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In this textbook written by Euclid, born in Alexandria, Egypt around 300 BC. He gathers all the known axioms at the time and uses them to create new theorems by combining them in such a way that he was able to extrapolate these new theorems. It was a new form of proving axioms to be either true or false, this technique is still used today. This was the first time in the ancient world that anyone had done this and it brought about a new way of thinking and problem solving. The advancements made by his works pales in comparison too the advancement made by humanity using Euclid's technique illustrated by his work the element. This textbook is still used too this day and is the quintessential guide to finding proof and demonstrating that proof.

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The Elements ( Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions ( theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century.

Euclid's Elements has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.

Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. The standard textbook for this purpose was none other than Euclid's The Elements.

Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians.

Proclus (412-485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".

Pythagoras ( c. 570-495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( c. 470-410 BC, not the better known Hippocrates of Kos) for book III, and Eudoxus of Cnidus ( c. 408-355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. Other similar works are also reported to have been written by Theudius of Magnesia, Leon, and Hermotimus of Colophon.

In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition.

Although Euclid was known to Cicero, for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760; this version was translated into Arabic under Harun al Rashid ( c. 800). The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete.

The first printed edition appeared in 1482 (based on Campanus of Novara's 1260 edition), and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533[citation needed ]. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley.

Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available).

Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text.

Also of importance are the scholia, or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study.

The Elements is still considered a masterpiece in the application of logic to mathematics. In historical context, it has proven enormously influential in many areas of science. Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, Albert Einstein and Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. Mathematicians and philosophers, such as Thomas Hobbes, Baruch Spinoza, Alfred North Whitehead, and Bertrand Russell, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight". Edna St. Vincent Millay wrote in her sonnet " Euclid alone has looked on Beauty bare", "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".

The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain the cornerstone of mathematics.

One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate. In Book I, Euclid lists five postulates, the fifth of which stipulates

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate ( elliptic geometry). If one takes the fifth postulate as a given, the result is Euclidean geometry.[citation needed ]

Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by Eudoxus, and proves properties such as "alternation" (if a : b :: c : d, then a : c :: b : d).

Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures.

Book 7 deals with elementary number theory: divisibility, prime numbers and their relation to composite numbers, Euclid's algorithm for finding the greatest common divisor, finding the least common multiple.

Book 8 deals with the construction and existence of geometric sequences of integers.

Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even perfect numbers.

Book 10 proves the irrationality of the square roots of non-square integers (e.g. ) and classifies the square roots of incommensurable lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples.[20]

Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of parallelepipeds.

Book 12 studies the volumes of cones, pyramids, and cylinders in detail by using the method of exhaustion, a precursor to integration, and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a sphere is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.

Book 13 constructs the five regular Platonic solids inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

Euclid's method and style of presentation

* "To draw a straight line from any point to any point."
* "To describe a circle with any center and distance."

Euclid, Elements, Book I, Postulates 1 & 3.

Euclid's axiomatic approach and constructive methods were widely influential.

Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge. His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.

As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases.

Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles, the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals.

The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.

No indication is given of the method of reasoning that led to the result, although the does provide instruction about how to approach the types of problems encountered in the first four books of the Elements. Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.

Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.

For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.2 and I.3 can be proved trivially by using superposition.

Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."

It was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being

The spurious Book XV was probably written, at least in part, by Isidore of Miletus. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.

1460s, Regiomontanus (incomplete)

1482, Erhard Ratdolt (Venice), first printed edition

1533, editio princeps by Simon Grynäus

1557, by Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)

1572, Commandinus Latin edition

1574, Christoph Clavius

1505, Bartolomeo Zamberti (Latin)

1543, Niccolò Tartaglia (Italian)

1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)

1558, Johann Scheubel (German)

1562, Jacob Kündig (German)

1562, Wilhelm Holtzmann (German)

1564-1566, Pierre Forcadel de Béziers (French)

1570, Henry Billingsley (English)

1572, Commandinus (Latin)

1575, Commandinus (Italian)

1576, Rodrigo de Zamorano (Spanish)

1594, Typographia Medicea (edition of the Arabic translation of The Recension of Euclid's "Elements"

1604, Jean Errard de Bar-le-Duc (French)

1606, Jan Pieterszoon Dou (Dutch)

1607, Matteo Ricci, Xu Guangqi (Chinese)

1613, Pietro Cataldi (Italian)

1615, Denis Henrion (French)

1617, Frans van Schooten (Dutch)

1637, L. Carduchi (Spanish)

1639, Pierre Hérigone (French)

1651, Heinrich Hoffmann (German)

1651, Thomas Rudd (English)

1660, Isaac Barrow (English)

1661, John Leeke and Geo. Serle (English)

1663, Domenico Magni (Italian from Latin)

1672, Claude François Milliet Dechales (French)

1680, Vitale Giordano (Italian)

1685, William Halifax (English)

1689, Jacob Knesa (Spanish)

1690, Vincenzo Viviani (Italian)

1694, Ant. Ernst Burkh v. Pirckenstein (German)

1695, Claes Jansz Vooght (Dutch)

1697, Samuel Reyher (German)

1702, Hendrik Coets (Dutch)

1705, Charles Scarborough (English)

1708, John Keill (English)

1714, Chr. Schessler (German)

1714, W. Whiston (English)

1720s, Jagannatha Samrat (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)

1731, Guido Grandi (abbreviation to Italian)

1738, Ivan Satarov (Russian from French)

1744, Mårten Strömer (Swedish)

1749, Dechales (Italian)

1749, Methodios Anthrakitis (Μεθόδιος Ανθρακίτης) (Greek)

1745, Ernest Gottlieb Ziegenbalg (Danish)

1752, Leonardo Ximenes (Italian)

1756, Robert Simson (English)

1763, Pibo Steenstra (Dutch)

1768, Angelo Brunelli (Portuguese)

1773, 1781, J. F. Lorenz (German)

1780, Baruch Schick of Shklov (Hebrew)

1781, 1788 James Williamson (English)

1781, William Austin (English)

1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)

1795, John Playfair (English)

1803, H.C. Linderup (Danish)

1804, François Peyrard (French). Peyrard discovered in 1808 the Vaticanus Graecus 190, which enables him to provide a first definitive version in 1814-1818

1807, Józef Czech (Polish based on Greek, Latin and English editions)

1807, J. K. F. Hauff (German)

1818, Vincenzo Flauti (Italian)

1820, Benjamin of Lesbos (Modern Greek)

1826, George Phillips (English)

1828, Joh. Josh and Ign. Hoffmann (German)

1828, Dionysius Lardner (English)

1833, E. S. Unger (German)

1833, Thomas Perronet Thompson (English)

1836, H. Falk (Swedish)

