Four Elements in Plato’s Timaeus

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Plato’s treatment, in Timaeus, of the geometry of the four elements is explored. The claim that the elements are connected by a geometric proportion has been variously interpreted as either playful or obscurantist, but there has not yet been a treatment which both takes the claim seriously and grounds it in the essential structure of the elements, conceived in the Timaean manner as consisting of atoms of the same shape as four of the five regular solids.


“John Black does valuable service in his extensive analysis by attempting to understand one of its more puzzling protoscientific claims, that the four constituitive elements of the cosmos are ultimately related and bound to each other by some sort of geometric proportion. More than this, he seeks to comprehend Plato’s intended meaning not in terms of later developments in natural philosophy but rather, and quite properly, in terms of the ideas available to Plato himself from those earlier traditions he was expert at bringing together: in this case, Pythagorean ideas about the relations of substances to numbers and geometric shapes. . . . Black’s treatment of the physics and geometry of the elements in Timaeus is a discussion of great value. . . for its ability to illuminate important Platonic themes in the subsequent history of science. The essay is also accessibly and engagingly presented, a feature of no small significance on topics of such inherent difficulty.” – Stephen Strake

“One does not need to be a classicist, or philosopher, nor an adroit mathematician, to follow Black’s reasoning. There is a simplicity and elegance in this exposition of the Timaean puzzle that recommends this work in teaching. . . . a highly satisfying unpacking of ancient possibilities. Black’s application of Proclus’ ‘powers’ within the rubric of a ‘shared factor’ proportion amongst the elements is intellectually satisfying and thoroughly in keeping with Pythagorean tradition. The book is a keeper.” – Russell McNeil

Table of Contents

Table of contents:
Foreword; Preface
Timaean Elements
Geometric Means
Traditional Explanations of the Geometric Proportion
Polygonal Numbers
Polyhedral Numbers
The Solution of the Puzzle
Appendix 1: Algebraic Treatment of the Figurate Numbers
Appendix 2: A Perfect Geometric Proportion
Appendix 3: The Chemistry of Timaeus by E. M. Bruins (translated)
Bibliography; Index

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