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Work #1816

Conics

The definitive ancient treatise on conic sections — parabola, ellipse, and hyperbola defined, named, and explored across eight books

Apollonius of Perga · c. 200 BCE · Ancient Greek (Books I-IV); Arabic translation (Books V-VII) · Mathematical treatise (definitions, propositions, proofs)

Tradition: Greek higher geometry

Parabola, ellipse, hyperbola — the curves that would become the orbits of planets, defined here in pure geometry

The Conics (Kōnika) of Apollonius of Perga is, with Euclid's Elements and Archimedes's collected works, one of the three supreme monuments of Greek mathematics. In eight books (seven surviving: four in Greek, three in 9th-century Arabic translation by Thābit ibn Qurra), Apollonius develops the theory of conic sections — the curves produced by cutting a cone with a plane — with a thoroughness and generality not surpassed until Descartes. He introduces the names parabola ("equal application"), ellipse ("deficiency"), and hyperbola ("excess"), derived from the Pythagorean theory of application of areas. He unifies the treatment of all three curves by varying the cutting plane rather than the type of cone, and proves over 400 propositions on their properties: tangents, normals, foci, conjugate diameters, and the theory of poles and polars. Books V-VII contain the most original material, including the theory of evolutes (normals and curvature). The Conics was foundational for Kepler's discovery that planetary orbits are ellipses (1609) and for Newton's derivation of conic trajectories from the inverse-square law (1687).

Author

Editions cited

  • T.L. Heath, Apollonius of Perga: Treatise on Conic Sections (Cambridge, 1896)
  • Gerald J. Toomer, Apollonius: Conics, Books V to VII — the Arabic Translation (2 vols., Springer, 1990)
  • Michael N. Fried and Sabetai Unguru, Apollonius of Perga's Conica: Text, Context, Subtext (Brill, 2001)
Compare this work

School Embodiments

Classical Greek Thought · 30%
Rationalism · 25%
Realism · 15%
Platonism (Classical) · 15%
Formalism (Mathematical) · 10%
Mechanism · 5%
Classical Greek Thought 30%

The supreme achievement of Greek higher geometry — the standard reference on conics for two millennia.

"If a cone is cut by a plane through the axis, and also cut by another plane …" (Conics I, Definition 4)
Rationalism 25%

Pure deductive geometry in the Euclidean style, carried to unprecedented abstraction.

Over 400 propositions proved by rigorous deduction across seven surviving books.
Realism 15%

Objective mathematical structures described with mind-independent properties — vindicated by Kepler and Newton.

Apollonius's geometrical parabola, ellipse, and hyperbola turned out to describe real trajectories.
Platonism (Classical) 15%

Mathematics studied for its own sake, not for application — the Platonic valuation of pure theory.

Apollonius addresses fellow geometers and pursues conic properties far beyond practical need.
Formalism (Mathematical) 10%

Systematic, axiomatic, self-contained — anticipates formalist approaches.

The Conics forms a deductive system readable without physical reference.
Mechanism 5%

Though Apollonius did not apply conics to physics, his curves became the mathematical language of celestial mechanics.

Kepler (1609): planetary orbits are ellipses. Newton (1687): inverse-square law yields conic trajectories.

Internal Tensions

The gap between pure geometry and physical application: Apollonius had no inkling that conics describe planetary orbits, yet his work provided exactly the apparatus Kepler needed. The "unreasonable effectiveness" of pure mathematics is the unresolved puzzle Apollonius embodies.

⧖

I. Time

Implicit: mathematical truths are timeless. Deterministic: properties follow necessarily from definitions.

Attributes
Extent: Infinite Ontological Status: Substantival Grain: Continuous Freedom: Deterministic Traversability: Linear Direction: Uni-directional Dimensionality: One
✦

II. Space

Infinite, flat (Euclidean), three-dimensional, local. The cone is 3D; sections are 2D curves within it.

Attributes
Extent: Infinite Ontological Status: Substantival Curvature: Flat Dimensionality: Three Locality: Local
◉

III. Matter

Unaddressed: pure geometry, not physics.

Attributes
Extent: not engaged Ontological Status: not engaged Conservation: not engaged Dimensionality: not engaged Locality: not engaged
◎

IV. Observer

The geometer who constructs, proves, and communicates with fellow mathematicians.

Attributes
Time Instance: Single Space Instance: Single Knowledge Extent: Mediated Knowledge Retainment: Total Physicality: Embodied Agency: Active Number: Plural Metaphysical Agency: not engaged
⚡

V. Energy

Unaddressed: no physical content. (Kepler and Newton later supplied the physics.)

Attributes
Extent: not engaged Ontological Status: not engaged Conservation: not engaged Dispersibility: not engaged
⧉

VI. Information

Mathematical information is substantival, conserved, and continuous — eternal truths discovered and proved.

Attributes
Ontological Status: Substantival Cosmic Conservation: Conserved Personal Conservation: not engaged Granularity: Continuous

Personas with the nearest attribute fingerprint

Historical figures whose own classification on the same six-dimensional grid lands closest to this work's. Computed by attribute-agreement on coordinates both address.

100%
Apollonius of Perga
23/23
100%
Aristarchus of Samos
23/23
96%
Archimedes of Syracuse
22/23
96%
Ctesibius of Alexandria
22/23
96%
David Lewis
22/23
96%
Gaius Plinius Secundus (Pliny the Elder)
22/23

Computed school proximity

The work's attribute fingerprint scored against all schools using the same quiz scorer. Useful as a sanity check on the hand-curated embodiments above.

96%
Scientism
18/19
96%
Newtonianism
18/19
94%
Deism
18/19
94%
Philosophy of Science
18/19
94%
Analytic Philosophy
18/19
90%
Mechanism
17/19
89%
Milesian School
17/19
89%
Deontological Ethics
18/19

How Conics resolves each dilemma

12 resolved positions across 4 dimensions, including 3 distinctive where the majority of schools go the other way · 45 unaligned.

