GREEK GEOMETRY TO ARCHIMEDES.
In order to enable the reader to arrive at a correct understanding of
the place of Archimedes and of the significance of his work it is
necessary to pass in review the course of development of Greek geometry
from its first beginnings down to the time of Euclid and Archimedes.
Greek authors from Herodotus downwards agree in saying that geometry was
invented by the Egyptians and that it came into Greece from Egypt. One
account says:--
"Geometry is said by many to have been invented among the Egyptians, its
origin being due to the measurement of plots of land. This was necessary
there because of the rising of the Nile, which obliterated the
boundaries appertaining to separate owners. Nor is it marvellous that
the discovery of this and the other sciences should have arisen from
such an occasion, since everything which moves in the sense of
development will advance from the imperfect to the perfect. From
sense-perception to reasoning, and from reasoning to understanding, is a
natural transition. Just as among the Phoenicians, through commerce and
exchange, an accurate knowledge of numbers was originated, so also among
the Egyptians geometry was invented for the reason above stated.
"Thales first went to Egypt and thence introduced this study into
Greece."
But it is clear that the geometry of the Egyptians was almost entirely
practical and did not go beyond the requirements of the land-surveyor,
farmer or merchant. They did indeed know, as far back as 2000 B.C., that
in a triangle which has its sides proportional to 3, 4, 5 the angle
contained by the two smaller sides is a right angle, and they used such
a triangle as a practical means of drawing right angles. They had
formulae, more or less inaccurate, for certain measurements, e.g. for the
areas of certain triangles, parallel-trapezia, and circles. They had,
further, in their construction of pyramids, to use the notion of similar
right-angled triangles; they even had a name, _se-qet_, for the ratio of
the half of the side of the base to the height, that is, for what we
should call the _co-tangent_ of the angle of slope. But not a single
general theorem in geometry can be traced to the Egyptians. Their
knowledge that the triangle (3, 4, 5) is right angled is far from
implying any knowledge of the general proposition (Eucl. I., 47) known
by the name of Pythagoras. The science of geometry, in fact, remained to
be discovered; and this required the genius for pure speculation which
the Greeks possessed in the largest measure among all the nations of the
world.
Thales, who had travelled in Egypt and there learnt what the priests
could teach him on the subject, introduced geometry into Greece. Almost
the whole of Greek science and philosophy begins with Thales. His date
was about 624-547 B.C. First of the Ionian philosophers, and declared
one of the Seven Wise Men in 582-581, he shone in all fields, as
astronomer, mathematician, engineer, statesman and man of business. In
astronomy he predicted the solar eclipse of 28 May, 585, discovered the
inequality of the four astronomical seasons, and counselled the use of
the Little Bear instead of the Great Bear as a means of finding the
pole. In geometry the following theorems are attributed to him--and
their character shows how the Greeks had to begin at the very beginning
of the theory--(1) that a circle is bisected by any diameter (Eucl. I.,
Def. 17), (2) that the angles at the base of an isosceles triangle are
equal (Eucl. I., 5), (3) that, if two straight lines cut one another,
the vertically opposite angles are equal (Eucl. I., 15), (4) that, if
two triangles have two angles and one side respectively equal, the
triangles are equal in all respects (Eucl. I., 26). He is said (5) to
have been the first to inscribe a right-angled triangle in a circle:
which must mean that he was the first to discover that the angle in a
semicircle is a right angle. He also solved two problems in practical
geometry: (1) he showed how to measure the distance from the land of a
ship at sea (for this he is said to have used the proposition numbered
(4) above), and (2) he measured the heights of pyramids by means of the
shadow thrown on the ground (this implies the use of similar triangles
in the way that the Egyptians had used them in the construction of
pyramids).
After Thales come the Pythagoreans. We are told that the Pythagoreans
were the first to use the term [Greek: mathemata] (literally "subjects
of instruction") in the specialised sense of "mathematics"; they, too,
first advanced mathematics as a study pursued for its own sake and made
it a part of a liberal education. Pythagoras, son of Mnesarchus, was
born in Samos about 572 B.C., and died at a great age (75 or 80) at
Metapontum. His interests were as various as those of Thales; his
travels, all undertaken in pursuit of knowledge, were probably even more
extended. Like Thales, and perhaps at his suggestion, he visited Egypt
and studied there for a long period (22 years, some say).
