GEOMETRY IN ARCHIMEDES.
The famous French geometer, Chasles, drew an instructive distinction
between the predominant features of the geometry of the two great
successors of Euclid, namely, Archimedes and Apollonius of Perga (the
"great geometer," and author of the classical treatise on Conics). The
works of these two men may, says Chasles, be regarded as the origin and
basis of two great inquiries which seem to share between them the domain
of geometry. Apollonius is concerned with the _Geometry of Forms and
Situations_, while in Archimedes we find the _Geometry of Measurements_,
dealing with the quadrature of curvilinear plane figures and with the
quadrature and cubature of curved surfaces, investigations which gave
birth to the calculus of the infinite conceived and brought to
perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.
In geometry Archimedes stands, as it were, on the shoulders of Eudoxus
in that he applied the method of exhaustion to new and more difficult
cases of quadrature and cubature. Further, in his use of the method he
introduced an interesting variation of the procedure as we know it from
Euclid. Euclid (and presumably Eudoxus also) only used _inscribed_
figures, "exhausting" the figure to be measured, and had to invert the
second half of the _reductio ad absurdum_ to enable approximation from
below (so to speak) to be applied in that case also. Archimedes, on the
other hand, approximates from above as well as from below; he approaches
the area or volume to be measured by taking closer and closer
_circumscribed_ figures, as well as inscribed, and thereby
_compressing_, as it were, the inscribed and circumscribed figure into
one, so that they ultimately coincide with one another and with the
figure to be measured. But he follows the cautious method to which the
Greeks always adhered; he never says that a given curve or surface is
the _limiting form_ of the inscribed or circumscribed figure; all that
he asserts is that we can approach the curve or surface _as nearly as we
please_.
The deductive form of proof by the method of exhaustion is apt to
obscure not only the way in which the results were arrived at but also
the real character of the procedure followed. What Archimedes actually
does in certain cases is to perform what are seen, when the analytical
equivalents are set down, to be real _integrations_; this remark applies
to his investigation of the areas of a parabolic segment and a spiral
respectively, the surface and volume respectively of a sphere and a
segment of a sphere, and the volume of any segments of the solids of
revolution of the second degree. The result is, as a rule, only obtained
after a long series of preliminary propositions, all of which are links
in a chain of argument elaborately forged for the one purpose. The
method suggests the tactics of some master of strategy who foresees
everything, eliminates everything not immediately conducive to the
execution of his plan, masters every position in its order, and then
suddenly (when the very elaboration of the scheme has almost obscured,
in the mind of the onlooker, its ultimate object) strikes the final
blow. Thus we read in Archimedes proposition after proposition the
bearing of which is not immediately obvious but which we find infallibly
used later on; and we are led on by such easy stages that the difficulty
of the original problem, as presented at the outset, is scarcely
appreciated. As Plutarch says, "It is not possible to find in geometry
more difficult and troublesome questions, or more simple and lucid
explanations". But it is decidedly a rhetorical exaggeration when
Plutarch goes on to say that we are deceived by the easiness of the
successive steps into the belief that any one could have discovered them
for himself. On the contrary, the studied simplicity and the perfect
finish of the treatises involve at the same time an element of mystery.
Although each step depends upon the preceding ones, we are left in the
dark as to how they were suggested to Archimedes. There is, in fact,
much truth in a remark of Wallis to the effect that he seems "as it were
of set purpose to have covered up the traces of his investigation as if
he had grudged posterity the secret of his method of inquiry while he
wished to extort from them assent to his results".
A partial exception is now furnished by the _Method_; for here we have
(as it were) a lifting of the veil and a glimpse of the interior of
Archimedes's workshop. He tells us how he discovered certain theorems in
quadrature and cubature, and he is at the same time careful to insist on
the difference between (1) the means which may serve to suggest the
truth of theorems, although not furnishing scientific proofs of them,
and (2) the rigorous demonstrations of them by approved geometrical
methods which must follow before they can be finally accepted as
established.
Writing to Eratosthenes he says: "Seeing in you, as I say, an earnest
student, a man of considerable eminence in philosophy and an admirer of
mathematical inquiry when it comes your way, I have thought fit to write
out for you and explain in detail in the same book the peculiarity of a
certain method, which, when you see it, will put you in possession of a
means whereby you can investigate some of the problems of mathematics by
mechanics. This procedure is, I am persuaded, no less useful for the
proofs of the actual theorems as well. For certain things which first
became clear to me by a mechanical method had afterwards to be
demonstrated by geometry, because their investigation by the said method
did not furnish an actual demonstration. But it is of course easier,
when we have previously acquired by the method some knowledge of the
questions, to supply the proof than it is to find the proof without any
previous knowledge. This is a reason why, in the case of the theorems
the proof of which Eudoxus was the first to discover, namely, that the
cone is a third part of the cylinder, and the pyramid a third part of
the prism, having the same base and equal height, we should give no
small share of the credit to Democritus, who was the first to assert
this truth with regard to the said figures, though he did not prove it.
I am myself in the position of having made the discovery of the theorem
now to be published in the same way as I made my earlier discoveries;
and I thought it desirable now to write out and publish the method,
partly because I have already spoken of it and I do not want to be
thought to have uttered vain words, but partly also because I am
persuaded that it will be of no little service to mathematics; for I
apprehend that some, either of my contemporaries or of my successors,
will, by means of the method when once established, be able to discover
other theorems in addition, which have not occurred to me.
"First then I will set out the very first theorem which became known to
me by means of mechanics, namely, that _Any segment of a section of a
right-angled cone_ [_i.e. a parabola_] _is four-thirds of the triangle
which has the same base and equal height_; and after this I will give
each of the other theorems investigated by the same method. Then, at the
end of the book, I will give the geometrical proofs of the
propositions."