1844, 1845, 1859, P. R. Bråkenhjelm (Swedish)

1850, F. A. A. Lundgren (Swedish)

1850, H. A. Witt and M. E. Areskong (Swedish)

1862, Isaac Todhunter (English)

1865, Sámuel Brassai (Hungarian)

1873, Masakuni Yamada (Japanese)

1880, Vachtchenko-Zakhartchenko (Russian)

1897, Thyra Eibe (Danish)

1901, Max Simon (German)

1907, František Servít (Czech)

1908, Thomas Little Heath (English)

1939, R. Catesby Taliaferro (English)

1953, 1958, 1975, Evangelos Stamatis (Ευάγγελος Σταµάτης) (Modern Greek)

1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian)

2009, Irineu Bicudo (Portuguese)

2019, Ali Sinan Sertöz (Turkish)

2022, Ján Čižmár (Slovak)

Euclid's Elements - All thirteen books complete in one volume, Based on Heath's translation, Green Lion Press ISBN 1-888009-18-7.

The Elements: Books I-XIII - Complete and Unabridged, (2006) Translated by Sir Thomas Heath, Barnes & Noble ISBN 0-7607-6312-7.

The Thirteen Books of Euclid's Elements, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3)

Euclid's Elements Redux, Volume 1, contains books I-III, based on John Casey's translation.

Euclid's Elements Redux, Volume 2, contains books IV-VIII, based on John Casey's translation.

1. ^ Wilson 2006, p. 278 states, "Euclid's Elements subsequently became the basis of all mathematical education, not only in the Roman and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written."

2. ^ Boyer 1991, p. 100 notes, "As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written - the Elements (Stoichia) of Euclid".

3. ^ Boyer 1991, p. 119 notes, "The Elements of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the Elements appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's Elements".

4. ^ Bunt, Jones & Bedient 1988, p. 142 state, "the Elements became known to Western Europe via the Arabs and the Moors. There, the Elements became the foundation of mathematical education. More than 1000 editions of the Elements are known. In all probability, it is, next to the Bible, the most widely spread book in the civilization of the Western world."

5. One older work claims Adelard disguised himself as a Muslim student to obtain a copy in Muslim Córdoba. However, more recent biographical work has turned up no clear documentation that Adelard ever went to ^ Muslim-ruled Spain, although he spent time in Norman-ruled Sicily and Crusader-ruled Antioch, both of which had Arabic-speaking populations. Charles Burnett, Adelard of Bath: Conversations with his Nephew (Cambridge, 1999); Charles Burnett, Adelard of Bath (University of London, 1987).

6. ^ Boyer 1991, pp. 118-119 writes, "In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's Elements include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, . It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [...] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A.D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge.

1. ^ The Earliest Surviving Manuscript Closest to Euclid's Original Text (Circa 850); an image Archived 2009-12-20 at the Wayback Machine of one page

2. ^ Murdoch, John E. (1967). "Euclides Graeco-Latinus: A Hitherto Unknown Medieval Latin Translation of the Elements Made Directly from the Greek". Harvard Studies in Classical Philology. 71: 249. doi:10.2307/310767.

3. ^ Andrew., Liptak (2 September 2017). "One of the world's most influential math texts is getting a beautiful, minimalist edition". The Verge.

4. ^ Herschbach, Dudley. "Einstein as a Student" (PDF). Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA. p. 3. Archived from the original (PDF) on 2009-02-26.: about Max Talmud visited on Thursdays for six years.

5. ^ Prindle, Joseph. "Albert Einstein - Young Einstein". www.alberteinsteinsite.com. Archived from the original on 10 June 2017.

6. ^ "JNUL Digitized Book Repository". huji.ac.il. 22 June 2009. Archived from the original on 22 June 2009.

Alexanderson, Gerald L.; Greenwalt, William S. (2012), "About the cover: Billingsley's Euclid in English", Bulletin of the American Mathematical Society, New Series, 49 (1): 163-167, doi:10.1090/S0273-0979-2011-01365-9

Artmann, Benno: Euclid - The Creation of Mathematics. New York, Berlin, Heidelberg: Springer 1999, ISBN 0-387-98423-2

Boyer, Carl B. (1991). "Euclid of Alexandria". A History of Mathematics (Second ed.). John Wiley & Sons. ISBN 0-471-54397-7.

Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1988). The Historical Roots of Elementary Mathematics. Dover.

Busard, H.L.L. (2005). "Introduction to the Text". Campanus of Novara and Euclid's Elements. Stuttgart: Franz Steiner Verlag. ISBN 978-3-515-08645-5.

Callahan, Daniel; Casey, John (2015). Euclid's "Elements" Redux.

Hartshorne, Robin (2000). Geometry: Euclid and Beyond (2nd ed.). New York, NY: Springer. ISBN 9780387986500.

Heath, Thomas L. (1956a). The Thirteen Books of Euclid's Elements. Vol. 1. Books I and II (2nd ed.). New York: Dover Publications. OL 22193354M.

Heath, Thomas L. (1956b). The Thirteen Books of Euclid's Elements. Vol. 2. Books III to IX (2nd ed.). New York: Dover Publications. OL 7650092M.

Heath, Thomas L. (1956c). The Thirteen Books of Euclid's Elements. Vol. 3. Books X to XIII and Appendix (2nd ed.). New York: Dover Publications. OCLC 929205858. Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.

Ketcham, Henry (1901). The Life of Abraham Lincoln. New York: Perkins Book Company.

Nasir al-Din al-Tusi (1594). Kitāb taḥrīr uṣūl li-Uqlīdus [The Recension of Euclid's "Elements"] (in Arabic).

Reynolds, Leighton Durham; Wilson, Nigel Guy (9 May 1991). Scribes and scholars: a guide to the transmission of Greek and Latin literature (2nd ed.). Oxford: Clarendon Press. ISBN 978-0-19-872145-1.

Sarma, K.V. (1997). Selin, Helaine (ed.). Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. ISBN 978-0-7923-4066-9.

Servít, František (1907). Eukleidovy Zaklady (Elementa) [Euclid's Elements] (PDF) (in Czech).

Sertöz, Ali Sinan (2019). Öklidin Elemanlari: Ciltli [Euclid's Elements] (in Turkish). Tübitak. ISBN 978-605-312-329-3.

Toussaint, Godfried (1993). "A new look at euclid's second proposition". The Mathematical Intelligencer. 15 (3): 12-24. doi:10.1007/BF03024252. ISSN 0343-6993. S2CID 26811463.

Waerden, Bartel Leendert (1975). Science awakening. Noordhoff International. ISBN 978-90-01-93102-5.

