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8/18/2019 M Ozanam s Introduction to the Mathemati
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8/18/2019 M Ozanam s Introduction to the Mathemati
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ALOZ N s
Introduction
TOTHE
M THEM TI KS
ORHIS
LGEBR
Wherein the
Rudiments
of
that
most
Useful
Science
are made Plain to a mean Capacity.
Done
out
of FRENCH.
LONDON:
Printed for
R.
SARE
at
Gra/s-Inn-Gate
in
Holbortt. MDCCXI.
1
8/18/2019 M Ozanam s Introduction to the Mathemati
9/89
W
l
is
8/18/2019 M Ozanam s Introduction to the Mathemati
10/89
TOTHE
Growing
Hopes
O F
MrEdwardNortbey,
SecondSONf
SirEdwardNorthey,
Her
MajestiesAttorney
General ;
THIS
TR NSL TION
A S
Suitable to his Youthful Studies,
Is with due
Respect
DEDIC TED
BYTHE
TR NSLOR
A
z
. . . J
8/18/2019 M Ozanam s Introduction to the Mathemati
11/89
-
- - - \
v
- - -,-V\\ *
Vi *V. ^ /s A { f
e r f ' ;
- - . >
. . . - w
*
. . . . - - - :
j .
8/18/2019 M Ozanam s Introduction to the Mathemati
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THE
PREF
CE
. . : TO THE
RE DER
MOSTersons
who care
not
t o b e at
t h e fains
of studying t h i s u s e f u l S c i e n c e , excuse them
selves b y pr
Mending
i t
i s t o o d i f f i c u l t .
I t i s t r u e
t h e r e
are v a s t
ascents
in i t s Ptogrefs^yet
the
Paths
are slain and
e a s i l y
p r e c e f t i b l e j and Cu
stom and Asp l i c a t i o n mil h e l p t h e Diligent even t o
ascend t h o s e
f l u p e n d i o u s
Heights.
To
begin
with them,
i s
indeed d i f f i c u l t
;
but Alge-
b ra
h a s
i t s
pleasant and
d e l i g h t f u l V a l l e y s ,
a s weU a s
i t s
craggy
Mountains
;
and t h e r e are S e a t s o f Plea
sure
and
P r o f i t
b e b w- HiB
a s
w e l l
a s
above.
This,I h o s e , w i l l be
evinedfrom
t h i s T r e a t i s e
; t h e
Author needs no Recommendation
;
i f b i s f a u l t b e
JWmhV
i t i f most e a s i l y
born, (where
t h e
Prejudice
i s
a g a i n s t D i f f i c u l t y ) ,
and
whoever p r o f i t s but
a l i t t l e
b y
reading
him,
cannot
complain that h i s Time i s
m i f p e n t ,
I amsure I Tranflattd him with a great d e a l o f
D e l i g h t s and no
l e s s P r o f i t ,
a l t h o I
h ad
read
our
I n .
i u f i r i o m Mir•Kersy
:
Plainness i s b o t h
t h e i r
Excel
l e n c i e s , and t h e r e f o r e
herein I [ o f t e njnake t h em g o
Hand
in
HanJ*
/
A
}
l , e t
8/18/2019 M Ozanam s Introduction to the Mathemati
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The
Preface
t o t h e
Reader.
Le t
me
b e understood
a l s o ,
That the Recommenda
t i o n of t h i s
Stranger
t a k e s not o f f from t h e du e Cha
r a c t e r o f our Countrymen, who have
out-gone
already
a l l t h e World in t h i s
S c i e n c e .
- - .
Bu t d i f f e r e n t
Men
have d i f f e r e n t Methods ;
and
a i
t o e
t a k e t h e i r Town's, wh y
n o t t h e i r
Methods
a l s o ,
i f we l i k ? them ? In s h o r t , with some
Ment
some Me
thods take where o t h e r s w i l l not ; and that t h i s may d o
with
you,
i s
my
hearty
D e s i r e .
1 i n t e r f e r e n o t with t h o s e I n g e n i o u s Men who have
undertaken t o Translate h i s whole c o u r s e of Mathema
tics ; I own 1 t h i n k , t h i s Part t h e B e s t , and t h e r e
fore Translated i t , and f i n i s h e d t h e Translation six
Tears a g o .
If
t h i s
h e
n o t
enough,
you
may
soon
have
more
j
however i f y ou learn t h i s w e l l ,
y o u
w i l l
the
b e t t e r b e
able t o comprehend that
. - Which
that y o u - may d t , i t
t h e h e a r t y
D e s i r e o f
.
,
Epsom,
New- , - „
- Your well
Wisher,
years
Day,
171 1 .
Daniel
Kilman.
INTRO-
»
8/18/2019 M Ozanam s Introduction to the Mathemati
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a
INTRODU TION
. i , TOT HE
:
:::Mxhjsmatc
k
:
1 - .
M
m
H
E
Mathematicks
i s
a
Science
which
takes
I Cognizance of whatsoever c an b e Counted
8/18/2019 M Ozanam s Introduction to the Mathemati
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f i Introductionto the
Matbematicks.
A l t h o u g h the
Mathematicks do
only consider
Magnitude,' nevertheless < f h e y
4 6 ) n o j c £ d r j s i d e j i j > f
ab
s o l u t e l y ,
and'iri
Tts
- f i l f , b u t
i f a e e r l y
tie
relation
that
i t
hath to another Magnitude of the fame
kind,
by
homogeneally
comparing them, i n Order to the
finding out
some hidden
Truth ;
whi ch
afterward ,
by Reasons
founded
on other known T/whs, such
as are
naturally known
by everyone, tiey demon
s t r a t e i t .
TMse;known
3J/»th».Mc
BorrimMilSi
known
by
, t h e Names of
common; Notions
or P r i n c i p l e s , of
which there are
three
s o r t s , D e f i n i t i o n s , Axi
oms, and P o s t u l a t e s - .
. ; / - . ; : - . . ,
-
-
J&efinitjms,
are
the Explications of
such.
Words
and Terms which pQncewui - P r o p o s i t i o n , towards
ithe^endfing
l i t
more
8/18/2019 M Ozanam s Introduction to the Mathemati
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Introduction t o
the
Mathematicks.
i i i
a s co require
neither direction
or Demonstra
t i o n .
These
three
Principles
being
granted,
are
those
which Mathematicians make use of t o demonstrate
t h e i r Propositions, whi c h are of two f o r t s j name
l y P r i n c i p a l s ,
whi c h
are e i t h e r Problems or Theo
rems j or l e s s Principal, which a r e , C o r o l l a r i e s or
Lemmas ;
whi c h
a l t e r having b e e n
demonstrated,
do i n t h e i r turn
conduce
to
the Proof
of
other
Pro
p o s i t i o n s
whi c h
de pe nd on
them.
AProblem i s a
Question
w hi c h proposes to do
somewhat, and (b y precedent Principles) sliews the
manner
how
i t
i s to
b e done and
constructed,
and
r e l a t e s
to some Action
necessary
i n
Demonstration :
Thus ; To find the Center of a C i r c l e given, t h e r e -
are
divers
s o r t s
of
Problems,
of
which,
a f t e r
we
have
explained what
i { meant b y the Word
Givent
we will explain some.
By the
Word
Given,
Mathematicians understand
tobe meant, of that w hi c h the Magnitude,
the Po
s i t i o n , the Kind, or the Proportion i s known j so
that i f the Magnitude b e known, i t i s called the
Magnitude
given
j
i f
the
P o s i t i o n ,
the
P o s i t i o n
given;
i f i t s
Magnitude and
P o s i t i o n ,
the
Magnitude and
P o s i t i o n
given
: As i n describing a Circle on a P l a n e , '
the Center i s the Position given, the Diameter the
Magnitude given, the who l e
Circle
t h e
Magnitude
and P o s i t i o n given.
Again,
If a Diameter i s
drawn
a t
Pleasure,
t h i s
Diameter
i s
i n
a
Magnitude
and
P o s i t i o n
given a t the fame time j that i s , whilst the
Circle s u b s i s t s only i n imagination, of which the
Diameter only
i s
knowrij that Circle c an only b e
i n
Magnitude given;
likewise when only the Kind
i s known, as that i t i s to b e a
C i r c l e ,
i t i s i n
*
Kind given j and when the r e l a t i o n of two Mag
8/18/2019 M Ozanam s Introduction to the Mathemati
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Introduction t o the Mathematicks.
nirudes
are
known,, they are i n a Proportion
given.
Problems
are
Direct
or
I n d i r e c t ;
Determinate
o r
Indeterminate, Simple, Plane, Solid, S u r - s o l i d ,
that
i s
t o fay, more than S o l i d .
A
Limited Problem
i s
such a
one
as
c an
b e done
only i n one manner^ as to
Cause j
The Circumfe*
r e n t e of a C i r c l e t o go through t h r e e Points, there b e
ing b u t one Circle w hi c h c an go through those
three
Points
given.