It is difficult to disentangle from the body of Pythagorean doctrines
the portions which are due to Pythagoras himself because of the habit
which the members of the school had of attributing everything to the
Master ([Greek: autos epha], _ipse dixit_). In astronomy two things at
least may safely be attributed to him; he held that the earth is
spherical in shape, and he recognised that the sun, moon and planets
have an independent motion of their own in a direction contrary to that
of the daily rotation; he seems, however, to have adhered to the
geocentric view of the universe, and it was his successors who evolved
the theory that the earth does not remain at the centre but revolves,
like the other planets and the sun and moon, about the "central fire".
Perhaps his most remarkable discovery was the dependence of the musical
intervals on the lengths of vibrating strings, the proportion for the
octave being 2 : 1, for the fifth 3 : 2 and for the fourth 4 : 3. In
arithmetic he was the first to expound the theory of _means_ and of
proportion as applied to commensurable quantities. He laid the
foundation of the theory of numbers by considering the properties of
numbers as such, namely, prime numbers, odd and even numbers, etc. By
means of _figured_ numbers, square, oblong, triangular, etc.
(represented by dots arranged in the form of the various figures) he
showed the connexion between numbers and geometry. In view of all these
properties of numbers, we can easily understand how the Pythagoreans
came to "liken all things to numbers" and to find in the principles of
numbers the principles of all things ("all things are numbers").
We come now to Pythagoras's achievements in geometry. There is a story
that, when he came home from Egypt and tried to found a school at Samos,
he found the Samians indifferent, so that he had to take special
measures to ensure that his geometry might not perish with him. Going to
the gymnasium, he sought out a well-favoured youth who seemed likely to
suit his purpose, and was withal poor, and bribed him to learn geometry
by promising him sixpence for every proposition that he mastered. Very
soon the youth got fascinated by the subject for its own sake, and
Pythagoras rightly judged that he would gladly go on without the
sixpence. He hinted, therefore, that he himself was poor and must try
to earn his living instead of doing mathematics; whereupon the youth,
rather than give up the study, volunteered to pay sixpence to Pythagoras
for each proposition.
In geometry Pythagoras set himself to lay the foundations of the
subject, beginning with certain important definitions and investigating
the fundamental principles. Of propositions attributed to him the most
famous is, of course, the theorem that in a right-angled triangle the
square on the hypotenuse is equal to the sum of the squares on the sides
about the right angle (Eucl. I., 47); and, seeing that Greek tradition
universally credits him with the proof of this theorem, we prefer to
believe that tradition is right. This is to some extent confirmed by
another tradition that Pythagoras discovered a general formula for
finding two numbers such that the sum of their squares is a square
number. This depends on the theory of the _gnomon_, which at first had
an arithmetical signification corresponding to the geometrical use of it
in Euclid, Book II. A figure in the shape of a _gnomon_ put round two
sides of a square makes it into a larger square. Now consider the number
1 represented by a dot. Round this place three other dots so that the
four dots form a square (1 + 3 = 2^2). Round the four dots (on two
adjacent sides of the square) place five dots at regular and equal
distances, and we have another square (1 + 3 + 5 = 3^2); and so on. The
successive odd numbers 1, 3, 5 ... were called _gnomons_, and the
general formula is
1 + 3 + 5 + ... + (2n - 1) = n^2.
Add the next odd number, i.e. 2n + 1, and we have n^2 + (2n + 1) = (n +
1)^2. In order, then, to get two square numbers such that their sum is a
square we have only to see that 2n + 1 is a square. Suppose that 2n + 1
= m^2; then n = 1/2(m^2 - 1), and we have {1/2(m^2 - 1)}^2 + m^2 =
{1/2(m^2 + 1)}^2, where m is any odd number; and this is the general
formula attributed to Pythagoras.
Proclus also attributes to Pythagoras the theory of proportionals and
the construction of the five "cosmic figures," the five regular solids.
One of the said solids, the dodecahedron, has twelve pentagonal faces,
and the construction of a regular pentagon involves the cutting of a
straight line "in extreme and mean ratio" (Eucl. II., 11, and VI., 30),
which is a particular case of the method known as the _application of
areas_. How much of this was due to Pythagoras himself we do not know;
but the whole method was at all events fully worked out by the
Pythagoreans and proved one of the most powerful of geometrical methods.
