On Spirals

On Spirals

On Spirals is a treatise by Archimedes in 225 BC. Although Archimedes did not discover the Archimedean spiral, he employed it in this book to square the circle and trisect an angle.[1]

Contents

Contents

Preface

Archimedes begins On Spirals with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of On the Sphere and Cylinder and On Conoids and Spheroids. He continues to state his results of On Spirals.

Archimedean spiral

The Archimedean spiral with three 360° turnings on one arm

The Archimedean spiral was first studied by Conon and was later studied by Archimedes in On Spirals. Archimedes was able to find various tangents to the spiral.[2] He defines the spiral as:

If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving , a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.[3]

Trisecting an angle

Example of how Archimedes trisected an angle in On Spirals.

The construction as to how Archimedes trisected the angle is as follows:

Suppose the angle ABC is to be trisected. Trisect the segment BC and find BD to be one third of BC. Draw a circle with center B and radius BD. Suppose the circle with center B intersects the spiral at point E. Angle ABE is one third angle ABC.[4]

Squaring the circle

The circle and the triangle are equal in area.

To square the circle, Archimedes gave the following construction:

Let P be the point on the spiral when it has completed one turn. Let the tangent at P cut the line perpendicular to OP at T. OT is the length of the circumference of the circle with radius OP.

Archimedes had already proved as the first proposition of Measurement of a Circle that the area of a circle is equal to a right-angled triangle having the legs' lengths equal to the radius of the circle and the circumference of the circle. So the area of the circle with radius OP is equal to the area of the triangle OPT.[5]

References

  1. ^ "Spiral". Encyclopædia Britannica. 2008. http://original.britannica.com/eb/article-9069158/spiral#823770.hook. Retrieved 2008-07-29. 
  2. ^ Weisstein, Eric W., "Archimedes' Spiral" from MathWorld.
  3. ^ Heath, Thomas Little (1921), A History of Greek Mathematics, Boston: Adamant Media Corporation, p. 64, ISBN 0543968774, http://books.google.com/?id=zGIYbEtzD-QC&printsec=frontcover, retrieved 2008-08-20 
  4. ^ Tokuda, Naoyuki; Chen, Liang (1999-03-18), Trisection Angles, Utsunomiya University, Utsunomiya, Japan, pp. 5–6, http://azalea.sunflare.co.jp/rd_dir/papers/trisection-ENG.pdf, retrieved 2008-08-20 
  5. ^ "History topic: Squaring the circle". http://www-history.mcs.st-and.ac.uk/PrintHT/Squaring_the_circle.html. Retrieved 2008-08-20. 

Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Spirals Boracay Resort — (Боракай,Филиппины) Категория отеля: 1 звездочный отель Адрес: Balabag, Station 1, Mala …   Каталог отелей

  • Spirals in Hyperspace — Infobox Album Name = Spirals in Hyperspace Type = studio Artist = Ozric Tentacles Released = 2004 Recorded = Genre = Psychedelic rock, space rock Length = 69:36 Label = Magna Carta Producer = Reviews = * Allmusic Rating|3|5… …   Wikipedia

  • Spirals (album) — Infobox Album Name = Spirals Type = Studio Artist = Blood Has Been Shed Released = March 11, 2003 Recorded = Genre = Mathcore Length = Label = Roadrunner Records Producer = Reviews = *Allmusic Rating|2|5 Last album = Novella of Uriel (2001) This… …   Wikipedia

  • spirals — spi·ral || spaɪərÉ™l n. curve that twists around a fixed center point; three dimensional curve turning around an axis, helix; something having a spiral like curve; progressive upward or downward movement v. wind, coil, twist around an axis,… …   English contemporary dictionary

  • spirals — plural of spiral present third singular of spiral …   Useful english dictionary

  • Love Spirals Downwards — Infobox musical artist Name = Love Spirals Downwards Img capt = Ryan Lum and Suzanne Perry Background = group or band Birth name = Alias = Born = Died = Origin = Los Angeles, California Instrument = Vocals, acoustic guitar, electric guitar,… …   Wikipedia

  • Figure skating spirals — A spiral is an element in figure skating where the skater glides on one foot while raising the free leg above hip level. Spiral positions are classified according to the skating leg (left or right), edge (outside or inside), direction the skater… …   Wikipedia

  • Curschmann's spirals — Curschmann s spiral Curschmann s spirals refers to a finding in the sputum of spiral shaped mucus plugs. These may occur in several different lung diseases.[1] The term can refer to parts of the desquamated epithelium seen in biopsies from asthm …   Wikipedia

  • Poinsot's spirals — In mathematics, Poinsot s spirals are two spirals represented by the polar equations:r=a,operatorname{csch}(n heta):r=a,operatorname{sech}(n heta)where csch is the hyperbolic cosecant, and sech is the hyperbolic secant …   Wikipedia

  • Curschmann's Spirals — refer to parts of the desquamated epithelium seen in biopsies from asthmatic patients. They are named after German physician Heinrich Curschmann (1846 1910). They are often seen in association with eosinophilic infiltration and Charcot Leyden… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”