The following description will, I hope, give an idea of the general
features of the mechanical method employed by Archimedes. Suppose that X
is the plane or solid figure the area or content of which is to be
found. The method in the simplest case is to weigh infinitesimal
elements of X against the corresponding elements of another figure, B
say, being such a figure that its area or content and the position of
its centre of gravity are already known. The diameter or axis of the
figure X being drawn, the infinitesimal elements taken are parallel
sections of X in general, but not always, at right angles to the axis or
diameter, so that the centres of gravity of all the sections lie at one
point or other of the axis or diameter and their weights can therefore
be taken as acting at the several points of the diameter or axis. In the
case of a plane figure the infinitesimal sections are spoken of as
parallel _straight lines_ and in the case of a solid figure as parallel
_planes_, and the aggregate of the infinite number of sections is said
to _make up_ the whole figure X. (Although the sections are so spoken of
as straight lines or planes, they are really indefinitely narrow plane
strips or indefinitely thin laminae respectively.) The diameter or axis
is produced in the direction away from the figure to be measured, and
the diameter or axis as produced is imagined to be the bar or lever of a
balance. The object is now to apply all the separate elements of X at
_one point_ on the lever, while the corresponding elements of the known
figure B operate at different points, namely, _where they actually are_
in the first instance. Archimedes contrives, therefore, to move the
elements of X away from their original position and to concentrate them
at one point on the lever, such that each of the elements balances,
about the point of suspension of the lever, the corresponding element of
B acting at its centre of gravity. The elements of X and B respectively
balance about the point of suspension in accordance with the property of
the lever that the weights are inversely proportional to the distances
from the fulcrum or point of suspension. Now the centre of gravity of B
as a whole is known, and it may then be supposed to act as one mass at
its centre of gravity. (Archimedes assumes as known that the sum of the
"moments," as we call them, of all the elements of the figure B, acting
severally at the points where they actually are, is equal to the moment
of the whole figure applied as one mass at one point, its centre of
gravity.) Moreover all the elements of X are concentrated at the one
fixed point on the bar or lever. If this fixed point is H, and G is the
centre of gravity of the figure B, while C is the point of suspension,
X : B = CG : CH.
Thus the area or content of X is found.
Conversely, the method can be used to find the centre of gravity of X
when its area or volume is known beforehand. In this case the elements
of X, and X itself, have to be applied where they are, and the elements
of the known figure or figures have to be applied at the one fixed point
H on the other side of C, and since X, B and CH are known, the
proportion
B : X = CG : CH
determines CG, where G is the centre of gravity of X.
The mechanical method is used for finding (1) the area of any parabolic
segment, (2) the volume of a sphere and a spheroid, (3) the volume of a
segment of a sphere and the volume of a right segment of each of the
three conicoids of revolution, (4) the centre of gravity (a) of a
hemisphere, (b) of any segment of a sphere, (c) of any right segment of
a spheroid and a paraboloid of revolution, and (d) of a half-cylinder,
or, in other words, of a semicircle.
Archimedes then proceeds to find the volumes of two solid figures, which
are the special subject of the treatise. The solids arise as follows:--
(1) Given a cylinder inscribed in a rectangular parallelepiped on a
square base in such a way that the two bases of the cylinder are
circles inscribed in the opposite square faces, suppose a plane drawn
through one side of the square containing one base of the cylinder and
through the parallel diameter of the opposite base of the cylinder. The
plane cuts off a solid with a surface resembling that of a horse's hoof.
Archimedes proves that the volume of the solid so cut off is one sixth
part of the volume of the parallelepiped.
(2) A cylinder is inscribed in a cube in such a way that the bases of
the cylinder are circles inscribed in two opposite square faces. Another
cylinder is inscribed which is similarly related to another pair of
opposite faces. The two cylinders include between them a solid with all
its angles rounded off; and Archimedes proves that the volume of this
solid is two-thirds of that of the cube.
Having proved these facts by the mechanical method, Archimedes concluded
the treatise with a rigorous geometrical proof of both propositions by
the method of exhaustion. The MS. is unfortunately somewhat mutilated at
the end, so that a certain amount of restoration is necessary.
I shall now attempt to give a short account of the other treatises of
Archimedes in the order in which they appear in the editions. The first
is--
_On the Sphere and Cylinder._
Book I. begins with a preface addressed to Dositheus (a pupil of Conon),
which reminds him that on a former occasion he had communicated to him
the treatise proving that any segment of a "section of a right-angled
cone" (i.e. a parabola) is four-thirds of the triangle with the same
base and height, and adds that he is now sending the proofs of certain
theorems which he has since discovered, and which seem to him to be
worthy of comparison with Eudoxus's propositions about the volumes of a
pyramid and a cone. The theorems are (1) that the surface of a sphere
is equal to four times its greatest circle (i.e. what we call a "great
circle" of the sphere); (2) that the surface of any segment of a sphere
is equal to a circle with radius equal to the straight line drawn from
the vertex of the segment to a point on the circle which is the base of
the segment; (3) that, if we have a cylinder circumscribed to a sphere
and with height equal to the diameter, then (a) the volume of the
cylinder is 1-1/2 times that of the sphere and (b) the surface of the
cylinder, including its bases, is 1-1/2 times the surface of the sphere.
Next come a few definitions, followed by certain _Assumptions_, two of
which are well known, namely:--
1. _Of all lines which have the same extremities the straight line is
the least_ (this has been made the basis of an alternative definition of
a straight line).