The most elementary case appears in Euclid, I., 44, 45, where it is
shown how to apply to a given straight line as base a parallelogram
having a given angle (say a rectangle) and equal in area to any
rectilineal figure; this construction is the geometrical equivalent of
arithmetical _division_. The general case is that in which the
parallelogram, though _applied_ to the straight line, overlaps it or
falls short of it in such a way that the part of the parallelogram which
extends beyond, or falls short of, the parallelogram of the same angle
and breadth on the given straight line itself (exactly) as base is
similar to another given parallelogram (Eucl. VI., 28, 29). This is the
geometrical equivalent of the most general form of quadratic equation ax
[+-] mx^2 = C, so far as it has real roots; while the condition that the
roots may be real was also worked out (= Eucl. VI., 27). It is important
to note that this method of _application of areas_ was directly used by
Apollonius of Perga in formulating the fundamental properties of the
three conic sections, which properties correspond to the equations of
the conics in Cartesian co-ordinates; and the names given by Apollonius
(for the first time) to the respective conics are taken from the theory,
_parabola_ ([Greek: parabole]) meaning "application" (i.e. in this case
the parallelogram is applied to the straight line exactly), _hyperbola_
([Greek: hyperbole]), "exceeding" (i.e. in this case the parallelogram
exceeds or overlaps the straight line), _ellipse_ ([Greek: elleipsis]),
"falling short" (i.e. the parallelogram falls short of the straight
line).
Another problem solved by the Pythagoreans is that of drawing a
rectilineal figure equal in area to one given rectilineal figure and
similar to another. Plutarch mentions a doubt as to whether it was this
problem or the proposition of Euclid I., 47, on the strength of which
Pythagoras was said to have sacrificed an ox.
The main particular applications of the theorem of the square on the
hypotenuse (e.g. those in Euclid, Book II.) were also Pythagorean; the
construction of a square equal to a given rectangle (Eucl. II., 14) is
one of them and corresponds to the solution of the pure quadratic
equation x^2 = ab.
The Pythagoreans proved the theorem that the sum of the angles of any
triangle is equal to two right angles (Eucl. I., 32).
Speaking generally, we may say that the Pythagorean geometry covered the
bulk of the subject-matter of Books I., II., IV., and VI. of Euclid
(with the qualification, as regards Book VI., that the Pythagorean
theory of proportion applied only to commensurable magnitudes). Our
information about the origin of the propositions of Euclid, Book III.,
is not so complete; but it is certain that the most important of them
were well known to Hippocrates of Chios (who flourished in the second
half of the fifth century, and lived perhaps from about 470 to 400
B.C.), whence we conclude that the main propositions of Book III. were
also included in the Pythagorean geometry.
Lastly, the Pythagoreans discovered the existence of incommensurable
lines, or of _irrationals_. This was, doubtless, first discovered with
reference to the diagonal of a square which is incommensurable with the
side, being in the ratio to it of [root]2 to 1. The Pythagorean proof of
this particular case survives in Aristotle and in a proposition
interpolated in Euclid's Book X.; it is by a _reductio ad absurdum_
proving that, if the diagonal is commensurable with the side, the same
number must be both odd and even. This discovery of the incommensurable
was bound to cause geometers a great shock, because it showed that the
theory of proportion invented by Pythagoras was not of universal
application, and therefore that propositions proved by means of it were
not really established. Hence the stories that the discovery of the
irrational was for a time kept secret, and that the first person who
divulged it perished by shipwreck. The fatal flaw thus revealed in the
body of geometry was not removed till Eudoxus (408-355 B.C.) discovered
the great theory of proportion (expounded in Euclid's Book V.), which is
applicable to incommensurable as well as to commensurable magnitudes.
By the time of Hippocrates of Chios the scope of Greek geometry was no
longer even limited to the Elements; certain special problems were also
attacked which were beyond the power of the geometry of the straight
line and circle, and which were destined to play a great part in
determining the direction taken by Greek geometry in its highest
flights. The main problems in question were three: (1) the doubling of
the cube, (2) the trisection of any angle, (3) the squaring of the
circle; and from the time of Hippocrates onwards the investigation of
these problems proceeded _pari passu_ with the completion of the body of
the Elements.
Hippocrates himself is an example of the concurrent study of the two
departments. On the one hand, he was the first of the Greeks who is
known to have compiled a book of Elements. This book, we may be sure,
contained in particular the most important propositions about the circle
included in Euclid, Book III. But a much more important proposition is
attributed to Hippocrates; he is said to have been the first to prove
that circles are to one another as the squares on their diameters (=
Eucl. XII., 2), with the deduction that similar segments of circles are
to one another as the squares on their bases. These propositions were
used by him in his tract on the squaring of _lunes_, which was intended
to lead up to the squaring of the circle. The latter problem is one
which must have exercised practical geometers from time immemorial.
