Ratio and proportion

Use in the book

The arithmetic passage connects harmony with numerical ratios. Aristoxenus’ excerpt says that numbers stand in a ratio to one another and presents number as extending beyond commercial calculation. The report under Pythagoras’ name identifies the proportions among numbers as harmonies and assigns them a role among principles. In the Timaeus quotation, proportion brings the world’s constituents into agreement and friendship.

Places to read

1. Arith.2aNumerical ratios ground the interpretation of harmony.
2. Arith.6Numbers stand in relations to one another.
3. I.10.12.10Proportions are called harmonies and treated as principles.
4. I.20.9aProportion brings the world’s constituents into agreement.

Translation in this edition

“Ratio” is used when a numerical relation is meant. It should not be replaced by the usual philosophical rendering “reason” solely because the Greek word is logos.

Proportion and commensurability

Summetros means proportionate or commensurate; summetria names the corresponding relation. The Pythagorean report at I.10.12.10 makes numbers and their proportions, called harmonies, principles. “Proportions” fits this occurrence. A stricter geometrical claim about a common measure should be made only where the context establishes that technical sense.

Use in the book

The painters in B 232 take more of some pigments and less of others to generate varied likenesses. Proportion is therefore integral to the explanatory analogy, not an optional embellishment. B 96 then gives a specific eight-part mixture: two parts of Nestis' contribution and four of Hephaestus in the earth's crucibles. Reading the remaining two as earth yields the familiar two–two–four account, but the arithmetic completion must be distinguished from the words explicitly assigning the named shares.

B 98 describes a different case. Earth meets Hephaestus, rain, and aither in near-equal measure, with slight variations permitted; blood and other flesh result. This is not a universal requirement that every successful compound have exactly equal quantities. It is a local explanation of bodily materials.

The water-catcher passage uses measure dynamically. As air or aither enters, a due measure of water moves, and the body's outgoing breath is said to equal what enters. The comparison makes balanced replacement intelligible without furnishing a modern equation or experiment. Taken together, these passages show why proportion is distinct from the equal standing of the roots and from mixture considered only as a process.

Places to read

1. B 23.3–8Painters vary the amounts of pigments to produce many forms.
2. B 96The bone account names two of eight parts for Nestis and four for Hephaestus; earth supplies the remaining ingredient.
3. B 98Near-equal ingredients with small variation produce blood and flesh.
4. B 100.15–252Due measures and equal outgoing breath organise the water-catcher and bodily comparison.

Translation in this edition

The edition renders the numerical words plainly and does not call the account stoichiometry. The eight-part total makes a remainder calculable, but “two parts earth” should be presented as an interpretation of the recipe rather than an additional verbatim clause.

Use in the book

Aristotle's On the Parts of Animals says that Empedocles, led by the truth, sometimes reaches an account in which a thing's being and nature are its logos. Bone is the example: it is not any one constituent, two, three, or all merely listed, but the proportion of their mixture. This is substantial recognition of an explanatory achievement, yet it is recognition within Aristotle's own argument that nature is a principle more fundamental than matter. It does not establish that Empedocles formulated Aristotle's theory of form. The tissue recipes in Aëtius give the claim concrete scope: flesh, sinew, and bone are differentiated by their composition, with nails and secretions additionally explained by processes acting on prior materials. Their numerical details are textually repaired, so they cannot all be treated as independent exact confirmations.

In On the Soul, Aristotle explores the cost of turning such proportions into a definition of soul. Flesh and bone do not share one ratio; if ratio is soul, a creature would possess many souls distributed through its parts. He then asks whether soul is the ratio or something present in limbs, whether Love causes any mixture or the mixture in accord with ratio, and whether Love is itself that ratio or something besides it. These are questions and conditional consequences. Reporting them as Empedocles' settled doctrine that soul, Love, and ratio are identical would turn an objection into a confession. Pseudo-Aristotle's On Breath presents a further problem about how common compositional proportions distinguish kinds of animal.

Ratio also does not settle the separate dispute about fusion. At A43 Aristotle compares preserved small constituents to a wall of bricks and stones. A determinate wall can have proportions too. Thus the introduction's assertion that a brick wall has no ratio is logically insufficient to dissolve the criticism. What remains to be established is whether the organized relation among constituents explains a genuine new compound, mere juxtaposition, or some account not adequately captured by Aristotle's alternatives. The evidence makes proportion central; it does not eliminate every interpretive difficulty.

The same conditional argument explicitly calls the ratio of mixture harmonia, harmony or attunement, and soul. Aristotle is testing the consequences of that identification; the clause does not independently establish an unqualified Empedoclean equation. This bodily harmony is also distinct in context from the literary arrangement discussed by Dionysius at A26.

Places to read

1. A 78 (testimony)2Aristotle, On the Parts of Animals. Recognizes bone's ratio of mixture as more explanatory than a mere inventory of constituents.
2. A 78 (polemical)2Aristotle, On the Soul, first passage. Argues that different tissue ratios would multiply souls if soul were ratio.
3. A 78 (polemical)2Aristotle, On the Soul, second passage. Asks how ratio, soul, and Love relate and whether Love causes chance or proportionate mixture.
4. A 78 (polemical)2Pseudo-Aristotle, On Breath. Questions how the same mixture ratio could differentiate animal kinds.
5. A 43 (polemical)Aristotle, De gen. et corr. B 7. Raises the logically distinct issue of juxtaposition rather than genuine mixture.
6. A 70 (testimony)2Aëtius. Uses logos for the male and female principle present in proportionately mixed plants.
7. A 61 (testimony)Aëtius, ps.-Plutarch and Stobaeus branches. The contradictory terms in a two-to-one distance comparison illustrate how a ratio depends on its relata.
8. A 26 (reception)Dionysius of Halicarnassus, De compositione verborum 22. Uses harmony in a literary context, distinct from Aristotle’s conditional discussion of bodily mixture at A782.

Translation in this edition

The ratios in the Aëtian recipes and Aristotle's interpretation are different kinds of evidence. The first is often reconstructed wording; the second is a critical conceptual appropriation. The edition's “number” slogan and anti-brick-wall argument should be recognized as modern interpretations rather than ancient conclusions.