2. _Of unequal lines, unequal surfaces and unequal solids the greater
exceeds the less by such a magnitude as, when (continually) added to
itself, can be made to exceed any assigned magnitude among those which
are comparable_ [_with it and_] _with one another_ (i.e. are of the same
kind). This is the _Postulate of Archimedes_.
He also assumes that, of pairs of lines (including broken lines) and
pairs of surfaces, concave in the same direction and bounded by the same
extremities, the outer is greater than the inner. These assumptions are
fundamental to his investigation, which proceeds throughout by means of
figures inscribed and circumscribed to the curved lines or surfaces that
have to be measured.
After some preliminary propositions Archimedes finds (Props. 13, 14) the
area of the surfaces (1) of a right cylinder, (2) of a right cone. Then,
after quoting certain Euclidean propositions about cones and cylinders,
he passes to the main business of the book, the measurement of the
volume and surface of a sphere and a segment of a sphere. By
circumscribing and inscribing to a great circle a regular polygon of an
even number of sides and making it revolve about a diameter connecting
two opposite angular points he obtains solids of revolution greater and
less respectively than the sphere. In a series of propositions he finds
expressions for (a) the surfaces, (b) the volumes, of the figures so
inscribed and circumscribed to the sphere. Next he proves (Prop. 32)
that, if the inscribed and circumscribed polygons which, by their
revolution, generate the figures are similar, the surfaces of the
figures are in the duplicate ratio, and their volumes in the triplicate
ratio, of their sides. Then he proves that the surfaces and volumes of
the inscribed and circumscribed figures respectively are less and
greater than the surface and volume respectively to which the main
propositions declare the surface and volume of the sphere to be equal
(Props. 25, 27, 30, 31 Cor.). He has now all the material for applying
the method of exhaustion and so proves the main propositions about the
surface and volume of the sphere. The rest of the book applies the same
procedure to a segment of the sphere. Surfaces of revolution are
inscribed and circumscribed to a segment less than a hemisphere, and the
theorem about the surface of the segment is finally proved in Prop. 42.
Prop. 43 deduces the surface of a segment greater than a hemisphere.
Prop. 44 gives the volume of the sector of the sphere which includes any
segment.
Book II begins with the problem of finding a sphere equal in volume to a
given cone or cylinder; this requires the solution of the problem of the
two mean proportionals, which is accordingly assumed. Prop. 2 deduces,
by means of 1., 44, an expression for the volume of a segment of a
sphere, and Props. 3, 4 solve the important problems of cutting a given
sphere by a plane so that (a) the surfaces, (b) the volumes, of the
segments may have to one another a given ratio. The solution of the
second problem (Prop. 4) is difficult. Archimedes reduces it to the
problem of dividing a straight line AB into two parts at a point M such
that
MB : (a given length) = (a given area) : AM^2.
The solution of this problem with a determination of the limits of
possibility are given in a fragment by Archimedes, discovered and
preserved for us by Eutocius in his commentary on the book; they are
effected by means of the points of intersection of two conics, a
parabola and a rectangular hyperbola. Three problems of construction
follow, the first two of which are to construct a segment of a sphere
similar to one given segment, and having (a) its volume, (b) its
surface, equal to that of another given segment of a sphere. The last
two propositions are interesting. Prop. 8 proves that, if V, V' be the
volumes, and S, S' the surfaces, of two segments into which a sphere is
divided by a plane, V and S belonging to the greater segment, then
S^2 : S'^2 > V : V' > S^(3/2) : S'^(3/2).
Prop. 9 proves that, of all segments of spheres which have equal
surfaces, the hemisphere is the greatest in volume.
_The Measurement of a Circle._
This treatise, in the form in which it has come down to us, contains
only three propositions; the second, being an easy deduction from Props.
1 and 3, is out of place in so far as it uses the result of Prop. 3.
In Prop. 1 Archimedes inscribes and circumscribes to a circle a series
of successive regular polygons, beginning with a square, and continually
doubling the number of sides; he then proves in the orthodox manner by
the method of exhaustion that the area of the circle is equal to that of
a right-angled triangle, in which the perpendicular is equal to the
radius, and the base equal to the circumference, of the circle. Prop. 3
is the famous proposition in which Archimedes finds by sheer calculation
upper and lower arithmetical limits to the ratio of the circumference
of a circle to its diameter, or what we call [pi]; the result obtained
is 3-1/7> [pi] > 3-10/71. Archimedes inscribes and circumscribes
successive regular polygons, beginning with hexagons, and doubling the
number of sides continually, until he arrives at inscribed and
circumscribed regular polygons with 96 sides; seeing then that the
length of the circumference of the circle is intermediate between the
perimeters of the two polygons, he calculates the two perimeters in
terms of the diameter of the circle. His calculation is based on two
close approximations (an upper and a lower) to the value of [root]3,
that being the cotangent of the angle of 30 deg., from which he begins
to work. He assumes as known that 265/153 < [root]3 < 1351/780. In the
text, as we have it, only the results of the steps in the calculation
are given, but they involve the finding of approximations to the square
roots of several large numbers: thus 1172-1/8 is given as the
approximate value of [root](1373943-33/64), 3013-3/4 as that of
[root](9082321) and 1838-9/11 as that of [root](3380929). In this way
Archimedes arrives at 14688/(4673-1/2) as the ratio of the perimeter of
the circumscribed polygon of 96 sides to the diameter of the circle;
this is the figure which he rounds up into 3-1/7. The corresponding
figure for the inscribed polygon is 6336/(2017-1/4), which, he says, is
> 3-10/71. This example shows how little the Greeks were embarrassed in
arithmetical calculations by their alphabetical system of numerals.
