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Aristaeus the Elder
Aristaeus the Elder (Ancient Greek: Ἀρισταῖος ὁ Πρεσβύτερος; 370 – 300 BC) was an ancient Greek geometer and mathematician who worked on conic sections. He was a contemporary of Euclid.
Very little is known of his life, even his birthplace is unknown. The mathematician Pappus of Alexandria refers to him as Aristaeus the Elder. He seems to have been an outstanding geometer, on par with or even inspiring powerhouses like Euclid and Apollonius, but our knowledge of his achievements are incredibly limited, especially for someone apparently so important.
His contributions to geometry were set down in propositional axiomatic treatises akin to Euclid's Elements. His mathematical techniques must be surmised by fragments of his work and by later commentators.
Aristaeus was one of the main contributors to the field of ancient geometric analysis, standing beside Euclid and Apollonius. Pappus collected 32 of these mens’ works and put them in a curriculum known as the Treasury of Analysis. Aristaeus’s investigations amounted to five volumes of Solid Loci, or intermediate treatises on conic sections. Analysis sets out to deduce properties of pre-existing figures, one purpose being to find out how they are constructed. It is the opposite of synthesis, which sets out to construct objects with certain properties from nothing.
The analysis of Aristeus goes like this. Imagine you want to find a point on a line that satisfies a specific condition with respect to the line. Depending on the difficulty, attacking this problem through synthesis feels like shooting arrows in the dark. So instead of directly synthesizing the point and proving it has the property desired (which is the habit of Euclid), Aristaeus instead assumes the desired point is given. Then, he analyzes its properties to see if the point resides on a hyperbola or some other conic section. If so, then a construction of the necessary conic section will intersect the initial line and you will obtain the desired point. Now these theorems can be rewritten as synthesis proofs, but it hides the process of how the solution was found, and geometers will wonder how the theorem was discovered if they are not well versed in analysis. Archimedes was famous for obscuring his analytical techniques which is why his treatises appear so streamlined, thus his constructions and discoveries look like they were made through successive strokes of genius.
Aristaeus had a rudimentary view of conic sections. Instead of treating all the different ways a plane can intersect with a cone, he only considered the intersection of a cone with a plane perpendicular to its surface. Thus, he discovered three types of conic sections. One he called "the section of an acute-angled cone" (which is an ellipse), another he called "the section of a right-angled cone" (which is a parabola), and the last one he called "the section of an obtuse-angled cone" (which is a hyperbola).
When Apollonius took up the study of conics, this terminology perplexed him since every type of cone give rises to every type of conic section. Thus, he had to rename the conic sections "ellipse", "parabola", and "hyperbola", naming them after their properties, and not the type of cone that made them. By generalizing the definition, the circle became included among the sections of the cone when previously Aristaeus did not give recognize them.
Aristaeus made useful contributions to conics, for Pappus explains people used his loci to solve the duplication of the cube. Now this may have been Aristaeus himself, since Aristaeus seems to have solved the angle trisection problem using a hyperbola.
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Aristaeus the Elder
Aristaeus the Elder (Ancient Greek: Ἀρισταῖος ὁ Πρεσβύτερος; 370 – 300 BC) was an ancient Greek geometer and mathematician who worked on conic sections. He was a contemporary of Euclid.
Very little is known of his life, even his birthplace is unknown. The mathematician Pappus of Alexandria refers to him as Aristaeus the Elder. He seems to have been an outstanding geometer, on par with or even inspiring powerhouses like Euclid and Apollonius, but our knowledge of his achievements are incredibly limited, especially for someone apparently so important.
His contributions to geometry were set down in propositional axiomatic treatises akin to Euclid's Elements. His mathematical techniques must be surmised by fragments of his work and by later commentators.
Aristaeus was one of the main contributors to the field of ancient geometric analysis, standing beside Euclid and Apollonius. Pappus collected 32 of these mens’ works and put them in a curriculum known as the Treasury of Analysis. Aristaeus’s investigations amounted to five volumes of Solid Loci, or intermediate treatises on conic sections. Analysis sets out to deduce properties of pre-existing figures, one purpose being to find out how they are constructed. It is the opposite of synthesis, which sets out to construct objects with certain properties from nothing.
The analysis of Aristeus goes like this. Imagine you want to find a point on a line that satisfies a specific condition with respect to the line. Depending on the difficulty, attacking this problem through synthesis feels like shooting arrows in the dark. So instead of directly synthesizing the point and proving it has the property desired (which is the habit of Euclid), Aristaeus instead assumes the desired point is given. Then, he analyzes its properties to see if the point resides on a hyperbola or some other conic section. If so, then a construction of the necessary conic section will intersect the initial line and you will obtain the desired point. Now these theorems can be rewritten as synthesis proofs, but it hides the process of how the solution was found, and geometers will wonder how the theorem was discovered if they are not well versed in analysis. Archimedes was famous for obscuring his analytical techniques which is why his treatises appear so streamlined, thus his constructions and discoveries look like they were made through successive strokes of genius.
Aristaeus had a rudimentary view of conic sections. Instead of treating all the different ways a plane can intersect with a cone, he only considered the intersection of a cone with a plane perpendicular to its surface. Thus, he discovered three types of conic sections. One he called "the section of an acute-angled cone" (which is an ellipse), another he called "the section of a right-angled cone" (which is a parabola), and the last one he called "the section of an obtuse-angled cone" (which is a hyperbola).
When Apollonius took up the study of conics, this terminology perplexed him since every type of cone give rises to every type of conic section. Thus, he had to rename the conic sections "ellipse", "parabola", and "hyperbola", naming them after their properties, and not the type of cone that made them. By generalizing the definition, the circle became included among the sections of the cone when previously Aristaeus did not give recognize them.
Aristaeus made useful contributions to conics, for Pappus explains people used his loci to solve the duplication of the cube. Now this may have been Aristaeus himself, since Aristaeus seems to have solved the angle trisection problem using a hyperbola.