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THE GIFT OF
Prof .William H.Eutta
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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. PROi»£RTIBS 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* €•
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 » '
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