The topic of conic sections has been around for many centuries and actually came from exploring the problem of doubling a cube. King Minos wanted to build a tomb and said that the current dimensions were sub-par and the cube should be double the size, but not the lengths. Many mathematicians of that time tried to determine a solution to the problem, but it wasn’t until the idea came to Plato’s Academy that it was solved by a mathematician named Menaechmus (Schmarge, 1999).
Around 360-350 B.C., Menaechmus discovered the curves parabola, hyperbola, and ellipse; however, he did not use these terms. Instead of those terms, he called a parabola a “section of a right-angled cone,” a hyperbola a “section of an obtuse-angled cone,” and an ellipse a “section of an acute-angled cone” (Schmarge, 1999). He found these by examining the intersection of “a right circular cone of a varying vertex angle” and “a plane perpendicular to [an] element of the cone” (Conic Sections, n.d.). The cone itself was a “single-napped cone in which the plane was perpendicular to the axis of symmetry of the cone” (Allen, 2009). Many other mathematicians, including Aristaeus and Euclid, continued to investigate conics, although the next major contributors to the topic were Archimedes and Apollonius.
Archimedes never published an entire work dedicated to conics, but he did publish several books that mentioned it for example Quadrature of the Parabola, Conoids and Spheroids, Floating Bodies, and Plane Equilibrium. It is said that Archimedes described the different sections of the cones with names and Apollonius just built of that idea from Archimedes. Even though there is no evidence of Euclid’s research into conic sections, many believe that he inspired Archimedes’ published works on Conics (Schmarge, 1999).
Apollonius is known as The Great Geometer. He extended the knowledge of conic sections through his studies and is most well known for his eight books that contain 487 propositions entitled Conic Sections. The first four books were discovered in the original Greek, five to seven were found in the Arabic translation, and the eighth book has never been recovered (Conic Sections, n.d.). When introducing conics he showed that it is not required for a plane that is intersecting the cone to be perpendicular to it. Furthermore, he showed that the cone could be a right, oblique, or scalene. He also disproved the idea that each conic section comes from a different cone and proved that they can be determined from the same cone. Apollonius gave the terms parabola, hyperbola, and ellipse derived from the Greek words for ellipsis, paraboli, and hyperboli.
He also began to use a double-napped cone instead of a single-napped cone because it had a better use of defining the conic sections. The way that he created the double-napped cone is as follows: “If a straight line indefinite in length, and passing always through a fixed point, be made to move round the circumference of a circle which is not in the same plane with the point, so as to pass successively through every point of that circumference, the moving straight line will trace out the surface of a double cone, or two similar cones lying in opposite directions and meeting in the fixed point, which is the apex of each cone” (Schmarge, 1999).
After Apollonius was the mathematicians Pappus and Proclus, who are the reasons why previous mathematicians are credited with examining conic sections. They did this by providing many commentaries on their predecessors’ work on conics in the fifth century. After these two mathematics historians, it wasn't until the fifteenth century, around the era of the Renaissance, that there was a revival in the interest of Greek culture and knowledge.
With the era of the Renaissance came the works of Johannes Kepler. In the year 1605 he was the first person to show the elliptic path of Mars revolving around the sun (Wilson, 2005). This new discovery fueled new motivation in the study of conic sections; however, it dealt more with a focus on the mechanics and astronomy. Kepler also discussed five types of conics rather than Apollonius’s three. Here are the “five types of conic sections: circle, ellipse, hyperbola and line and states that one figure can be obtained from the other by continuous change” (Koudela, 2005). He also claimed that the line and parabola are extreme types of a hyperbola while the parabola and the circle are extreme types of an ellipse. In addition to these pioneers, there were many other mathematicians that contributed to conics such as Newton, Poncelet, Steiner, Dandelin, Dupin, Gergonne, Brianchon, and Chasles.
References
Allen, A. (Fall 2009). Apollonius of Perga: Historical Background and Conic Sections. Retrieved May 19, 2014, from http://digitalcommons.liberty.edu/cgi/viewcontent.cgi?article=1184&context=honors&sei-nredir=1&referer
=hyyp%3A%2F%@Fwww.google.com%2Furl%3Fq%3Dhttp%253A%252F%252Fdigitalcommons.liberty.edu%252Fcgi%
252Fviewcontent.cgi%253Farticle%253D1184%2526context%253Dhonors%26sa%3DD%26sntz%3D1%26usg%3DAFQ
jCNGeW6ms-q4cY0VGymV62obxmdMqnw#search=%22http%3A%2F%2Fdigitalcommons.liberty.edu%2Fcgi%
2Fviewcontent.cgi%3Farticle%3D1184%26context%3Dhonors%22
Conic Sections. (n.d.). Conic Sections. Retrieved May 19, 2014, from http://xahlee.info/SpecialPlaneCurves_dir/
ConicSections_dir/conicSections.html
Koudela, L. (2005). Curves in the History of Mathematics: The Late Renaissance. Retrieved May 19, 2014, from
http://www.mff.cuni.cz/veda/konference/wds/proc/pdf05/WDS05_035_m8_Koudela.pdf
Schmarge, K. (Spring 1999). Conic Sections in Ancient Greece. Retrieved May 19, 2014, from
https://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html
Wilson, J. (2005). Conic Sections: A Resource for Teachers and Students of Mathematics. Retrieved May 19, 2014, from
http://jwilson.coe.uga.edu/EMAT6680/Sloan/emat%20project%20site/Conic%20Section%20Project.htm
Around 360-350 B.C., Menaechmus discovered the curves parabola, hyperbola, and ellipse; however, he did not use these terms. Instead of those terms, he called a parabola a “section of a right-angled cone,” a hyperbola a “section of an obtuse-angled cone,” and an ellipse a “section of an acute-angled cone” (Schmarge, 1999). He found these by examining the intersection of “a right circular cone of a varying vertex angle” and “a plane perpendicular to [an] element of the cone” (Conic Sections, n.d.). The cone itself was a “single-napped cone in which the plane was perpendicular to the axis of symmetry of the cone” (Allen, 2009). Many other mathematicians, including Aristaeus and Euclid, continued to investigate conics, although the next major contributors to the topic were Archimedes and Apollonius.
