history
Halle,
1839.
14.
Marie,
allgemeinen
Principien
der
Mechanik.
Leipzig,
1887.
33.
Brewster,
D.
The
Memoirs
of
Newton.
Edinburgh,
1860.
33.
Prospects
of
to
failure
a
circle
to
its
diameter
is
incom-ensurable.
Some
quadrature
of
the
circle,
y
means
of
the
ruler
and
compass
only,
is
impos-
will
be
pleased
to
hear
the Arabic
ener-ies
on
the
are
given
as
the
squares
in
the
sexagesimal
nota-ion
of
integers
the
'^principle
of
position
was
employed.
Thus,
in
1.4
(=64),
the
sexagesimal
fractions
parts
by
its
radius,
and
into
360
degrees,they
had
some
knowledge
of
geometrical
figures,
uch
as
the
triangle
and
quadrangle,
which
they
Ibis
was
sacred
to
him,
inscriptions,
simplest
geo-etrical
truths
were
divided
used
the
term
fraction
in
a
restricted
sense,
closed
the
gates
to
progress.
THE
GEEEKS.
About
the
seventh
century
b.c.
an
active
commercial
inter-ourse
sprang
up
between
Greece
and
Egypt.
Naturally
there
pyramids.
Egyptian
ideas
were
thus
transplanted
across
the
sea
and
there
stimulated
Greek
thought,
our
most
school,
falls
inferred
that
had
observances
approaching
masonic
peculiarity.They
were
forbidden
to
divulge
the
discoveries
Neo-Pythago-
reans
this
objection
the
magnitudes.
The
term
number
was
used
by
them
in
a
restricted
sense.
What
we
call
a
tortoise;
for
the last drawn
unconsciously
inquiries
was
epoch-making.
Menaechmus,
an
we now
solution of the
astronomical observation. From
fifth
book.
and
works
we
have
no
details
is
Aristaeus,
the
elder,probably
a
senior
contempo-ary
wrote
a
work
on
conic
disposition,
ar-icularly
at
all
times
rested
mainly
upon
his
book
on
geometry,
called
the
Elements.
a
Exhaustions
(Book
XII.),
that
Thesetetus
contributed
much
light
of
strict
mathematical
logic,
it
proof
to
exercises in
detection of
The
following
are
the
extant
a
straight
line
can
equal
a
curved
one.
The
finding
by
a
part
Archimedes
prized
most
highly
those
straight
line
a
sphere
are
f
of
in the
is known
:
curve
; now,
through
any
point
whatever
the
characteristic
property
which
Apolloniusrecognises
a
Zenodorus.
He
wrote
an
interesting
treatise
Euclid,
but
recent
critics
are
of
opinion
us
that
Hippar-hus
originated
the
science
of
trigonometry,
manuscript
copies,
quite
dissimilar.
But
M.
Marie
a
work
so
ancient
as
no
reli-ble
Dioptra
is
a
treatise
on
peoples
of
the
;
Carnot
makes
this
proposition,
nown
as
the
great
predecessors.
The
Almagest
forms
system,
the
Ptole-aic
System,
created,
for
astronomical
use,
a
trigonometry
remarkably
perfect
in
form.
was
laid
by
the
illustrious
Hipparchus.
The
Almagest
into
360
degrees,
any
arc
school. His
contemporaries
is the author of
it
gives
on
various
treatises
by
the
by Pap us
cylinder passingthrough
sphere
was
known
since
Archimedes'
time,
to
measure
portions
thereof,
such
as
spher-cal
triangles,
as
three instances
not
furnish
any
The
abacris,
as
900 1000 2000 3000
at
1000,
the
alphabet
is
begun
over
again,
but,
to
prevent
confusion,
a
line
drawn
(supposed
to
extend
to
the
sun)
1,000,000
stadia,
Archimedes
finds
a
number
which
this
calcula-ion
was
the
improvement
structure
of
things.
or
analogies
between
that he
scale,
and
also
seven
from
1
to
2
w
first incommensurable ratio known
ninth
book,
the
same
subject
is
Only
two
are
known
to
of
terms
multiplied
by
ill
arranged
and
cows.'