An
Unlimited
Problem
i s
such a one as
c an
b e - done
a f t e r
sundry manners ; a s t o d e s c r i b e t h e Circumfe
r e n c e of a C i r c l e , which shall go through two Points
given,
i t being apparent, that
'through
any two
Points an
i n f i n i t e
variety of Circles may b e drawn.-
,
A
determinate
Problem
i s
that
which
hath a
deter
minate,
or a
certain
number of Solutions ; as t o di
vide a Line i n t o
two
equal Parts
whi c h
hath b u t one
Solution,
or to find two w ho l e numbers, the
d i f f e r
rence of whose Squares JhaU be equal t o 4S, which
hath b u t two Solutions, v i \ . 8 , 4 , and 7 , 1 , are
a l l the Numbers s o q u a l i f i e d . - 1 - ' - i \ l - . < ,
An
indeterminate
or
l o c a l
Problem,
i s
that which
i f
c apab l e of an i n f i n i t e variety of d i f f e r e n t Solutions,
so that the Point w hi c h contributes to the Resolu
t i o n of i t 6
when
i n
Geometry
J
be
at Pleasure with
i n a
certain
Place, whi c h
i s c a l l e d
she Geometrical
-W**,',
w hi c h may i b e
ejtheriaJLinei a Plane,
arSor
l i d
:
And
therefore
when the
s a i d
Place
i s
i n
a
r i g h t
Line, i t i s called Simple Place, o r PUce of a right
Linen when
on
the
Circumference
of
a
C i r c l e ,
Plane,
or
Place,
Place
-
of a C i r c l e
^
when on the Cir
c umference
of any
Conick
Section, as a
Parabola,
H y p er bo la or E l y p s i s , Solid P i e c e . -
- : - . -j
' s o : j Y - ~ u , : \ u - - i v /
I , : - ;
-ih b-r-X
8/18/2019 M Ozanam s Introduction to the Mathemati
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Introduction t o the Matbetrtathks.
v
Asimplex*
l i n e a r Problem
i s such a
one
as c an b e
refolv'd by the i n t e r s e c t i o n of two Right Lines j i t
i s
true,
a l l
such
Prob lems
are
a l s o d i r e c t ,
because
capable
of
b u t
One Solution,
since two Right
Lines
c an
c u t each
other b u t
i n
one Point only.
A Plane Problem
i s
such a one as
c an
b e r e f o l v ' d
i n
Geomttty by
the
Intersections
of
two
C i r c l e s , or
the
Intersection of a Right Line wit h the Circum
ference of'a Circle
j
i t i s also here evident, such
Prob lems
are
-only
of
two
Solutions,
because
two
Circumferences, or a
Right
Line and a Circumfe
rence c an
c ut each
other
b u t i n
two
Points
only.
The Solid Problem i s ( i n Geometry) that which
c an
be r e s o l v e d 1
by
the Intersection of two Conick
Sections, other
than
two
Circles
; i t i s
evident, that
such
Problem
at the most
c an
have b u t four Soluti
ons,
because
two
Conick
Sections
c an
c ut
each
o-
ther
b u t
in
four P o i n t s .
The Sur-Solid
Problem i s
that which cannot b e
resolved by Geometry, without making use of some
Curved
Line of a
more exalted
kind than
what
c o m e s from1 Conick Sections ; i t
i s
evident
such a
Problem
i s
capable
of
more
than
four
Solutions,
s i n c e such a
Curved
Line may b e c ut by another
Curved
Line in more than four
Points.
APro b l e m which i s
exrreamly
easy, and almost
demonstrates i t s e l f , and
serves only to demon
s t r a t e others more d i f f i c u l t ,
i s
called a
Porima,
from sh e GreeiWotd P o r i t f i o t , whi ch s i g n i f i e s easy
t o
b e
apprehended,
and
whi ch
.opens
the
way
to
something
more d i f f c u l t j
as from a Line
given,
t o
cut
« j s
4 l e f t Line given.
A
Problem
which
i s
p o s s i b l e ,
b u t has
not yet
been
resolved,
because
i t
has appear'd too d i s f i c u l t
i s c a l l e d an Aporimeas, as i s
now
the squaring the
Circle,
8/18/2019 M Ozanam s Introduction to the Mathemati
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Vi Introduction t o the Matbematicks.
Circle,
and before Archimedes the squaring the
Parabola.
: i
-
By t h i s word Squaring, i s meant i n the Mathe-
ticks
the
manner
of
reducing
a
Curved
Lined
Fi
gure, into a Righ t Lined Figure i by Curved Li
ned, I
mean a Figure bounded with Curved Lines,
for
a l l Right Lined Figures are
e a s i l y
deduc'd into
Squares j
thus
the
squaring
the
Parabola
i s
the
way
of finding a Right Lined Figure
equal
to a Para
bola
;
the
squaring
the
C i r c l e ,
the
way
of
describing
a Rig h t Lined
Figure equal
to a
C i r c l e .
The Theorem
i s
a
determinate Proposition
con
cerning the nature and propriety of Things, which
shews how
to f i n d
out
a
hidden Truth, and deduce
i t from
i t s proper Principles j
of
which
s o r t i s t h i s
Proposition
which a s s e r t s , That when
t h e two f i d e s
of
a
Triangle
are
e q u a l ,
the
two
Angles
at
t h e
Base
are a l s o e q u a l .
Ageneral Theorem,
which discovers i t
s e l f i n any
Place
found,
i s
c a l l e d
a Porisma
j
s o (whether b y
the Ancient or Modern
Analysis, the
Construction
of
any local Problem
i s
found
out
j
from
which
Construction
a
Theorem
i s
drawn,)
such
Theorem
i s
c a l l e d a Porisma, and therefore a Porisma
i s
no
other than a Corollary discovered i n i t s Place,
with
i t s
Construction and Demonstration declared
by way of
Theorem,
serving
( f a i t h Pappus)
t o
wards the Construction of the most
general
and
d i f f i c u l t Problems : This word Porisma c o m es from
P o r i s o ,
which,
according
t o P r o c l u s ,
s i g n i f i e s
an
E-
f t a b l i s l i m e n t or Conclusion of what hath b e e n done
and
demonstrated,
which made him
define a
Poris
ma, ATheorem drawn by reason of another The-;
orem,
done
and
demonstrated.
The
8/18/2019 M Ozanam s Introduction to the Mathemati
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Introduction-
t o
the
Mathematicks.
vii
The
C o r o l l a r y
i s
a
necessary
and
evident Truth,
that i s to f a y , a Consequence
drawn
from wh«
hath
b e en
done and
demonstrated,
as
i f
from
the
precedent Theorem, v i \ . The two Angles of a Tri
angle
are
equal when t h e o p p o s i t e Sides
are equal ;
i t
Jhould b e concluded that the t h r e e Angles of an E q ui
l a t e r a l Triangle a r e
equal.
The Lemma i s
a
Proposition
made use of to c on
tribute to the
Demonstration
of a Theorem, or Re
solution
of
a
Prob lem
j
i t
i s
most
commonly
pu t
before
the Demonstration of a Theorem,to
the e nd
i t s Demonstration
should b e l e s s incumbred
j
or b e
fore the Resol u tion of a Theorem, t o the
end
i t
should b e
the
s h o r t e r .
Thus
i t happens,
that
Euclid
i n
h i s Elements,
b e
fore
h e
shews
how
from
a Point
given,
t o
draw
a
Line equal
t o
a Line
given
j shews
how t o
make an
Equilateral Triangle, and
that he
always
demon
s t r a t e s
a Theorem before i t s Inverse, which i n an
other
Place we c a l l a Reciprocal Theorem.
Amongst the number
of l e s s Principal
P r o p o s i t i
ons
may
a l s o be'mentioned the
Scholium,
which (af
t e r
we
have
shewed
what
Demonstration
i s ,
and
explained i t s
d i f f e r e n t
Kinds) we s h a l l fay some
thing o f .
Demonstration
i s one or several Syllogisms, or
suc
c e s s i v e Reasonings drawn one
from
the o t h e r , which
c l e a r l y
and invincibly
demonstrate
a
Proposition
;
t h a t
i s ,
whi ch
convince
the
Mind
of
the
Truth
or
F a l f l i o o d ,
the P o s s i b i l i t y
or Impossibility of
i t ;
and
without Demonstration, ( u n l e s s i t
b e a
Principle)
there
i s great reason to doubt of
any
Proposition
;
f o r i t often
happens, that
what appears
true to the
Sense
as well as
the Mind, i s F a l s e , for the Sense
B . - often
8/18/2019 M Ozanam s Introduction to the Mathemati
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- v i i i Introduction t o the Matbematicks.
often imposes upon the Mind, e s p e c i a l l y
where
t h e
Thing i s
not
s u f f i c i e n t l y
examined.
These
Reasonings
are
founded
on
three
f o r t s
of
Principles
mentioned
before, indiscreetly applying
them the one
t o the
other j
that
i s ,
i n
applying one
Truth
to
the o t h e r ,
and from these two
Truths
drawing
a Third
j and
thus b y
continuing
to
draw
Truths
from
Truths
i n Choice, Discretion andOr
der, not only
by
Definitions, Axioms and Postulates,
which are agreed t o , b u t b y Th eore ms, Lemmas',
Problems and Corollaries we a t t a i n the Truth
fought,
which
i s
called a Conclusion, because i t
concludes and
comphrats the
vanquishing the Mind
5 0 . what was demonstrated.
