souls,  Oxford. 

K. 

Kenyon,  The  Right  Hon.  Lord •*  sir  copies. 

Kaye,  Charles,  Esq.  Tdienhouse  Yard,  London    •     .four  Copies. 
.  Key,-.Alexander,  Esq.  Hanover  Square,  London^ 
Knatchbull,  Rev.  W.  All  Souls,  Oxfoni. 
Kendal,  G.  Esq.  St.  John's  CoU.  Cambridge. 
Killick^  William,  Esq.  Cheam. 

L.' 

Larpent,  Francis  Seymour,  Esq.  M.  A.  Deputy  Judge  Advocate. 
Larpent,  John  James,  Esq. 

Larpent,  George  Gefard,  Esq.  tit»  Fallmall}  London. 
Lacon,  Henry,  Esq.  Norwich!  * 


t   . 


LIST  OF  SUBSCRIBERS.  606 

Lane,  Riduurd  K.  Esq.  Guildford*  Street,  London. 

Lawl^,  BeUby,  Esq.  Ml  Souls,  Oxford. 

Lawley,  Francis,  Esq.  All  Souls,  Oxford. 

Lloyd,  Charl^,  Esq.  M.  A.  Christ  Church,  Oxford. 

Lock,  The  Rev.  G. 

Longley,  W.  Esq.  M.  A.  Fellow  of  St.  John's  Coll.  Cambridge. 

Longley,  Charles,  Esq.  B.  A.  Christ  Church,  Oxford. 

Longley,  George,  Esq.  East  India  House,  London. 

Longley,  Capt.  Joseph,  Royal  Engineers. 

Lushington,  Stephen,  Esq.  AH  Souls,  Oxford. 

Lys,  Geoige,  Esq.  Clapluun  Terrace two  copiesi 

M. 
Macbride,  John  David,  Esq.  LL.  D.  Principal  of  Magdalen  Hall, 

^  Oxford two  copies. 

Marsh,  The  Rev.  W.  M.  A.  Vicar  of  Basilden. 

Mayd,  John  W.  Esq.  Epsom. 

Menshull,  William,  Esq.  Finchley. 

Mercer,  William,  Esq.  Basinghall  Street,  London. 

Mercer,  George,  Esq. 

Mercer,  Warren,  Esq. 

Mercer,  John  Ambrose,  Esq. 

Mercer,  Mrs. 

Metcalfe, ,  Esq.  B.  A.  Trinity  ColL  Cambridge. 

Miller,  Michael,  Esq.  B.  A.  St.  John's  CoU.  Cambridge  two  copies. 

Miller,  Giles,  Esq.  Sutton two  copies. 

Miner,  Richard,  Esq.  Highbury  Place. 

Miller,  John,  Esq.  Red  Lion  Square,  London  .     .     .  two  copies. 

Millet,  Charles,  Esq.  Canton. 

Millet,  George,  Esq.  B.  A.  Fellow  of  Christ's  CoU.  Cambridge. 

Millet,  Henry,  Esq.  Bengal. 

Monger,  Mr.  EweU  Academy. 

Montagu,  John,  Esq.  Devizes. 

Morris,  J.  Esq.  London  Docks. 

Murray,  Sir  Archibald,  Bart.  York  Place,  London. 

N. 
Nayler,  Sir  George,  F.  S.  A.  Coll.  of  Arms. 
Neale,  William,  Esq.  Cheam. 

Nicholson,  General,  York  Place,  Portman  Square,  London. 
Nisbet,  Robert  Parry,  Esq.  Gower  Street,  Londdn. 
Nott,  Rev.  Dr.  All  Souls,  Oxford. 


M6  LIST  OP  SUBSCRIBERS. 

O. 

Oakes,  Lieutenant  Henry  Thomas,  52nd  Reg. .    .    .  ttoo  copies! 
Oakes,  Hildebrand  Gordon,  Esq.  East  India  Coll. 
Oswdl,  The  Rev.  Thomas,  M.  A.  Westbury,  Salop« .  two  copies. 
OsweH,  William,  Esq.  Leyton two  copies. 

F. 

Pairemain,— -— >£Bq.  Sutton. 

Palmer,  Thomas,  Esq.  Cheam  . two  copies. 

Pattenson,  The  Rev.         ,  M.  A. 

Peach,  The  Rey.  Henzy,  B.D.  late  Rector  of  Cheam. 

Penfold,  Thomas,  Esq.  Croydon. 

Fenfold,  Thomas,  Jun.  Esq. 

Penfold,  Mrs.  ... 

Penfold,  James,  Esq.  Cheam. 

Pennington,  Mr.  Ewell. 

Perring,  Jackson,  Esq.  Brunswick  Square,  London. 

Pickfbrd,  Francis^,  Esq.  Midhurst. 

Pontardent,  Edward  B.  Esq. 

Poole, ,  Esq.  B.  A.  St.  John*s  Coll.  Cambridge. 

Post,  Beale,  Esq.  B.  A.  Trinity  Coll.  Cambridge, 

Pratt,  The  Rev.  John,  M.  A. 

PreQCott,  Sir  George  B.  Bart. 

Prescott,  W.  Willoughby.  Esq. 

Pritchard,  The  Rev.  William,  M.  A,  St.  John*s  Coll.  Cambridge . 

five  copies. 
Puckle,  Henry,  Esq.  I>octor*s  Commons,  London. 