Another
problem
in
the
Anthology
is
quite
familiar
to
school-boys
on
algebra
now
extant.
In
only
for
adding
the
particularproducts,
rules
were
probably
Greek
certain
philosophic
and
theologic
teachings
hand,
evidences
have
Brahmagupta
(born
598).
In
general
we
may
safely
infer
that
our
with six
religion,
had
to
submit,
when
a
youth,
in
order
to
to
right,
as
in
writing.
Thus,
they
added
the
:
'
people
assumption
of
Diophantus.
Hindostan
are
the
real
equations.
Cantor
of
mathematics
belongs
to
analysis
differs
quadratic
equa-ion
cy^
great
areas
of
given
polygon,
circle
was
therefore
made
up
of
21,600
equal
parts.
Indians
never
reckoned,
angles
by
subdued
Syria
and
Mesopotamia.
Distant
used
by
the
Saracens
those
West Africa
numerals,
tongue.
The
translations
of
mathematical
works
must
have
been
very
deficient
at
first,
s
it
was
difficult
to
secure
translation
of
Euclid's
Elements
was
ordered
by
Al
Mamun.
to
his
knowledge
of
algebra
the
area
of
triangleexpressed
in
terms
other)
sine
or
half
the
chord,
in
place
of
Abbasides
lost
power.
One
province
moon,
an
inequality
usually
supposed
to
with
a
single
opening
indeterminate
analysis.
equation
a^
roots
thirteenth
century,
they
had
centuries.
Hovarezmi,
written
nearly
800
years
before.
Wonderful
eleventh
century.
It
was
formerly
to
trigonometry.
In
his
treatment
of
sphericaltrigonometry,
he
exercises
great
independence
of
thought.
spherical
right
triangles.
To
sweep
Europe.
Eoman
encyclopae-ias
119
year
in
which
Alcuin
(735-804)
was
born.
Alcuin
was
educated
in
Ireland,
and
was
in
which
were
taught
the
psalms,writing,
singing,
computation
(computus),
and
grammar.
By
computus
was
here
meant,
probably,
not
merely
There is
Scientific
pursuits
were
abandoned,
century,
when
subject.
direction,
John
Arabs the
centuries
state
an
example
apices
were
Seville,drawing
from
Arabic
works,
use
the
term
algorism,
calculate
roots,
the
abacists
do
not;
they
flourished
science.
He
employed
a
century.
f mat ematics
business
pursuits
found
time
Fibonacci,
i.e.
son
unquestionably
the
knowledge
Ages,
a
apices.
The
reckoning
Italy
used
it
as
early
as
the
thirteenth
century,
arts
of
the
fifteenth
centuries.
About
(1200
a.d.),
century,
are
were
required
not
only
capture
of
Constantinopleby
began.
In
1453,
the
Turks
THE EENAISSAITCE.
splendour.
In
fact,
Italy
was
the
had
not
remained
In
addition
to
the
ApoUonius,
of
Archi-edes,
employment
several
calculators.
The
work
was
completed
by
his
pupil,
Valentine
Otho,
equations
a^
in the
ten
days
before
the
appointed
date,
as
so
long;
a ter this
Magna
equation
is
82.319...
For
this
process,
Vieta
was
greatly
admired
by
his
contemporaries.
It
was
and
others..
Its
principle
studying,
sometimes
several
days
in
succession,
without
eating
and
sleeping
more
than
was
necessary
to
by
cor-esponding
equations
solutions,
is
equations
change
which
they
were
making.
negative
roots
of
equations.
Indeed,
we
find
few
algebraists
efore
.and
during
gives
plus,
but
applies
it
reallyonly
to
the
development
of
the
product
of
(a
of the
masterly
and
original
treatment
superiority
began
to
be
studied.
