View source for Quadrature of the Parabola

{{Short description|Geometric treatise by Archimedes}}
{{italic title}}
[[File:Parabolic Segment.svg|thumb|upright=1.05|A parabolic segment.]]
'''''Quadrature of the Parabola''''' ({{langx|el|Τετραγωνισμὸς παραβολῆς}}) is a [[treatise]] on [[geometry]], written by [[Archimedes]] in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding [[parabola]]s, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a [[line (geometry)|line]]) is <math>\tfrac43</math> that of a certain inscribed [[triangle]].

It is one of the best-known works of Archimedes, in particular for its ingenious use of the [[method of exhaustion]] and in the second part of a [[geometric series]]. Archimedes  dissects the area into infinitely many [[triangles]] whose areas form a [[geometric progression]].<ref>{{Cite journal|last1=Swain|first1=Gordon|last2=Dence|first2=Thomas|date=1998|title=Archimedes' Quadrature of the Parabola Revisited|url=https://www.jstor.org/stable/2691014|journal=Mathematics Magazine|volume=71|issue=2|pages=123–130|doi=10.2307/2691014|jstor=2691014|issn=0025-570X}}</ref> 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]]'' argument in ancient [[Greek mathematics]], and Archimedes' solution remained unsurpassed until the development of [[integral calculus]] in the 17th century, being succeeded by [[Cavalieri's quadrature formula]].<ref>{{Cite journal|last=Cusick|first=Larry W.|date=2008|title=Archimedean Quadrature Redux|url=https://www.jstor.org/stable/27643090|journal=Mathematics Magazine|volume=81|issue=2|pages=83–95|doi=10.1080/0025570X.2008.11953535|jstor=27643090|s2cid=126360876|issn=0025-570X}}</ref>

== 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 (geometry)|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 <math>\tfrac43</math> that of the inscribed triangle.
== Structure of the text ==
[[File:ArchimedePese2.svg|thumb|upright=1.05|Archimedes' first proof of the area of a parabolic segment.]]
[[Conic sections]] such as the parabola were already well known in Archimedes' time thanks to [[Menaechmus]] a century earlier. However, before the advent of the [[differential calculus|differential]] and [[integral calculus]], 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.<ref>{{Cite journal|last=Towne|first=R.|date=2018|title=Archimedes in the Classroom|url=https://collected.jcu.edu/mastersessays/91|journal=Master's Thesis|volume=[[John Carroll University]]}}</ref>

Archimedes gives two proofs of the main theorem: one using abstract [[mechanics]] 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]] at specific distances from the fulcrum.<ref>{{Cite web|title=Quadrature of the parabola, Introduction|url=https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/QuadraturaParabolae/QP.contents.html|access-date=2021-07-03|website=web.calstatela.edu}}</ref> 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.<ref>{{Cite web|title=The Illustrated Method of Archimedes|url=https://www.scribd.com/doc/200723036/The-Illustrated-Method-of-Archimedes|access-date=2021-07-03|website=Scribd|language=en}}</ref> Archimedes here deviates from the procedure found in ''[[On the Equilibrium of Planes]]'' in that he has the centers of gravity at a level below that of the balance.<ref>{{Cite web|last=Dijksterhuis|first=E. J.|date=1987|title=Quadrature of the Parabola|url=https://muse.jhu.edu/chapter/1319251|publisher=Archimedes|pages=336–345|language=en}}</ref> 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]]'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 ==
[[File:Parabolic Segment Dissection.svg|thumb|upright=1.05|Archimedes' second proof dissects the area using an arbitrary number of triangles.]]
=== 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 <math>\tfrac18</math> the area of the blue triangle, so that both green triangles together sum to <math>\tfrac14</math> the area of the blue triangle. From a modern point of view, this is because the green triangle has <math>\tfrac12</math> the width and <math>\tfrac14</math> the height of the blue triangle:<ref>The green triangle has <math>\tfrac12</math> the width of blue triangle by construction. The statement about the height follows from the geometric properties of a parabola, and is easy to prove using modern [[analytic geometry]].</ref>

{{clear|right}}
{{bi|left=1.6|1= [[File:Quadrature Parabola Relative Sizes.svg|frameless|left|upright=1.1]] }}

Following the same argument, each of the <math>4</math> yellow triangles has <math>\tfrac18</math> the area of a green triangle or <math>\tfrac1{64}</math> the area of the blue triangle, summing to <math>\tfrac4{64} = \tfrac1{16}</math> the area of the blue triangle; each of the <math>2^3 = 8</math> red triangles has <math>\tfrac18</math> the area of a yellow triangle, summing to <math>\tfrac{2^3}{8^3} = \tfrac1{64}</math> the area of the blue triangle; etc. Using the [[method of exhaustion]], it follows that the total area of the parabolic segment is given by

:<math>\text{Area}\;=\;T \,+\, \frac14T \,+\, \frac1{4^2}T \,+\, \frac1{4^3}T \,+\, \cdots.</math>

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

:<math>\text{Area}\;=\;\left(1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots\right)T.</math>

=== Sum of the series ===
[[File:GeometricSquares.svg|thumb|upright=1.05|Archimedes' proof that {{math|1= {{sfrac|1|4}} + {{sfrac|1|16}} + {{sfrac|1|64}} + ⋯ = {{sfrac|1|3}}}}]]
To complete the proof, Archimedes shows that

:<math>1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots\;=\; \frac{4}{3}.</math>

The formula above is a [[geometric series]]—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the [[geometric series#Sum|sum formula for a geometric series]].