'
Besides the Conclusion, there
belongs to a
De
monstration
the
H y p o t h e s i s ,
which
i s
a
supposition
of
the
Things
known
or given i n
t h e
Proposition t o b e
demonstrated
j
as a l s o the Preparation, which i s a
Construction made before-hand,
i n
drawing some
Lines, whet h er
Effectually
or b y the Head j to the
*nd the Demonstration may
b e
performed
with
the
greater
Ease,
and
the
more
e a s i e
t o
entice
the
Mind
to ' t h e
knowledge
of the
Truth intended
f o r the
Demonstration. There are s e v e r a l s o r t s of Demon
s t r a t i o n s , of
which
the
two
most
considerable
are
those
which are c a l l e d P o s i t i v e ,
Affirmative
or Di
r e c t , and Negative, I m p o J J t b l e o r I n d i r e c t .
The, P o s i t i v e . , Affirmative or Direct Demonstra
tion
i s
t h a t ,
whi ch
b y
Affirmative
and
Evident
Pro
p o s i t i o n s , directly drawn one s c o r n t h e . - other, a t
the
bottom, discovers
the Truth sought
j
t h i s c on
cludes what i t pretends
to demonstrate, i n such a
-manner, That i t f o r c e s
the Reason
t o , consent t o - i t s
♦
Truth
j
of
which
f o r t i s , that i n P r o f .
1 .
Boel^the
1
s t .
8/18/2019 M Ozanam s Introduction to the Mathemati
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Introduction t o the Matbematicks. i%
of the Elements of Euclid, upon a F i n i t e Rig h t Line
given t o describe an
Equilateral
Triangle, and di
vers
o t h e r s .
The
Negative, Impossible, or
Indirect
Demon
s t r a t i o n , i s that which demonstrates a Proposition
by some Absurdity,
whi c h
must of n e c e s s i t y follow,
i f the Proposition proposed
and
contested should
not
b e t r u e .
Thus
Euclid, t o
demonstrate,
That a 7W»
angle that hath
two
Angles e c p u a l , hath a l s o two
Sides
equal,
shews
that the
Part
must
b e
equal
to i t s
Whole,
i f one of the
two Sides
were
greater
than
the o t h e r ,
from
whence
h e concludes they must b e
equal. , -
Either of
these two
ways of Demonstrating e-
qually
convince the
Mind, and oblige i t
to agree
to
the
Truth
of
what
i s
demonstrated
j
b u t
i t
must
b e
confess'd
they do not equally
enlighten
i t , for
i t
i s
most
Manifest,
that the D i r e c t , abundantly more
enlightens, s a t i s f i e s , and c l e a r s the Mind than the
Indirect
; Wherefore the l a t t e r i s not to b e u sed
b u t i n default of the former, Euclid, indeed, d o t f t -
i n several Places make use of Indirect Demonstra
t i o n s ,
b ut we
ought t o do
our
utmost
endeavour
to demonstrate t h e m d i r e c t l y .
Th?
Scholium
i s
a Remark made on
the
Demon
s t r a t i o n of a Theorem, dr the Construction of a
Prob l em
j
a s i f a f t e r having
demonstrated
a
The
orem b y S y n t h e s i s , i t b e observed that the Demon
s t r a t i o n
might
hav e b ee h
performed
by
A n a l y s i s ,
or
having found
the
Resolution
of a
Problem,
i t i s
observed
i t
might have b e e n found out a shorter
way b y
Abridgments
drawn from the general Re
solution $ and now i t
b ehoves
us to explain whac
i s S y n t h e s i s , and-wiias Anotyfii,-
B i
S y n t h e s i s
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X Introduction t o the Mathematicks.
S y n t h e s i s
or Composition,
i s the Art
of
finding
the Truth of a Proposition b y Consequences drawn
from
e s t a b l i f h ' d
P r i n c i p l e s ,
i n
t h e i r
Order,
or
from
Propositions which demonstrate one the other
,
beginning
i t the most
Simple,
and
continuing
t o
wardsthe
more
Compound, u n t i l arrived a t the l a s t ;
a t that which compleats
the convincing the
Mind
of the Truth fought, and commands the Consent,
without having made any
Digression
from
the Pu r
pose
:
So
that
whosoever
s l u l l
attentively
consider
t h e i r
Consequence,
s h a l l b e
invincibly convinced j
and i t
s l i a l l
not b e in
h i s Power
t o deny the Truth
found,
of
which
before he
was
i n
doubt,
or abso
l u t e l y
ignorant o f .
1
n
- - :
A n a l y s i s
or
Resolution, i s
the
A rt
of discovering
the
Tru t hs of
a
Proposition
b y
away
d i s f e r e n t from
the
former j
namely, b y
taking the
Proposition for
granted, and ex amining
i t s
Consequences u n t i l ar
rived
unto some
clear Truth , being
a
necessary
Consequence
of
the Proposition
from whence
i t s
Truth
i s
agreed unto or concluded
:
This may
b e c a l l e d a making use of Composition i n a retro
grade
Marnier,
that
i s ,
beginning
where
the
other
ended.
You hav e an
Example
of S y n t h e s i s and
Analysis i n the a r f Part, Chap. 3 . of Geometry.
N. B. This r e f e r s t o the Continuation of our Authors
Cours de Mathematiques.
.
A n a l y s i s ,
when
concerned
only i n pure
Geome
t r y ,
as
practised
by
the Ancients,
depends
more
on
the
Judgment
and strength of
Thou g h t ,
than any
particular Rusesj b u t at present i t i s made use of
i n Algebra,
which i s
a s o r t of Arithmetick of Le t
t e r s , b y the hel p of which, hidden Truths are more
e a s i l y and methodically
sound out.
. '
r
-
, .
Hear
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Introduction t o the Matbematick, xi
tieit wiia* MonC- P r t f t e t
says in
h i s
New Ele--
tnents of Mathetnaticks; - - - j ' a o V '
. « *
Never
dould
the S ynthesis
of
Geometricians
have arrived t o s o high a point as i t hath i n t h i s
A g e , h ad i t
not
b e en supported b y
the A n a l y f i s -
of the Moderns, which has brough t to
l i g h t
an
i n f i n i t e
Number
of noble Discoveries, altogether
unknown to the most knowing
Men
of former
Times
j
i n
s h o r t ,
i t
i s
impossible
t o
A r g u e
in
any other Manner,
more Ingeniously, Methodi-
c a l l y ,
Profoundly,
Learnedly
and Short ; i t s eri
p r e s s i o n s b y Letters (of w hi c h i t makes u s e ) are
altogether Simple and Familiar ; and there i s
nothing with w hi c h we c an supply our
Minds,
of s o great Strength and S k i l l to find out hid-
den
Truths,
f o r
i t
diminishes
the
Labour,
dt-
r e c t s the Application aright ; i t f i x e s i t
and
ren-
ders i t attentive towards the O b j e c t of i t s search}
i t marks and distinguishes a l l the P a r t s , i t sup-
ports
the Imagination; i t renews, spares and-
improves the Memory a s much as p o s s i b l e ; in
s h o r t ,
i t
r u l e s
and
perfectly
guides
the
Mind
j
and
although
i t
works
and
imploys i t , yet sub -
jugates
i t s o very l i t t l e to
Sense,
that
i t
leaves i t
i n i n t i r e l y Liberty to
employ
a l l i t s Vigour and
Activity
i n
i t s search
a f t e r Truth, s o that
no-
thing may escape i t s Penetration: And by reason
of the neatness and exactness of i t s Reasonings ;
i t
f o r
the
most
part discovers the
s h o r t e s t
way
to
-
the
Truth
i t i s i n
quest
o f , or at
l e a s t
the distance
i t
i s from
i t , when
i t
c omes short
i n i t s attempt.
These, and many
other
Reasons have
made
me
of Opinion, That s i n c e A g e b r n
i s
at present more
Esteem'd
and Cultivated
than ever
j
f i t would not
b e
amiss)
before
we
goto
any
thing
e l s e ,
for
the
Bj
sake-
e
-
8/18/2019 M Ozanam s Introduction to the Mathemati
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\ n
ItitroduStionto
the Mathematicks.
1 ■ ■
■ -
* ~
sake of
young Beginners,
to begin with an
Abridg
ment of that Noble Science, at l e a s t as much of i t
as
may
b e
of
Use
t o
us
i n
the
Elements
of
Euclid
and others j and that to s o f t e n the Demonstrations
which would seem more
d i f f i c u l t any
other way
than b y
t h i s
Analysis.
8/18/2019 M Ozanam s Introduction to the Mathemati
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I
BRIDGEMENT
OF . ,
LGEBR
ALGEBRA
s a S c i e n c e ,
b y means of which
we
may endeavour
t o r e s o l v e any
Problem
p o s s i b l e
i n
the Mathematicks ; and
t h i s b y
the means of a s o r t of L i t t e r a l Arithmetick,
which f o r
that
Cause has
been
c a l l e d S p e c i o u s , b e c a u s e
i t s Reasonings
a c t
b y the S p e c i e s or
Forms
of Things,
V i % .
the
Alphabet.
This
i s
extreamly
a s s i s t a n t
t o
the
Memory, a s well a s
the Imagination
of
a l l t h o s e who
study t h i s noble Science
; f o r
without
i t ,
whatsoever
i s
n e c e s s a r y t o a t t a i n the Truth s o u g h t , must a t once b e
contained in the
Mind, which
r e q u i r e s both a v a s t
Memory, and a
s t r o n g Imagination ; which
cannot b e
obtained but b y a v a s t labour of the Brain.