_On Conoids and Spheroids._
The preface addressed to Dositheus shows, as we may also infer from
internal evidence, that the whole of this book also was original.
Archimedes first explains what his conoids and spheroids are, and then,
after each description, states the main results which it is the aim of
the treatise to prove. The conoids are two. The first is the
_right-angled conoid_, a name adapted from the old name ("section of a
right-angled cone") for a parabola; this conoid is therefore a
paraboloid of revolution. The second is the _obtuse-angled conoid_,
which is a hyperboloid of revolution described by the revolution of a
hyperbola (a "section of an obtuse-angled cone") about its transverse
axis. The spheroids are two, being the solids of revolution described by
the revolution of an ellipse (a "section of an acute-angled cone") about
(1) its major axis and (2) its minor axis; the first is called the
"oblong" (or oblate) spheroid, the second the "flat" (or prolate)
spheroid. As the volumes of oblique segments of conoids and spheroids
are afterwards found in terms of the volume of the conical figure with
the base of the segment as base and the vertex of the segment as vertex,
and as the said base is thus an elliptic section of an oblique circular
cone, Archimedes calls the conical figure with an elliptic base a
"segment of a cone" as distinct from a "cone".
As usual, a series of preliminary propositions is required. Archimedes
first sums, in geometrical form, certain series, including the
arithmetical progression, a, 2a, 3a, ... na, and the series formed by
the squares of these terms (in other words the series 1^2, 2^2, 3^2, ...
n^2); these summations are required for the final addition of an
indefinite number of elements of each figure, which amounts to an
_integration_. Next come two properties of conics (Prop. 3), then the
determination by the method of exhaustion of the area of an ellipse
(Prop. 4). Three propositions follow, the first two of which (Props. 7,
8) show that the conical figure above referred to is really a segment of
an oblique _circular_ cone; this is done by actually finding the
circular sections. Prop. 9 gives a similar proof that each elliptic
section of a conoid or spheroid is a section of a certain oblique
_circular_ cylinder (with axis parallel to the axis of the segment of
the conoid or spheroid cut off by the said elliptic section). Props.
11-18 show the nature of the various sections which cut off segments of
each conoid and spheroid and which are circles or ellipses according as
the section is perpendicular or obliquely inclined to the axis of the
solid; they include also certain properties of tangent planes, etc.
The real business of the treatise begins with Props. 19, 20; here it is
shown how, by drawing many plane sections equidistant from one another
and all parallel to the base of the segment of the solid, and describing
cylinders (in general oblique) through each plane section with
generators parallel to the axis of the segment and terminated by the
contiguous sections on either side, we can make figures circumscribed
and inscribed to the segment, made up of segments of cylinders with
parallel faces and presenting the appearance of the steps of a
staircase. Adding the elements of the inscribed and circumscribed
figures respectively and using the method of exhaustion, Archimedes
finds the volumes of the respective segments of the solids in the
approved manner (Props. 21, 22 for the paraboloid, Props. 25, 26 for the
hyperboloid, and Props. 27-30 for the spheroids). The results are stated
in this form: (1) Any segment of a paraboloid of revolution is half as
large again as the cone or segment of a cone which has the same base and
axis; (2) Any segment of a hyperboloid of revolution or of a spheroid is
to the cone or segment of a cone with the same base and axis in the
ratio of AD + 3CA to AD + 2CA in the case of the hyperboloid, and of 3CA
- AD to 2CA - AD in the case of the spheroid, where C is the centre, A
the vertex of the segment, and AD the axis of the segment (supposed in
the case of the spheroid to be not greater than half the spheroid).
_On Spirals._
The preface addressed to Dositheus is of some length and contains,
first, a tribute to the memory of Conon, and next a summary of the
theorems about the sphere and the conoids and spheroids included in the
above two treatises. Archimedes then passes to the spiral which, he
says, presents another sort of problem, having nothing in common with
the foregoing. After a definition of the spiral he enunciates the main
propositions about it which are to be proved in the treatise. The spiral
(now known as the Spiral of Archimedes) is defined as the locus of a
point starting from a given point (called the "origin") on a given
straight line and moving along the straight line at uniform speed, while
the line itself revolves at uniform speed about the origin as a fixed
point. Props. 1-11 are preliminary, the last two amounting to the
summation of certain series required for the final addition of an
indefinite number of element-areas, which again amounts to integration,
in order to find the area of the figure cut off between any portion of
the curve and the two radii vectores drawn to its extremities. Props.
13-20 are interesting and difficult propositions establishing the
properties of tangents to the spiral. Props. 21-23 show how to inscribe
and circumscribe to any portion of the spiral figures consisting of a
multitude of elements which are narrow sectors of circles with the
origin as centre; the area of the spiral is intermediate between the
areas of the inscribed and circumscribed figures, and by the usual
method of exhaustion Archimedes finds the areas required. Prop. 24 gives
the area of the first complete turn of the spiral (= 1/3[pi](2[pi]a)^2,
where the spiral is r = a[theta]), and of any portion of it up to OP
where P is any point on the first turn. Props. 25, 26 deal similarly
with the second turn of the spiral and with the area subtended by any
arc (not being greater than a complete turn) on any turn. Prop. 27
proves the interesting property that, if R1 be the area of the first
turn of the spiral bounded by the initial line, R2 the area of the ring
added by the second complete turn, R3 the area of the ring added by the
third turn, and so on, then R3 = 2R2, R4 = 3R2, R5 = 4R2, and so on to
R_n = (n - 1)R2, while R2, = 6R1.