Anaxagoras for instance (about 500-428 B.C.) is said to have worked at
the problem while in prison. The essential portions of Hippocrates's
tract are preserved in a passage of Simplicius (on Aristotle's
_Physics_), which contains substantial fragments from Eudemus's _History
of Geometry_. Hippocrates showed how to square three particular lunes of
different forms, and then, lastly, he squared the sum of a certain
circle and a certain lune. Unfortunately, however, the last-mentioned
lune was not one of those which can be squared, and so the attempt to
square the circle in this way failed after all.
Hippocrates also attacked the problem of doubling the cube. There are
two versions of the origin of this famous problem. According to one of
them, an old tragic poet represented Minos as having been dissatisfied
with the size of a tomb erected for his son Glaucus, and having told the
architect to make it double the size, retaining, however, the cubical
form. According to the other, the Delians, suffering from a pestilence,
were told by the oracle to double a certain cubical altar as a means of
staying the plague. Hippocrates did not, indeed, solve the problem, but
he succeeded in reducing it to another, namely, the problem of finding
two mean proportionals in continued proportion between two given
straight lines, i.e. finding x, y such that a : x = x : y = y : b, where
a, b are the two given straight lines. It is easy to see that, if a : x
= x : y = y : b, then b/a = (x/a)^3, and, as a particular case, if b =
2a, x^3 = 2a^3, so that the side of the cube which is double of the cube
of side a is found.
The problem of doubling the cube was henceforth tried exclusively in the
form of the problem of the two mean proportionals. Two significant early
solutions are on record.
(1) Archytas of Tarentum (who flourished in first half of fourth century
B.C.) found the two mean proportionals by a very striking construction
in three dimensions, which shows that solid geometry, in the hands of
Archytas at least, was already well advanced. The construction was
usually called mechanical, which it no doubt was in form, though in
reality it was in the highest degree theoretical. It consisted in
determining a point in space as the intersection of three surfaces: (a)
a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0.
(2) Menaechmus, a pupil of Eudoxus, and a contemporary of Plato, found
the two mean proportionals by means of conic sections, in two ways,
([alpha]) by the intersection of two parabolas, the equations of which
in Cartesian co-ordinates would be x^2 = ay, y^2 = bx, and ([beta]) by
the intersection of a parabola and a rectangular hyperbola, the
corresponding equations being x^2 = ay, and xy = ab respectively. It
would appear that it was in the effort to solve this problem that
Menaechmus discovered the conic sections, which are called, in an
epigram by Eratosthenes, "the triads of Menaechmus".
The trisection of an angle was effected by means of a curve discovered
by Hippias of Elis, the sophist, a contemporary of Hippocrates as well
as of Democritus and Socrates (470-399 B.C.). The curve was called the
_quadratrix_ because it also served (in the hands, as we are told, of
Dinostratus, brother of Menaechmus, and of Nicomedes) for squaring the
circle. It was theoretically constructed as the locus of the point of
intersection of two straight lines moving at uniform speeds and in the
same time, one motion being angular and the other rectilinear. Suppose
OA, OB are two radii of a circle at right angles to one another.
Tangents to the circle at A and B, meeting at C, form with the two
radii the square OACB. The radius OA is made to move uniformly about O,
the centre, so as to describe the angle AOB in a certain time.
Simultaneously AC moves parallel to itself at uniform speed such that A
just describes the line AO in the same length of time. The intersection
of the moving radius and AC in their various positions traces out the
_quadratrix_.
The rest of the geometry which concerns us was mostly the work of a few
men, Democritus of Abdera, Theodorus of Cyrene (the mathematical teacher
of Plato), Theaetetus, Eudoxus, and Euclid. The actual writers of
Elements of whom we hear were the following. Leon, a little younger than
Eudoxus (408-355 B.C.), was the author of a collection of propositions
more numerous and more serviceable than those collected by Hippocrates.
Theudius of Magnesia, a contemporary of Menaechmus and Dinostratus, "put
together the elements admirably, making many partial or limited
propositions more general". Theudius's book was no doubt the geometrical
text-book of the Academy and that used by Aristotle.