Wilson, Nigel Guy (2006). Encyclopedia of Ancient Greece. Routledge.

Euklid (1999). Elementi I-VI. Translated by Hudoletnjak Grgić, Maja. KruZak. ISBN 953-96477-6-2.

Clark University Euclid's elements

Multilingual edition of Elementa in the Bibliotheca Polyglotta

Euclid (1997) [c. 300 BC]. David E. Joyce (ed.). "Elements". In HTML with Java-based interactive figures.

Heath's English translation (HTML, without the figures, public domain) (accessed February 4, 2010)

Web adapted version of Byrne's Euclid designed by Nicholas Rougeux

Video adaptation, animated and explained by Sandy Bultena, contains books I-VII.

The First Six Books of the Elements by John Casey and Euclid scanned by Project Gutenberg.

Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)

Sir Thomas More's manuscript

Latin translation by Aethelhard of Bath

Euclid Elements - The original Greek text Greek HTML

Clay Mathematics Institute Historical Archive - The thirteen books of Euclid's Elements copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD

Kitāb Taḥrīr uṣūl li-Ūqlīdis Arabic translation of the thirteen books of Euclid's Elements by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by Islamic Heritage Project.

Euclid's Elements Redux, an open textbook based on the Elements

1607 Chinese translations reprinted as part of Siku Quanshu, or "Complete Library of the Four Treasuries."

Elements with Highlights Interactive rendering of all 13 books

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The Elements ( Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions ( theorems and constructions), and mathematical proofs of the propositions.

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Euclid's postulate on how to make a hexagon

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Euclid's postulate on how to make a hexagon

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This is how Euclid proposes that one should make a hexagon with only a compass and a straight edged ruler. Euclid was the master of two dimensional geometry of his time, he had all known previous knowledge of planar geometry and used those axioms to test others and create new postulates.

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Though this is not a depiction of Euclid teaching his students, it is however an illustration of a woman teaching her students using Euclid's Elements. More than a thousand years later his textbook would still be used, all the way till today, using tools crafted after the same ones Euclid would have used to teach his own students.

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Born 287 BC, in Sicily, Greece. In the image above is Archimedes in his bath during the moment he had his famous "Eureka" moment. It was then that he had a series of realizations about how to solve a problem the king had set him out too solve, little did he know how useful was the method he had created and all using a widely known and understood too, the fulcrum and leaver. He discovered by placing an object on either side of a leaver and changing the position of the fulcrum, he could understand the relationship between two objects and their mass. First he used it to solve the kings problem and was able too deduce that the crown he had a craftsman make for him was cut by silver and the rest of the gold pocketed by the craftsman. He then went on to discover the relationship between a cone, a sphere and a rectangle. For the fist time ever man could understand the relationship of those shapes and their mass. This was something that we can easily solve with algebra but he did it without in a brute force method. He preformed algebraic equations without the use of algebra. His method was lost throughout the ages several times but eventually in 1800s his writings about it were discovered and lost yet again. In the early 90's the writings were found again and reconstructed using advanced technologies so now humanity now knows for sure what Archimedes method was, in his own words.

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One famous quote of Archimedes is "Give me a leaver long enough and a fulcrum on which to place it and I shall move the world." Indeed while the idea is completely impossible in reality, the concept of using leavers and fulcrums to move things that one would believe to be impossible. To Archimedes, the leaver and fulcrum were his most prized tools.

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Born in France, 1596. Before I explain the math of Descartes it is important to first explain a little about the man too better understand how he came about his mathematical discoveries. He would become a very multifaceted individual. A philosopher,  scientist, biologist and mathematician. When he was in twenties he suffered an episode while dreaming where he believed he had been given a vision where a divine spirit revealed to him a new philosophy. This philosophy was a marriage between mathematical method and philosophy, he believed that by finding some core truth, all secrets of the universe could be laid bare. His search for these truths lead him to many discoveries in all fields of science. His most prized discovery to the field of algebra and geometry was Cartesian Geometry and the Cartesian graph.  What was very important was he finally brought algebraic equations into geometry, this was a turning point for geometry as a discipline. This provided a basis for the development of calculus. His works would be the guiding force behind Newton. He even created the rule of signs and made strides in the science of optics. Truly we would not be where we are without the accomplishments of Descartes, one would be hard pressed to find someone with more influence over such a wide array of disciplines.

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René Descartes ( day-KART or DAY-kart; French: [ʁənedekaʁt] ; Renatus Cartesius; 31 March 1596 - 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, later becoming a central intellectual of the Dutch Golden Age. Although he served a Protestant state and was later counted as a deist by critics, Descartes was Roman Catholic.

Many elements of Descartes' philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into matter and form; second, he rejected any appeal to final ends, divine or natural, in explaining natural phenomena. In his theology, he insists on the absolute freedom of God's act of creation. Refusing to accept the authority of previous philosophers, Descartes frequently set his views apart from the philosophers who preceded him. In the opening section of the Passions of the Soul, an early modern treatise on emotions, Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before." His best known philosophical statement is "cogito, ergo sum" ("I think, therefore I am"; French: Je pense, donc je suis), found in Discourse on the Method (1637, in French and Latin, 1644) and Principles of Philosophy (1644, in Latin, 1647 in French). The statement has either been interpretated as a logical syllogism or as an intuitive thought.

Descartes has often been called the father of modern philosophy, and is largely seen as responsible for the increased attention given to epistemology in the 17th century. He laid the foundation for 17th-century continental rationalism, later advocated by Spinoza and Leibniz, and was later opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, and Hume. The rise of early modern rationalism-as a systematic school of philosophy in its own right for the first time in history-exerted an influence on modern Western thought in general, with the birth of two rationalistic philosophical systems of Descartes ( Cartesianism) and Spinoza ( Spinozism). It was the 17th-century arch-rationalists like Descartes, Spinoza, and Leibniz who have given the " Age of Reason" its name and place in history. Leibniz, Spinoza, and Descartes were all well-versed in mathematics as well as philosophy, and Descartes and Leibniz contributed to science as well.

Descartes' Meditations on First Philosophy (1641) continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system was named after him. He is credited as the father of analytic geometry-used in the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution.

Coat of arms of the Descartes family.

The house where Descartes was born in La Haye en Touraine

René Descartes was born in La Haye en Touraine, Province of Touraine (now Descartes, Indre-et-Loire), France, on 31 March 1596. In May 1597, his mother Jeanne Brochard, died a few days after giving birth to a still-born child. Descartes' father, Joachim, was a member of the Parlement of Rennes at Rennes. René lived with his grandmother and with his great-uncle. Although the Descartes family was Roman Catholic, the Poitou region was controlled by the Protestant Huguenots. In 1607, late because of his fragile health, he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche, where he was introduced to mathematics and physics, including Galileo's work. While there, Descartes first encountered hermetic mysticism. After graduation in 1614, he studied for two years (1615-16) at the University of Poitiers, earning a Baccalauréat and in canon and civil law in 1616, in accordance with his father's wishes that he should become a lawyer. From there, he moved to Paris.