_Quadrature of the Parabola._
The title of this work seems originally to have been _On the Section of
a Right-angled Cone_ and to have been changed after the time of
Apollonius, who was the first to call a parabola by that name. The
preface addressed to Dositheus was evidently the first communication
from Archimedes to him after the death of Conon. It begins with a
feeling allusion to his lost friend, to whom the treatise was originally
to have been sent. It is in this preface that Archimedes alludes to the
lemma used by earlier geometers as the basis of the method of exhaustion
(the Postulate of Archimedes, or the theorem of Euclid X., 1). He
mentions as having been proved by means of it (1) the theorems that the
areas of circles are to one another in the duplicate ratio of their
diameters, and that the volumes of spheres are in the triplicate ratio
of their diameters, and (2) the propositions proved by Eudoxus about the
volumes of a cone and a pyramid. No one, he says, so far as he is aware,
has yet tried to square the segment bounded by a straight line and a
section of a right-angled cone (a parabola); but he has succeeded in
proving, by means of the same lemma, that the parabolic segment is equal
to four-thirds of the triangle on the same base and of equal height, and
he sends the proofs, first as "investigated" by means of mechanics and
secondly as "demonstrated" by geometry. The phraseology shows that here,
as in the _Method_, Archimedes regarded the mechanical investigation as
furnishing evidence rather than proof of the truth of the proposition,
pure geometry alone furnishing the absolute proof required.
The mechanical proof with the necessary preliminary propositions about
the parabola (some of which are merely quoted, while two, evidently
original, are proved, Props. 4, 5) extends down to Prop. 17; the
geometrical proof with other auxiliary propositions completes the book
(Props. 18-24). The mechanical proof recalls that of the _Method_ in
some respects, but is more elaborate in that the elements of the area of
the parabola to be measured are not straight lines but narrow strips.
The figures inscribed and circumscribed to the segment are made up of
such narrow strips and have a saw-like edge; all the elements are
trapezia except two, which are triangles, one in each figure. Each
trapezium (or triangle) is weighed where it is against another area hung
at a fixed point of an assumed lever; thus the whole of the inscribed
and circumscribed figures respectively are weighed against the sum of an
indefinite number of areas all suspended from one point on the lever.
The result is obtained by a real _integration_, confirmed as usual by a
proof by the method of exhaustion.
The geometrical proof proceeds thus. Drawing in the segment the
inscribed triangle with the same base and height as the segment,
Archimedes next inscribes triangles in precisely the same way in each of
the segments left over, and proves that the sum of the two new triangles
is 1/4 of the original inscribed triangle. Again, drawing triangles
inscribed in the same way in the four segments left over, he proves that
their sum is 1/4 of the sum of the preceding pair of triangles and
therefore (1/4)^2 of the original inscribed triangle. Proceeding thus,
we have a series of areas exhausting the parabolic segment. Their sum,
if we denote the first inscribed triangle by [Delta], is
[Delta]{1 + 1/4 + (1/4)^2 + (1/4)^3 + . . . .}
Archimedes proves geometrically in Prop. 23 that the sum of this
infinite series is 4/3[Delta], and then confirms by _reductio ad
absurdum_ the equality of the area of the parabolic segment to this
area.
CHAPTER V.
THE SANDRECKONER.
The _Sandreckoner_ deserves a place by itself. It is not mathematically
very important; but it is an arithmetical curiosity which illustrates
the versatility and genius of Archimedes, and it contains some precious
details of the history of Greek astronomy which, coming from such a
source and at first hand, possess unique authority. We will begin with
the astronomical data. They are contained in the preface addressed to
King Gelon of Syracuse, which begins as follows:--
"There are some, King Gelon, who think that the number of the sand is
infinite in multitude; and I mean by the sand not only that which exists
about Syracuse and the rest of Sicily but also that which is found in
every region whether inhabited or uninhabited. Again, there are some
who, without regarding it as infinite, yet think that no number has been
named which is great enough to exceed its multitude. And it is clear
that they who hold this view, if they imagined a mass made up of sand in
other respects as large as the mass of the earth, including in it all
the seas and the hollows of the earth filled up to a height equal to
that of the highest of the mountains, would be many times further still
from recognising that any number could be expressed which exceeded the
multitude of the sand so taken. But I will try to show you, by means of
geometrical proofs which you will be able to follow, that, of the
numbers named by me and given in the work which I sent to Zeuxippus,
some exceed not only the number of the mass of sand equal in size to
the earth filled up in the way described, but also that of a mass equal
in size to the universe.
"Now you are aware that 'universe' is the name given by most astronomers
to the sphere the centre of which is the centre of the earth, while the
radius is equal to the straight line between the centre of the sun and
the centre of the earth. This is the common account, as you have heard
from astronomers. But Aristarchus of Samos brought out a book consisting
of some hypotheses, in which the premises lead to the conclusion that
the universe is many times greater than that now so called. His
hypotheses are that the fixed stars and the sun remain unmoved, that the
earth revolves about the sun in the circumference of a circle, the sun
lying in the centre of the orbit, and that the sphere of the fixed
stars, situated about the same centre as the sun, is so great that the
circle in which he supposes the earth to revolve bears such a ratio to
the distance of the fixed stars as the centre of the sphere bears to its
surface."
Here then is absolute and practically contemporary evidence that the
Greeks, in the person of Aristarchus of Samos (about 310-230 B.C.), had
anticipated Copernicus.
By the last words quoted Aristarchus only meant to say that the size of
the earth is negligible in comparison with the immensity of the
universe. This, however, does not suit Archimedes's purpose, because he
has to assume a definite size, however large, for the universe.