Each dimension is sorted so minority positions come first. Mainstream positions are folded into an expandable list.

Time · 9 dilemmas · 3 distinctive

Persistence, the future, and the direction of becoming.

Distinctive · only 10% of schools agree (20/208)
Do you really choose?
If the brain is a physical system and physical systems are governed by laws, then every choice is also a chain of causes — which raises the question of what was really left to choose.
Choice is real within a determined order — agency and determinism aren’t opposites.
On this view, the future is determined and you are genuinely choosing. Those aren't contradictory because the determination runs through you rather than around you: your reasoning, deliberation, and assent are the way the determined outcome gets settled. Choice is what it feels like from …
Roads not taken The future is open and you are a genuine origin of it. (69%) · Choice is structural illusion — every event is fixed by the prior state. (10%) · Even if the universe is undetermined, you are not the chooser. (6%)
Distinctive · only 10% of schools agree (20/208)
Are addicts responsible for their addiction?
Addiction looks from one angle like the textbook case of agency failing — a person doing what they don't, in any meaningful sense, want to do. From another angle it looks like agency at work in hard conditions. Which it is depends on what agency is.
The addict is genuinely responsible within a determined order.
On this view, the addict is acting within a determined order but is genuinely acting — making decisions, endorsing or resisting urges, seeking or refusing help. Responsibility attaches not because some uncaused choice happened, but because the addict is the kind of agent through which …
Roads not taken The addict could have chosen otherwise — that's why recovery is real. (69%) · The addict's behaviour is the outcome of causes; 'responsibility' is a useful fiction, not a metaphysical fact. (10%) · Even if the universe is undetermined, the addict isn't the chooser. (6%)
Distinctive · only 10% of schools agree (20/208)
Should we hold AI systems responsible for what they do?
When an autonomous AI takes an action that harms someone, the question of who or what is responsible — the developer, the operator, the model itself — turns on whether the model is the kind of thing that can be a responsible agent.
The AI can be a genuine agent within determined conditions — and therefore genuinely responsible.
On this view, what makes a being responsible is not indeterminism but the kind of process the being is. An AI that deliberates, considers consequences, can be given reasons, and modifies its behaviour on reflection is doing what responsible agency is, even if its underlying …
Roads not taken An AI without a free will is not the kind of thing that can be responsible. (69%) · An AI's behaviour is fully determined by training and input; 'responsibility' applies if at all to its makers. (10%) · Neither AIs nor anyone else are the locus of free agency; the question is the wrong one. (6%)
3 mainstream positions
How much weight do future people deserve? Future people are as real as you are — and their interests count for as much. 47% Is regret rational? The past is as real as the present; regret is a real attitude toward a real thing. 47% Do we owe extinct species something we cannot give them? Extinct species are as real as we are; they have standing. 47%
3 unaligned
Are natural resources fundamentally finite, or only practically so? Schools split: 47% / 26% / 15% — Could we owe future generations more than is materially possible to provide? Schools split: 47% / 26% / 15% — Is the universe running out of usable energy? Schools split: 47% / 26% / 15% —
Matter · 7 dilemmas, all mainstream
Does matter have intrinsic moral standing? — — Is the physical world fully real? — — Is the world created from nothing? — — Should we edit the human germline? — — What is a nation? — — What is money? — — What makes someone male or female? — —
Observer · 37 dilemmas, all mainstream
Could causation work backwards? Causation runs one way — the arrow of time is real and structural. 68% Is the asymmetry between memory and anticipation a real feature of time, or just of us? The asymmetry is real because time itself has a real direction. 68% Is the arrow of time a real feature of the cosmos, or only of how we describe it? The arrow is real and structural; the asymmetry isn't an artifact of description. 68% Should we trust expert testimony when we can't verify it? Trust expertise whose conclusions a competent mind can in principle reproduce. 31% Is religious revelation a real source of knowledge? Revelation is evaluable by reason — and not above it. 31% Does an LLM 'know' the things it correctly produces? An LLM can produce correct outputs but not reason to them; useful, not knowing. 31% Are coincidences ever more than coincidence? — — Are the dead morally present to the living? — — Are there indivisible units of experience? — — Can a civilization recover from collapse? — — Can prayer for someone far away affect them? — — Could a fetal brain organoid in a petri dish be conscious? — — Could an AI have a mind that matters? — — Do animals have moral standing comparable to humans? — — Does environmental harm in another country bind me morally? — — Does history have a direction or meaning? — — Does meditation reveal something genuinely timeless? — — Does prayer change God's mind? — — Does the second law of thermodynamics mean something morally? — — How is knowledge of reality produced? — — If a teleporter copied and destroyed you, would you have survived? — — Is divine omniscience compatible with human freedom? — — Is environmental damage ever truly permanent? — — Is genetic engineering of food stewardship or domination? — — Is memory stored or reconstructed? — — Is reality fundamentally digital? — — Is salvation, liberation, or fulfillment individual or communal? — — Is the late-stage dementia patient still the person their spouse married? — — Is truth universal, tradition-bound, situated, or constructed? — — Should we colonize space? — — What happens to "you" when you die? — — What is marriage? — — What is our place in nature? — — What kind of religious-theological authority does the tradition recognize? — — What makes someone the same person over time? — — When does a person begin? — — Who is the moral primary — the individual, the community, the cosmos, the class, or the species? — —
Information · 4 dilemmas, all mainstream
Could the dead, in principle, be brought back? — — Does deleting your data online destroy something real? — — Is anything truly lost when someone forgets? — — Is forgiveness ontologically possible? — —
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