TBese
Letters
do
i n
each
p a r t i c u l a r
r e p r e s e n t e i t h e r
Lines or Numbers,
according
t o what the Problem be
longs
u n t o ,
e i t h e r
t o Geometry or
Arithmetick ; and
b e
ing applyed
to each
o t h e r , r e p r e s e n t P l a n e s , S o l i d s ,
o r
more
e l e v a t e d
Powers, accdPUing t o
the Number
of
them;
f o r i f there are t wo Letters together a s ( a b ) they r e
p r e s e n t a Parallelogram, whose t w o . Dimensions are r e
presented
b y
the
Letters
a
and
b
; namely,-
one
s i d e
b y
the Letter a , and the other s i d e b y the Letter b ,
t o the
end
that
being Multiplied
i n t o each other
they may,form the Plane a b
;
In the fame man-
n e r ^ i f the
two
Letters a r e the same a s (a a) the
Pl ane a a s h a l l b e a Square, whose Side a i s c a l l e d the
Square Root
;
b ut i f
there
a r e three L e t t e r s t o g e t h e r , a s
(abc)
they
r e p r e s e n t
a
S o l i d ;
that
i s
t o s a y , a Re
c t o
ngular
Parallelopipedon,
whole
three
Dimensions
. ' - - „ '
i
a r e
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A bridgement
of Algebra.
are represented b y the Letters
( a )
( b ) ( e )
t o wit,
t h e
Leng th b y
the
Letter ( a ) , the Brdadfh b y
the
Letter
( b ) ,
and
the
D e p t h
Or
HeiglitWtrie^Lotter
( e ) ;
and
t h e r e f o r e t h e s e
three
Multiplied
one into
the
other,
produce
the
Solid
(abc).
In
the
same
manner,
i f
the three Letters are the
same a s aaa, the
Solid
aaa
/ h a l l represent
a Qi b e ,
whose
Side
i s
c a l l e d
the
Cube Rooh
Further,' If
there a r e
more than
three
L e t t e r s , they
r e p r e s e n t a
moreexalted
Powe r? and
of
a s
many Di
mensions
a s t h e s e ' a r e
Letters ; whic h Powers
are c a l l
ed
Imaginary,
because i n Nature there i s no s e n s i b l e
Quantity capable of more than three Dimensions ; this
Power,
or imaginary Magnitude i s c a l l e d Planes Pla*e9
or a Power of f o u r Dimensions,
when
c o n s i s t i n g of four
Letters, a s (abed), a ' n d w h en the Letters a r e . the
fame a s ( a a a a)
' t i s
c a l l e d the Square
Squared, whose
Side
i s
( a ) ,
which
i s
c a l l e d
the
Root
of
the
Square
Squared.
The next Power i s c a l l e d S u r - S o l i d , when c o n s i s t
ing of f i v e Letters; and when a l i k e , a s (aaa a a) i s
c a l l e d
S u r f o l i d , whose
Side ( a )
i s c a l l e d
the S u r s o l i d
R o o t .
Thus
you s e e t h e s e Powers g o on I n c r e a s i n g , i n 3
continual
Addition
of
L e t t e r s ,
which
i s
an
equivalent
t o a continual Multiplication, which w h i l s t they con-
l i s t
of
the
fame L e t t e r s ,
a r e
c a l l e d Regulars y
and
V i e t a
c a l l s
t h e m
Gradual Magnitudes,
because they I n c r e a s e
b y a S c a l e conformable t o the
number
of t h e i r
Letters :
Thus
( a a )
i s known
t o
b e
a
Power of
the
second De
gree b e c a u s e i t
c o n s i s t s
of t wo Letters; (aaa) of the
Third,
b e c a u s e -
i t
c o n s i s t s
of
t h r e e ,
S 2 f < \
From
whence
i t
a l s o f o l l o w s ,
that the common Root Of a l l t h e s e
Powers, the
Side a
i s t h e Power
of
t h e
f i r s t
Degtee,
or
T o r v e r .
Bu t
s e e i n g
b y
c o n t i n u a l l y
augmenting
the
gradual
Magnitudes,' b y o f t e n
annexing the
fame
L e t t e r ,
the
Number may happen t o b e s o great a s t o make i t dif
f i c u l t
t o
number
them,
and
even
t o
d e s c r i b e
t h e m
on
Paper ;
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A bridgement
of Atgebra. j
Paper ; l e t i t
then
s u f f i c e
t o
write
only the
Root,
that
i s to s a y , only
one Letter,
and towards
the
Right
Hand
t o
add
a Figure equal t o the
Number
of
Letters
the Power c o n t a i n s ; which-
Number
i s
c a l l e d an
Ex
ponent of
the
fame
Power, and
shews the
Number
of
i t s Dimensions;
they
are ordinarily written
a
l i t t l e
higher than the L e t t e r s , f o r s e a r of confounding them
with
the
Numbers when
there a r e
any, or any other
Letters
that
may
b e annexed. As t o e x p r e s s a Sur-
s o l i d a s a
Power
of
the f i f t h Degree, whose
Side or
Root
i s
( a ) ,
i n s t e a d
of
r e p r e s e n t i n g
i t
b y
f i v e
L e t t e r s ,
i t may
b e
r e p r e s e n t e d
b y a 5
; the
same
f o r expressing
the Cube of a , a ' ,
f o r the Square
Squared, a 4 ;
and
s o of
the
r e s t .
From hence may e a s i l y b e s e e n ,
that
a l l graduate
Magnitudes or Powers that have any Root, a s (a) have
t h i s natural
Consequence ;
a1 a1 a> a4 a' a4 a7 a» a9 a'°,
Ve.
and that they a r e i n a Geometrical P r o g r e s s i o n , w h i l s t
their
Exponents
a r e i n an Arithmetical
one ;
f o r a s
the
Powers
a r i s e
b y
continual
Multiplication of
the
same
Root; s o
the
Exponents, b y the
continual
Addition of
the
same
Root,
which
l e t
here
b e
1 ,
which
i s
not
men
tioned i n the
f i r s t
P l a c e , because a
i s
equal t o a *
.
Or
i f f o r ( a ) any Number a t Pleasure b e
taken ;
a s
suppose
( 2 ) ,
i t
w i l l
then appear
a1
i s equal
t o 4 , 4 ' to
8 , and
the
r e s t of
the
Powers- w i l l b e s u c h a s t h e s e ,
a1 a? o f a* af a* a1 a*
2
4
8
16 32
64
128
Z56
which s l i e w s that the Powers are
i n
a Geometrical
P r o g r e s s i o n , and the Exponents i n an
Arithmetical-
one ; and
that
i s the Reason
why
t h e s e Exponents may
be c o n s i d e r e d a s
Logarithms
t o
the Powers,
t o which-
they b e l o n g . .
1 . From
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4 A bridgement of Algebra.
From
whence
i t
a l s o
f o l l o w s , That
the Exponent of
a Power produced from the Multiplication of t wo c i
ther
Powers, i s
equal t o the
Sum
of
the
Exponents of
the
two
s a i d
Powers.
Thus
the
S u r s o l i d
32
has
f o r i t s
Exponent 5 , namely the
Sum
of
the
Exponents
1
and 4 ,
of the
Powers
2 and
1
6 , or of
the
Exponents 2 , 3
,
of
the Powers
4 and 8
which produced i t .
And hereby you may f e e the v a s t d i s f e r e n c e between
3
a
and
a ' ;
fora' s i g n i f i e s the
Cube of
the
Root < » , b ut
3 a the t r i p l e of that Root; s o that i f ( a ) b e eq ual t o 2 ,
the
Cube
s h a l l
be
8 ,
b ut
the
t r i p l e
6
;
l i k e w i s e ,
3
a*
ex
p r e s s e s the t r i p l e of the Squared Square of ( a ) , which
s u p p o s e equal to 2 , s h a l l b e equal t o 48
;
and
s o
of the
r e s t . - -
CHAP L
' - -
OfMonomes. ; ' .
AMonome i s a l i t e r a l Quantity b y i t s e l f c o n s i s t
ing ;
that
i s ,
s u c h a one a s
i s not accompanied
with
any
other
Magnitude,
joyned
b y
t h i s
Character
- r - ;
which s i g n i f i e s More,
or b y t h i s
—which
s i g n i
f i e s L e s s . .
- i . .
PROBLEMK
.
To a Quantity, add a Quantity.
t
A s
a l l
Homogeneal Quantities a f f e c t not t h o s e who
a r e Heterogeneal
;
that
i s , one
Quantity
cannot.
au g
ment another of a
d i f f e r e n t
kind b y being Added t o i t ,
or diminish
i t
b y being Substracted from
i t ; i t
f o l
l o w s , that such Quantities a s
may
b e added
t o g e t h e r ,
ought t o be of
the
same kind
;
i f they
a r e , -
then Unity
may
be
added
t o
Unity,
and
the
same
L e t t e r s ,
and
s a m e .
t
8/18/2019 M Ozanam s Introduction to the Mathemati
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Abridgment
of
A l ge bra.
