The Elements: definition, axiom, proof
The textbook that gathered all Greek geometry and number theory into a single chain of deduction: first twenty-three definitions, then five postulates and five common notions accepted without proof, and after that every statement derived only from what has already been proved. This was a full written record of what would later be called the axiomatic method (the building of a whole theory from a few accepted starting points).
Its author, Euclid, worked around 300 BCE. From this moment an argument about correctness stops being a clash of opinions and becomes an inspection of the steps of the derivation. The book remained the principal textbook for more than two thousand years.
It set the form in which physics and philosophy were later written: Newton opened his work with definitions and with axioms, or laws of motion, and Spinoza set out his ethics in geometrical order. From here European science inherited the habit of holding that a statement is worth exactly as much as the proof behind it.
Why this date
approximate. There are no exact dates for Euclid's life. The single early testimony about his time, Proclus (c. 450 CE), places him under the first Ptolemy; Britannica gives a floruit of c. 300 BCE, MacTutor cautiously proposes a life of about 325 to 265 BCE. The axis value is 300 BCE. The surviving text passed through later recensions, above all that of Theon of Alexandria (4th century), so the Elements as we know them are in part a reconstruction.
What grew out of it
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Analytic geometry
influenced · Confidence: medium
Analytic geometry grew on the Greek problem of construction that the Elements codified: Descartes divides problems into the same classes as the ancient geometers, refers to Euclid's constructions, and sets algebraic operations in correspondence with geometric constructions, lifting the old requirement that the product of two line segments must be an area.
Dispute: Historians differ on whether this continues ancient geometry or breaks with it. Descartes himself insisted that the ancients had wrongly restricted geometry to compass and straightedge, and demanded that any curve traced by a continuous motion be admitted; on that reading the Elements were for him a problem he outgrew rather than a foundation. Bos, in his study of Descartes, calls this outright a transformation of the very concept of construction. On the opposite reading the link stands: one can only outgrow what is already there, and Descartes' analysis is conducted in terms set by the Greek tradition.
Link sources
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
geometric problems, previously unsolved since ancient Greek times, to be tackled using algebraic techniques
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
established equivalences between algebraic operations and geometric constructions
- Domski M. «Descartes' Mathematics», Stanford Encyclopedia of Philosophy Verified 20 Sep 2026
The problem of bisecting a given angle is counted among planar problems, because, as detailed by Euclid in Elements I.9, to construct the line segment that divides a given angle into two equal parts, we construct (by compass) three circles of equal radius, and then (by straightedge) join the vertex of the angle with the point at which the circles intersect
- Domski M. Verified 20 Sep 2026
Moreover, in a 13 November 1629 letter written to Mersenne, Descartes refers to the second (1589) edition of Clavius's annotated version of Euclid's Elements
- Mahoney M. S. «Descartes: Mathematics and Physics», Dictionary of Scientific Biography (via Encyclopedia.com) Verified 20 Sep 2026
the Greek mathematicians had established the correspondence between addition and the geometrical operation of laying line lengths end to end
- Descartes R. «Discours de la méthode», 2 (1637) Verified 20 Sep 2026
Ces longues chaînes de raisons, toutes simples et faciles, dont les géomètres ont coutume de se servir pour parvenir à leurs plus difficiles démonstrations, m'avoient donné occasion de m'imaginer que toutes les choses qui peuvent tomber sous la connoissance des hommes s'entresuivent en même façon
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- «La Géométrie», Encyclopaedia Britannica Verified 20 Sep 2026
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Mathematical Principles of Natural Philosophy
influenced · Confidence: high
The Mathematical Principles of Natural Philosophy is built in Euclidean form: first a section of definitions, then the axioms, or laws of motion, and after them propositions derived from these and labelled, as in Euclid, either theorems or problems.
Dispute: The title of the book alludes not to Euclid but to Descartes. According to the Stanford Encyclopedia, Newton changed the title in 1686 to Philosophiae Naturalis Principia Mathematica in allusion to Descartes' Principia Philosophiae, that is as a challenge to Cartesian physics. What is Euclidean here is the architecture of the book, definitions, axioms, theorems and problems, not its name; the common claim that the title too points back to Euclid's Elements is not supported by the sources.
Link sources
- Grabiner J. «How Euclid once ruled the world», Plus Magazine, 9 2016 (2016) Verified 20 Sep 2026
In science, Isaac Newton's famous work Principia Mathematica clearly demonstrates Euclid's influence.
- Grabiner J. (2016) Verified 20 Sep 2026
Newton called his famous laws of motion 'axioms' and deduced his law of gravitation in the form of two mathematical theorems.
- Smith G. «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy Verified 20 Sep 2026
In keeping with the Euclidean tradition, the propositions mathematically derived from the laws of motion are labeled either as theorems or as problems.