Graduation registry for Descartes at the University of Poitiers, 1616

I entirely abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way to derive some profit from it.

In accordance with his ambition to become a professional military officer in 1618, Descartes joined, as a mercenary, the Protestant Dutch States Army in Breda under the command of Maurice of Nassau, and undertook a formal study of military engineering, as established by Simon Stevin. Descartes, therefore, received much encouragement in Breda to advance his knowledge of mathematics. In this way, he became acquainted with Isaac Beeckman, the principal of a Dordrecht school, for whom he wrote the Compendium of Music (written 1618, published 1650).

While in the service of the Catholic Duke Maximilian of Bavaria from 1619, Descartes was present at the Battle of the White Mountain near Prague, in November 1620.

According to Adrien Baillet, on the night of 10-11 November 1619 ( St. Martin's Day), while stationed in Neuburg an der Donau, Descartes shut himself in a room with an "oven" (probably a cocklestove) to escape the cold. While within, he had three dreams, and believed that a divine spirit revealed to him a new philosophy. However, it is speculated that what Descartes considered to be his second dream was actually an episode of exploding head syndrome. Upon exiting, he had formulated analytic geometry and the idea of applying the mathematical method to philosophy. He concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life's work. Descartes also saw very clearly that all truths were linked with one another, so that finding a fundamental truth and proceeding with logic would open the way to all science. Descartes discovered this basic truth quite soon: his famous " I think, therefore I am."

In 1620, Descartes left the army. He visited Basilica della Santa Casa in Loreto, then visited various countries before returning to France, and during the next few years, he spent time in Paris. It was there that he composed his first essay on method: Regulae ad Directionem Ingenii (Rules for the Direction of the Mind). He arrived in La Haye in 1623, selling all of his property to invest in bonds, which provided a comfortable income for the rest of his life. Descartes was present at the siege of La Rochelle by Cardinal Richelieu in 1627. In the autumn of that year, in the residence of the papal nuncio Guidi di Bagno, where he came with Mersenne and many other scholars to listen to a lecture given by the alchemist, Nicolas de Villiers, Sieur de Chandoux, on the principles of a supposed new philosophy, Cardinal Bérulle urged him to write an exposition of his new philosophy in some location beyond the reach of the Inquisition.

In Amsterdam, Descartes lived at Westermarkt 6 (Maison Descartes, left).

Title page of "Principia philosophiae" (Principles of Philosophy), 1656

Descartes returned to the Dutch Republic in 1628. In April 1629, he joined the University of Franeker, studying under Adriaan Metius, either living with a Catholic family or renting the Sjaerdemaslot. The next year, under the name "Poitevin", he enrolled at Leiden University, which at the time was a Protestant University. He studied both mathematics with Jacobus Golius, who confronted him with Pappus's hexagon theorem, and astronomy with Martin Hortensius. In October 1630, he had a falling-out with Beeckman, whom he accused of plagiarizing some of his ideas. In Amsterdam, he had a relationship with a servant girl, Helena Jans van der Strom, with whom he had a daughter, Francine, who was born in 1635 in Deventer. She was baptized a Protestant and died of scarlet fever at the age of 5.

Unlike many moralists of the time, Descartes did not deprecate the passions but rather defended them; he wept upon Francine's death in 1640. According to a recent biography by Jason Porterfield, "Descartes said that he did not believe that one must refrain from tears to prove oneself a man." Russell Shorto speculates that the experience of fatherhood and losing a child formed a turning point in Descartes' work, changing its focus from medicine to a quest for universal answers.

Despite frequent moves, he wrote all of his major work during his 20-plus years in the Netherlands, initiating a revolution in mathematics and philosophy. In 1633, Galileo was condemned by the Italian Inquisition, and Descartes abandoned plans to publish Treatise on the World, his work of the previous four years. Nevertheless, in 1637, he published parts of this work in three essays: "Les Météores" (The Meteors), " La Dioptrique" (Dioptrics) and La Géométrie ( Geometry), preceded by an introduction, his famous Discours de la méthode (Discourse on the Method). In it, Descartes lays out four rules of thought, meant to ensure that our knowledge rests upon a firm foundation:

The first was never to accept anything for true which I did not know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

In La Géométrie, Descartes exploited the discoveries he made with Pierre de Fermat. This later became known as Cartesian Geometry.

Descartes continued to publish works concerning both mathematics and philosophy for the rest of his life. In 1641, he published a metaphysics treatise, Meditationes de Prima Philosophia ( Meditations on First Philosophy), written in Latin and thus addressed to the learned. It was followed in 1644 by Principia Philosophiae (Principles of Philosophy), a kind of synthesis of the Discourse on the Method and Meditations on First Philosophy. In 1643, Cartesian philosophy was condemned at the University of Utrecht, and Descartes was obliged to flee to the Hague, settling in Egmond-Binnen.

Between 1643 and 1649 Descartes lived with his girlfriend at Egmond-Binnen in an inn. Descartes became friendly with Anthony Studler van Zurck, lord of Bergen and participated in the design of his mansion and estate. He also met Dirck Rembrantsz van Nierop, a mathematician and surveyor. He was so impressed by Van Nierop's knowledge that he even brought him to the attention of Constantijn Huygens and Frans van Schooten.

Christia Mercer suggested that Descartes may have been influenced by Spanish author and Roman Catholic nun Teresa of Ávila, who, fifty years earlier, published The Interior Castle, concerning the role of philosophical reflection in intellectual growth.

Descartes began (through Alfonso Polloti, an Italian general in Dutch service) a six-year correspondence with Princess Elisabeth of Bohemia, devoted mainly to moral and psychological subjects. Connected with this correspondence, in 1649 he published Les Passions de l'âme (The Passions of the Soul), which he dedicated to the Princess. A French translation of Principia Philosophiae, prepared by Abbot Claude Picot, was published in 1647. This edition was also dedicated to Princess Elisabeth. In the preface to the French edition, Descartes praised true philosophy as a means to attain wisdom. He identifies four ordinary sources to reach wisdom and finally says that there is a fifth, better and more secure, consisting in the search for first causes.

Descartes in conversation with Queen Christina in Stockholm

By 1649, Descartes had become one of Europe's most famous philosophers and scientists. That year, Queen Christina of Sweden invited him to her court to organize a new scientific academy and tutor her in his ideas about love. Descartes accepted, and moved to the Swedish Empire in the middle of winter. Christina was interested in and stimulated Descartes to publish The Passions of the Soul.