Archimedes evaluates the sum using an entirely geometric method,<ref>Strictly speaking, Archimedes evaluates the [[partial sum]]s of this series, and uses the [[Archimedean property]] to argue that the partial sums become arbitrarily close to <math>\tfrac43</math>. This is logically equivalent to the modern idea of summing an infinite series.</ref> 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

:<math>\frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots.</math>

However, the purple squares are congruent to either set of yellow squares, and so cover <math>\tfrac13</math> of the area of the unit square. It follows that the series above sums to <math>\tfrac43</math> (since {{nobr|<math>1 + \tfrac13 = \tfrac43</math>).}}

== See also ==
{{commons category|Quadrature of the Parabola}}
* [[Quadrature (geometry)]]
* [[History of calculus]]

{{clear}}

== Notes ==
{{reflist}}

== Further reading ==
* {{cite journal|doi=10.2307/2690617|title=Proof without Words: Geometric Series|author=Ajose, Sunday and Roger Nelsen|journal=Mathematics Magazine|volume=67|issue=3|date=June 1994|pages=230|jstor=2690617}}
* {{cite journal|title=Quadrature of the parabola with the square pyramidal number|author=Ancora, Luciano|journal=Archimede|volume=66|issue=3|date=2014|url=https://www.zentralblatt-math.org/matheduc/en/?q=an%3A2016f.01058}}
* {{Cite book| first=David M.| last=Bressoud|author-link=David Bressoud| title=A Radical Approach to Real Analysis| edition = 2nd| publisher=[[Mathematical Association of America]]| year=2006| isbn=0-88385-747-2}}.
* Dijksterhuis, E.J. (1987) "Archimedes", Princeton U. Press {{isbn|0-691-08421-1}}
* {{Cite book| first=C. H.| last=Edwards Jr.| title=The Historical Development of the Calculus| publisher=Springer| year=1994| isbn=0-387-94313-7| edition=3rd}}.
* {{cite book| last=Heath| first=Thomas L.| title=The Works of Archimedes|year=2011| edition= 2nd| publisher=CreateSpace| isbn=978-1-4637-4473-1}}
* {{Cite book| first=George F.| last=Simmons| title=Calculus Gems| publisher=Mathematical Association of America| year=2007| isbn=978-0-88385-561-4| url-access=registration| url=https://archive.org/details/calculusgemsbrie0000simm}}.
* {{cite book|first=Sherman K.|last=Stein|author-link=Sherman K. Stein|title=Archimedes: What Did He Do Besides Cry Eureka?|publisher=Mathematical Association of America|year=1999|isbn=0-88385-718-9|url-access=registration|url=https://archive.org/details/archimedeswhatdi00stei}}
* {{Cite book| first=John| last=Stillwell| title=Mathematics and its History| year=2004| publisher=Springer| edition=2nd| isbn=0-387-95336-1}}.
* {{cite journal|doi=10.2307/2691014|author=Swain, Gordon and Thomas Dence|title=Archimedes' Quadrature of the Parabola Revisited|journal=Mathematics Magazine|volume=71|issue=2|date=April 1998|pages=123–30|jstor=2691014}}
* {{Cite book| first=Alistair Macintosh| last=Wilson| title=The Infinite in the Finite| publisher=[[Oxford University Press]]| year=1995| isbn=0-19-853950-9}}.

== External links ==
{{wiktionary|quadrature}}
* {{cite web| last=Casselman| first=Bill| title=Archimedes' quadrature of the parabola| url=http://www.math.ubc.ca/~cass/archimedes/parabola.html| archive-url=https://web.archive.org/web/20120204050048/http://www.math.ubc.ca/~cass/archimedes/parabola.html| archive-date=2012-02-04}} Full text, as translated by T.L. Heath.
* {{cite web| last=[[Xavier University – Ateneo de Cagayan|Xavier University]] Department of Mathematics and Computer Science| title=Archimedes of Syracuse| url=http://www.cs.xu.edu/math/math147/02f/archimedes/archpartext.html |url-status=dead |archiveurl=https://web.archive.org/web/20160113100044/http://www.cs.xu.edu/math/math147/02f/archimedes/archpartext.html |archivedate=2016-01-13}}. Text of propositions 1–3 and 20–24, with commentary.
* http://planetmath.org/ArchimedesCalculus

{{Archimedes}}
{{Ancient Greek mathematics}}

{{DEFAULTSORT:Quadrature Of The Parabola}}
[[Category:Archimedes]]
[[Category:Ancient Greek mathematical works]]
[[Category:Works by Archimedes]]
[[Category:History of mathematics]]