(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "An Easy Introduction to the Mathematics: In which the Theory and Practice are Laid Down and ..."

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, 







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 di fferenc e, 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 M a thiwa atics 

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 mathe m a t ica l 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^— 28 168_ 
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 pay ment, 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:=t ke 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 + 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 1 1 ■ ■■ 

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 $he fifHj r^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 squa re root of the sum and di fference^ 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 ; 
when ce by pro ceeding ai in th e foregoing p roblem, 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 p il ing 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±Mz J'''^''^ .^ 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 th e 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 ^ 

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 

■ ■■ ■' ■ l lll H llll | | ,1 III IWI »—— 1M I I f 

, « 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 

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 «yw ao>«f oat thii^ wiU» 
aaother,) and in the latter, according to its mathematical import. The 
leaxncv «n§^ to bo eau tio a e d 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 

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- 

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 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 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 harmoni eal 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 pro po rtionals, 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 



ni l 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 6at l.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 g r eates t 
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 =(99 94-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 t he 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 ; 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 xH h<= 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 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 =s y^— 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 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 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 





-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 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 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 





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 


^ 


+ 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 





-20 


-1 


+ 5 


-2 


+210 



Subst 


Results. 


2 


70 


1 


144 





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 

putting m=:a-l , n=r&-| — r-, 4c. we obtain x* — mx-h 1, x* — nx 

a 

+ 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 th e 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, ma king 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 tt e propose d 
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 

likewi se be e qual 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 eem me tm n riiief 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. 



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 + w p_ ^^^^^ 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 3.0 41 is th e 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 simila r 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 =V 2+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 iq0Oaa 3.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. 

Whe refore ^= 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 L adlam' 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'+2a5 4-&' ^. , . .' 

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, mu st 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 ; 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 II U 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-f 4:]»~» + 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. t o 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 * +nn F* + , &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+, Sf c. ... -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 

^ 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, 
Si nce #^>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* 
-f O-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=s l+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 expre ssion 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 H I I t^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 um eq^ 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, pt wh i rin g o stniglit 
line, or describiiig; a drde: henee a |M»fla te to that effect 
secoM o ccttoju y • 

' 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 de m aO iUa Me 
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« a a dhftr . , 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 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, the n if the a bove 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) A C 
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?. 

n il " ' . 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 A B=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 DA szx, 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 cir cle i n 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=D C^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 itk r ea e a 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 eo mp gf e 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 p roportion 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 c ompar ed 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." 



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 hal f of A BC. 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 trian gle fiv e 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 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 c h f canife f e nce 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 
dr awn 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 bi sected in E, S 2[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 p r odtt eed, 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 ; the n sinc e 
*»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 neces aary^ as' in % g.) 
in D, then w i U 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. Becau se DBssPE , and B£=2 BD, '.' AB^E=^(AB,ABT 
B £=:) AB^Bt^ B PszASI^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 
wo m fe r ence^ 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. Th en '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 m a tja i ri 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 re p r e e e ni 
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 d i ti m$e e from A m 
above direeied, md U mill give the loigth propoied, 

3. Tb draw a line thai shall r epre ac nt 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 
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 LetA Br ^ayA C^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.LC l^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 fi nd 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 mnltipT y 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«t fae 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 Mw my 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 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 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 sin e ^r-sin 
fi the Mppleme ntal versed si ne ^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. 

• • 


. 


7- rad. 





Co-sine + rod. O 


•5- rad. 


O 


rhrad. 


Tangent inf. 





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»s ine 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^ 

shal l 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 j g 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 (= 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^) co sJg; 
••• 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 t an 9B) in the lastego a- 
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)^ .0008 9088890 86)^ .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-'A C (cor. 5.2.); al so log. 

C W^'^^^^^^^^- ^^-^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 logarithm s, log. C^^^ 
\i^:AB-\-A€^\og. AB—AC 



that is, to log. A J^-^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 a ngle AC B=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 

A B-^A C (:: 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. ^:=(9 O0-4(y>=:) 50°; this b eing k noway 
in ^rder to find 4 B~C, w e hav e log, ta n 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. 5 0a± . ...... 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, lo g, 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, ta n, i- B ^CzsJo g. AC-^A B -^-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 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 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 


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 


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 A rt.- 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»« =) N P' : 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 =: Fj E* .-. 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, •.• P C; Ctzi FS 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, th en 

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 produ ced 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« + A TPg a n4 PF^^lf F^ ■¥ 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, / b y 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 <gy w (2 vg. 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::8PS Ni 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 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 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« : 4C E Cr :; 
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 l atter from the form e r .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, '.' (1 6. 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« : : F C* : £( » . 

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 
'.' coropo unding 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 i s (sec Art. liS.) %CT r gjF ; ; 2F C : 5P+ PF But 
(Art. 113.) SN^NF.SN^NF^SF:Iff.sFTpF, -.• since 
SN^NF=.SF, SP^PF^^rC (Art. 104.)> a nd 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\'/ M r^ : 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.