Aristaeus the Elder | |
|---|---|
| Born | c. 370 BC |
| Died | c. 300 BC |
Aristaeus the Elder (Ancient Greek:Ἀρισταῖος ὁ Πρεσβύτερος; 370 – 300 BC) was an ancientGreekgeometer andmathematician who worked onconic sections. He was a contemporary ofEuclid.
Very little is known of his life, even his birthplace is unknown. The mathematicianPappus of Alexandria refers to him as Aristaeus the Elder.[1] He seems to have been an outstanding geometer, on par with or even inspiring powerhouses likeEuclid andApollonius, 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 toEuclid'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 theTreasury of Analysis. Aristaeus’s investigations amounted to five volumes ofSolid Loci, or intermediate treatises on conic sections.[2] 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).[3]
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.[4] 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 theduplication of the cube.[5] Now this may have been Aristaeus himself, since Aristaeus seems to have solved the angle trisection problem using a hyperbola.[6]
These curves were called solid loci, or lines that can only be generated by means of solids. The problems that could only be solved through solid loci were called solid problems. For instance, trisecting the angle and duplicating the cube required these special curves.[7] To Plato's display, it would be later proven these problems cannot be solved by compass and straight-edge alone.
Unfortunately, all of his works are lost, although some theorems may have been preserved by Pappus and Archimedes.
Aristaeus's lost work on the Solid Loci was an intermediate treatise on conic sections. Because it was not introductory this inspired Apollonius to fully treat the basics of conics. Pappus lists the fiveSolid Loci as the next set of book sto read after Apollonius's eightConics in hisTreasury of Analysis. Even though only the title of this work survives, we can be sure pertained to conics because Pappus in another place describes it as "five books of conic elements".[8]
A lost book that compared the regular polehedra when inscribed within the same spheres.Thomas Heath in his history of Greek mathematics notes, "Hypsicles (who lived in Alexandria) says also that Aristaeus, in a work entitledComparison of the five figures, proved that the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere; whether this Aristaeus is the same as the Aristaeus of theSolid Loci, the elder contemporary of Euclid, we do not know."[9] Current scholarship suggests that it was a genuine work of his.[10] This book may have inspired Book XIII of Euclid's Elements. Apollonius also wrote a work on the same subject.
Both Pappus and Archimedes appear to quote an ancient treatise on conics. It is postulated that they either come from Euclid or Aristaeus. Some argue Euclid did not write at all on conics, but if he did, it would have been a derivative work of Aristaeus.[11]
The first three propositions in Archimedes'sQuadrature of the Parabola are enunciations of previously proven theorems in the so-calledConic Elements which surely go back to Aristaeus.
Proposition 1. If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, asPV, and ifQQ′ be a chord parallel to the tangent to the parabola atP and meeting inPV inV, then
QV =VQ′
Conversely, ifQV =VQ', the chordQQ′ will be parallel to the tangent atP.
Proposition 2. If in a parabolaQQ′ be a chord parallel to the tangent atP, and if a straight line be drawn throughP which is either itself the axis or parallel to the axis, and which meetsQQ′ inV and the tangent atQ to the parabola inT, then
PV =PT
Proposition 3. If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as PV, and if from two other pointsQ,Q′ on the parabola line be drawn parallel to the tangent atP and meetingPV inV,V′ respectively, then
PV :PV′ =QV2 :Q′V′2"And these propositions are proved in the elements of conics."
The first statement of the third proposition in Archimedes'sOn Conoids and Spheroids again refers to a theorem in theConic Elements.
Proposition 3. IfTP,TP′ be two tangents to any conic meeting inT, and ifQq,Q′q′ be any two chords parallel respectively toTP,TP′ and meeting inO, then
QO.Oq :Q′O.Oq′ =TP2 :TP′2"And this is proved in the elements of conics."
Some say an alternative to theorem 34 in Pappus'sCollection IV goes back to Aristaeus. It is also one of the three times in all of preserved ancient Greek mathematics that preserves the focus directrix construction of conics.[12]
Theorem 34b. Given an arc, its trisection point lies on a certain hyperbola.
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