or
MAGDALEN
COLLF.fiE,
OXFORD
7-18
of the Lyceum from
tarchus
formulated
to have
hypothesis
to the
Aristarchus
and
centre
{v.
Ecphantus,
a
Pytha-
earth
a
motion
sphere and having the
same centre as the
sphere; as the Greek
»
and
Ptolemy
a
little
less
than
2f
for
of the Zodiac"
; as he believed
;
we
obtain
4050
111
88
the
approximation
1 +
Archimedes
We
?),
L.
Leipzig, 1910-1915).
of
of
it.
Tzetzes, Chil. xii. 995
the Measure-
and
Romans.
round and,
the
that,
Hiero
men
hand
at
coming
"trap
as
though
she
because he
from war and
set by Nature
fled
spirit,
so
pro-
found
a
inventions
a
name
intelligence,
he
Regarding the
art as
proof,
credit that he
;
body
was
anointed,
being
overcome
elegant
death
a
cylinder
enclosing
impression
that
a
polluted
person,
and
sought
out
form of
cum eius rei rationem
Let
the
weight
volume
are
the whole
written
finitions.
43
but having
the
same
extremities
with
it,
or
is
,
TO
For let there
magnitude
Let
and
is
equivalent
than
the
surface
of
the
cone
between
the
triangle
and
the
[Prop.
10].
From
in a cylinder, the
surface of the prism
cylinder excluding
the bases
the
bases.
Prop.
13
The
surface
of
rvhose
raditcs
bases.
this and
the next
possible,
;
the surface of
of the rectilineal
circle
inscribed
the
in
;
regards
the explanation to this effect in the text as an inter-
polation.
of
prism.
But
equal
the
radius
of
radius of the circle B.
75
has
to
the
76
to
[Prop.
12].
interpolation.
77
either
greater
and the circle B,
about
the
cone
is
equal
the
of
a ratio less than that which
the surface of the cone
has
to
the
circle
"
,
\11 have
as
to the
surface of
to the
inscribed
in
through
sum
of
the
rectangle
contained
by
,
of
to
be
are
parallel
to
the
circle ;
bounded by
diameter
the other
be
imagined
a
out
a
circle
H,
the rectangle
equal
to
the
rect-
angle
contained
by
+,
and
let
the
contained
by
AE
EZ,
EZ,
EZ,
99
surface
of
the
figure
figure
sphere.
Prop.
28
Let
be
sphere, and
and
smaller
circle]
will
and
the
angles
circumscribed
figure
circles
in
the
lesser
sphere,
K,
being
having the same
revolution of the circular
revolution
of
the
portion ...
has
4«
the
figure
circumscribed
of the
of
the
of the
figure in-
of whose
straight
hnes
subtended
by
equal
to
the
sum
of
angles
than the circle.
,
in the circle
.
.
B,
be
similarly
the
circle
A.
greatest circle.
Other
a
seg-
ment
of
segment of
a parabola.
He also
finds the
not
by
a
method
equivalent
to
integration.
The
circumscribed
figure
has
to
that
scribed
a
28],
but
the
base e ual
of
to
the cylinder
one
ms.
125
of
the
various stages of the proof.
The problem
have a
:
problem,
problem,
so
to
cut
a
given
straight
line
at
the greatest of
—
greatest
when
real solution
be the work
of the
polated,
that
to
the
radius
given
same
is
necessary
to
investigate
the
limits
of
possibility,
•For
investigation
a
proof
of
what
cannot
be
found
approached
turn
it solves the
the equation
x%a
-x)= 6c*
and
Let
it
then
be
lo
nius's
Elements
of
Conies,
and
it
will
be
given
in
But
it
lies
to
the
square
the
given
straight
;
as
was
shown
in
the
analysis
soUds
in
two
[equal]
solids
the
bases
are
inversely
proportional
let
it
meet
at
N,
and
through
let
the same straight
instead
on
the
left.
sought, it is per-
either the point
to
H,
that
will determine the
let
there
a
dia-
on .
possibility."''
