Euclid's Optics, is a work on the geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 of vision written by the Greek mathematician Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 around 300 BC. The earliest surviving manuscript of Optics is in Greek and dates from the 10th century AD.

The work deals almost entirely with the geometry of vision, with little reference to either the physical or psychological aspects of sight. No Western scientist had previously given such mathematical attention to vision. Euclid's Optics influenced the work of later Greek, Islamic, and Western European Renaissance scientists and artists.

Historical significance

Writers before Euclid had developed theories of vision. However, their works were mostly philosophical in nature and lacked the mathematics that Euclid introduced in his Optics. Efforts by the Greeks prior to Euclid were concerned primarily with the physical dimension of vision. Whereas Plato

Plato

Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

 and Empedocles

Empedocles

Empedocles was a Greek pre-Socratic philosopher and a citizen of Agrigentum, a Greek city in Sicily. Empedocles' philosophy is best known for being the originator of the cosmogenic theory of the four Classical elements...

 thought of the visual ray as "luminous and ethereal emanation", Euclid’s treatment of vision in a mathematical way was part of the larger Hellenistic trend to quantify a whole range of scientific fields.

Because Optics contributed a new dimension to the study of vision, it influenced later scientists. In particular, Ptolemy

Ptolemy

Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

 used Euclid's mathematical treatment of vision and his idea of a visual cone in combination with physical theories in Ptolemy's Optics, which has been called "one of the most important works on optics written before Newton". Renaissance artists such as Brunelleschi

Filippo Brunelleschi

Filippo Brunelleschi was one of the foremost architects and engineers of the Italian Renaissance. He is perhaps most famous for inventing linear perspective and designing the dome of the Florence Cathedral, but his accomplishments also included bronze artwork, architecture , mathematics,...

, Alberti, and Dürer

Albrecht Dürer

Albrecht Dürer was a German painter, printmaker, engraver, mathematician, and theorist from Nuremberg. His prints established his reputation across Europe when he was still in his twenties, and he has been conventionally regarded as the greatest artist of the Northern Renaissance ever since...

 used Euclid's Optics in their own work on linear perspective

Perspective (graphical)

Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

.

Structure and method

Similar to Euclid's much more famous work on geometry, Elements

Euclid's Elements

Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

, Optics begins with a small number of definitions and postulates

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

, which are then used to prove

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

, by deductive reasoning

Deductive reasoning

Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

, a body of geometric propositions (theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s in modern terminology) about vision.

The postulates in Optics are:

Let it be assumed

1. That rectilinear rays proceeding from the eye diverge indefinitely;

2. That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;

3. That those things are seen upon which visuals rays fall and those things are not seen upon which visual rays do not fall;

4. That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;

5. That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;

6. That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;

7. That thing seen under more angles are seen more clearly.

The geometric treatment of the subject follows the same methodology as the Elements.

Content

According to Euclid, the eye sees objects that are within its visual cone. The visual cone is made up of straight lines, or visual rays, extending outward from the eye. These visual rays are discrete, but we perceive a continuous image because our eyes, and thus our visual rays, move very quickly. Because visual rays are discrete, however, it is possible for small objects to lie unseen between them. This accounts for the difficulty in searching for a dropped needle. Although the needle may be within one's field of view, until the eye's visual rays fall upon the needle, it will not be seen. Discrete visual rays also explain the sharp or blurred appearance of objects. According to postulate 7, the closer an object, the more visual rays fall upon it and the more detailed or sharp it appears. This is an early attempt to describe the phenomenon of optical resolution

Optical resolution

Optical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged.An imaging system may have many individual components including a lens and recording and display components...

.

Much of the work considers perspective, how an object appears in space relative to the eye. For example, in proposition 8, Euclid shows that the perceived size of an object is not related to its distance from the eye by a simple proportion.