- Smith G. Verified 20 Sep 2026
The Principia opens with a section called 'Definitions' that includes Newton's discussion of absolute space, time, and motion.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Grabiner J. «How Euclid once ruled the world», Plus Magazine, 9 2016 (2016) Verified 20 Sep 2026
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Non-Euclidean geometry: denial of the fifth postulate
influenced · Confidence: high
The fifth postulate of the Elements and the centuries of attempts to derive it from the other axioms set the very problem that Lobachevsky and Bolyai solved in the 1820s by the opposite move, adopting the denial of the postulate and meeting no contradiction.
Dispute: Priority is contested, and priority of publication diverges from priority of discovery. Lobachevsky printed first (1829-1830), Bolyai in 1832, while Gauss kept his conclusions to himself from the 1790s, which the Stanford Encyclopedia of Philosophy explains thus: 'he refrained from publishing for fear of scandal'. The Dictionary of Scientific Biography awards precedence to Lobachevsky on the fact of printing: 'Since Gauss did not publish his work on the subject, and since Bolyai published only at a later date, Lobachevsky clearly holds priority'. Wolfe separates the two things: Lobachevsky published earlier, but on Wolfe's account made the discovery later than Bolyai did, 'But there is ample evidence that he made his discoveries later than Bolyai made his', where the pronoun refers to Lobachevsky and not to Gauss. Awareness of one another needs a date attached to it. At the time of the Kazan publication of 1829-1830 it was unknown outside Kazan: MacTutor writes 'Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication'. Later this changed: the German exposition of 1840 reached Gauss, 'This last publication greatly impressed Gauss', while Bolyai read Lobachevsky in 1848 and suspected Gauss stood behind it. Lobachevsky, on Wolfe's account, never learned of Bolyai at all: 'there seems to be no evidence that Lobachewsky ever heard of Bolyai'. The role of the precursors is disputed as well: Saccheri (1733) and Lambert (1766) derived a large part of the theorems of this geometry, and Papadopoulos and Théret call Lambert's text the closest to hyperbolic geometry among everything written before Lobachevsky, yet neither of them accepted it as a system of its own.
Link sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
In the 1820's, Nikolai I. Lobachevsky (b. 1793, d. 1856) and Janos Bolyai (b. 1802, d. 1860) independently tackled this question in a radically new way.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Girolamo Saccheri (b. 1667, d. 1733) tried reductio. He inferred a long series of propositions from the negation of Euclid's Postulate, until he reached one which he pronounced 'repugnant to the nature of the straight line'.
- Harold E. Wolfe, Introduction to Non-Euclidean Geometry, ch. 3 (Dryden Press) (1945) Verified 20 Sep 2026
Slowly but surely they had directed the speculations of geometers to the point where the discovery of Non-Euclidean Geometry could not long be delayed.
- Mathematicians Reconsider Euclid's Parallel Postulate, Science and Its Times (via Encyclopedia.com) (2001) Verified 20 Sep 2026
Work on the parallel postulate in the eighteenth century set the stage for the development of non-Euclidean geometries in the nineteenth century.
- Nikolai Ivanovich Lobachevsky, MacTutor History of Mathematics Archive (2000) Verified 20 Sep 2026
This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non euclidean geometry which is just simply false.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Node sources
- «Elements», Encyclopaedia Britannica Verified 20 Sep 2026
Elements, treatise on geometry and mathematics written by the Greek mathematician Euclid (flourished 300 bce).
- «Elements», Encyclopaedia Britannica Verified 20 Sep 2026
a foundation that Euclid began in Book I with 23 definitions (such as "A point is that which has no part" and "A line is a length without breadth"), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions.
- «Elements», Encyclopaedia Britannica Verified 20 Sep 2026
It set a standard for deductive reasoning and geometric instruction that persisted, practically unchanged, for more than 2,000 years.
- MacTutor History of Mathematics, «Euclid of Alexandria» Verified 20 Sep 2026, quotation assembled from fragments
Not much younger than these [pupils of Plato] is Euclid, who put together the 'Elements'... This man lived in the time of the first Ptolemy
- MacTutor History of Mathematics, «Euclid of Alexandria» Verified 20 Sep 2026
The Elements begins with definitions and five postulates.
- Grabiner J. «How Euclid once ruled the world», Plus Magazine, 9 2016 (2016) Verified 20 Sep 2026
In science, Isaac Newton's famous work Principia Mathematica clearly demonstrates Euclid's influence.
- Grabiner J. (2016) Verified 20 Sep 2026
even wrote an Ethics Demonstrated in Geometrical Order, with explicitly labelled axioms and definitions
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.