%
same
Exponents
a r e
retained ;
b ut when of
d i f f e r e n t
kinds they
may
b e
added
b y the Sign o r more, a s
. w e l l a s b y the Sign—or
l e s s j
t h i s Addition
i s
e a s y
to
b e
apprehended
b y
the
following
Examples.
ia 2 < t ' iabb la laab
4
< » 4 .
a *
4 . abb yb: ^abb
3 a 8 < t ' 10 abb — 4 - * *
— . . 2«+3& .
sQa 14a' 16 abb
i ^aab-\-
y z a b - ^ - ^ a 1
Where you may
s e e ,
b y the Addition of s e v e r a l Mag
nitudes of
the
same kind,
the Aggregate i s
one only
Magnitude, v i \ . a Monome
;
and b y the Addition of
several
Quantities of
d i f f e r e n t
kinds a Polynome
i s
formed, which when composed of
two
Monomes
i s
c a l l e d
a B.inome, a s 2 a
- f -
3 b
; the Monomes
a r e here
c a l l e d
Terms,
and
w h en
composed
of
three
Terms
or
Monomes,
a
Trinome, iaab-\-iabb~\-\a*,
& c .
PROBLEM
I .
Ifrom
a Magnitude, t o t a k e away a Magnitude.
S u b s t r u c t i o n
s u p p o s e s
i t s -
Magnitudes
t o
be
Homoge-
n e a l
; s o r i t i s
evident,
t h a t ,
a Plane cannot b e dimi
nished b y
the
S u b s t r a c t i o n of a Liney
f o r
a Plane
i s
com>
posed
of i n f i n i t e L i n e s .
Neither
can a . S o l i d b e dimir
n i s t i e d b y
S u b s t r a c t i o n
of a Line or
Plane,
b e c a u s e a So-
; l i d i s composed of i n f i n i t e
Lines and
i n f i n i t e Planes.
. . Whereas w e * have b e s p r e observed, that t h e . Sign—
( l e s s )
. d o t h
n p f c
make
that
Magnitude
i t
i s
annexed
u n
t o , of a d i f f e r e n t kind from what hath the Sign , or
more;
now a
Magnitude, i s taken
from a
Magnitude
e x p r e s s e d b y the same L e t t e r , b y taking the Units of
tie L e f t .
from the
Greater, and retaining the
same Let
t e r s with t h e i r Exponents
;
b ut i f they a r e e x p r e s s e d b y
d i f f e r e n t L e t t e r s ,
the Less
i s
s e t a f t e r
the Greater
t o
wards the Right Hand
with
the
Sign—
l e s s ) annexed,
which
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6
Abridgment
of Algebra.
which Quantity i s
c a l l e d a Negative
Q u a n t i t y , which
although i n i t s s e l f
i t i s
Affirmative, yet i t
i s
other
wise i n
r e s p e c t
t o the Quantity
i t i s
t o b e taken from.
See t h e f o l l o w i n g EXAMPLES*
From
6 a
From
8 a a From
\ iabb From
3 a
Take
2 a
Take
3 a a
Take
4 a b b
Take
2 b
44
5 < » < » 8*£6 3
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Abridgment of Algebra.
PROBLEM I I I .
To
Multiply
a
Quantity
b y
a
Q u a n t i t y .
M u l t i p l i c a t i o n a s w e l l a s D i v i s i o n doth not require i t s
Quantities t o b e Homogeneal; f o r a Plane may b e
Multiplied b y a Line,
and
i t becomes a S o l i d ; a So
l i d b y a Line,
and
i t
becomes
a
P l a n e ' s Plane
; s o that
the
Multiplication of d i f f e r e n t kinds changes and ex
a l t s
them,
except
when
e x p r e s t
b y
Numbers,
and
i n
t h a t y C a f e the kind
remains.
F i r s t then t o Multiply a l i t t e r a l Quantity
b y
a
Num»
b e r , the Units of t h i s l i t t e r a l Quantity a r e t o be Mul
t i p l i e d
b y t h i s Number, retaining the lame
Letters
with t h e i r Exponents: Thus t o Multiply t h i s l i t t e r a l
Quantity 3 a a b b b y 4 , Multiply 3 b y 4 ,
and the
Pro
duct
i s
12
a
ab b.
Bu t to
Multiply
one l i t t e r a l
Quantity
b y
another*
Multiply
together the
Units
on the Left-Hand, ana
i f the Letters i n each are the fame, add together t h e i r
Exponents, i f d i f f e r e n t , a s t e r the Product of the Num
b e r s annex
towards
the rig ht Hand the Letters of
each p a r t i c u l a r Quantity, with t h e i r
Exponents,
a s i s
t o
b e
s e e n
i n
the
following
EXAMPLES.
Multiply
2a 2«« 3
a gaa
lSaabc
By 3£ 4«« 3a 3a \aacd
6ab
8«4
o««
27a'
72a*bccA
H e r e
you
may
f e e the Exponent
of a Square i s dou-
l »
t o i t s
Root,
of a C u b e Triple, of a
Square
S q u a - '
1
Quadruple.
PRO
B,
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8 Abridgment of Algebra.
PROBLEMIV.
To
Divide
a
Quantity
b y
a
Q u a n t i t y ,
D i v i s i o n , which V i e t a c a l l s
Application, a s we have a l
ready Noted, r e q u i r e s not i t s
Quantities t o b e Homo-
- g e n e a l ,
f o r
oftentimes a Quantity of a
greater
Power
i s Divided
b y a Quantity of a l e f l e r , a s a Plane b y a
Line, when the Quotient w i l l b e a Line; - a Solid b y
a
Line,
when
the
Quotient
w i l l
b e
a
Plane
;
b ut
no
continual Magnitude can fGeometrically speaking; b e
Divided b y
another
c o n t i n u a l
Magnitude
os a
higher
Power, begause i t i s a g a i n s t the nature of Magnitude ;
h v t t any s u c h Magnitude may b e Divided b y another
of
the
same
kind, and
then
the:Quotient i s
an absolute
Number,
g e n e r a l l y speaking. , . . ' - - ' - . > £ - . . - - >
F i r s t
then
i f
the
Divisor
b e
a
Number,
l e t
the
Units
on the Left of the Dividend b e divided b y t h i s Num
b e r ,
retaining the
same
Letters
and
t h e i r Exponents :
Thus,
i f
I
divide 8
a .
b b b y 4 ,
the Quotient s h a l l
b e
2 « j £ and dividing 3 2 a
b y - 8 , the Quotient
s h a l l
b e 4«*.
- v
Bu t i f to the Divisor there a r e oae or more
Letters
annexed,
and
that the
same
Letters
a r e
sound
i n
the
Dividend,
(which
i s
here supposed
of
a
higher
Power
- t h a n the D i v i s o r - ) l e t the Units of the Dividend be
divided
b y t h o s e o f
the
Divisor,
and the
Exponent of
the Divisor s u b s t r a c t e d from that of the Dividend,
and t h o s e
which
remain without Exponent w i l l
ex«
p^nge
each o t h e r , and
the
remaining Letters w i l l b e
the Quotient s whic h f i f the Divisor have no
d i f f e r e n t
Letters
from
the
Dividend,
or
i f
the
Exponents
of
the
Divisor
may
be s u b s t r a c t e d from the l i k e Exponents i n
the
Dividend)
s h a l l b e an e n t i r e
Quotient,
otherwise
i t h i s e d i f f e r e n t L e t t e r s ,
a s
well
a s
the d i f f e r e n c e - of the
Exponents
of the same
Letters found
b y
s u b s t r a c t i n g
t h e
Lester
from
the Greater, t r u s t
be
p l a c e d under
a
Line,
*»
you may
i e e i n t h e l a s t of
the ensuing
Examples.
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Abridgment of A l ge bra'
E XAMPL B $
a o- o
3abb{fib'
6a-bb\jj»ib.
o o a
7 b Extra& a Root from a
Quantity
given; -
We have
taken
Notice i n , what
we
f a i d of
MukipK-
e t t i o n , that t h e * Exponent of the Square i s double t o - '
that of i t s Root, the Exponent of a Cube t r i p l e to i t s
Root ; wherefore t o Extract the Square Root of any
Magnitude
proposed,
take the
Square
Root
of
i t s
U-
n j t i e s and
the
halfof
i t s Exponent,and f o r
the Cube, the
Cube of
i t s U n i t i e s , and the third
of
i t s Exponent ;
thu»
the
Square
Root of t h i s Power 64 a6b , w i l l b e 8 a'
b %%
and
the Cu b e Root 4 a a b b , wh ose S qu are Root will
b e
2 - a
b , the Root of the
s i x t h
Power.
A'Power , that hath neither the Signs
-f-or
—e
f o r e
i t ,
i s
look'd
upon
a s
Affirmative,
and
i f
i t
b e *
preceded b y a Number that contains the Root fought
and
i t s Exponent may b e
commensured
b y
the
Ex
ponent
of the Root;- namely
f o r the Square Root-
b y 2 ,
f o r
the Cu b e b y 3 , £ 5 V . ) i t - w i l l
contain
the Root
s o u g h t .
Thus i t appears, that the
Square
Root of
4«8
£ris
2*?b+,
and
the
Cube
Root
of
a6,
b
i s
a a
b
j >
Unity
being
understood
i n
the
Root a s well a s i n
the
Power ;
f o r i t i s evident that a6 bs i s the fame with 1 a 6 b 6 , anct
ite'Cube Root a a b b ,
the
fame with
1
a abb.
