The Method of Archimedes
Author(s): S. H. Gould
Reviewed work(s):
Source: The American Mathematical Monthly, Vol. 62, No. 7 (Aug. - Sep., 1955), pp. 473-476
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2307356 .
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THE METHOD OF ARCHIMEDES*
S. H. GOULD, WilliamsCollege
The worksof Archimedeshave come down to us in two streamsof tradition,
one of them continuous,the otherbrokenby a gap of a thousand years between
the tenth centuryand the year 1906, when the discovery of a manuscriptin
Constantinoplebroughtto light an importantwork called the Method,on the
subject of integration.
Newton and his contemporariesin the seventeenth century were much
puzzled by one aspect of the integrationsto be found in the continuous tradition. In the books on the Sphere and Cylinder,forexample, it is clear that the
somewhat complicated method employed there for findingthe volume of a
sphere representsmerelya rigorousproof of the correctnessof the result and
gives no indication how Archimedeswas led to it originally.The discovery of
1906 removes the veil, at least to some extent.
F
C
N
P
,H
A
E
S
M
,K
B
FIG. 1
The newlydiscoveredMethodconsistsof imaginingthe desiredvolume as cut
up into a very large numberof thin parallel slices or discs, which are then suspended at one end of an imaginarylever in such a way that they are in equilibrium with a solid whose volume and center of gravity are known. Thus, in
Proposition4 of the Method,Archimedesshows that the volume of a paraboloid
of revolutionis one-halfof the volume of the circumscribingcylinderby slicing
the two solids (see Figure 1 which representsa plane section throughtheircommon axis AD) at rightangles to AD. For let us take HAD to be the bar of a
balance with HA =AD and with the fulcrumat A, and imagine the circle PO
to be removedfromthe paraboloid and suspendedat H. Since AD/AS= DB2/SO2
in the parabola BA C, we have
* An addressto the MathematicalAssociationof Americaat the 1953SummerMeetingin
Kingston,Ontario,Canada.
473
474
[September
THE METHOD OF ARCHIMEDES
HA
AS
AD
MS2
~~=
~~=
AS
S02
=
(circlein cylinder)
.
(circlein paraboloid)
so that, by the law of the lever,the circlein the cylinder,remainingwhereit is,
is in equilibriumwith the circle fromthe paraboloid restingin its new position.
If we deal in the same way with all the circlesmakingup the paraboloid, we find
that the cylinder,restingwhere it is with its centerof gravityat the midpoint
K of AD, is in equilibriumabout A with the paraboloid placed with its centerof
gravityat H. Since HA =AD = 2AK, the volume of the paraboloid is therefore
one-halfof that of the cylinder,as desired.
Many accounts of the Methodhave been given since its discoveryin 1906; for
example, by T. L. Heath in his Supplementto the Worksof Archimedes,Cambridge,1912. In all of them,as in the originalworkofArchimedeshimself,we are
invited to imagine the lever and the objects suspended fromit. But if we constructan actual lever and actual discs, the various figures,whichmay be spheres,
cones, etc.,see below, will be observed to balance, slice by slice, as successive
slices are added. The whole procedurethen becomes a picturesqueand effective
illustrationof the concept of an integralas the limit of a sum.
To findthe volume of a sphere,a problemwhich Archimedesconsideredso
importantthat he asked to have the result engraved on his tombstone,a cone
and a sphere are togetherweighed against a cylinder (see Figure 2 and the
accompanying sketch). Here the circle NM, resting where it is in the large
cylinderGLEF is in equilibriumabout A with two circlesplaced at H, the one
circle PO being taken fromthe given sphere and the other RQ fromthe cohe
FAE. For we have
OS2 + QS2
= OS2
+ AS2 =
A02
CA-AS
=
MS*SQ
and therefore
HA
MS
MS2
AS
SQ
MS.SQ
MS2
OS2 + QS2
Thus, by the law of the lever as before,
one-halfof cylinderequals cone plus sphere
fromwhich,since the cone is one-thirdof the cylinder,
sphereequals one-sixthcylinder.
Thus the cylindercircumscribedabout the sphere, being one-quarteras great
as the large cylinderGLEF, is three-halvesas great as the sphere,which is the
resultstated on the tombstoneof Archimedes.
If squares are substitutedforthe circlesof cross-sectionin these figures,the
argument remains unchanged and we have the solution of another famous
problem (Proposition 15 in the Method),namely to findthe volume common to
1955]
475
THE METHOD OF ARCHIMEDES
G
F
N
D
P
,H
AS
_
C
0
L
FIG. 2
M
E
476
EFFICIENCY
IN THE ELEMENTARY THEORY OF EQUATIONS
[September
two rightcircularcylindersintersectingat rightangles.
The actual models were constructed by D. A. Eberle of the Psychology
Workshop at Purdue University.The various slices were cut froma piece of
white pine 1/2" thick and 7" wide. Thus the cylinder GLEF is composed of
seven slices, each with a diameter of 7". The seven slices for the cone, being
firstcut as stepwise increasing cylindrical discs with easily calculated radii,
were placed all togetheron a mandrel passing through a 3/16" hole through
their centers and were then shaped down on a lathe, a procedure found to be
especially necessaryforthe square cross-sectionsin the problemof the intersecting cylinders.The lever itselfis a piece of steel 9" by 1/2" by 1/32', placed so
that its 1/2" face is vertical. In each disc a thinslit was cut with a finehacksaw
fromedge to centerso that the disc could be slipped onto the lever.
SIMPLE
DEVICES
FOR EFFICIENCY IN THE ELEMENTARY
THEORY OF EQUATIONS
ofIllinois,Chicago
FLORADINKINES,University
who attemptsto teachhis studentsto find
The instructor
1. Introduction.
the "best" method forsolvingeach problemfrequentlyfindsthat the process of
producing several solutions, fromwhich the "best" is to be chosen, tends to
stimulate ingenuityand produce a habit of efficiencyin most students. Not infrequently,it is necessary to consider whether the author has indicated the
"best" procedurein the directionsgiven, or whetherthe ultimate aim-such as
findingall of the roots of a given equation-can be more readily attained by
disregardingthe author's directions involving some specific method-such as
testingfor rational roots. In such cases shall we disregard the directionsand
obtain the results more efficiently;shall we consider the problem satisfactorily
worked only when we have followed the author's directions; or shall we work
and then followthe author's instructions?The answer
the problem efficiently
probably depends upon how frequentlysuch directionsare given, and whether
or not there are enough problems remainingto teach the students the desired
standard techniques.
We shall consider here the "best" methods forworkingsome of the college
algebra problems ordinarilycontained in the chapter on theory of equations,
and the frequencywith which these methods can be applied to problems in
certain text books.*
Beforeenteringupon the main points of discussion the author would like to
say that no special methodwas used in selectingthe books referredto. Those at
hand were merelyused to decide whetheror not the devices to be given here
*
to by number.
Thesebooksare listedat theend of thisarticleand referred