He was a guest at the house of Pierre Chanut, living on Västerlånggatan, less than 500 meters from Castle Tre Kronor in Stockholm. There, Chanut and Descartes made observations with a Torricellian mercury barometer. Challenging Blaise Pascal, Descartes took the first set of barometric readings in Stockholm to see if atmospheric pressure could be used in forecasting the weather.

Descartes arranged to give lessons to Queen Christina after her birthday, three times a week at 5 am, in her cold and draughty castle. However, by 15 January 1650 the Queen had actually met with Descartes only four or five times. It soon became clear they did not like each other; she did not care for his mechanical philosophy, nor did he share her interest in Ancient Greek language and literature. On 1 February 1650, he contracted pneumonia and died on 11 February at Chanut.

"Yesterday morning about four o'clock a.m. has deceased here at the house of His Excellency Mr. Chanut, French ambassador, Mr. Descartes. As I have been informed, he had been ill for a few days with pleurisy. But as he did not want to take or use medicines, a hot fever appears to have arisen as well. Thereupon, he had himself bled three times in one day, but without operation of losing much blood. Her Majesty much bemoaned his decease, because he was such a learned man. He has been cast in wax. It was not his intention to die here, as he had resolved shortly before his death to return to Holland at the first occasion. Etc."

The cause of death was pneumonia according to Chanut, but peripneumonia according to Christina's physician Johann van Wullen who was not allowed to bleed him. (The winter seems to have been mild, except for the second half of January which was harsh as described by Descartes himself; however, "this remark was probably intended to be as much Descartes' take on the intellectual climate as it was about the weather.")

E. Pies has questioned this account, based on a letter by the Doctor van Wullen; however, Descartes had refused his treatment, and more arguments against its veracity have been raised since. In a 2009 book, German philosopher Theodor Ebert argues that Descartes was poisoned by Jacques Viogué, a Catholic missionary who opposed his religious views. As evidence, Ebert suggests that Catherine Descartes, the niece of René Descartes, made a veiled reference to the act of poisoning when her uncle was administered "communion" two days before his death, in her Report on the Death of M. Descartes, the Philosopher (1693).

As a Catholic in a Protestant nation, he was interred in the churchyard of what was to become Adolf Fredrik Church in Stockholm, where mainly orphans had been buried. His manuscripts came into the possession of Claude Clerselier, Chanut's brother-in-law, and "a devout Catholic who has begun the process of turning Descartes into a saint by cutting, adding and publishing his letters selectively." In 1663, the Pope placed Descartes' works on the Index of Prohibited Books. In 1666, sixteen years after his death, his remains were taken to France and buried in Saint-Étienne-du-Mont. In 1671, Louis XIV prohibited all lectures in Cartesianism. Although the National Convention in 1792 had planned to transfer his remains to the Panthéon, he was reburied in the Abbey of Saint-Germain-des-Prés in 1819, missing a finger and the skull. His skull is on display in the Musée de l'Homme in Paris.

René Descartes at work

In his Discourse on the Method, he attempts to arrive at a fundamental set of principles that one can know as true without any doubt. To achieve this, he employs a method called hyperbolical/metaphysical doubt, also sometimes referred to as methodological skepticism or Cartesian doubt: he rejects any ideas that can be doubted and then re-establishes them in order to acquire a firm foundation for genuine knowledge. Descartes built his ideas from scratch which he does in The Meditations on First Philosophy. He relates this to architecture: the top soil is taken away to create a new building or structure. Descartes calls his doubt the soil and new knowledge the buildings. To Descartes, Aristotle's foundationalism is incomplete and his method of doubt enhances foundationalism.

Initially, Descartes arrives at only a single first principle: he thinks. This is expressed in the Latin phrase in the Discourse on Method " Cogito, ergo sum" (English: "I think, therefore I am"). Descartes concluded, if he doubted, then something or someone must be doing the doubting; therefore, the very fact that he doubted proved his existence. "The simple meaning of the phrase is that if one is skeptical of existence, that is in and of itself proof that he does exist." These two first principles-I think and I exist-were later confirmed by Descartes' clear and distinct perception (delineated in his Third Meditation from The Meditations): as he clearly and distinctly perceives these two principles, Descartes reasoned, ensures their indubitability.

Descartes concludes that he can be certain that he exists because he thinks. But in what form? He perceives his body through the use of the senses; however, these have previously been unreliable. So Descartes determines that the only indubitable knowledge is that he is a thinking thing. Thinking is what he does, and his power must come from his essence. Descartes defines "thought" () as "what happens in me such that I am immediately conscious of it, insofar as I am conscious of it". Thinking is thus every activity of a person of which the person is immediately conscious. He gave reasons for thinking that waking thoughts are distinguishable from dreams, and that one's mind cannot have been "hijacked" by an evil demon placing an illusory external world before one's senses.

And so something that I thought I was seeing with my eyes is grasped solely by the faculty of judgment which is in my mind.

In this manner, Descartes proceeds to construct a system of knowledge, discarding perception as unreliable and, instead, admitting only deduction as a method.

Mind-body dualism

L'homme (1664)

Descartes, influenced by the automatons on display at the Château de Saint-Germain-en-Laye near Paris, investigated the connection between mind and body, and how they interact. His main influences for dualism were theology and physics. The theory on the dualism of mind and body is Descartes' signature doctrine and permeates other theories he advanced. Known as Cartesian dualism (or mind-body dualism), his theory on the separation between the mind and the body went on to influence subsequent Western philosophies. In Meditations on First Philosophy, Descartes attempted to demonstrate the existence of God and the distinction between the human soul and the body. Humans are a union of mind and body; thus Descartes' dualism embraced the idea that mind and body are distinct but closely joined. While many contemporary readers of Descartes found the distinction between mind and body difficult to grasp, he thought it was entirely straightforward. Descartes employed the concept of modes, which are the ways in which substances exist. In Principles of Philosophy, Descartes explained, "we can clearly perceive a substance apart from the mode which we say differs from it, whereas we cannot, conversely, understand the mode apart from the substance". To perceive a mode apart from its substance requires an intellectual abstraction, which Descartes explained as follows:

The intellectual abstraction consists in my turning my thought away from one part of the contents of this richer idea the better to apply it to the other part with greater attention. Thus, when I consider a shape without thinking of the substance or the extension whose shape it is, I make a mental abstraction.