The
foundations
were
laid
by
Fermat
and
Pascal
for
the
theory
of
improvements
5912
or
tables.
a
quadrant
upon
the
new
plan.
Napier
that table
bi-quadratic
equa-ions.
All
attempts
at
solving
algebraically
quations
of
higher
degrees
remaining
fruitless,
new
Kepler
and
Cavalieri
in
following
of the Greeks
as
composed
of
an
infinite
area
to
be
triple
to
Method of
Developing
this
idea,
Fermat
obtained
his
rule
x
calculus is that
Pascal's
genius
for
geometry
performing
arithmetical
operations
mechanically.
This
triangles,
discovered
on
this
subject,
o
his
writings.
over
400
corol-aries,
embracing
only.
In
some
cases
later
analysts
a
truly
wonderful
proof,
a
long-
sought-for
+
was
unable
to
prove
it
rigor-usly.
an
inductive
method,
called
by
him
la
descente
infinie
u
indefinie.
He
says
probabilities
ngaged
the
slept
nearlytwenty
centuries,
;
independence
eigh-
.teenth
centuries,
who
employed
same
key.
a
given
ratio
to
the
product
of
in substance.
at
Stockholm
one
year
later.
an
ingenious
rule
for
finding
equal
roots.
the
hyperbola
back
to
the
quadrature
of
the
hyperbola.
The
semi-cubical
parabola
2/^
ax'
was
The
majority
parabolic
and
hyperbolic
conoid,
and
exponents,
which,
zero,
then
the
area
must
be
more
only
to
omit
expression.
In
a
letter
to
Oldenburg
(June
13,
1676),
Newton
states
the
theorem
as
follows
positive
hole
expo-ents
were
known
to
some
grand property
to
BA,
then
we
Since
XIV.
empire
and
brutalised
the
people.
the advances of the future. Such
was
the
age
a
tract,
entitled
De
Analysi
per
^quationes
Numero
Terminorum
Infinitas,
hich
was
sent
New-on
does
not
of
Flvxions
(as
well
as
object
of
determination.
The
difference
of the
infinitely
mall
quan-
:
a
different
purpose.
The first lemma of the first book has been made the
foundation of the
method of limits
that
time
approach
nearer
the
one
to
as
deduced
gravitation.
The
first
book
was
completed
on
April
28,
proof
of
casu-lly
ascertained
at
a
meeting
squares
was
verified.
In
a
a
spherical
hell
upon
an
external
point
of
determining
the
infinite
branches
to
curves,
or
their
figure
at
multiplepoints.
Newton
gave
no
proof
for
it,
nor
any
clue
as
to
then
existing.
In
his
fifteenth
year
he
1676,
existing
new
calculus
came
to
be,
a
Tangents.
He
tangents
and
reasoning
backwards
to
the
originalsupposition.
to
know
could be
to
three
variables,
y
which
to
only
under
Optica.
Evidently
and
a
variety
Binomial
Theorem,
fluxions and
fluents in
form of
sentence
communicated
were
placed
in
alphabetical
order.
Thus
Newton
says
to
also fallen
notions of fluxions
calculus differen-
de
Duillier,
Swiss,
who
plagiarism.
Commercium
Epistolicum
was
had
been
towards
Leibniz.
Keill
replied,
challenged
the
best
mathematicians
in
Europe
was
injudicious,
for
a
controversy
with
Varignon
on
the
subject.
Among
the
most
vigorous
promoters
of
the
calculus
on
the
Continent
were
survivors
at
a
given
age
from
a
given
the
method
of
finding
the
limiting
value
of
a
fraction
whose
two
terms
was
Pierre
sums
of
a
considerable
to
problem,
called
Eiooati's
equation,published
in
the
there
was
in
Germany
not
a
single
mathematician
of
note.
form
of
Euclid,
of
course
only
in
outward
form,
will
be
a
straight
on
geometric
demonstrations
way
of
distinguishing
between
maxima
and
minima,
and
explained
their
use
in
the
theory
of
multiple
points.
problems
were
usually
reduced
to
geometric
form.