Consequently he takes a liberty with Aristarchus. He says that the
centre (a mathematical point) can have no ratio whatever to the surface
of the sphere, and that we must therefore take Aristarchus to mean that
the size of the earth is to that of the so-called "universe" as the size
of the so-called "universe" is to that of the real universe in the new
sense.
Next, he has to assume certain dimensions for the earth, the moon and
the sun, and to estimate the angle subtended at the centre of the earth
by the sun's diameter; and in each case he has to exaggerate the
probable figures so as to be on the safe side. While therefore (he says)
some have tried to prove that the perimeter of the earth is 300,000
stadia (Eratosthenes, his contemporary, made it 252,000 stadia, say
24,662 miles, giving a diameter of about 7,850 miles), he will assume it
to be ten times as great or 3,000,000 stadia. The diameter of the earth,
he continues, is greater than that of the moon and that of the sun is
greater than that of the earth. Of the diameter of the sun he observes
that Eudoxus had declared it to be nine times that of the moon, and his
own father, Phidias, had made it twelve times, while Aristarchus had
tried to prove that the diameter of the sun is greater than eighteen
times but less than twenty times the diameter of the moon (this was in
the treatise of Aristarchus _On the Sizes and Distances of the Sun and
Moon_, which is still extant, and is an admirable piece of geometry,
proving rigorously, on the basis of certain assumptions, the result
stated). Archimedes again intends to be on the safe side, so he takes
the diameter of the sun to be thirty times that of the moon and not
greater. Lastly, he says that Aristarchus discovered that the diameter
of the sun appeared to be about 1/720th part of the zodiac circle, i.e.
to subtend an angle of about half a degree; and he describes a simple
instrument by which he himself found that the angle subtended by the
diameter of the sun at the time when it had just risen was less than
1/164th part and greater than 1/200th part of a right angle. Taking this
as the size of the angle subtended at the eye of the observer on the
surface of the earth, he works out, by an interesting geometrical
proposition, the size of the angle subtended at the centre of the earth,
which he finds to be > 1/203rd part of a right angle. Consequently the
diameter of the sun is greater than the side of a regular polygon of 812
sides inscribed in a great circle of the so-called "universe," and _a
fortiori_ greater than the side of a regular _chiliagon_ (polygon of
1000 sides) inscribed in that circle.
On these assumptions, and seeing that the perimeter of a regular
chiliagon (as of any other regular polygon of more than six sides)
inscribed in a circle is more than 3 times the length of the diameter of
the circle, it easily follows that, while the diameter of the earth is
less than 1,000,000 stadia, the diameter of the so-called "universe" is
less than 10,000 times the diameter of the earth, and therefore less
than 10,000,000,000 stadia.
Lastly, Archimedes assumes that a quantity of sand not greater than a
poppy-seed contains not more than 10,000 grains, and that the diameter
of a poppy-seed is not less than 1/40th of a _dactylus_ (while a stadium
is less than 10,000 _dactyli_).
Archimedes is now ready to work out his calculation, but for the
inadequacy of the alphabetic system of numerals to express such large
numbers as are required. He, therefore, develops his remarkable
terminology for expressing large numbers.
The Greek has names for all numbers up to a myriad (10,000); there was,
therefore, no difficulty in expressing with the ordinary numerals all
numbers up to a myriad myriads (100,000,000). Let us, says Archimedes,
call all these numbers numbers of the _first order_. Let the _second
order_ of numbers begin with 100,000,000, and end with 100,000,000^2.
Let 100,000,000^2 be the first number of the _third order_, and let this
extend to 100,000,000^3; and so on, to the _myriad-myriadth_ order,
beginning with 100,000,000^(99,999,999) and ending with
100,000,000^(100,000,000), which for brevity we will call P. Let all the
numbers of all the orders up to P form the _first period_, and let the
_first order_ of the _second period_ begin with P and end with
100,000,000 P; let the _second order_ begin with this, the _third order_
with 100,000,000^2 P, and so on up to the _100,000,000th order_ of the
_second period_, ending with 1,000,000,000^(100,000,000) P or P^2. The
_first order_ of the _third period_ begins with P^2, and the _orders_
proceed as before. Continuing the series of _periods_ and _orders_ of
each _period_, we finally arrive at the _100,000,000th period_ ending
with P^(100,000,000). The prodigious extent of this scheme is seen when
it is considered that the last number of the first period would now be
represented by 1 followed by 800,000,000 ciphers, while the last number
of the 100,000,000th period would require 100,000,000 times as many
ciphers, i.e. 80,000 million million ciphers.
As a matter of fact, Archimedes does not need, in order to express the
"number of the sand," to go beyond the _eighth order_ of the _first
period_. The orders of the _first period_ begin respectively with 1,
10^8, 10^16, 10^24, ... (10^8)^(99,999,999); and we can express all the
numbers required in powers of 10.
Since the diameter of a poppy-seed is not less than 1/40th of a
dactylus, and spheres are to one another in the triplicate ratio of
their diameters, a sphere of diameter 1 _dactylus_ is not greater than
64,000 poppy-seeds, and, therefore, contains not more than 64,000 X
10,000 grains of sand, and _a fortiori_ not more than 1,000,000,000, or
10^9 grains of sand. Archimedes multiplies the diameter of the sphere
continually by 100, and states the corresponding number of grains of
sand. A sphere of diameter 10,000 _dactyli_ and _a fortiori_ of one
stadium contains less than 10^21 grains; and proceeding in this way to
spheres of diameter 100 stadia, 10,000 stadia and so on, he arrives at
the number of grains of sand in a sphere of diameter 10,000,000,000
stadia, which is the size of the so-called universe; the corresponding
number of grains of sand is 10^51. The diameter of the real universe
being 10,000 times that of the so-called universe, the final number of
grains of sand in the real universe is found to be 10^63, which in
Archimedes's terminology is a myriad-myriad units of the _eighth order_
of numbers.