Theodorus of Cyrene and Theaetetus generalised the theory of irrationals,
and we may safely conclude that a great part of the substance of
Euclid's Book X. (on irrationals) was due to Theaetetus. Theaetetus also
wrote on the five regular solids (the tetrahedron, cube, octahedron,
dodecahedron, and icosahedron), and Euclid was therefore no doubt
equally indebted to Theaetetus for the contents of his Book XIII. In the
matter of Book XII. Eudoxus was the pioneer. These facts are confirmed
by the remark of Proclus that Euclid, in compiling his Elements,
collected many of the theorems of Eudoxus, perfected many others by
Theaetetus, and brought to irrefragable demonstration the propositions
which had only been somewhat loosely proved by his predecessors.
Eudoxus (about 408-355 B.C.) was perhaps the greatest of all
Archimedes's predecessors, and it is his achievements, especially the
discovery of the _method of exhaustion_, which interest us in connexion
with Archimedes.
In astronomy Eudoxus is famous for the beautiful theory of concentric
spheres which he invented to explain the apparent motions of the
planets, and, particularly, their apparent stationary points and
retrogradations. The theory applied also to the sun and moon, for which
Eudoxus required only three spheres in each case. He represented the
motion of each planet as compounded of the rotations of four
interconnected spheres about diameters, all of which pass through the
centre of the earth. The outermost sphere represents the daily rotation,
the second a motion along the zodiac circle or ecliptic; the poles of
the third sphere, about which that sphere revolves, are fixed at two
opposite points on the zodiac circle, and are carried round in the
motion of the second sphere; and on the surface of the third sphere the
poles of the fourth sphere are fixed; the fourth sphere, revolving about
the diameter joining its two poles, carries the planet which is fixed at
a point on its equator. The poles and the speeds and directions of
rotation are so chosen that the planet actually describes a _hippopede_,
or _horse-fetter_, as it was called (i.e. a figure of eight), which lies
along and is longitudinally bisected by the zodiac circle, and is
carried round that circle. As a _tour de force_ of geometrical
imagination it would be difficult to parallel this hypothesis.
In geometry Eudoxus discovered the great theory of proportion,
applicable to incommensurable as well as commensurable magnitudes, which
is expounded in Euclid, Book V., and which still holds its own and will
do so for all time. He also solved the problem of the two mean
proportionals by means of certain curves, the nature of which, in the
absence of any description of them in our sources, can only be
conjectured.
Last of all, and most important for our purpose, is his use of the
famous _method of exhaustion_ for the measurement of the areas of curves
and the volumes of solids. The example of this method which will be most
familiar to the reader is the proof in Euclid XII., 2, of the theorem
that the areas of circles are to one another as the squares on their
diameters. The proof in this and in all cases depends on a lemma which
forms Prop. 1 of Euclid's Book X. to the effect that, if there are two
unequal magnitudes of the same kind and from the greater you subtract
not less than its half, then from the remainder not less than its half,
and so on continually, you will at length have remaining a magnitude
less than the lesser of the two magnitudes set out, however small it is.
Archimedes says that the theorem of Euclid XII., 2, was proved by means
of a certain lemma to the effect that, if we have two unequal magnitudes
(i.e. lines, surfaces, or solids respectively), the greater exceeds the
lesser by such a magnitude as is capable, if added continually to
itself, of exceeding any magnitude of the same kind as the original
magnitudes. This assumption is known as the Axiom or Postulate of
Archimedes, though, as he states, it was assumed before his time by
those who used the method of exhaustion. It is in reality used in
Euclid's lemma (Eucl. X., 1) on which Euclid XII., 2, depends, and only
differs in statement from Def. 4 of Euclid, Book V., which is no doubt
due to Eudoxus.
The method of exhaustion was not discovered all at once; we find traces
of gropings after such a method before it was actually evolved. It was
perhaps Antiphon, the sophist, of Athens, a contemporary of Socrates
(470-399 B.C.), who took the first step. He inscribed a square (or,
according to another account, an equilateral triangle) in a circle, then
bisected the arcs subtended by the sides, and so inscribed a polygon of
double the number of sides; he then repeated the process, and maintained
that, by continuing it, we should at last arrive at a polygon with
sides so small as to make the polygon coincident with the circle. Though
this was formally incorrect, it nevertheless contained the germ of the
method of exhaustion.