According to Descartes, two substances are really distinct when each of them can exist apart from the other. Thus, Descartes reasoned that God is distinct from humans, and the body and mind of a human are also distinct from one another. He argued that the great differences between body (an extended thing) and mind (an un-extended, immaterial thing) make the two ontologically distinct. According to Descartes' indivisibility argument, the mind is utterly indivisible: because "when I consider the mind, or myself in so far as I am merely a thinking thing, I am unable to distinguish any part within myself; I understand myself to be something quite single and complete."

Moreover, in The Meditations, Descartes discusses a piece of wax and exposes the single most characteristic doctrine of Cartesian dualism: that the universe contained two radically different kinds of substances-the mind or soul defined as thinking, and the body defined as matter and unthinking. The Aristotelian philosophy of Descartes' days held that the universe was inherently purposeful or teleological. Everything that happened, be it the motion of the stars or the growth of a tree, was supposedly explainable by a certain purpose, goal or end that worked its way out within nature. Aristotle called this the "final cause," and these final causes were indispensable for explaining the ways nature operated. Descartes' theory of dualism supports the distinction between traditional Aristotelian science and the new science of Kepler and Galileo, which denied the role of a divine power and "final causes" in its attempts to explain nature. Descartes' dualism provided the philosophical rationale for the latter by expelling the final cause from the physical universe (or res extensa) in favor of the mind (or res cogitans). Therefore, while Cartesian dualism paved the way for modern physics, it also held the door open for religious beliefs about the immortality of the soul.

Descartes' dualism of mind and matter implied a concept of human beings. A human was, according to Descartes, a composite entity of mind and body. Descartes gave priority to the mind and argued that the mind could exist without the body, but the body could not exist without the mind. In The Meditations, Descartes even argues that while the mind is a substance, the body is composed only of "accidents". But he did argue that mind and body are closely joined:

Nature also teaches me, by the sensations of pain, hunger, thirst and so on, that I am not merely present in my body as a pilot in his ship, but that I am very closely joined and, as it were, intermingled with it, so that I and the body form a unit. If this were not so, I, who am nothing but a thinking thing, would not feel pain when the body was hurt, but would perceive the damage purely by the intellect, just as a sailor perceives by sight if anything in his ship is broken.

Descartes' discussion on embodiment raised one of the most perplexing problems of his dualism philosophy: What exactly is the relationship of union between the mind and the body of a person? Therefore, Cartesian dualism set the agenda for philosophical discussion of the mind-body problem for many years after Descartes' death. Descartes was also a rationalist and believed in the power of innate ideas. Descartes argued the theory of innate knowledge and that all humans were born with knowledge through the higher power of God. It was this theory of innate knowledge that was later combated by philosopher John Locke (1632-1704), an empiricist. Empiricism holds that all knowledge is acquired through experience.

In The Passions of the Soul, published in 1649, Descartes discussed the common contemporary belief that the human body contained animal spirits. These animal spirits were believed to be light and roaming fluids circulating rapidly around the nervous system between the brain and the muscles. These animal spirits were believed to affect the human soul, or passions of the soul. Descartes distinguished six basic passions: wonder, love, hatred, desire, joy and sadness. All of these passions, he argued, represented different combinations of the original spirit, and influenced the soul to will or want certain actions. He argued, for example, that fear is a passion that moves the soul to generate a response in the body. In line with his dualist teachings on the separation between the soul and the body, he hypothesized that some part of the brain served as a connector between the soul and the body and singled out the pineal gland as connector. Descartes argued that signals passed from the ear and the eye to the pineal gland, through animal spirits. Thus different motions in the gland cause various animal spirits. He argued that these motions in the pineal gland are based on God's will and that humans are supposed to want and like things that are useful to them. But he also argued that the animal spirits that moved around the body could distort the commands from the pineal gland, thus humans had to learn how to control their passions.

Descartes advanced a theory on automatic bodily reactions to external events, which influenced 19th-century reflex theory. He argued that external motions, such as touch and sound, reach the endings of the nerves and affect the animal spirits. For example, heat from fire affects a spot on the skin and sets in motion a chain of reactions, with the animal spirits reaching the brain through the central nervous system, and in turn, animal spirits are sent back to the muscles to move the hand away from the fire. Through this chain of reactions, the automatic reactions of the body do not require a thought process.

Above all, he was among the first scientists who believed that the soul should be subject to scientific investigation. He challenged the views of his contemporaries that the soul was divine, thus religious authorities regarded his books as dangerous. Descartes' writings went on to form the basis for theories on emotions and how cognitive evaluations were translated into affective processes. Descartes believed the brain resembled a working machine and that mathematics, and mechanics could explain complicated processes in it. In the 20th century, Alan Turing advanced computer science based on mathematical biology as inspired by Descartes. His theories on reflexes also served as the foundation for advanced physiological theories, more than 200 years after his death. The physiologist Ivan Pavlov was a great admirer of Descartes.

Descartes denied that animals had reason or intelligence. He argued that animals did not lack sensations or perceptions, but these could be explained mechanistically. Whereas humans had a soul, or mind, and were able to feel pain and anxiety, animals by virtue of not having a soul could not feel pain or anxiety. If animals showed signs of distress then this was to protect the body from damage, but the innate state needed for them to suffer was absent. Although Descartes's views were not universally accepted, they became prominent in Europe and North America, allowing humans to treat animals with impunity. The view that animals were quite separate from humanity and merely machines allowed for the maltreatment of animals, and was sanctioned in law and societal norms until the middle of the 19th century. The publications of Charles Darwin would eventually erode the Cartesian view of animals. Darwin argued that the continuity between humans and other species suggested the possibility of animal suffering.

For Descartes, ethics was a science, the highest and most perfect of them. Like the rest of the sciences, ethics had its roots in metaphysics. In this way, he argues for the existence of God, investigates the place of man in nature, formulates the theory of mind-body dualism, and defends free will. However, as he was a convinced rationalist, Descartes clearly states that reason is sufficient in the search for the goods that individuals should seek, and virtue consists in the correct reasoning that should guide their actions. Nevertheless, the quality of this reasoning depends on knowledge and mental condition. For this reason, he said that a complete moral philosophy should include the study of the body. He discussed this subject in the correspondence with Princess Elisabeth of Bohemia, and as a result wrote his work The Passions of the Soul, that contains a study of the psychosomatic processes and reactions in man, with an emphasis on emotions or passions. His works about human passion and emotion would be the basis for the philosophy of his followers (see Cartesianism), and would have a lasting impact on ideas concerning what literature and art should be, specifically how it should invoke emotion.

