Apollonius of Perga

apollonius-of-perga

Apollonius of Perga is famous for his work in geometry, particularly on conics. He was known as “The Great Geometer” and wrote a number of books on a number of subjects. Unfortunately, most of these books are lost, although, a few did survive.

Early Life

There is not a lot of information that has survived regarding the life of Apollonius of Perga. Most of the information comes from his own work where he mentions details about his life in the prefaces of the books of his only surviving major work, Conics.

Apollonius was born in Perga, Pamphylia (modern day Antalya in Turkey). He was born between the years 246 and 221 B.C. during the reign of Egyptian king Ptolemy Euregetes. He did his most famous work during the reign of Egyptian king Ptolemy Philopater during the years 221 to 201 B.C.

Apollonius lived in Alexandria and there is some dispute as to whether he studied with students of Euclid. Pappus, another mathematician who lived in Alexandria around the 4th century A.D., claims that Apollonius did, but this is the only proof for the statement. Apollonius also wrote about visiting Pergamum and Ephesus. Pergamum is now known as Bergama and is in Izmir, Turkey.

While in Pergamum, Apollonius met a man by the name of Eudemus. Apollonius writes to Eudemus in the prefaces of his first three books. In the preface of his second book, Apollonius states that he is giving the book to his son (also named Apollonius) to send to Eudemus. This is the basis for the assumption that Apollonius was a mature man when he wrote his bookConics

In the preface of the second book, Apollonius mentions introducing Eudemus to a man named Philonideswhen they were all at the city of Ephesus. This helps set the time period when Apollonius wrote Conics. Philonides was Eudemus’s first student. He was a philosopher as well as a mathematician and knew the Seleucid kings Antiochus IV Epiphanes (reigned 175–163 b.c.) and Demetrius I Soter. Antiochus reigned between 175 and 163 B.C. while Demetrius reigned between 162 and 150 B.C.

This means that the introduction occurred sometime in the mid-100s B.C. and that the Conics were written around the same time. Further dating of the work can be done on the basis of Apollonius having a full-grown son. Since his son was old enough to deliver the second book to Eudemus, Apollonius must have been working around the second and third centuries. There is some internal evidence in the Conics to support this timeline. Apollonius writes about Archimedes, who died around 212 to 211 B.C., when Archimedes was an old man.

Apollonius writes in the preface to the fourth book that Eudemus has died and the rest of the books in Conics are addressed to a man named Attalus. Some scholars believe this is King Attalus I of Pergamum but this is in dispute since Apollonius did not use the title “king”.

Writings

Apollonius wrote a number of books but only two of them still exist today. The reason we know about the books is that in the 4th century A.D., another mathematician, Pappus wrote about Apollonius and Apollonius’s works in his own writings.

 

Conics

Apollonius is best known for his work Conics. Conics consists of eight different books but only seven still survive. The first four have survived in the original Greek but there is an Arabic translation of seven of the eight books. Although we don’t have access to the eighth book, scholars have an idea of what was in the eighth book because of the work of Pappus who referenced the material.

Apollonius wrote the book at the request of Naucrates, another mathematician who had visited him in Alexandria. Apollonius had to finish the book quickly because Naucrates was leaving Alexandria. After writing the book and giving it to Naucrates, Apollonius spent more time on the book and revised some of the material. The revised edition is what make up the book.

The definition of a conic states that it is the curve one gets at the intersection of a cone and a plane. By changing the place and angle of the intersection, different conic sections are created. This basically just means that you can cut a cone in a specific angle to get different curves. Apollonius’s work was abot these conics and he introduced the terms parabola, ellipse, and hyperbola.

The first four books that make up Apollonius’s Conics is an introduction to the topic. To some extent, the first four books utilize the work of other mathematicians although Apollonius claims that he was expanded on the work and developed the work more than previous writers. In fact, any mathematicians in Apollonius’s time (or earlier) would not be very surprised by anything they read in the first four books of Conics. Most of the material would be well known to them.

Book one looks at conics and their properties. He develops ways to create the three conics or sections, which he identifies as parabola, ellipse, and hyperbola. He then describes the three sections. Apollonius also looks at the basic properties of these three sections. Apollonius states that he has developed the concepts to a higher degree than previous writers. Pappus agrees that the first four books are additions to the Euclid’s work on conics.

Book two looks at diameters and axes of the conic sections as well as asymptotes. An axis is simply a straight line the cuts an object into two. An asymptote is a straight line that comes close to a curve but does not meet it.

Book three contains some original material and does not simply restate the work of other mathematicians. Apollonius states that he discovered new ideas on how to create solid loci (a locus is another conic section). He writes that after he came up with some of these ideas, he realized that Euclid had not figured out how to create locus using three and four lines. Apollonius stated that this was impossible to do without using the theorems that he had discovered. Apollonius also dealt with focal properties and with rectangles found in conic segments.

