Euclid
Greek · Philosophy · Other Treatise · Mathematics · Poetry
8 works · 37,676 aligned sentences
Catoptrics
This work is a geometrical treatise on the science of vision and light, focusing on the laws of reflection in plane, convex, and concave mirrors. Written in a structured format of mathematical propositions, the text begins by establishing the fundamental law of reflection, stating that the angle of incidence equals the angle of reflection. In the opening sections, the author defines the pathways of reflected visual rays and explains how images appear erect, inverted, or reversed. The middle section provides rigorous geometric proofs regarding the characteristics of images—such as their size, distance, and orientation—across different types of mirrors. The discussion then progresses to more complex phenomena, including multiple reflections using arrays of mirrors and the specific conditions of visibility when an eye is placed at various positions relative to a concave mirror. Finally, the work concludes with practical applications, demonstrating how to construct composite mirrors that produce multiple images and geometrically proving how a concave mirror can concentrate sunlight to ignite a fire.
Philosophy11 chunks · §pr-2–§29-301,363 aligned sentencesRead →Data
This work is a mathematical treatise that explores what it means for geometric elements to be "given" (data) and how their determination leads to the determination of other elements. Through a series of rigorous propositions and proofs, it demonstrates that if certain lengths, angles, areas, ratios, or positions are known, then other related geometric components are also uniquely determined. The discussion begins with foundational definitions and basic properties of lines and angles, gradually moving toward complex relationships in plane figures such as triangles and quadrilaterals. In the latter sections, the work addresses more intricate problems, proving that when the angle between two lines, the area they contain, and the difference of the squares on them are given, the lines themselves can be found. Ultimately, the treatise systematically categorizes the solvability of various geometric problems, establishing a robust framework for geometric analysis.
Philosophy1 chunks · §18.1-18.2128 aligned sentencesRead →Division of the Canon
This ancient Greek treatise on music theory presents a mathematical approach to defining intervals as numerical ratios and constructing a musical scale on a monochord (canon). It begins by establishing that pitch is determined by the frequency of movement and defines consonances as multiple or epimoric (superparticular) ratios. Through a series of geometric-style propositions, the work mathematically demonstrates the relations of major intervals, showing that the octave is a duple ratio (2:1), the fifth is 3:2, the fourth is 4:3, and the whole tone is 9:8, while proving that six whole tones are slightly larger than an octave. Finally, it provides practical instructions for dividing the monochord by geometric proportion to mark the fixed and movable notes of the "immutable system" (systema ametabolon). The work achieves a rigorous mathematical foundation for the sensory harmony of music.
Philosophy5 chunks · §pr-3–§18-20797 aligned sentencesRead →Elements
This work is a mathematical treatise that systematically and logically organizes the fundamental principles of geometry, both in planes and in three-dimensional space, as well as the theory of numbers. Throughout the text, the discussion proceeds in a rigorous demonstrative format, starting from a minimal set of definitions, postulates, and common notions. The early books deal with the properties of plane figures, the Pythagorean theorem, and the construction of inscribed and circumscribed polygons, followed by an exploration of similarity based on a general theory of proportion. In the middle section, the focus shifts to number theory—covering divisibility, prime numbers, geometric progressions, and perfect numbers—and leads into an intricate classification of incommensurable irrational magnitudes. The later books transition to solid geometry, establishing the properties of planes, three-dimensional solids, and the volume ratios of cones and spheres using the method of exhaustion. Finally, the work culminates in the geometric construction of the five regular Platonic solids inscribed in a sphere using the properties of the golden ratio, proving that no other regular polyhedra can exist.
Philosophy316 chunks · §1.def.1-1.def.23–§13.prop.18#332,783 aligned sentencesRead →Epigrams
This short poem (epigram) presents an algebraic riddle addressed to geometers, featuring a mule and a donkey comparing the amounts of their respective loads. In the work, the two animals carrying heavy burdens converse with each other, asking the reader to determine the original quantity of each load based on how the ratio changes when a portion of the load is transferred from one to the other. It is a work that expresses a classical mathematical puzzle in poetic form, combining brevity with intellectual amusement. Additionally, the end of the text includes philological notes in Latin regarding the transmission history of the epigram and the collated manuscripts.
Poetry1 chunks · §185 aligned sentencesRead →Fragments
This work is a collection of fragments that reconstructs several lost geometrical treatises by the ancient Greek mathematician Euclid, based on quotations from ancient commentators and surviving manuscript fragments. The text begins with an introduction to "On Divisions," which deals with dividing figures, and "Pseudaria" (Book of Fallacies), an educational work designed to help beginners detect geometrical errors. It then transitions to the core of the collection, detailing the "Porisms"—a concept positioned between theorems and problems—accompanied by numerous auxiliary proofs preserved by Pappus. These proofs demonstrate advanced propositions concerning collinearity, the composition of ratios, and the geometrical properties of circles and lines. In the latter section, the focus shifts to "Surface Loci," which extends two-dimensional relations into three-dimensional space, providing constructions and proofs of loci that trace conic sections such as parabolas and ellipses. The work concludes with propositions from "Conics," drawing on testimonies from Archimedes and Apollonius, thereby laying bare the sophisticated analytical and synthetic methods of ancient geometry.
Fragmentary Texts29 chunks · §1.1–§13.12,002 aligned sentencesRead →Phaenomena (alternative proof of b recension)
This work is a mathematical treatise on spherical astronomy, discussing the geometric proofs related to the motion of the ecliptic and parallel circles on the celestial sphere. The primary subject is the rigorous proof of how different arcs of the ecliptic rise and set over varying durations of time. The treatise begins by demonstrating through contradiction that when one point of a diameter on the ecliptic rises, the opposite point sets. It then establishes the relations between the horizon, the equator, and the tropics to compare the setting times of equal arcs along the ecliptic. By offering alternative, clearer proofs for existing astronomical propositions, it demonstrates that arcs closer to the summer solstice take longer to set than those closer to the equator. Finally, the work concludes with a geometric proof showing that for two equal and opposite arcs, the time one takes to leave the visible hemisphere equals the time the other takes to leave the invisible hemisphere.
Philosophy6 chunks · §1–§4389 aligned sentencesRead →Phaenomena (b recension)
This scientific work analyzes the movement of the ecliptic based on the ancient model of the celestial sphere through geometrical methods. The central theme of the treatise is to clarify the variation in time required for equal arcs of the ecliptic to pass below the horizon, specifically through the invisible hemisphere. The author demonstrates how this passage time relates to the distance of these arcs from the solstices and the celestial equator. Utilizing geometrical figures and theorems representing the rotation of the celestial sphere and its relationship with the horizon, the text develops rigorous proofs step by step. Ultimately, the work successfully explains the seemingly irregular movements of celestial bodies, such as the rising and setting of zodiacal signs, by anchoring them in precise mathematical and geometrical laws.
Philosophy1 chunks · §1129 aligned sentencesRead →