If
Archimedes set
p.
book
a
segment
2A,
3A
. .
the
geometrical,
being
is
equivalent
let
the
section
be
will
the
circum-
about the diameter A
BI
be
circumscribed
Chords KA
inside and outside
by
."
its
axis,
bears
to
the
ratio
its axis
having
for
its
3.
ylinder
axis
half
between the
straight lines
cylinders in
the
lines
cut
and
the
cylinders
in
the
whole
cylinder
is
less
than
double
of
the
cylinders
in
the
circum-
scribed
figure
.
.
+(«
-
revolve uni-
move uni-
revolves
be
called
the
9-52.
15
If,
from
the
origin
of
as
the
arcs
of
the
circle
between
the
extremity
of
let
be
the
a
line
from
the
centre
meet the
in Prop.
the
positions
ratio
is
greater
than
that
which
half
of
In the
be
the
acute.
of Prop.
been
described
is true
bases,
After these
sixty-two
bases,
the
ninety-
If
certain
equal
lengths
be
cut
off
is
and
equally
is less
communicated
at Alex-
stranger, compute
white, another
a glossy
were bulls, mighty
in number according
equal to a
whole
all the yellow.
the whole herd of
to the
"
203
*
sum
the number of
epigram states that
Equations
(1)
to
(7)
give
the
following
as
the
values
of
(4149387
+=&
Archimedes
solved
it.
A,
is
is
com-
mensurate
with
be
placed
on
a
magnitude
equal
to
Z,
the
middle
of
the
be equal
figure
com-
pounded
side of
on
a
of gravity
of the
magnitude compounded
the point . Therefore if
+
is
parts—
A,
when it is taken away, the
remainder
A
equilibrium.
is
on
EZ
proved.
(J)
Mechanical
zealous
student
and
principle of
particularly
his
mechanical
method
by
mechanical
rigorous, so
works
of
are
un-
for
you
and
same
no less
useful even
for the
this
of
Conies
TH,
MH,
and
of the
"
"
been demonstrated,
of course,
because the
But
I
do
not
know
segment,
and
for
now
published
by
me
trapezium
in
the
equal
 
Bodies,
the
earliest
extant
treatise
on
should
Z, H,
to
the
than the
not
to
the
taken from William's transla-
unrecognizable in
figures are
maneat
non
are
borne
upwards
in
a
fluid,
each
of
gravit}'."
(ii.)
Surface
of
earth
be
K,
and
maior,
angulos,
muntur quae quidem
drawn from
line
HO
0
are
unmoved
with
centre
K.
Similarly
it
may
be
while the weight of
the fluid having the
tour
de
force
manner
that
the
segment
is
not
segment
extremity
which
is
that each
along the
the fluid," it
line
provided
a
to the harmonic
the
solid.
suppose,
in
this
case
also,
has
to
of
10-12)
also
corrected
250000
to
On this
edition
of
his
work.
*
the
ellipse
for the
by
side ofthe
of
cones
same way
of
the
pp.
278-283,
and
more
particularly
p.
;
Geminus, but
he
may
have
vol. i.
Geminus
sections
in
im
Altertum
(1886)
and
so I was not
Halley
(Oxford,
1710)
first four
(Leipzig,
1891-1893)
circumference
of
a
circle
I
call
the
into mathematical
which
scalene those
as
axis,
and
if
it
be
also
cut
the
base
of
the
duced
only
but
if
it
be
sea•
is
perpendicular
to
the
base
of
the
cone.
Let
there
be
a
cone
whose
vertex
is
the
is
in
the
plane
containing
the
other
part
triangle
is
perpendicular
to
the
circle
let
the plane
perpendicular
to
ZH.
Corollary
the
cutting
plane
i. 36.
of
the
axial
triangle,
and
further
the
intercept
of
called
a
parabola.
plane
through
diameter
straight
line
subtending
parallel
to
rectum), and
the
line
is
such
and let suck
another
plane
drawn
and
the
tangent
at
This form is,
B, E,
is
and circle
as well.
The general
Prop. 43.
of
Euclid's
the
he
proceeds
a
given
ratio,
in
this
to
at
and
the
(1)
three
points
or
(2)
the
thing
the
in order
the remaining
of
670.
4-672.
3
of
these
first
those
them in
position,
which
will
sometimes
be
of
made by Fermat,
perpendicular
to
straight
and
a
straight
line,
two
rhombus
Hypsicles [Euclid,
can
this proved
wrote
proved
work
On
the
Comparison
of
bears
whom
v.
infra,
pp.