R. 

Roseberry,  The  Right  Hon.  the  Earl  of 

Reid,  Thomas,  Esq.  EweU  Grove. 

Richardson,  Charl^  Esq.  Covent  Garden,  London. 

Rjgaud,  Stephen,  Esq.  M.  A.  Professor  of  Geometry,  Oxford. 

Robertson,  Colin,  Esq.  Russel  Square,  London     .    .  ttoo  copies. 

Rodney,  The  Hon.  and  Rev.  S.  All  Souls,  Oxford. 

Rogers,  John,  Esq.  10,  Park  Place,  Islington. 

Rogers,  John,  Esq.  B.  A.  St.  John's  Coll.  Cambridge. 

Rooke,  The  Rev.  George,  M.  A.  Rector  of  Yardley  Hastings, 

Northamptonshire * two  copies. 

Rose,  The  Rev.  Joseph,  M.  A.  Carshalton    .     .    .    •    ten  copies. 
Rose,  William,  Esq.  London. 


LIST  OF  SUBSCRIBERS^  507 

Ruding,  The  Rev.  Rogers,  M.  A.  F.  A.  S.  Rector  of  Maiden  and 

Chessington. 
Ruding,  J.  C.  £6q.  Gower  Street. 
Ruding,  R.  S.  Esq.  Maiden. 

*S. 
Sidmouth,  Thie  Right  Hon.  Lord  Viscount,  Principal  Secretary 

of  State  for  the  Home  Department. 
Sydney,  The  Right  Hon.  Lord  Viscoimt 
Sandys,  Hannibal,  Esq.  Queen  Street,  Westminster. 
Sanxay,  Miss,  Epsom. 

Saigent,  John,  Esq.  Montpelier  Row,  Twickenham  .  two  copies. 
Sawyer,  Robert,  Esq.  Red  Lion  Square^  London. 
Sell^y  Frideaux,  Esq.  London. 
Skelton,  Jonathan,  Esq.  Hammersmith. 

Sketchley,  Alexander,  Esq.  Clapham  Rise    .J     .     .  two  copies, 
Smalley,  The  Rev.  Cornwall,  B.  A.  St.  John*s  Coll.  Cambridge. 
Smalley,  Edward,  Esq.  India. 
Smart,  Richard,  Esq.  London. 
Smelt,  The  Rev.  C.  M.  A.  Christ  Church,  Oxford. 
Smith,  Kennardy  Esq.  Cheam. 

Smith,  William  Adams,  Esq.  6,  Park  Street,  Westminster. 
Stones,  Henry,  Esq.  B.  A.  Kentish  Town. 
Streatfield,  John,  Esq.  Long  Ditton. 
Sutton,  Robert,  Esq.  London    . two  copies, 

T. 

Taylor,  Sir  Simon,  Bart six  copies, 

Teasdale,  R.  Esq.  Merchant  Taylor*s  Hall,  London     two  copies. 

Thomas,  The  Rev.  Matthew,  M.  A.  Sutton  Lodge. 

Thomas,  Rees  Goring,  Esq.  Tooting  Lodge. 

Thompson,  Henry,  Esq.  Cheltenham. 

Turner,  The  Very  Reverend  Joseph,  D.  D.  Dean  of  Norwich, 

and  Master  of  Pembroke  Hall,  Cambridge. 
Tustian,  Mr.  John,  %  Prince's  Square,  Ratclifie  Highway,  London. 
Twopeny,  Edward,  Esq.  Rochester. 

V.  ^ 

Vansittart,  The  Right  Hon.  Nicholas,  M.  P.  Chancellor  of  the 

Exchequer two  copies. 

Van  Cooten,  Lucius,  Esq.  Petersham two  copies, 

Vaux,  Edward,  Esq.  Clapham.  _ ,  _ 


v' 


V 


508  LIST  OF  SUBSCRIBERS; 

W. 

Waddilove^  — -.^  Esq.  B.  A.  St  John's  ColL  Cambridge. 

Walker^  I>ean>  Esq. 

Wallace,  John  Rowland,  Esq.  Canhalton. 

Ward,  Edward,  Esq.  M.  A.  St.  Peter's  Coll.  Cambridge,  Secretary 

of  Embassy  at  the  Court  of  Wmtembui^g. 
Wathen,  Captain,  Fetcham. 
.  AVhite,  Edward,  Esq.  East  India  House. 
Whitmore,  John,  Jun.  Esq.  Old  Jewry,  London. 
Whitmore,  Edward,  Esq.  24,  Lombaid  Street,  London. 
Whitmore,  Robert,  Esq. 
Whitmore,  Frederic,  Esq. 

Wilding,  The  Rev.  James,  M.  A.  Cheam      •     .     .    •  ivio  c(fpie$. 
Wilding,  Thomas^  Esq.  High  Ercall,  Salop. 
Wilding,  Miss 

Wilson,  The  Rev.  Joseph,  M.  A.  Guildford  ....  <t£ro  copies, 
Wright,  Robert,  Esq.  6,  Kettisford  Place,  Hackney  Road. 

Y. 
Young,  The  Rev.  Thomas^  M.  A.  Richmond. 


Baitlett  and  Newmui,  Printort,  Oxford.