greek mathematics ii loeb
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ancient greek and english translationTRANSCRIPT
THE LOEB CLASSICAL LIBRARYFOUNDED BY JAMES LOEB,EDITED BYfT. E. PAGE,O.H., LiTT.D.
LL.D.
tE. CAPPS,L. A.
PH.D., LL.D.
tW. H.
D.
ROUSE,
litt.d.
POST,
M.A.
E. H.
WARMINGTON,
m.a., f.b.hist.soo.
GKEEK MATHEMATICSII
SELECTIONSILLUSTRATING THE HISTORY OF
GREEK MATHEMATICSWITH AN ENGLISH TRANSLATION BY
IVOR
THOMAS
FORMERLY SCHOLAR OF ST, JOHN'S AND SENIOR DEMY or MAGDALEN COLLF.fiE, OXFORD
IN
TWO VOLUMESII
FROM ARISTARCHUS TO PAPPUS
LONDON
WILLIAM HEINEMANN LTDCAMBRIDGE, MASSACHUSKITS
HARVARD UNIVERSITY PRESS
First printed 1941 Reprinted 1951, 1957
mPrinted in Oreat Britain
.
CONTENTS OF VOLUMEXVI. Aristarchus of Samos(a)(6)(c)
II
. .
General
.
.
.,
Distances of the sun and
moon
^ 414
Continued fractions
(?)
#
XVII. Archimedes(a)(b)
General
Surface and volume of the cylinder and sphere (c) Solution of a cubic equation . , (d) Conoids and spheroids(i)(ii)(iii)
......
18
40126
Preface
.
..
Two lemmas
,164 .164,
Volume of a segment of aparaboloid of revolution170
(e)
The(i)(ii)
spiral of
Archimedes. .
Definitions
,,
Fundamental property
(iii)
(iv)
A verging Property of the subtangent,,
182 184 186 190194.
(J")
Semiregular solids
V
CONTENTS
PAOB
(g)(A)
of System numbers
expressing. .
large.
Indeterminate analysis Problem
....:
.198202
the Cattlegravity
(i)
Mechanics(i)(ii)
:
centres.
of.
(iii)
Postulates . Principle of the lever . Centre of gravity of a parallelo.
.206208
gram(j)(^)
Mechanical methodHydrostatics(i)(ii)
(iii)
(iv)
Postulates Surface of fluid at rest Solid immersed in a fluid Stability of a paraboloid . revolution ..
.... ....in
216
geometry
.
220 242 244 248252
..
of.
XVIII. Eratosthenes(a)(b)(c)
.
General
On means
(d)
The Plaionicus Measurement of the earth
..... ..... .
.
.
260 262 264 266
XIX. Apollonius of Peroa(a)
...
The(i)(ii)
conic sections
Relation to previous works
Scope of the workDefinitions
(iii)
.....
(iv)
Construction of the sections
(v)(vi)
Fundamental propertiesTransition to
. .
new diameter
276 280 284 288 304 328
CONTENTS()
CONTENTSXXI. Trigonometry1.
..
2.
Hipparchus and Menelaus Ptolemy(a)(6)
406408
General Table of sines(i)(ii)(iii)
.....
Introductionsin 18
,. ,.
(iv)
and sin 36 =1 sin2 ^+cos2 Ptolemy's theorem
.412 .414 .420
(v) sin
() = 5cossin sinsin
cos,