If the Power, whose Root i s
proposed
t o b e extracted
b e
Negative,
o r preceded b y—i t s h a l l not have
any
s u c h Root, (although
under the
above-mentioned Qua
l i f i c a t i o n ,
u n l e s s
the
Exponent
of
the
Root
fought
b e
an
C 2
odd
»
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jo abridgment of A lge bra*
odd Number, and then the Root
a l s o w i l l
b e Negative;
Thusthe Cube Root of—8 *' h f i t— a
b ,
and the
Sursolid o r
^th Power
of
—3 2 « ' °
b '
i s—
a b ,
T a r t
—
a
0
b
hath
no
Square
Root,
s u c h a
Root i s
only
Imaginary, and e x p r e s s i b l e t h u s , y— aab, the Cha
r a c t e r t f ,
s i g n i f i e s Root.
When
a Quantity proposed hath no
Root,
the Cha
r a c t e r
V i s anne x ed on the
Left
with the Exponent
of
the Root
;
which t o prevent Consusion with the Unity-
of
the Powers, may
b e
e n c l o s e d
with P a r e n t h e s i s , a s
(2)
f o r
the
Square
Root,
(3)
f o r
the
Cube
Root,
C S V .
Thus
t o e x p r e s s the Cube Root of 12 i t
i * W3)
12
a* b * , and to e x p r e s s the Square Root of 24 a a b b , i t
i s
V
( 2 > 2 4
a ab b , or
V
2 4 a a b b , the Exponent 2 b e
ing
understood, which
( f o r the most
part)
i n
express
ling
Square Roots
i s neglected
;
such Roots a s
t h e s e arc
c a l l e d I r r a t i o n a l Q u a n t i t i e s .
These
I r r a t i o n a l
Quantities,
or
t h e i r
Roots,
may.
more
simply
b e e x p r e s t
when
the Power
i s d i v i s i b l e
by
another
Power which
h a s 5
the Root- sought ^
namely,,
in f c t t i r i g the Character V between* the Root of this
dther Power,
and tho Quotient
of s u c h Division
: Thos
to e x p r e s s the Cafes Root of t h i s Power 12 «' b3, * B
May be
s e t
down a
b t ( l )
12,
b e c a u s e
12 *' b *
i s
d i
v i s i b l e
b y
*'
b * ,
which
hath
f o r
i t s
Cube
Root
ab ,
and
f o r i t s Quotient 1 2 .
In
the
fame
manner
t o
e x p r e s s
the Square Root of the Power 6a
abb;
i n s t e a d of
writing
V 6 a ab b , i t
may
b e a b V 6 ,
b e c a u s e the
Power propos'd 6 aabb, i s _ d i v i s i b l e b y aabb, whose
5quare Root
i s
ab , and i t s Quotient 6 .
CHAP.
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Abridgment of Algebra. n
C
H
A
P.
II.
0/Polynomes.
BY
he precedent
Chapter
hath heen
shewn, how
b y
the Addition or Subtraction of s e v e r a l
Quanti
t i e s
of d i f f e r e n t k i n d s , a Polynome i s formed, whofe
Terms that - i s , the
Monomes
that
compose i t , may
b e
d i f f e r e n t l y
a f f e c t e d
;
that
i s ,
may
b e
Asfirmative
or
Negative, according a s they
may
have g one t hrou g h
Addition or
Substraction; wherefore, l e s t the d i f f e r
rente between
- \ -
and
which are
c a l l e d
S i g n s , s h o u l d
r a i s e some D i f f i c u l t i e s , e ' e r we venture upon Practice
we
/ h a l l
Propose the foll owing Theorems*
THEOREM
.
The Summ of
tw>
Quantities a l i k e a f f e S e d ,
i s
of
l i k e df-
f e t t i o n with
them.
That i s
t o f a y , i f
any t wo Quantities
a r e Asfirma«
t i v e , namely p r e c e d e d ' b y the Sign +,
their Summ.-
s t a l l
b e
Affirmative
;
i f
Negative
,
t h e i r
Sumnr-
s t a l l b e Negative
;
f o r i t
i s
evident the Summ a - + - by
of
the two Quantities
a
and
b , o r - r - r t - ^ r £ , f which a r e .
alike
a f f e c t e d ,
that
i s
preceded b y
the
fame Sign, and
whish we f e e here a r e both A f f i r m a t i v e , ) must a l s o b e
Asfirmative
; f o r
i f
i t
were Negative, a s ——y
each of the s a i d t wo
Quantities
should a l s o b e Nega
t i v e ,
contrary
t o
the
S u p p o s i t i o n . .
I t
i s
a l s o
evident,
that the Summ
—.—
, of
t wo Negative Quantities
—.and—y i s Negative;
b e c a u s e
i f i t were- A s f i r
mative, a s a - + - B , each of the s a i d t wo Quantities
must a l s o b e
Asfirmative, which i s
a l s o
contrary
t o t h e
S u p p o s i t i o n .
Thus
we f e e+
dded to
- 4 - makes
and
—dded
t o
—
akes
—
2^
E.
D .
C
3
TH
E»
o
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12 Abridgment of
Algebra.
THEOREM.
II.
The
Summ of two unequal Quantities d i v e r f f y a f f e S e d ,
i s
of
t h e
fame JffeSion mtb t h e g r t a t e r , and
i t
e q u a l
t o t h e i r D i f f e r e n c e s .
Pb r s e e i n g b y S t i ^ p o f r c i o n they are d i v e r f l y a f f e c t e d ,
t he one
ought
t o h e X f f l r r r f a t ' i v e ,
the
other Negative J
amd t h e i r Summ Being compounded of one Affirmative
and
one
Negative
Quantity,
shews that
the
Negative
Q uantit y ou g ht t o b e takenfrom the Affirmative, b e
c a u s e Negation i s a Mark of Subtraction : Wherefore,
I f the
Negative
b e l e s s than the Affirmative, i t may
Be t a l c e n from i t , and then a part of the A f f i r m a f t i v e
w i l l remain, and b y Consequence an Affirmation ;
t h T a t
i s t o f a y , the Difference w i l l be Affirmative,
and
of t h e
fame
A s s e c t i o n
with
the
g r e a t e r ,
whith
i s
one
of
t h e
Truths we were t o demonstrate.
Bu t i f the Negative Qtrantfty
i s
greater t h a n the Af
f i r m a t i v e , s e e i n g the Negative cannot be
taken
fromthe
Affirmative,
which
i s supposed
l e f t
than i t , the
l e f t
f r r t r f t b e t a f c e n from the g r e a t e r , that
i s ,
the
Affirma
t i v e from the
Negative,
and
there
s h a l l remain a part
«f
the
Negative;
s o
that
the
Difference
s h a l l
b e
Nega--
t i v e , and b y Gonsetjuence o f the fame A f f e c t i o n with
the g r e a i t e s t . g . E . D.
Thus the Summ of— a and+ * s h a l l b e found
t o b e - r - 3 * ,
and
the
Summ
of
- j -
2 a and
—
a , s h a l l
he— a ; from whence i t appears, that t wo equal
Quantities d i v e r s l y A f f e c t e d , eipunge each o t h e r , o r
a r e
e c | i i a l
t o P ,
dr
nothing.
THEO*
_'i
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A bridgment of Algebra.
THEOREMI I .
To
t a k e
ahq a
Qaanihy
f r o t h
a
Quantity,
i s
t h e
fame
mitb adding i o t h e l a t t e r t h e s i r f l y ntti i t s c o n t r a r y
Sign.
For i f ,
s e r
Example,- -
2 * f e e to
f e e
taken from-4-
* ,
a ,
i t
i s
the fame with adding— * t o 3 a , b e c a u s e the
privation
of
Asfirmation i s the r e s t o r i n g
Negation, a r i d
the
Summ
+
a
s h a l l
b e
the
Remainder
a f t e r
^ u b s t r k -
c t i o n .
In l i k e manner f f « * - »
*beto
b e
taken
from
—
a ,
i t
i s
the fame a s i f unto
—} < r - r -
2 *
should
b e
added,
because
the privation
of
Negation i s
the
r e s t o r i n g
o s
Affirmation, and the Summ— a s h a l l be the Remain*
der a f t e r S u b s t r a c t i o n .
Bu t
i f
from
—
a
you
would
take
■ + -
2
a ,
i t
i s
the
feme a s i f t o—
>
a you s h o u l d add— a , and the Summ
7 * s h a l l f e e the Remainder a f t e r S u b s t r a c t i o n .
And i f from 5 * you
would
take—2 a ,
i t
i s the
feme
with
adding 4-
5 * to
. + ■
2 ay
and the Summ - \ -
7
*
s t a l l be the Remainder a f t e r S u b s t r a c t i o n .
B. tie Examples of
i h f t
Theorem may h e T U u s t r a .
t e d
a f t e r
t b i t
manner.
that i s , i f being
worth
5 / . Iam o- sp4„|
f c l i e e d
t o
pay
2
J .
the
Remainder < t , u s
( 2 . ; —2««—«-)*=— 30;
that
i s ,
i f
I
am
i n
debt
< ;
/ .
and
take
from
i t
a
debt
of
2 J . the Remainder of
my
debt w i l l be 3 / . or= a .
i s ,
i f I o i
- . ) .Asia
&— —
—
7a; thit
^ / .
and
am
obliged t o pay 2 / . I s h a l l b e i n debt 7 / .
that i s ,—
a .
( 4 . J
— < »
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14
A bridgement
of
Algebra.
