![Parabolic Segment](https://upload.wikimedia.org/wikipedia/commons/b/be/Parabolic_Segment.svg)
''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, written by
Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
s, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a
line) is
that of a certain inscribed
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
.
It is one of the best-known works of Archimedes, in particular for its ingenious use of the
method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
and in the second part of a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
. Archimedes dissects the area into infinitely many
triangles
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
whose areas form a
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of a ''
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'' argument in ancient
Greek mathematics, and Archimedes' solution remained unsurpassed until the development of
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
in the 17th century, being succeeded by
Cavalieri's quadrature formula.
Main theorem
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given
chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord. Proposition 1 of the work states that a line from the third vertex drawn parallel to the axis divides the chord into equal segments. The main theorem claims that the area of the parabolic segment is
that of the inscribed triangle.
Structure of the text
Conic sections such as the parabola were already well known in Archimedes' time thanks to
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.''
Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
a century earlier. However, before the advent of the
differential and
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
, there were no easy means to find the area of a conic section. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord.
Archimedes gives two proofs of the main theorem: one using abstract
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity, with weighted segments of a parabola and a triangle suspended along the arms of a
lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
at specific distances from the fulcrum. When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height. Archimedes here deviates from the procedure found in ''
On the Equilibrium of Planes
''On the Equilibrium of Planes'' ( grc, Περὶ ἐπιπέδων ἱσορροπιῶν, translit=perí epipédōn isorropiôn) is a treatise by Archimedes in two volumes. The first book contains a proof of the law of the lever and culminate ...
'' in that he has the centers of gravity at a level below that of the balance. The second and more famous proof uses pure geometry, particularly the sum of a geometric series.
Of the twenty-four propositions, the first three are quoted without proof from
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's ''Elements of Conics'' (a lost work by Euclid on
conic sections). Propositions 4 and 5 establish elementary properties of the parabola. Propositions 6–17 give the mechanical proof of the main theorem; propositions 18–24 present the geometric proof.
Geometric proof
![Parabolic Segment Dissection](https://upload.wikimedia.org/wikipedia/commons/d/d7/Parabolic_Segment_Dissection.svg)
Dissection of the parabolic segment
The main idea of the proof is the dissection of the parabolic segment into infinitely many triangles, as shown in the figure to the right. Each of these triangles is inscribed in its own parabolic segment in the same way that the blue triangle is inscribed in the large segment.
Areas of the triangles
In propositions eighteen through twenty-one, Archimedes proves that the area of each green triangle is
the area of the blue triangle, so that both green triangles together sum to
the area of the blue triangle. From a modern point of view, this is because the green triangle has
the width and
the height of the blue triangle:
Following the same argument, each of the
yellow triangles has
the area of a green triangle or
the area of the blue triangle, summing to
the area of the blue triangle; each of the
red triangles has
the area of a yellow triangle, summing to
the area of the blue triangle; etc. Using the
method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
, it follows that the total area of the parabolic segment is given by
:
Here ''T'' represents the area of the large blue triangle, the second term represents the total area of the two green triangles, the third term represents the total area of the four yellow triangles, and so forth. This simplifies to give
:
Sum of the series
![GeometricSquares](https://upload.wikimedia.org/wikipedia/commons/e/e6/GeometricSquares.svg)
To complete the proof, Archimedes shows that
:
The formula above is a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the
sum formula for a geometric series.
Archimedes evaluates the sum using an entirely geometric method,
[Strictly speaking, Archimedes evaluates the ]partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s of this series, and uses the Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typica ...
to argue that the partial sums become arbitrarily close to . This is logically equivalent to the modern idea of summing an infinite series. illustrated in the adjacent picture. This picture shows a unit square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, with the total purple area being the sum
:
However, the purple squares are congruent to either set of yellow squares, and so cover
of the area of the unit square. It follows that the series above sums to
(since
See also
*
History of calculus
Notes
Further reading
*
*
* .
* Dijksterhuis, E.J. (1987) "Archimedes", Princeton U. Press
* .
*
* .
*
* .
*
* .
External links
* Full text, as translated by T.L. Heath.
* Text of propositions 1–3 and 20–24, with commentary.
* http://planetmath.org/ArchimedesCalculus
{{DEFAULTSORT:Quadrature Of The Parabola
Archimedes
Ancient Greek mathematical works
Works by Archimedes
History of mathematics