.
42242 4281 AH HA, for HA>HN. Therefore AM MA exceeds > AH HA that is,bysimilartriangles,: : : ::
,
;
:
the ratio of the polygon to the surface of the pyramid.
88
GREEK MATHEMATICS
7
ive^av
,8.
' dpae^etrj
iv
Be
6
6
iv
A
iv
icrrlv
iv
iLava
['\}
. '.ovhk
, iviv
il
iSe
ilv
8,
iavea
,
jLterafu
,iiK
" , , ,84
ihv
i8o.
8
, iB
,
ARCHIMEDESscribed in the circle A has to the polygon inscribed in the circle a ratio greater than that which the same polygon [inscribed in the circle A] has to the surface of the pyramid therefore the surface of the pyramid is greater than the polygon inscribed in B. Now;
the polygon circumscribed about the circle has to the inscribed polygon a ratio less than that which has to the surface of the cone the circle by much more therefore the polygon circumscribed about the has to the surface of the pyramid inscribed circle in the cone a ratio less than that which the circle which is impossible." has to the surface of the cone Therefore the circle is not greater than the surface of the cone. And it was proved not to be less ; therefore it is equal.; ;
Prop. 16
If an
isosceles cone be cut by
a plane parallel
to the
base, the portion
of
the surface
of
the cone between the
parallel planes
is equal to a circle whose radius is a mean proportional between the portion of the side of the cone between the parallel planes and a straight line equal to
sum of the radii of the circles in the parallel planes. Let there be a cone, in which the triangle through and let it be cut by a plane the axis is equal to parallel to the base, and let [the cutting plane] make and let BH be the axis of the cone, the sectionthe
,TO
,
For the circumscribed polygon is greater than the but the surface of the inscribed pyramid is less than the explanation to this the surface of the cone [Prop. 12] effect in the text is attributed by Heiberg to an interpolator.circle B,;
*
^
...
TotJ
om. Heiberg.
VOL.
D
85
rov
GREEK MATHEMATICSrrjs
Kevrpov
,
,
re
6
,6
^ ,6
)"
Se
.,
8
,IS!Z,
^
,
,. ,, " , , ,rfj Tjj
, he
ttj
,,,
6 86
, ,
?.
ARCHIMEDESandlet there
be set out a
circle
mean proportional between HA, and let be the circleis
whose radius and the sum of
is
,
a
;
I
say that the
circle
equal to the portion of the surface of the cone
betweenFor
let
square of the radius ofcontained by
,. ,,.
the circles A,
be set out, and let the be equal to the rectanglelet
andis
the square of the radius
of A be equal to the rectangle containedtherefore the circle
by BA,
AH
cone
,
A
equal to the surface of theis
while the circle[Prop. 14].
equal to the surfacesince
of the cone
BAbecause
=is
.
+(...
And
+AH)
parallel to AH,'* while the square of
the radius ofradius of
A isisis
equal to
AB AH, the square of the
equal to equal to
the radius of
, (.
and the square of + AH), therefore the
square on the radius of the circle
A
is
equal to the
sum
of the squares on the radii of the circles K,
;
so that the circle
A
is
equal to the
sum
of the circles
The proof is
*
,
GREEK MATHEMATICS
ju,ia
, , , ? . " , ^ ^^, ?els
.
8 ,iarl
^
rfj
^evdeiai
*

,,,, , ,, ,. , , , U , [ ]^ , ,, , . , ? ,
, 'etvatjuia
6
iv
'^,
^
otl
^,
'
00,
*
iva
.
,
.
om. Heiberg.
88
ARCHIMEDES
is equal to the surface of the cone But BE ; while is equal to the surface of the cone therefore the remainder, the portion of the surface is of the cone between the parallel planes equal to the circle
,.
,
.
,,
Prop. 21
If a
regular polygon
tvith
an even number of sides be
inscribed in a circle,the angles "
and
straight lines be
drawn joiningto be
of
the polygon, in such
a manner as
parallel to any one whatsoever
of the lines subtended by two sides of the polygon, the sum of these connecting lines bears to the diameter of the circle the same ratio as thehalf the sidesless
straight line subtended bythe side
one bears
to
of the polygon. Let be a circle, and in it let the polygon be inscribed, and let EK, HN, be joined then it is clear that they are parallel to a straight line subtended by two sides of the polygon * I say therefore that the sum of the aforementioned straight lines bears to the diameter of the circle, the same ratio as bears to EA. For let ZK, AB, be joined then ZK is
,
,
;
;
parallel to
A,
BA;
and
to
to ZK, also therefore
,:
,
;
to
BA,
to
But
ES:HA=KH:HO. KH HO =Zn 0,:
[Eucl.
vi.
4
" Sides " according to the Archimedes probably wrote*
text,
= lKZA For, because the arcs EZ are equal, [Eucl. iii. 27] ; therefore EK is parallel to and so on. ; " For, as the arcs AK, EZ are equal, ^AEK = ^EKZ, and therefore AE is parallel to ZK ; and so on.89
,
but Heiberg thinks where we have nXevpas.
ju,ev
7/30
, ^^ ? ,, ,? , , ,," ^
GREEK MATHEMATICS
,?cos Setj
Be
tJ

^
/cat
en,
?, ?, ?,Tf
NT
he 
NT
ert,
MX
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