Book four looks at the different ways that conic sections or the circumference of a circle can meet each other. Apollonius states that a lot of the material in book four has not been addressed by other mathematicians.

The first four books are a result of Apollonius organizing the work of other mathematicians into a more organized whole. Previously, this work was a set of various theorems that were not connected in any way. Apollonius was a good enough mathematician to see how the various theorems could be connected according to his general method.

Apollonius’s most original work is presented in books five to seven. Book five is the most famous of the books that make up the Conic and the one that has received the most praise. In books five to seven, Apollonius looks at normals to conics. Normals ae the mathematical name for lines that are perpendicular to an object, in this case, perpendicular to a conic. Apollonius is able to prove that on either side of a conic’s axis, there are a number of points. From these points, it is only possible to draw one normal to the other side of the conic. Apollonius also describes how to make these points which will form a curve known as an evolute (although this term was not used in Apollonius’s time). An evolute is a curve in geometry that describes a set of points at the centre of curvature for another curve.

Apollonius also looks at propositions dealing with the inequalities between functions of conjugate diameters. Conjugate diameters are two diameters of ellipses or hyperbolas that cut across lines (chords) that are parallel to one another.

Book eight has been lost but there has been an attempt to restore it using the work of Pappus. Pappus wrote a number of lemmas based on Apollonius’s work and these lemmas can be used to figure out what Apollonius wrote. A lemma is a helping theorem. It is a proposition that is used to help prove a larger proposition or theorem. A mathematician by the name of Haley attempted to reconstruct book eight using Pappus’s lemmas and if his work is correct, then book eight was about problems with conjugate diameters and functions with given values.

Apollonius’s work is not well known outside the mathematical world and even in the mathematical world, not a lot of people have read it. The main reason for this is that the work is very difficult to read, particularly given the lack of mathematical symbols that modern mathematicians use. Even though the text is difficult to read, it has been studied and praised by some of the greatest mathematicians, including Newton, Fermat, and Halley.

Other Works

Apollonius wrote other books but these have all been lost. Pappus, in his main book called Collection or Synagoge, covered a number of Apollonius’s works that have been lost. Pappus summarized six other books written by Apolloniusas well as summarizing Conics. Pappus’s work has allowed other mathematicians to reconstruct the material written by Apollonius, although there are some concerns over the reconstructions. In these six missing works, Apollonius took an in depth look at specific or general problems. Each book was divided into two books and according to Pappus, the works were important works that were studied by ancient mathematicians.

One book titled Cutting of a Ratio (De Rationis Sectione) is the only other book written by Apollonius that still exists, although the Arabic version is the only one that exists. The Arabic version was an adaptation of Apollonius’s work rather than a translation so Pappus’s summary of the work is still important.

In this book, Apollonius looked athow to draw a straight line through a point and two other straight lines in such a way that the cut off sections have a specific ratio. Apollonius looked at specific cases as well as more general cases.

Another book, Cutting of an Area(De Spatii Sectione), looked at the same problem as in Cutting of a Ratio but in this book, Apollonius used rectangles. The rectangle created by two intercepts needs to be equal to a specific rectangle.

The bookDeterminate Section(De Sectione Determinata) looked at how to find another point on a straight line if you already have two or more points on the line so that the new point’s distance from the other points meets a specific set of conditions.

According to Pappus, the bookTangencies (De Tactionibus) looked at the problem of how to describe a circle when you have three things (circles, straight lines, or points) in such a way so that the circle passes through the given points and touches the given circles or straight lines. A sixteenth century mathematician Vieta was able to develop solutions to this problem and from the solution, Vieta was able to reconstruct Apollonius’s treatise in a book called Apollonius Gallus.

In the lost book Inclinations (De Inclinationibus), Apollonius wanted to demonstrate how a specific, straight line moving towards a point can be placed between two straight or circular lines.

The last missing work is calledPlane Loci (De Locis Planis) and looks at a number of propositions about loci that are straight lines or circles.

Other writers besides Pappus have mentioned the work done by Apollonius. A number of writers mention Apollonius’s work in astronomy. According to these writers, Apollonius came up with the concept of eccentric orbits to explain the motion of the planets and the different speeds of the moon. Both Ptolemy and Hipparchus used and improved on Apollonius’s work. Ptolemy even used some of Apollonius’s work in his model of planetary movement.

Apollonius’s mastery of mathematics was recognized by his peers but there was no one able to take his place once he was gone. His work on conics was the main work on the subject and a number of later mathematicians wrote commentaries or annotations on his work. Unfortunately, Pappus was the only mathematician at the time who seemed capable of understanding Apollonius’s work and capable of expanding on the results.

Even in later centuries, Apollonius’s mathematical skill was recognized by such mathematicians as Descartes, Newton, Fermat, and Pascal.

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