Pappus of Alexandria facts for kids
Pappus of Alexandria (born around 290 AD, died around 350 AD) was one of the last great Greek mathematicians from ancient times. He is famous for his book called Synagoge (which means Collection). This book was written around 340 AD.
Pappus also contributed to projective geometry, including Pappus's hexagon theorem. We don't know much about his life. We only know from his writings that he had a son named Hermodorus and taught in Alexandria.
His Collection is his most famous work. It's a large book about mathematics, with eight parts. Most of it still exists today. It covers many topics like geometry, fun math puzzles, how to double a cube, and information about polygons and polyhedra.
Contents
A Special Mathematician
Pappus lived in the 4th century AD. This was a time when not much new math was being discovered. But Pappus was different; he was a very special exception.
The historian Thomas Little Heath said that Pappus was much smarter than others at his time. However, people didn't really understand or appreciate his work. This is why other Greek writers didn't mention him much. His work also didn't stop math from becoming less popular. Heath compared Pappus's situation to that of another mathematician, Diophantus.
When Did Pappus Live?
Pappus didn't write down exactly when he lived or when he wrote his books. We know he lived after Ptolemy (who died around 168 AD) because Pappus quoted him. We also know he lived before Proclus (born around 411 AD) because Proclus quoted Pappus.
An old book from the 10th century says Pappus lived at the same time as Theon of Alexandria. Theon was active when Emperor Theodosius I ruled (372–395 AD). Another old note says Pappus wrote during the time of Emperor Diocletian (284–305 AD).
But we have a more exact date from Pappus himself. In one of his writings, he talks about a solar eclipse. He calculated the time and place of this eclipse. It happened on October 18, 320 AD. This means Pappus was actively working around that time.
Pappus's Books
Pappus's most important work is his eight-book series called Synagoge or Collection. The first book is missing, and the others are partly damaged.
Other books Pappus wrote include:
- Description of the Inhabited World (Chorographia oikoumenike).
- A commentary on Ptolemy's Almagest.
- The Rivers in Libya.
- The Interpretation of Dreams.
Pappus also mentioned his commentary on the Analemma by Diodorus of Alexandria. He wrote notes on Euclid's Elements and Ptolemy's Harmonika.
Later, in 1588, Federico Commandino translated Pappus's Collection into Latin. Then, in the 1930s, Paul ver Eecke translated it into French. This was the first time it was available in a modern European language.
About the Collection
Pappus's Collection is special for two main reasons. First, it neatly organizes the most important math discoveries made by earlier mathematicians. Second, it adds explanations and new ideas to these old discoveries. Pappus basically used these older works as a starting point for his own discussions.
Thomas Little Heath thought the introductions to each book were very helpful. They clearly explained what each section was about. Heath also found Pappus's writing style to be excellent and even elegant. He said the Collection was a wonderful replacement for many valuable math books from earlier times that are now lost.
Here's a summary of the parts of the Collection that still exist:
Book II: Ancient Multiplication
The first part of Book II is missing. The part we have starts in the middle of a math problem. This book talks about an old way of multiplying numbers. It came from a book by Apollonius of Perga, but Pappus doesn't name it. The end of the book shows how to multiply the number values of Greek letters in two lines of poetry. This creates two very large numbers.
Book III: Geometry Problems
Book III is about geometry problems, both flat (plane) and 3D (solid). It has five main parts:
- Doubling the Cube: This section discusses a famous problem: finding two lines that are proportional between two given lines. This problem came from the challenge of doubling the size of a cube. Pappus offers several ways to solve it. He even shows a method to get closer and closer to the answer. He also adds his own solution for finding the side of a cube that has a specific volume compared to another cube.
- Different Means: Pappus explains arithmetic, geometric, and harmonic means between two lines. He shows how to draw all three in the same geometric picture. This leads to his general idea of means, where he describes ten different types.
- Euclid's Problem: This part looks at an interesting problem from Euclid's book.
- Shapes in a Sphere: Pappus shows how to fit the five regular 3D shapes (like a cube or a pyramid) inside a sphere. He noticed that a regular dodecahedron (12 faces) and a regular icosahedron (20 faces) could fit in the same sphere. Their corners would all lie on the same four circles around the sphere.