Descartes and Zeno both identified sovereign goods with virtue. For Epicurus, the sovereign good was pleasure, and Descartes says that, in fact, this is not in contradiction with Zeno's teaching, because virtue produces a spiritual pleasure that is better than bodily pleasure. Regarding Aristotle's opinion that happiness (eudaimonia) depends on both moral virtue and also on the goods of fortune such as a moderate degree of wealth, Descartes does not deny that fortunes contributes to happiness, but remarks that they are in great proportion outside one's own control, whereas one's mind is under one's complete control. The moral writings of Descartes came at the last part of his life, but earlier, in his Discourse on the Method, he adopted three maxims to be able to act while he put all his ideas into doubt. Those maxims are known as his "Provisional Morals".

In the third and fifth Meditation, Descartes offers proofs of a benevolent God (the trademark argument and the ontological argument respectively). Descartes has faith in the account of reality his senses provide him, since he believed that God provided him with a working mind and sensory system and does not desire to deceive him. From this supposition, however, Descartes finally establishes the possibility of acquiring knowledge about the world based on deduction and perception. Regarding epistemology, therefore, Descartes can be said to have contributed such ideas as a conception of foundationalism and the possibility that reason is the only reliable method of attaining knowledge. Descartes, however, was very much aware that experimentation was necessary to verify and validate theories.

Descartes invokes his causal adequacy principle to support his trademark argument for the existence of God, quoting Lucretius in defence: "Ex nihilo nihil fit", meaning " Nothing comes from nothing" ( Lucretius). Oxford Reference summarises the argument, as follows, "that our idea of perfection is related to its perfect origin (God), just as a stamp or trademark is left in an article of workmanship by its maker." In the fifth Meditation, Descartes presents a version of the ontological argument which is founded on the possibility of thinking the "idea of a being that is supremely perfect and infinite," and suggests that "of all the ideas that are in me, the idea that I have of God is the most true, the most clear and distinct."

Descartes considered himself to be a devout Catholic, and one of the purposes of the Meditations was to defend the Catholic faith. His attempt to ground theological beliefs on reason encountered intense opposition in his time. Pascal regarded Descartes' views as a rationalist and mechanist, and accused him of deism: "I cannot forgive Descartes; in all his philosophy, Descartes did his best to dispense with God. But Descartes could not avoid prodding God to set the world in motion with a snap of his lordly fingers; after that, he had no more use for God," while a powerful contemporary, Martin Schoock, accused him of atheist beliefs, though Descartes had provided an explicit critique of atheism in his Meditations. The Catholic Church prohibited his books in 1663.

Descartes also wrote a response to external world skepticism. Through this method of skepticism, he does not doubt for the sake of doubting but to achieve concrete and reliable information. In other words, certainty. He argues that sensory perceptions come to him involuntarily, and are not willed by him. They are external to his senses, and according to Descartes, this is evidence of the existence of something outside of his mind, and thus, an external world. Descartes goes on to argue that the things in the external world are material by arguing that God would not deceive him as to the ideas that are being transmitted, and that God has given him the "propensity" to believe that such ideas are caused by material things. Descartes also believes a substance is something that does not need any assistance to function or exist. Descartes further explains how only God can be a true "substance". But minds are substances, meaning they need only God for it to function. The mind is a thinking substance. The means for a thinking substance stem from ideas.

Descartes steered clear of theological questions, restricting his attention to showing that there is no incompatibility between his metaphysics and theological orthodoxy. He avoided trying to demonstrate theological dogmas metaphysically. When challenged that he had not established the immortality of the soul merely in showing that the soul and the body are distinct substances, he replied, "I do not take it upon myself to try to use the power of human reason to settle any of those matters which depend on the free will of God."

x for unknown; exponential notation

Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 2 used in x ² to indicate x squared.

A Cartesian coordinates graph, using his invented x and y axes

One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He was first to assign a fundamental place for algebra in the system of knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such as Pacioli, Cardano, Tartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three-dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity could represent length as well as an area. This was in opposition to the teachings of mathematicians such as François Viète, who insisted that a second power must represent an area. Although Descartes did not pursue the subject, he preceded Gottfried Wilhelm Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.

Influence on Newton's mathematics

Current popular opinion holds that Descartes had the most influence of anyone on the young Isaac Newton, and this is arguably one of his most important contributions. Descartes' influence extended not directly from his original French edition of La Géométrie, however, but rather from Frans van Schooten's expanded second Latin edition of the work. Newton continued Descartes' work on cubic equations, which freed the subject from fetters of the Greek perspectives. The most important concept was his very modern treatment of single variables.

Descartes' work provided the basis for the calculus developed by Leibniz and Newton, who applied the infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics. His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial.

Philosophy, metaphysics, and physics

Descartes is often regarded as the first thinker to emphasize the use of reason to develop the natural sciences. For him, philosophy was a thinking system that embodied all knowledge, as he related in a letter to a French translator:

Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principals, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.

The beginning to Descartes' interest in physics is accredited to the amateur scientist and mathematician Isaac Beeckman, whom he met in 1618, and who was at the forefront of a new school of thought known as mechanical philosophy. With this foundation of reasoning, Descartes formulated many of his theories on mechanical and geometric physics. It is said that they met when both were looking at a placard that was set up in the Breda marketplace, detailing a mathematical problem to be solved. Descartes asked Beeckman to translate the problem from Dutch to French. In their following meetings Beeckman interested Descartes in his corpuscularian approach to mechanical theory, and convinced him to devote his studies to a mathematical approach to nature. In 1628, Beeckman also introduced him to many of Galileo's ideas. Together, they worked on free fall, catenaries, conic sections, and fluid statics. Both believed that it was necessary to create a method that thoroughly linked mathematics and physics.

Although the concept of work (in physics) was not formally used until 1826, similar concepts existed before then. In 1637, Descartes wrote:

Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet.

-  René Descartes, Letter to Huygens

In Principles of Philosophy ( Principia Philosophiae) from 1644 Descartes outlined his views on the universe. In it he describes his three laws of motion. ( Newton's own laws of motion would later be modeled on Descartes' exposition.) Descartes defined "quantity of motion" (Latin: quantitas motus) as the product of size and speed, and claimed that the total quantity of motion in the universe is conserved.

If x is twice the size of y, and is moving half as fast, then there's the same amount of motion in each.

[God] created matter, along with its motion ... merely by letting things run their course, he preserves the same amount of motion ... as he put there in the beginning.

-  René Descartes, Principles of Philosophy, Part 2: The principles of material things

Descartes had discovered an early form of the law of conservation of momentum. He envisioned quantity of motion as pertaining to motion in a straight line, as opposed to perfect circular motion, as Galileo had envisioned it. Descartes' discovery should not be seen as the modern law of conservation of momentum, since had no concept of mass as distinct from weight or size, and since he believed that it is speed rather than velocity that is conserved.

Descartes' vortex theory of planetary motion was later rejected by Newton in favor of his law of universal gravitation, and most of the second book of Newton's Principia is devoted to his counterargument.