CHAPTER VI.
MECHANICS.
It is said that Archytas was the first to treat mechanics in a
systematic way by the aid of mathematical principles; but no trace
survives of any such work by him. In practical mechanics he is said to
have constructed a mechanical dove which would fly, and also a rattle to
amuse children and "keep them from breaking things about the house" (so
says Aristotle, adding "for it is impossible for children to keep
still").
In the Aristotelian _Mechanica_ we find a remark on the marvel of a
great weight being moved by a small force, and the problems discussed
bring in the lever in various forms as a means of doing this. We are
told also that practically all movements in mechanics reduce to the
lever and the principle of the lever (that the weight and the force are
in inverse proportion to the distances from the point of suspension or
fulcrum of the points at which they act, it being assumed that they act
in directions perpendicular to the lever). But the lever is merely
"referred to the circle"; the force which acts at the greater distance
from the fulcrum is said to move a weight more easily because it
describes a greater circle.
There is, therefore, no proof here. It was reserved for Archimedes to
prove the property of the lever or balance mathematically, on the basis
of certain postulates precisely formulated and making no large demand on
the faith of the learner. The treatise _On Plane Equilibriums_ in two
books is, as the title implies, a work on statics only; and, after the
principle of the lever or balance has been established in Props. 6, 7 of
Book I., the rest of the treatise is devoted to finding the centre of
gravity of certain figures. There is no dynamics in the work and
therefore no room for the parallelogram of velocities, which is given
with a fairly adequate proof in the Aristotelian _Mechanica_.
Archimedes's postulates include assumptions to the following effect: (1)
Equal weights at equal distances are in equilibrium, and equal weights
at unequal distances are not in equilibrium, but the system in that case
"inclines towards the weight which is at the greater distance," in other
words, the action of the weight which is at the greater distance
produces motion in the direction in which it acts; (2) and (3) If when
weights are in equilibrium something is added to or subtracted from one
of the weights, the system will "incline" towards the weight which is
added to or the weight from which nothing is taken respectively; (4) and
(5) If equal and similar figures be applied to one another so as to
coincide throughout, their centres of gravity also coincide; if figures
be unequal but similar, their centres of gravity are similarly situated
with regard to the figures.
The main proposition, that two magnitudes balance at distances
reciprocally proportional to the magnitudes, is proved first for
commensurable and then for incommensurable magnitudes. Preliminary
propositions have dealt with equal magnitudes disposed at equal
distances on a straight line and odd or even in number, and have shown
where the centre of gravity of the whole system lies. Take first the
case of commensurable magnitudes. If A, B be the weights acting at E, D
on the straight line ED respectively, and ED be divided at C so that A :
B = DC : CE, Archimedes has to prove that the system is in equilibrium
about C. He produces ED to K, so that DK = EC, and DE to L so that EL =
CD; LK is then a straight line bisected at C. Again, let H be taken on
LK such that LH = 2LE or 2CD, and it follows that the remainder HK = 2DK
or 2EC. Since A, B are commensurable, so are EC, CD. Let x be a common
measure of EC, CD. Take a weight w such that w is the same part of A
that x is of LH. It follows that w is the same part of B that x is of
HK. Archimedes now divides LH, HK into parts equal to x, and A B into
parts equal to w, and places the w's at the middle points of the x's
respectively. All the w's are then in equilibrium about C. But all the
w's acting at the several points along LH are equivalent to A acting as
a whole at the point E. Similarly the w's acting at the several points
on HK are equivalent to B acting at D. Therefore A, B placed at E, D
respectively balance about C.
Prop. 7 deduces by _reductio ad absurdum_ the same result in the case
where A, B are incommensurable. Prop. 8 shows how to find the centre of
gravity of the remainder of a magnitude when the centre of gravity of
the whole and of a part respectively are known. Props. 9-15 find the
centres of gravity of a parallelogram, a triangle and a
parallel-trapezium respectively.
Book II., in ten propositions, is entirely devoted to finding the centre
of gravity of a parabolic segment, an elegant but difficult piece of
geometrical work which is as usual confirmed by the method of
exhaustion.
CHAPTER VII.
HYDROSTATICS.
The science of hydrostatics is, even more than that of statics, the
original creation of Archimedes. In hydrostatics he seems to have had no
predecessors. Only one of the facts proved in his work _On Floating
Bodies_, in two books, is given with a sort of proof in Aristotle. This
is the proposition that the surface of a fluid at rest is that of a
sphere with its centre at the centre of the earth.
Archimedes founds his whole theory on two postulates, one of which comes
at the beginning and the other after Prop. 7 of Book I. Postulate 1 is
as follows:--
"Let us assume that a fluid has the property that, if its parts lie
evenly and are continuous, the part which is less compressed is expelled
by that which is more compressed, and each of its parts is compressed by
the fluid above it perpendicularly, unless the fluid is shut up in
something and compressed by something else."
Postulate 2 is: "Let us assume that any body which is borne upwards in
water is carried along the perpendicular [to the surface] which passes
through the centre of gravity of the body".
In Prop. 2 Archimedes proves that the surface of any fluid at rest is
the surface of a sphere the centre of which is the centre of the earth.