- Another Solution: A later writer added another way to solve the first problem in this book.
Book IV: Circles and Curves
The beginning of Book IV is lost, so we learn about its topics from the book itself. It starts with a well-known idea that expands on Euclid's geometry. Then, it has many theorems about circles. One big problem is how to draw a circle that goes around three other circles, where those three circles touch each other in pairs.
This book also looks at special curves like Archimedes's spiral, the conchoid of Nicomedes, and the quadratrix. Pappus describes how to make a 3D curve called a helix on a sphere. This is the first known time someone found the area of a curved surface. The rest of the book deals with dividing an angle into three equal parts (trisection). It also solves similar problems using the quadratrix and spiral. In one solution, Pappus uses a property of a conic (like a hyperbola) related to its focus and directrix. This is the first time this property was recorded.
Book V: Shapes and Honeycombs
Book V begins with an interesting introduction about regular polygons. It includes notes on why honeycombs have a hexagonal shape. Pappus then compares the areas of different flat shapes that all have the same perimeter. He also compares the volumes of different 3D shapes that all have the same surface area. Finally, he compares the five regular 3D shapes of Plato.
Pappus also describes the thirteen other 3D shapes discovered by Archimedes. These shapes have flat faces that are all equal and have equal angles, but the faces are not all the same shape. Pappus finds the surface area and volume of a sphere using a method similar to Archimedes.
Book VI: Astronomy Help
Book VI aims to explain difficult parts of the "Lesser Astronomical Works." These are other astronomy books besides the Almagest. So, Pappus comments on books by Theodosius, Autolycus, Aristarchus, and Euclid.
Book VII: Advanced Geometry
This book has received a lot of attention from mathematicians.
The introduction to Book VII explains what "analysis" and "synthesis" mean in math. It also explains the difference between a "theorem" (something to prove) and a "problem" (something to solve). Pappus then lists 33 books by mathematicians like Euclid and Apollonius of Perga. He plans to summarize these books and add helpful notes.
This introduction also includes:
- Pappus's Problem: A famous problem that asks you to find a point. The distances from this point to several given lines have a special relationship.
- Guldin's Theorems: Theorems about finding the volume and surface area of shapes by rotating them. These were later rediscovered by Paul Guldin.
Book VII also contains:
- Notes on Apollonius's De Sectione Determinata. These notes show ideas related to how six points can be arranged in a special way.
- Important notes on Euclid's Porisms, including what is now called Pappus's hexagon theorem.
- A note on Euclid's Surface Loci. This note says that if a point's distance from a fixed point has a constant ratio to its distance from a fixed line, then the point will form a conic shape. Pappus then proves that this conic is a parabola, ellipse, or hyperbola depending on the ratio. These are the first recorded proofs of these properties.
Book VIII: Mechanics and More Geometry
Book VIII mainly talks about mechanics, which is the study of how things move and forces. It covers the properties of the center of gravity and some simple machines. It also includes some pure geometry problems. For example, Proposition 14 shows how to draw an ellipse through five given points. Proposition 15 gives a simple way to find the axes of an ellipse if you know two of its conjugate diameters.
Pappus's Lasting Impact
Pappus's Collection was not well known to Arab or European scholars in the Middle Ages. But it became very important in the 17th century after it was translated into Latin.
His work, along with Diophantus's Arithmetica, was a major source for François Viète's book on analytic geometry. Pappus's problem led René Descartes to develop analytic geometry, which combines algebra and geometry. Pierre de Fermat also developed his own analytic geometry methods based on Pappus's summaries of lost works by Apollonius.
Many other mathematicians were influenced by Pappus, including Luca Pacioli, Leonardo da Vinci, Johannes Kepler, Blaise Pascal, Isaac Newton, Leonhard Euler, and Carl Friedrich Gauss.
See also
- Pappus's hexagon theorem
- Pappus's centroid theorem
- Pappus chain
- Pappus configuration
- Pappus graph
Images for kids
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In Spanish: Papo de Alejandría para niños