391-397.
called
is
(vvea
(where
he
20, 70,
1, 5, 4, 5, 5, 1, 5, 5, 5, 1,
represent numbers
for the
seventeen
tens,
it
10000®.•*
by
100
and
ApoUonius's
Erocedure
in
separate
powers
of
10,
and
then
dividing
by
the
111.
1-112.
11
Geminus
cissoid, and
line,
addi-
mathematics may
new sciences
of trigo-
be de-
and
lines
formed
round
formed
curves
by Menaechmus,
and Eratosthenes
by
Perseus,*
who
Perseus
thanked
contracts on each
side, a third
is elongated and
for
his
therefore be
Nicomedes appears to have
the
valent
to
equation {supra,
Schmidt
in
Bibliotheca
mathematica,
iv.
pp.
Postulate
(i.)
General
and
in
these
Tannery
sees
his
to place
elementare,
of
the
right
angles
teacher
of
Cicero
an astronomical work
pupil of
Posidonius, and
that he
wrote about
the
about
be
necessary
two
right
angles,
Let
the
two
straight
lines
be
AB,
is they are non-
to
if it had been
proved that the two
BZH
equal
to
two
angles.
Let
A,
meet,
then,
where
falls.
from
two
angles
to-
are
right
angles.
For
if
same straight
must
also
be
greater
than
two
four
make
two right
angles, the
on Eucl.
on
distance
there-
fore
at
the
vertex
than the
distance between
than
Archi-
medes
among
plane
figures
and
in
a
summary
to
the
of
that
if
a
circle
have
an
equal
perimeter
having
Hypsicles
38(.
Anaph.,
arcs
be
in
which
it has
moon,
the
argument
that
place
the earth,
. .
.
earth.
For
stood,
to
the
size
the shadow
Theon the
not under-
this
or readily, for it calls for
a very
fail
in
their
made by Theon in the years a.d.
127, 129,
to
science of sphaeric, the geometry of the sphere, for which
V.
was
founded,
by Ptolemy to
His greatest
and he made
present accepted
recorded
regular
hendecagon
—
;
presumably
Latin
Among the
(a
;
work in
came to
considerable
depths.
commentaries by
version,
the twelfth century
the last are
De
Speculis
shall aim
60»
its
on BZ
on the
the
while
while
Ptolemy
which is
given and
the
calculation,
that
the
chord
subtending
1^°
is
approximately
1»
to the
an angle
value
IP
2'
50"
for
arcs
increasing
by
half
degrees,
chords subtending the arcs
a
well
should
suspect
an
error
in
chords
degrees in the
the ap-
proximations for
in the
the
celestial
be able to prove
;
of
the
arc
AB
in the
right-angled tri-
angle will
be given,
BE
another at
the point
of the chord
HB
produced
chord
subtended
by
of
Definitions
on
point
as
has
no
parts,
are
prismatic
again
9
root,
we
-^^,
•
from
Proclus
cited
supra,
pp.
360-365
: In
a
triangle
tvhose
sides
are
given
to
find
the
area.
by drawing
generate a
the spirae on
of the
to
Heron had in
trated. I am
base of
the aforesaid
a
1131.
multiply
chords cutting
book
(iii.
20)
pendicular
drawn
perpendiculars cut off
tested
so
that
it
shall
to
them
the
other,
of
tion,
let
the
weight
if
the five
axle EZ,
the rope
holding the
load will
the teeth
40 talents. Again,
in
Book
the weight.
to sink
down and
lines
can
be
shown
by
equation
is
con-
hypotenuse, S
sixth
L.
Yieath,
Diophantus
of
Alexandria:
Comm. i.
10, ed.
Rome, Studi
38.
22-39.
1
is regularly used
Syntaxis
the
number
which
exceeds
Anatolius
collected
the
most
essential
parts
of
the
theory
as
stated
by
to
of
finding
the
area
of
a
circle
are
difficult books
forth the
nature and
they
with the
the index
so
of
the
unknovm
quantity
minate
and
a minus,
to
one
term.