THEOREM
V.
7 te ProduB
of
two
Quantities
a l i k e
a f f e B e d
» V .
Affirma
t i v e , and t h e ProduB of two Quantities d i v e r f l j as
feBed i s Negative.
I t i s already s u f f i c i e n t l y , evident i f the t wo Quanti
t i e s a r e
Affirmative, their Products / h a l l
b e
Affirma
t i v e
;
b e c a u s e w h en an
Affirmative
Quantity
i s Multi
p l i e d
j j y
an
Affirmative
Quantity,
the
one
i s
a s
o f t e n
added
a s
there are Unities
i n
the other ; f o r Affirmati
on
i s
a Mark of Addition, and i f s u c h Addition b e
made
osan Affirmative
Quantity, the
Summ
i t
produces lhaH
a l s o
b e Affirmative.
I t i s a l s o evident, that i f the t wo
Quantities M u l t i
p l i e d
a r e Negative,
t h e i r
Products
s h a l l n e v e r t h e l e s s
b e .
Affirmative
;
b e c a u s e
i n
Multiplying
a
Negative
Quantity b y a Negative
{
the
one i s
a s o f t e n
Substracted
a s
there
areUnities i n t h e - o t h e r , f o r
Negation
i s
aMark
ofSubstraction ; and whereas
t h i s S u b s t r a c t i o n
i s made
from a Negative Quantity, the Negation
i s
d e s t r o y e d ' ,
and b y consequence the Affirmation i s r e s t o r e d ; which
i s the
r e a s o n t$at
the
Remainder,
which i s the Product,
i s
a l s o
Affirmative.
To
Conclude,
I t
i s
evident
that
i&one
of
the
t wo
Quantities i s Negative , and the other
Affirmative,
t h e i r
Products
s h a l l
be Negative
:
Because i n Multiply
ing
the Negative b y
the
Affirmative, the
Negative i s
a s o f t e n added a s there a r e Unities i n the A f f i r m a t i v e . ;
and whereas t h i s Addition i s made b y a.Negation, the
Summ o r Product s h a l l be Negative; . i
In the
same
manner
when
the Affirmative
i s
Multi-
p l i e d b y the Negative, the Affirmative
i s
a s o f t e n S u b
s t r a c t e d a s
there a r e Unities
i n
the Negative ; and
f e e
ing
t h i s S u b s t r a c t i o n i s done b y Affirmation, i n destroy
ing
the
Affirmation
the Negation i s S u b s t i t u d e d ,
which
i s
the o c c a s i o n of the Remainder or Products being Ne?
g a t i v e . . ;
. , .
N..B..7I*
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A b ridg e m e nt of
Algthr*.
j f
N. B. This
Thesrem
majtbe further i l l u s t r a t e d i n t b i t
manner*
(i.) 4-6ax+24O=f I2«,
because i f I have 6 7 . - and i t b e dou- f t M u l t i p l y .
b l e d t o me, o r
augmented
a s
many O
rodub.
times a s there a r e Units
in 2 ,
i t
w i l l =Equal
t o .
then:
b e 12
7 .
t h a t i s , - \ - 12 a t
( ' 2 . )—6ax—
2*-D=-f-
12*, because
iflowe
6
1 ,
and
i t
i s
forgiven
me ,
or
taken
o f f
from
me
twice,
my Condition i s amended 12
1 .
that
i s ,
rt a >
( 3 . ) — a X
+
a
□
=—2 a , b e c a u s e i f I owe
6 1 , and I
i n c r e a s e
the D e b t twicej my Condition i s
worse b y 12 / . which i s , —12 a .
( 4 . )
-\-6aK
—2
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Abridgment of Algebra. 17
that Summ
s h a l l
be
the Remainder
a f t e r the
S u b s t r a -
ction
p r o p o s ' d , according to T b e o r . 3. a s
may
b e s e e n
in
the
following
Examples,
EXA
MPL
E L
From : 6 a abb— « ' £ -\- 4 a b b s
Take :—laabbT^I
a* b 6
abbcir^c
i
l^aab b < * ' b — abbs— c i
E
XAMP
L £ U. . ^N
From
:
8 ab
-\-
2 b b Z\~ 4
c f
.Take-— lab
7 s . %bb~. ic e
6abJ\-$
b b-\-
6 c
c
.
P
R
O
B . -
I I I .
- M u l t i p l i c a t i o n of Polynomes.
'Having s e t the Multiplicator under the Multipli
cand, a s i n
common Arithmetick, l e t the
Poh/nome
Multiplicator Multiply every part of the Pdlynome
Multiplicand,
according
t o
the
Rules
of
the
precedent
Chapter ; and
t h o s e of —f-
and—
a s they a r e s e t
forth
i n
T b e o r .
4 -
a f t e r which, l e t the s e v e r a l Products
b e
added t o g e t h e r ,
a s i n the
following Examples. Of
which
t h e .
l a i l b ut
one shews that the
Square
of
the
Bi
nomial a —( - b i s the Trinomial a a -\- 2 a b —\- b
b ,
and
which
may
s e r v e f o r a
Model
towards
the Extraction
of
Square
Roots.
EXA
M-
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1 8 Abridgment of Algebra.
Example i . Example
2 .
2
< »
-4-4
b
2
«
-4-3$
By
a a
-T^ft go
—36
4 at -\-%T>b '—6ab
—
$bb
8 a b
4.aa-\-6ab
4aa-\~i2ab-\-%bb~ 40a o —966'
Example
3 . . Example 4 .
2aa
—
s . i
—
^ - 2 < j 6
-f££
2 a a — 3 6
2ab-—6b
—
.
a a b b-\- ^ b+
—
aa££—2abbb—b*
4 a4— a a £ £
2a4--4-4aaÆ&-}-2a£££
4 a4
—
a a £ -4- 4 i
+ 2a4_)-2aa££
o
—
A
a —| - 6 Example
5 .
7aS one.
a
the
S i d e .
~ah Z^Tbb
a a
- | - r t £
a a —\-
2*i
- f -
Square. Example
6 . l a s t .
-\-b
S i d e .
aa
b
-\-
1
abb-\-b*
a > -4- laab -\-
abb
a*
-f-3
aab
-
iab
b
~\-bf
Cube
;
whether
i n L i t t e r a l
Quantities
o r
Numerical ; and Ex
ample t h e l a s t Ibews that the Cube of the s a i d Binomial
a b i s the Quadrinomiala' - f - 3 aa 3 a b b - ^ - b 1
;
which
a l s o
may s e r v e
a s a
Model f o r the Extraction
of
- t h e Cube
Root, whether
L i t t e r a l o r Numerical.
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Abridgment of A lgebra. 19
PROB. IV*
D i v i s i o n of Polynomes.
In the
f i r s t
Place t o divide a Polynome b y a
Mo-
nome.
Each Term of
the
Polynome,
one a f t e r
the o t h e r ,
ought to b e divided b y the Monome p r o p o s ' d , accord
ing to the Rules of
the precedent
Chapter ;
and
the
Quotients
placed
on
the
Right,
a s
i n
ordinary
Arith
metics with the Signs - j - and—according t o the
Ru le i n T b e o r . 5 . a s may b e s e e n b y
the
following
EXA
M
P L
ES.
o o o
—* jg«»— 1 2 «»
bb-4bbc-.{ra
'Ĥ flf—
00 o
Bu t
i f
the
Divisor
b e
a
Polynome,
l e t
i t
and
the
Dividend
b e s e t down a s i n ordinary
D i v i s i o n
; a f t e r
which l e t the Division
begin
with that
Quantity ;
which,
with regard t o
the D i v i s o r , c o n t a i n s i t s
Let
t e r s , most of
the
r e s t i s done a s in common Arithme-
t i c k , a s may b e s e e n b y the Examples f o l l o w i n g .
, ,\ l2ab-\-%bb/r -
• o - + - 8 ab-\-%bb
4-
%ab-\-%bb
O O
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Abridgment of Algebra. ai
o — a abb
—
a 6'—4
+ — & &
->t-ahb—*
a - o t
If a f t e r Division any thing remains, or i f (because
of the
d i f f e r e n c e
of
Letters
in the
Divisor
and Divi
dend)
the
Division
cannot
b e
performed,
the
f a i d
Po-
l ynome
may b e formed into a Fraction, b y putting
the Divisor under the
Dividend
j v i t h a s m a l l Line b e
tween.
Thus
the Quotient o f a a -f b b , divided b y a+
s h a l l b e
and
«*
—
> divided b y a
—
s h a l l have f o r Quotient l̂iL, the l i k e of o t h e r s .
/ < t
—
N. B. 1 have a
l i t t l e d i f f e r e d
from t h e
Author
i n t h e
manner of h i t D i v i s i o n , knowing b y Experience t h e
way I h e r e u s e , being
a c c o r d i n g
t o
Mr.
Kersey, it
b e
t h e
b e s t .
PROB. V.
To extraB t h e Hoot of a Polynome.
We
have already s a i d i n the Problem of Multipli
c a t i o n ,
that
the
T
rinome
a
a
-\-
2
a
b
-V-
b
b ,
whole
D
2 Square
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22 Abridgment of Algebra.