Hippocrates, as we have seen, is said to have proved the theorem that
circles are to one another as the squares on their diameters, and it is
difficult to see how he could have done this except by some form, or
anticipation, of the method. There is, however, no doubt about the part
taken by Eudoxus; he not only based the method on rigorous demonstration
by means of the lemma or lemmas aforesaid, but he actually applied the
method to find the volumes (1) of any pyramid, (2) of the cone, proving
(1) that any pyramid is one third part of the prism which has the same
base and equal height, and (2) that any cone is one third part of the
cylinder which has the same base and equal height. Archimedes, however,
tells us the remarkable fact that these two theorems were first
discovered by Democritus (who flourished towards the end of the fifth
century B.C.), though he was not able to prove them (which no doubt
means, not that he gave no sort of proof, but that he was not able to
establish the propositions by the rigorous method of Eudoxus).
Archimedes adds that we must give no small share of the credit for these
theorems to Democritus; and this is another testimony to the marvellous
powers, in mathematics as well as in other subjects, of the great man
who, in the words of Aristotle, "seems to have thought of everything".
We know from other sources that Democritus wrote on irrationals; he is
also said to have discussed the question of two parallel sections of a
cone (which were evidently supposed to be indefinitely close together),
asking whether we are to regard them as unequal or equal: "for if they
are unequal they will make the cone irregular as having many
indentations, like steps, and unevennesses, but, if they are equal, the
cone will appear to have the property of the cylinder and to be made up
of equal, not unequal, circles, which is very absurd". This explanation
shows that Democritus was already close on the track of infinitesimals.
Archimedes says further that the theorem that spheres are in the
triplicate ratio of their diameters was proved by means of the same
lemma. The proofs of the propositions about the volumes of pyramids,
cones and spheres are, of course, contained in Euclid, Book XII. (Props.
3-7 Cor., 10, 16-18 respectively).
It is no doubt desirable to illustrate Eudoxus's method by one example.
We will take one of the simplest, the proposition (Eucl. XII., 10) about
the cone. Given ABCD, the circular base of the cylinder which has the
same base as the cone and equal height, we inscribe the square ABCD; we
then bisect the arcs subtended by the sides, and draw the regular
inscribed polygon of eight sides, then similarly we draw the regular
inscribed polygon of sixteen sides, and so on. We erect on each regular
polygon the prism which has the polygon for base, thereby obtaining
successive prisms inscribed in the cylinder, and of the same height with
it. Each time we double the number of sides in the base of the prism we
take away more than half of the volume by which the cylinder exceeds the
prism (since we take away more than half of the excess of the area of
the circular base over that of the inscribed polygon, as in Euclid XII.,
2). Suppose now that V is the volume of the cone, C that of the
cylinder. We have to prove that C = 3V. If C is not equal to 3V, it is
either greater or less than 3V.
Suppose (1) that C > 3V, and that C = 3V + E. Continue the construction
of prisms inscribed in the cylinder until the parts of the cylinder left
over outside the final prism (of volume P) are together less than E.
Then C - P < E.
But C - 3V = E;
Therefore P > 3V.
But it has been proved in earlier propositions that P is equal to three
times the pyramid with the same base as the prism and equal height.
Therefore that pyramid is greater than V, the volume of the cone: which
is impossible, since the cone encloses the pyramid.
Therefore C is not greater than 3V.
Next (2) suppose that C < 3V, so that, inversely,
V > 1/3 C.
This time we inscribe successive pyramids in the cone until we arrive at
a pyramid such that the portions of the cone left over outside it are
together less than the excess of V over 1/3 C. It follows that the
pyramid is greater than 1/3 C. Hence the prism on the same base as the
pyramid and inscribed in the cylinder (which prism is three times the
pyramid) is greater than C: which is impossible, since the prism is
enclosed by the cylinder, and is therefore less than it.
Therefore V is not greater than 1/3 C, or C is not less than 3V.
Accordingly C, being neither greater nor less than 3V, must be equal to
it; that is, V = 1/3 C.
It only remains to add that Archimedes is fully acquainted with the main
properties of the conic sections. These had already been proved in
earlier treatises, which Archimedes refers to as the "Elements of
Conics". We know of two such treatises, (1) Euclid's four Books on
Conics, (2) a work by one Aristaeus called "Solid Loci," probably a
treatise on conics regarded as loci. Both these treatises are lost; the
former was, of course, superseded by Apollonius's great work on Conics
in eight Books.