If
per-
chance
there
be
until
in
1885,
of the verb Xeiireiv,
used, but there
one
power
that
be
middle
is
one-third
of
2
middle
as
^r
+2
the
simultaneous
equations
He
says,
of
=
of a
and
2
to
25
and
the
of the
to
^x^
is to
solve the
(
=
square to a
41/ -4
the
first
and
second
first,
is
j;'
note
square. But
that
X™
+y'"
are not the
form
4"
(24Jfc
10
into
three
of
approxima-
tion,**
each
unity
part which
—^
added
to
their
a number
into four
or the
ed.
has
as
side
is
the
number
is,
2.
Thus
3
will
be
a
and so
the
multiplica-
tion
was
thought
that
be a
how
a
p.
There
is
with the
way
would
benefit
9
the angles
the
the
triangles.**
(g)
the areas
of
opposite
sign
the theorem may
a
leather-
worker's
knife
are
both
series
Hultsch
264.
3-268.
21
a spiral is
he
makes
this
interesting
digression.
;
spiral first in
•
which
it
manner,
the
is proof
33]
and
the
sector
spiral and the base of the
hemisphere is
,
bears
bears
to
the
surface
cut
off
by
the
spiral
the
same
segment
(since
proposi-
tion
an
analytical
equivalent,
which
I
have
adapted
to
the
and
hexagonal
in
form.
That
they
which are
triangles and
would
be
chose for
greatest
of
(k)
The
the usual
from
it
through
its
the
opposite
problems
its
con-
that which
is sought
will
we suppose
that which
is set
something admitted
be
Treasury
of
Apollonius's
Cutting-off
of
a
Ratio,
two
books,
Cutting-
off
of
books,
has
arising in
position,
and
 
of
the
solid
figure
so
pp.
492-503).
through
the
points
the
second
mss.,
which
vary
accord•
to
the
of expressing
by
philosophers
highest
esteem,
and
is
[about their
motions
in
a
motion
contrary
geometry, arithmetic,
astronomy and
youth in
the aforesaid
impossible
for
 
to a great
such-
like
as
properly called makers
more
easily
On
n.
8.
seemed
n. a,
317 n.
Apollodorus the
277-
281
251
n.
a
a,
455
n.
6
;
a,
ii.
47
Bede, the Venerable, 31
589-
593
Benecke,
pothesis
n.
,
and n. a
a;
duplication
of
cube
Descartes, system
n.b
ii.
Division
n. 6
Elements
;
and n.
;
Pappus's lemmas, 363
a,
h,
23
n.
6,
145
nn.
a
and
b,
153
n.
a,
ii.
163
n.
a,
ii.
n. a,
;
ii.
467
n.
477-483
cubes,
129-131
;
;
387 and n.
Pseudaria :
ii.
462
n.
a
Spiral
: of
517
nn.
n.
6,
;
6
Wilamowitz
220)
De
Corpor.
Fluit.
editions
of
Encliii,
principal
part,
but
a
few
cross-
ments on
a
rectangle
be
,
a
drawn through a
solid
ad-
m,it
an
angle formed by
ii.
Baxter. (2nd Imp.)
Caesar
W. D.
Hooper. (3rd
(Vols.
Pro
Cvuvsiio,
Pbo
K.ABIRI0.
H.
Grose
Hodge.
Imp.)
J. C.
Bolfe. (2nd
III. 2nd
JirVENAL
LiVT. B.
Rouse.
(Sth
VoL
I.
(BNNirS
AND
3 Vols. (Vol.
Vols. I. and
. 2nd Imp. revised.)
IL
Srd
Imp.)
(Vol. I. 7th
(2nd
Imp.
revised.)
VoL
II.
(2nd
Imp.)
I.-V.
2
Vols.
(VoL
L
Srd
Imp.,
Vol.
Julian.
Wilmer
2nd
Imp.)
LuciAN.
V.
2nd
Imp.)
Lycophron.
Cf.
Callimachus.
Lyra
Graeca.
J.
M.
Edmonds.
Imp.,
Papyri. Non-Literary
3rd
;
M. Lamb. (2nd
Hippias.
(Vols.
I.,
II.,
Imp.
7 Vols.
(Vol. I.
Srd Imp.,
Plotinus
: A.
H.
Armetrong.
Latin
Authors
St.