m ozanam s introduction to the mathemati

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1/89

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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


9/89

W

l

is


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


11/89

-

- - - \

v

- - -,-V\\ *

Vi *V. ^ /s A { f

e r f ' ;

- - . >

. . . - w

*

. . . . - - - :

j .


12/89

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


13/89

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-

»


14/89

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


15/89

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


16/89

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


17/89

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


18/89

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,


19/89

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


20/89

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


21/89

- 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 .


22/89

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


23/89

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


24/89

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

-


25/89

\ 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.


26/89

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


27/89

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 ;


28/89

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


29/89

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


30/89

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


31/89

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


32/89

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,


33/89

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.


34/89

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

»


35/89

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.


36/89

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


37/89

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


38/89

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

— < »


39/89

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*


40/89

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


41/89


42/89

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-


43/89

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.


44/89

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


45/89


46/89

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


47/89

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


48/89

Abridgment of Algebra. »5


49/89

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


50/89

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-.


51/89


52/89

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


53/89

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

m ozanam s introduction to the mathemati

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