Square Root i s a -\- b , s e r v e s a s a Model
f o r
the
Square
Root ; which t o make yet more p l a i n l y appear, l e t u s
f i n d i t s
Square
Root a s t h o ' we knew i t not,,which w i l l
b e
f o u n d ,
i n
the
manner
f o l l o w i n g -
Seeing
; t h e
Terms
a a and b b a r e s q u a r e , i t matters
not which i s -
b e g un
with ; i f with (a a ) ,
- l e t
i t s Square
Root
( a )
be s e t towards the Right, i n manner of a
Ouotient,
f o r the
f i r s t Letter of
the
Root sought;
a s
a l s o under the Square { a a ) t o
the
end, that Multiply
ing ( a ) b y ( a ) , the Square a a be produced, which
being
taken
from
the
Trinome
a
a
- I -
2 a
b
T\-
b
b ,
l e t
the
Remainer 2 a b —\- b b b e l e t under the Line ; and
whereas
i n the f a i d Remainder there w i l l b e 2 a , a s
may be s e e n i n
the Term
( 2 a b ) , i t shews that the f a i d
Term (iab) ought t o b e divided b y ( 2 a ) double t o
the Letter f i r s t found, v i $ . ( a ) , whose
Quotient
i s -4- b ,
f o r the second Letter of the Root sought ; wherefore
the
f a i d
second
Letter
( f r )
ought
t o
b e
s e t
a f t e r
t h e
Letter s a ) on the Right, with i t s Sign a s a l s o
under i t s Square (bb) which i s the l a s t Term of t h e
Remainder
2 a b - , \ - i b , i n s u c h manner, that under
the
s a i d Remainder
there
w i l l b e ^ a
-\-
b f o r Divisor ;
and whereas
a f t e r
having
Multiplied and
Substracted i t
according t o the Rules of D i v i s i o n , there w i l l
remain
nothing
;
i t
f o l l o w s
that
t j j j e
Square
Root
as
the
a a - \ - 2
a l t
b b i s p r e c i s e l y a b - . m,
O
-\-iab
-\-b b
2 a b^y which- -- 2 » b -\- £ ' & -
2ab-\-b b-)'
O
O
In the fame manner the Square Root of any
other
Power
i s
drawn t o understand
; which
no more
i s need-
f u l l
than
the
fallowing Examples.
}
Dou b l e
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z4
Abridgment of
Algebra.
If i n t h i s second Example the Square Root had been
b e g un t o have been drawn from the l e a s t Term 36 £ 4 ,
the
Root
found
had
been
6b
b —\-
6
ah
—
a
a ,
i n
which the Signs -\- and—re contrary t o those o f
t he former Root found, 3 a*—6ab—b b , which
shews
that a Polynome hath
always
t wo
Square
Roots
a s - well a s a Monome and a l l - other Powers ; and t h e r e *
f o r e here we s h a l l i n general o b s e r v e , That a - Quantity
hath a s many R j o o t s a s the Exponent of
its-Roots con
t a i n s U n i t i e s .
We
have
f a i d i n T r o b . 3 .
that the
Quadrinome
a *
_j-
3 a a b -4- 7
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Abridgment
os Algebra. if
-
o
- J -
3
a .
a b . - \ - 3
ab
b b *
$aa b
- .
o %abb
Q
o
If the Polynome- propos'd have not s u c h a Root j s
i s required, the s a i d Root must be e x p r e s t b y the
Cha
r a c t e r
V,
which
must b e put on
the l e f t
s i d e of the
Po
lynome, with a Line over the s a i d Polynome, which i s
to shew that the s a i d
Character
extends u n i v e r s a l l y
over
the
whole
Polynome.
Thus
t o
e x p r e s s
the
Square
Root
of thisBinome*«££-r- £ } _ | - < » , ; < 7 *
, o r .
a s
well
t h u s ,
a
V
1
-\-cl,
b e c a u s e
the Bynome
a 1
b *
- + - < » ' f *
i s d i v i s i b l e
b y the C u b e
a ' , whose
Side
i s
a ,
and
whose Quotient
i s
£ » -4-
and
in l i k e manner
o t h e r s * . - » .
C
H
A
P-.
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Abridgement of Algebra. ay
Amongst
a l l the
Terms of an Equation, the f i r s t i s
that where tha unknown Quantity i s i n
i t s
h i g h e s t de
gree; the
f e c o n d , t h a t
where
the
f a i d
Quantity
i s
abated
one degree
below
the h i g h e s t , and s o forwards u n t i l the
l a s t
Term, a s i n t h i s
Equation.
x3 -\-axx — bbxtnacc^ the f i r s t Term isx'» the
second axx,
the
third b x x , and the
l a s t
acc.
A l t h o u g h amongst a l l the
Terms
of an Equation/the
degree of the
Quantity
unknown i s rot e q u a l l y abated or
l e s s e n e d , b y r e a s o n of some Term that i s wanting, which
very
o f t e n
happens,
i t
hinders
n o t ,
b ut
the
f a i d
unknown*
Quantity
o r Term may b e the third i n d i s t a n c e from
t h e - f i r s t , although i n order i t
s t a n d
next unto i t . Thus-
i n the Equation f o l l o w i n g , x 4
- f -
a ax x
- \ - b *
x < s > c * ;
Where the second Term i s wanting, the f i r s t Term i s
»4, the third i s a axx, the fourth i s £ ' x , the l a s t -
i s
c * .
N. B. Our
Authors Meaning i s ,
r o h e n t h e
Terms con
s t i t u t e f
me p a r t s
o f i
Power,
a s berex* and at XX
c o n s t i t u t e p a r t of
t h e
fourth
Power
of t h e
Binomial
Root w?,x44-4,x' a-j-o'xxa a-J-4*a»
- j -
4 a4; So t h a t a axx i f indeed t h e
t h i r d ,
and
b ' x
t h e
f o u r t h , a s b e i n g b u t owe m u l t i p l i e d - i n t o -
x,
a s
4xa',
*
A l l the
Terms
of a n - Equation ( e s p e c i a l l y i n Geome
t r i c a l Problems) ought to b e
Homogeneal;
and
t h o s e :
where
the unknown Quantity happens t o be
equally
e x - -
alted i n , ought t o be accounted a s one only Term. As-
i n t h i s
Equation,
x x - \ - a x 4- bx ad - \ -
b d,
where
xx
i s
the
f i r s t
Term,
a
x
- J -
b
x
the
s e c o n d ,
andai-^-
b d
the l a s t
; i n which
l a s t
the
unknown
Quantity x ,
i s
noo a t a l l
found,
and t h e r e f o r e a d - j - b d
i s
c a l l e d
b ut
one Term-.
An
Equation
i s
esteemed
t o b e of - a s many Dimensi
ons a s the Quantity unknown has i n the f i r s t Term ;
that i s of t wo Dimensions o r Squares, i f the Squares
of
the
unknown
i s
found
in
the
f i r s t
Term.
Of
three
Dimensions
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a
8
A bridgement
of
Algebra.
Dimensions
or Cubick, i f
the
Cube of the s a i d
unknown
Quantity
i s
found i n the
f i r s t
Term, O f r .
Thus
i t i s
known,
that the
following
Equation
x}
—
b
a
a
b
i s
of
three Dimensions
or
C u b i c a l ,
because the-Cube of
the unknown Quantity
i s
found i n the
f i r s t Term.
Also
when i n the Equation there
i s
only one Term unknown^
i t
i s c a l l e d
a pure
Equation,as
x} i n abb, or x x t o a
b .
The
unknown
Quantity of an Equation
may
have
a s
many
d i f f e r e n t Values or E q u a l i t i e s a s the
Equation
hath
Dimensions.
Thus
we know
that i n t h i s
Equati
on
of
two
Dimensions xx
=
2.xco
15
there
a r e
t wo
Roots,
v i % .
- \ -
3 , which, because i t i s Asfirmative,
i s -
c a l l e d a . true Root, and
— which i s
a
f a l s e
Root
;
that i s , x may b e supposed equal t o or—. ,
This indeed r e q u i r e s
Demonstration,
b ut here
i s
n o t ; -
the
Place t o
s a y
any
more. See t h e Geometry
of des
Cartes.
When
one
of
the
Roots
of an
Equation
i s
known
which
depends upoa some
Problem, the Problem a l s o -
i s r e s o l v e d : Bu t to f i n d t h i s Root, the Equation s h o u l d
b e s o
reduced,
that
the f i r s t Term
b e
Multiplied
b y no
ether
Quantity than Unity,
which
i s always under
s t o o d although
not
mentioned ;
or at
l e a s t
into
another
Quantity
which
hath a Root, whose Exponent
i s
equal
to
the
num b er
of
Dimensions
of
the
Equation.
Further, All unknown Terms ought
to b e i n o n e -
Member of the Equation
;
f o r
which
Reason i t
i s
c a l l e d
the
unknown,
or
f i r s t Member, because
i t
i s
commonly
wrote
on the
Left,
and
the
known on the
other Member,
which i s
commonly
p l a c ' d
on t h e -
Right,
a f t e r
the
Character t / i .
To
Conclude,
the
Equation
ought
to
b e b rou g ht
a s
fow
a s p o s s i b l e
;
that
i s ,
the
Equation ought
t o be s o
reduced, that the unknown Quantity be brought
t o t h e
l o w e s t
Degree p o s s i b l e , that the
Roots may
more
e a s i l y
b e
f o u n d s
This Reduction
i s performed
b y means of
the
f