A
change
now
took
place.
Euler
brought
about
an
emancipation
With,
perhaps,
less
exuberance
of
invention,
found
to
be
true
in
Gamma Func-ions
was
the
first
to
conies he
obtained five
observed
to
be
dependent
upon
the
integration
of
a
a
theorem
on
homogeneous
functions,
known
by
his
name,
and
contributed
largely
to
the
theory
of
differential
equations,
subject
which
of
arbitrary
constants.
By
repre-ents
any
arbitrary
function
of
a
variable.
in-ersely
proportional
attracting
hands
algebraic
expression
for
the
roots
of
a
cubic
equation
1766
Euler
left
given
equa-ion.
Lagrange
they
He
was
made
one
of
The
principles
of
theorem,
and
thus
to
avoid
all
reference
to
for
incapacity.
Said
Micanique
CMeste
is
prefixed
a
note
book,
that
most
precious
to
paper
was
the
beginning
of
a
series
of
profound
researches
by
Lagrange
and
Laplace
on
celebrated nebular
to
a
predecessor
are
really
his
own.
We
are
told
Micanique
Cileste.
It
gives
an
exhaustive
treatment
determinants,
the
Mecanique
Cileste
he
made
a
generalisation
f
Lagrange's
theorem
on
the
develop-ent
on
the
velocity
of
physi-al
problems.
The
true
result
being
once
reached,
he
spent
little
time
in
explaining
prepared
an
essay
on
the
curve
described
resigned
his
position
in
order
to
reserve
more
public
commissions.
were
the
elliptic
unctions
and
the
theoiy
EUments,
it
before
Budan's
publication.
arts
and
the
analytical
treatment
of
geometry
was
brought
into
was
closed
et.
Metiers
in
Paris,
published
conception
of
conjugate
tangents
of
a
point
of
a
surface,
chair
at
Edinburgh.
During
the
eighteenth
century
mathematics.
For
example,
or
the
theory
century
and
the
beginning
of
the
present
has
most
important
always possible
degree,
by
synthetic
methods
on
maxima
and
minima,
and
arrived
at
the
solution
of
problems
which
at
that
Steiner raised
success
by
Luigi
Cremona,
professor
spans.
Henry
T.
Eddy,
of
calls reaction
their attention
space
defined
by
Euclid's
thirty-
five
years
this
appendix,
as
also
Lobatchewsky's
researches,
same
popularised
the
subject
in
lectures,
analytic
geometry.
Julius
Pliicker
(1801-1868)
was
born
at
Elberfeld,
source
of
his
proofs.
The
System
der
AnalytischenGeometrie,1835,
con-ains
a
complete
to
his
first
love,
mathematics,
in
algebra
Hesse
applied
to
the
analyticstudy
of
curves
great
memoir
on
this
subject
(Crelle,
855)
was
published
at
the
same
time
as
helped powerfully
mechanics
at
the
Polytechnicum
in
Carlsruhe.
The
study
cubic surfaces. Oth r surfaces have been studied in the
same
way
by
recent
writers,
particularly
M.
Nother
of
Erlangen,
Armenante,
Felix
between
the
measure
of
curvature
progress
in
recent
years,
except
father,
who
represented
in
qua-ernions
product,
are
sub-ect
to
Lithographed copies
;
origin
incapacitated
him
from
taking
a
degree.
coefficients
in
regard
to
x,
which
remain
unaltered
by
the
interchange
of
x
and
y.
This
theory
is
more
general
than
the face in the
in
exhibit-ng
the
terms
Isaac
Tod-
hunter
(1820-1884)
necessity
f
inquiring
into
the
convergence
of
infinite
series,
found their culmination
if
two
would
dispose
criteria,
s
kind,
entirely
ew-
rest
mainly
on
the
consideration
consecu-ive
terms
or
coefficients,
n
a
pre-cribed
manner,
shall
reach
a
maximum
or
minimum
value,
demands,
differential
equations
which
can
be
on
by
Briot
and
Bouquet,
and
by
Poincar^.*^
The
subject
of
singular
of Boole
to centre
August
Leopold
Crelle
(1780-1856),
and
met
Sterner.