Archimedes never published an entire work dedicated to conics, but he did publish several books that mentioned it for example Quadrature of the Parabola, Conoids and Spheroids, Floating Bodies, and Plane Equilibrium. It is said that Archimedes described the different sections of the cones with names and Apollonius just built of that idea from Archimedes. Even though there is no evidence of Euclid’s research into conic sections, many believe that he inspired Archimedes’ published works on Conics (Schmarge, 1999).
Apollonius is known as The Great Geometer. He extended the knowledge of conic sections through his studies and is most well known for his eight books that contain 487 propositions entitled Conic Sections. The first four books were discovered in the original Greek, five to seven were found in the Arabic translation, and the eighth book has never been recovered (Conic Sections, n.d.). When introducing conics he showed that it is not required for a plane that is intersecting the cone to be perpendicular to it. Furthermore, he showed that the cone could be a right, oblique, or scalene. He also disproved the idea that each conic section comes from a different cone and proved that they can be determined from the same cone. Apollonius gave the terms parabola, hyperbola, and ellipse derived from the Greek words for ellipsis, paraboli, and hyperboli.
He also began to use a double-napped cone instead of a single-napped cone because it had a better use of defining the conic sections. The way that he created the double-napped cone is as follows: “If a straight line indefinite in length, and passing always through a fixed point, be made to move round the circumference of a circle which is not in the same plane with the point, so as to pass successively through every point of that circumference, the moving straight line will trace out the surface of a double cone, or two similar cones lying in opposite directions and meeting in the fixed point, which is the apex of each cone” (Schmarge, 1999).
After Apollonius was the mathematicians Pappus and Proclus, who are the reasons why previous mathematicians are credited with examining conic sections. They did this by providing many commentaries on their predecessors’ work on conics in the fifth century. After these two mathematics historians, it wasn't until the fifteenth century, around the era of the Renaissance, that there was a revival in the interest of Greek culture and knowledge.
With the era of the Renaissance came the works of Johannes Kepler. In the year 1605 he was the first person to show the elliptic path of Mars revolving around the sun (Wilson, 2005). This new discovery fueled new motivation in the study of conic sections; however, it dealt more with a focus on the mechanics and astronomy. Kepler also discussed five types of conics rather than Apollonius’s three. Here are the “five types of conic sections: circle, ellipse, hyperbola and line and states that one figure can be obtained from the other by continuous change” (Koudela, 2005). He also claimed that the line and parabola are extreme types of a hyperbola while the parabola and the circle are extreme types of an ellipse. In addition to these pioneers, there were many other mathematicians that contributed to conics such as Newton, Poncelet, Steiner, Dandelin, Dupin, Gergonne, Brianchon, and Chasles.
References
Allen, A. (Fall 2009). Apollonius of Perga: Historical Background and Conic Sections. Retrieved May 19, 2014, from http://digitalcommons.liberty.edu/cgi/viewcontent.cgi?article=1184&context=honors&sei-nredir=1&referer
=hyyp%3A%2F%@Fwww.google.com%2Furl%3Fq%3Dhttp%253A%252F%252Fdigitalcommons.liberty.edu%252Fcgi%
252Fviewcontent.cgi%253Farticle%253D1184%2526context%253Dhonors%26sa%3DD%26sntz%3D1%26usg%3DAFQ
jCNGeW6ms-q4cY0VGymV62obxmdMqnw#search=%22http%3A%2F%2Fdigitalcommons.liberty.edu%2Fcgi%
2Fviewcontent.cgi%3Farticle%3D1184%26context%3Dhonors%22
Conic Sections. (n.d.). Conic Sections. Retrieved May 19, 2014, from http://xahlee.info/SpecialPlaneCurves_dir/
ConicSections_dir/conicSections.html
Koudela, L. (2005). Curves in the History of Mathematics: The Late Renaissance. Retrieved May 19, 2014, from
http://www.mff.cuni.cz/veda/konference/wds/proc/pdf05/WDS05_035_m8_Koudela.pdf
Schmarge, K. (Spring 1999). Conic Sections in Ancient Greece. Retrieved May 19, 2014, from
https://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html
Wilson, J. (2005). Conic Sections: A Resource for Teachers and Students of Mathematics. Retrieved May 19, 2014, from
http://jwilson.coe.uga.edu/EMAT6680/Sloan/emat%20project%20site/Conic%20Section%20Project.htm
