return to Vignettes of Ancient Mathematics

Introduction
Propositions

Notes on translation: Euclid uses two expressions: διάστημα and ἀπόστημα, which will be translated as 'interval' and 'interval-away'. The scholia also use διάστασις and ἀπόστασις, which will be translated as 'distance' and 'distance-away'. It would be nice to use words with the same roots, but all alternatives, such as the 17th century 'distancy' would make the translation more obscure.

The word ὄψις will be translated as 'sight-line', while ἀκτίς will be translated as 'ray'.

Introduction

Optical Definitions of Euclid

Let it be hypothesized that straight-lines drawn out from the eye travel an interval1 of large magnitudes,

1Scholion: Or at distances and their sectionings off from one another

and that the figure enclosed by the sight-lines is a cone having its vertex at the eye and its base at the limits of the things seen,

and that these upon which the sight-lines fall are seen, and those upon which the sight-lines do not fall are not seen,

and that things seen enclosed by a larger angle appear larger, while things seen enclosed by a smaller angle appear smaller, and things seen enclosed by equal angles seem equal,

and that things seen by higher rays appear higher, while things seen by lower appear lower,

and similarly that things seen by rays more to the right appear more to the right, while those that appear by rays more to the left appear more to the left,

and those seen by more angles appear more precisely.

Propositions

Prop. 1. Of things seen none is seen together as a whole.

Prop. 2. Of equal magnitudes positioned at a distance those positioned nearer are seen more precisely.

Prop. 3. Each of things seen has some length of a distance-away where on coming-to-be there it is no longer seen.

Prop. 4. With equal distances also being on the same straight-line things seen from more distance appear smaller.

Prop. 5. Equal magnitudes unequally distant appear unequal, and always that positioned nearer to the eye appears larger.

Prop. 6. Of distances, parallels seen from a distance away appear unequally wide.

Prop. 7. Equal magnitudes that are on the same straight-line not positioned in succession to one another also appear unequal as unequally distant from the eye.

Prop 8. Equal and parallel magnitudes at an unequal distance from the eye are not seen proportionally to the distances.

Prop. 9. Rectangual magnitudes seen from a distance-away appear curved.

Prop. 10. Of planes positioned below the eye those further appear higher.

Prop. 11. Of planes positioned above the eye those further appear lower.

Prop. 47. There are certain locations, such that when the eye is place at them unequal magnitudes put together at the same point will appear equal to each of the unequal lines.

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