Encouraged
by
Abel
and
Steiner,
elliptic
ntegrals
are
were
applicable
to
many
other
functions,
periodicity.
The
papers
Gauss confirm
are
not
theorem has reference
it
may
In
1842
by Cauchy, Konigsberger
given
also
by
A.
used
only
in
mathematical
physics,
was
w
is
uniquely
a
complex
variable
embraces,
among
others,
functions
having
an
infinite
Gauss. Of these three
repaired
to
the
university
t
Helmstadt
to
work. In 1828 he
Disquisitiones
Arith-
meticoe,
a
work
(awarded
the
Grand
prix
of
1892),
are
among
the
latest
Glaisher.
The
printing,
by
the
Association,
of
marked
the
completion
of
tables,
to
the
preparation
Europe
remarks that the
squares
is
odd,
it
involves
processes
peculiar
to
the
theory
the
modern
1825
by
the
not recount
of
modern
practical
stron-my
and
geodesy.
As
an
observer
discovery
of
Neptune.
John
upon
other
computed
rings
was
taken
up
first
by
Laplace,
who
demonstrated
that
a
homogeneous
solid
ring
could
not
be
in
equilibrium,
and
in
1851
by
B.
Peirce,
who
proved
their
non-solidityy
showing
that
even
an
irregu-ar
conception
of
the
nature
of
rotary
motion
was
theory
of
on
the
The mathe-atical
on
clouds
point
his hands
to
protect
him
from
perishing
under
hardly
a
problem
in
elasticity
o
Tait in their
the
force
upon
a
As
a
result,
a
truer
theory
of
flexure
recentlyby
Simon
New-
comb,
assumptions
were
not
satisfactory;
hence
Laplace,
Poisson,
and
others
belonging
to
the
strictly
athematical
school,
at
first
disdained
to
theory
of
concave
gratings,
mathematical
theory
of
electricity
nd
magnetism.
Green
of
Konigsberg
developed
from
the
experimental
laws
were on
went
to
Trinity
College,
Cambridge,
and
came
out
Second
Wrangler,
E.
Eouth
being
Senior
Wrangler.
Maxwell
then
became
lecturer
at
Cambridge,
in
1856
professor
at
Aberdeen,
and
in
1860
professor
at
King's
College,
to
private
experiments
favoured
interpretation
nd
development
angles
of
contact
between
liquids
in
expres-ion
molecules
according
average
a
paper
of
1879,
intended
to
explain
mathematically
the
effects
observed
in
Crookes'
seem
to
support
assumption.
Among
68,
125.
Arithmetic:
Pythagoreans,
20,
67-70;
Platonists,
29
401.
348.
Biot,
275,288,
393.
Biquadratic
368,
45^9,
285,
291,
333,
325, 334,
329, 345, 346,
347-354, 363, 367,
57,
262,
241.
Factor-tables,
368.
Fagnano,
241.
278,
343.
Fink,
XII.
Fitzgerald,
394.
Flachenabbildung,
313.
Flamsteed,
218.
Floridas,
142,
144.
110, 113, 114;
349,
377, 378,
354.
Kleinian
groups,
345.
Kleinian
functions,
360.
Kohlrausch,
394.
Kohn,
337.
Konig,
401.
Konigsberger,
353;
ref.
to,
344,
350,
354,
355.
Kopcke,
382.
296,
364,
365,367.
Legendre's
function,
280.
Leibniz,
219-235;
ref.
to,
4,
208,
244,
178,
186.
392,
395,
396.
238, 242, 245, 249, 259; spherical,57,
116,
280,
294.
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Ele-entary
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A
HISTORY
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