Props. 3-7 deal with the behaviour, when placed in fluids, of solids (1)
just as heavy as the fluid, (2) lighter than the fluid, (3) heavier
than the fluid. It is proved (Props. 5, 6) that, if the solid is lighter
than the fluid, it will not be completely immersed but only so far that
the weight of the solid will be equal to that of the fluid displaced,
and, if it be forcibly immersed, the solid will be driven upwards by a
force equal to the difference between the weight of the solid and that
of the fluid displaced. If the solid is heavier than the fluid, it will,
if placed in the fluid, descend to the bottom and, if weighed in the
fluid, the solid will be lighter than its true weight by the weight of
the fluid displaced (Prop. 7).
The last-mentioned theorem naturally connects itself with the story of
the crown made for Hieron. It was suspected that this was not wholly of
gold but contained an admixture of silver, and Hieron put to Archimedes
the problem of determining the proportions in which the metals were
mixed. It was the discovery of the solution of this problem when in the
bath that made Archimedes run home naked, shouting [Greek: eureka,
eureka]. One account of the solution makes Archimedes use the
proposition last quoted; but on the whole it seems more likely that the
actual discovery was made by a more elementary method described by
Vitruvius. Observing, as he is said to have done, that, if he stepped
into the bath when it was full, a volume of water was spilt equal to the
volume of his body, he thought of applying the same idea to the case of
the crown and measuring the volumes of water displaced respectively (1)
by the crown itself, (2) by the same weight of pure gold, and (3) by the
same weight of pure silver. This gives an easy means of solution.
Suppose that the weight of the crown is W, and that it contains weights
w1 and w2, of gold and silver respectively. Now experiment shows (1)
that the crown itself displaces a certain volume of water, V say, (2)
that a weight W of gold displaces a certain other volume of water, V1
say, and (3) that a weight W of silver displaces a volume V2.
From (2) it follows, by proportion, that a weight w1 of gold will
displace w1/W . V1 of the fluid, and from (3) it follows that a weight
w2 of silver displaces w2/W . V2 of the fluid.
Hence V = w1/W . V1 + w2/W . V2;
therefore WV = w1V1 + w2V2,
that is, (w1 + w2)V = w1V1 + w2V2,
so that w1/w2 = (V2 - V)/(V - V1),
which gives the required ratio of the weights of gold and silver
contained in the crown.
The last two propositions of Book I. investigate the case of a segment
of a sphere floating in a fluid when the base of the segment is (1)
entirely above and (2) entirely below the surface of the fluid; and it
is shown that the segment will in either case be in equilibrium in the
position in which the axis is vertical, the equilibrium being in the
first case stable.
Book II. is a geometrical _tour de force_. Here, by the methods of pure
geometry, Archimedes investigates the positions of rest and stability of
a right segment of a paraboloid of revolution floating with its base
upwards or downwards (but completely above or completely below the
surface) for a number of cases differing (1) according to the relation
between the length of the axis of the paraboloid and the principal
parameter of the generating parabola, and (2) according to the specific
gravity of the solid in relation to the fluid; where the position of
rest and stability is such that the axis of the solid is not vertical,
the angle at which it is inclined to the vertical is fully determined.
The idea of specific gravity appears all through, though this actual
term is not used. Archimedes speaks of the solid being lighter or
heavier than the fluid or equally heavy with it, or when a ratio has to
be expressed, he speaks of a solid the weight of which (for an equal
volume) has a certain ratio to that of the fluid.
BIBLIOGRAPHY.
The _editio princeps_ of the works of Archimedes with the commentaries
of Eutocius was brought out by Hervagius (Herwagen) at Basel in 1544. D.
Rivault (Paris, 1615) gave the enunciations in Greek and the proofs in
Latin somewhat retouched. The _Arenarius_ (_Sandreckoner_) and the
_Dimensio circuli_ with Eutocius's commentary were edited with Latin
translation and notes by Wallis in 1678 (Oxford). Torelli's monumental
edition (Oxford, 1792) of the Greek text of the complete works and of
the commentaries of Eutocius, with a new Latin translation, remained the
standard text until recent years; it is now superseded by the definitive
text with Latin translation of the complete works, Eutocius's
commentaries, the fragments, scholia, etc., edited by Heiberg in three
volumes (Teubner, Leipzig, first edition, 1880-1; second edition,
including the newly discovered _Method_, etc., 1910-15).
Of translations the following may be mentioned. The Aldine edition of
1558, 4to, contains the Latin translation by Commandinus of the
_Measurement of a Circle_, _On Spirals_, _Quadrature of the Parabola_,
_On Conoids and Spheroids_, _The Sandreckoner_. Isaac Barrow's version
was contained in _Opera Archimedis_, _Apollonii Pergoei conicorum
libri_, _Theodosii Sphoerica_, _methodo novo illustrata et demonstrata_
(London, 1675). The first French version of the works was by Peyrard in
two volumes (second edition, 1808). A valuable German translation, with
notes, by E. Nizze, was published at Stralsund in 1824. There is a
complete edition in modern notation by T. L. Heath (_The Works of
Archimedes_, Cambridge, 1897, supplemented by _The Method of
Archimedes_, Cambridge, 1912).
CHRONOLOGY.
(APPROXIMATE IN SOME CASES.)
B.C.
624-547 Thales
572-497 Pythagoras
500-428 Anaxagoras
470-400 / Hippocrates of Chios
\ Hippias of Elis
470-380 Democritus
460-385 Theodorus of Cyrene
430-360 Archytas of Taras (Tarentum)
427-347 Plato
415-369 Theaetetus
408-355 Eudoxus of Cnidos
/ Leon
fl. about 350 < Menaechmus
| Dinostratus
\ Theudius
fl. 300 Euclid
310-230 Aristarchus of Samos
287-212 Archimedes
284-203 Eratosthenes
265-190 Apollonius of Perga