Descartes also made contributions to the field of optics. He showed by using geometric construction and the law of refraction (also known as Descartes' law, or more commonly Snell's law outside France) that the angular radius of a rainbow is 42 degrees (i.e., the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°). He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.

Within Discourse on the Method, there is an appendix in which Descartes discusses his theories on Meteorology known as Les Météores. He first proposed the idea that the elements were made up of small particles that join together imperfectly, thus leaving small spaces in between. These spaces were then filled with smaller much quicker "subtile matter". These particles were different based on what element they constructed, for example, Descartes believed that particles of water were "like little eels, which, though they join and twist around each other, do not, for all that, ever knot or hook together in such a way that they cannot easily be separated." In contrast, the particles that made up the more solid material, were constructed in a way that generated irregular shapes. The size of the particle also matters, if the particle was smaller, not only was it faster and constantly moving, it was more easily agitated by the larger particles, which were slow but had more force. The different qualities, such as combinations and shapes, gave rise to different secondary qualities of materials, such as temperature. This first idea is the basis for the rest of Descartes' theory on Meteorology.

While rejecting most of Aristotle's theories on Meteorology, he still kept some of the terminology that Aristotle used such as vapors and exhalations. These "vapors" would be drawn into the sky by the sun from "terrestrial substances" and would generate wind. Descartes also theorized that falling clouds would displace the air below them, also generating wind. Falling clouds could also generate thunder. He theorized that when a cloud rests above another cloud and the air around the top cloud is hot, it condenses the vapor around the top cloud, and causes the particles to fall. When the particles falling from the top cloud collided with the bottom cloud's particles it would create thunder. He compared his theory on thunder to his theory on avalanches. Descartes believed that the booming sound that avalanches created, was due to snow that was heated, and therefore heavier, falling onto the snow that was below it. This theory was supported by experience "It follows that one can understand why it thunders more rarely in winter than in summer; for then not enough heat reaches the highest clouds, in order to break them up,"

Another theory that Descartes had was on the production of lightning. Descartes believed that lightning was caused by exhalations trapped between the two colliding clouds. He believed that in order to make these exhalations viable to produce lightning, they had to be made "fine and inflammable" by hot and dry weather. Whenever the clouds would collide it would cause them to ignite creating lightning, if the cloud above was heavier than the bottom cloud it would also produce thunder.

Descartes also believed that clouds were made up of drops of water and ice, and believed that rain would fall whenever the air could no longer support them. It would fall as snow if the air was not warm enough to melt the raindrops. And hail was when the cloud drops would melt, and then freeze again because cold air would refreeze them.

Descartes did not use mathematics or instruments (as there were not any at the time) to back up his theories on Meteorology and instead used qualitative reasoning in order to deduce his hypothesis.

Cover of Meditations

Descartes has often been dubbed the father of modern Western philosophy, the thinker whose approaches has profoundly changed the course of Western philosophy and set the basis for modernity. The first two of his Meditations on First Philosophy, those that formulate the famous methodic doubt, represent the portion of Descartes' writings that most influenced modern thinking. It has been argued that Descartes himself did not realize the extent of this revolutionary move. In shifting the debate from "what is true" to "of what can I be certain?", Descartes arguably shifted the authoritative guarantor of truth from God to humanity (even though Descartes himself claimed he received his visions from God)-while the traditional concept of "truth" implies an external authority, "certainty" instead relies on the judgment of the individual.

In an anthropocentric revolution, the human being is now raised to the level of a subject, an agent, an emancipated being equipped with autonomous reason. This was a revolutionary step that established the basis of modernity, the repercussions of which are still being felt: the emancipation of humanity from Christian revelational truth and Church doctrine; humanity making its own law and taking its own stand. In modernity, the guarantor of truth is not God anymore but human beings, each of whom is a "self-conscious shaper and guarantor" of their own reality. In that way, each person is turned into a reasoning adult, a subject and agent, as opposed to a child obedient to God. This change in perspective was characteristic of the shift from the Christian medieval period to the modern period, a shift that had been anticipated in other fields, and which was now being formulated in the field of philosophy by Descartes.

This anthropocentric perspective of Descartes' work, establishing human reason as autonomous, provided the basis for the Enlightenment's emancipation from God and the Church. According to Martin Heidegger, the perspective of Descartes' work also provided the basis for all subsequent anthropology. Descartes' philosophical revolution is sometimes said to have sparked modern anthropocentrism and subjectivism.

In commercial terms, The Discourse appeared during Descartes' lifetime in a single edition of 500 copies, 200 of which were set aside for the author. Sharing a similar fate was the only French edition of The Meditations, which had not managed to sell out by the time of Descartes' death. A concomitant Latin edition of the latter was, however, eagerly sought out by Europe's scholarly community and proved a commercial success for Descartes.: xliii-xliv

Although Descartes was well known in academic circles towards the end of his life, the teaching of his works in schools was controversial. Henri de Roy ( Henricus Regius, 1598-1679), Professor of Medicine at the University of Utrecht, was condemned by the Rector of the university, Gijsbert Voet (Voetius), for teaching Descartes' physics.

According to John Cottingham-whose translation of Meditations is considered to be "authoritative"-Descartes's Meditations on First Philosophy is considered to be "one of the key texts of Western philosophy". Cottingham said that the Meditations is the "most widely studied of all Descartes' writings".

According to Anthony Gottlieb, a former senior editor of The Economist, and the author of The Dream of Reason and The Dream of Enlightenment, one of the reasons Descartes and Thomas Hobbes continue to be debated in the second decade of the twenty-first century, is that they still have something to say to us that remains relevant on questions such as, "What does the advance of science entail for our understanding of ourselves and our ideas of God?" and "How is government to deal with religious diversity."

In her 2018 interview with Tyler Cowen, Agnes Callard described Descartes' thought experiment in the Meditations, where he encouraged a complete, systematic doubting of everything that you believe, to "see what you come to". She said, "What Descartes comes to is a kind of real truth that he can build upon inside of his own mind." She said that Hamlet's monologues-"meditations on the nature of life and emotion"-were similar to Descartes' thought experiment. Hamlet/Descartes were "apart from the world", as if they were "trapped" in their own heads. Cowen asked Callard if Descartes actually found any truths through his thought experiment or was it just "an earlier version of the contemporary argument that we're living in a simulation, where the evil demon is the simulation rather than Bayesian reasoning?" Callard agreed that this argument can be traced to Descartes, who had said that he had refuted it. She clarified that in Descartes' reasoning, you do "end up back in the mind of God"-in a "universe God has created" that is the "real world"...The whole question is about being connected to reality as opposed to being a figment. If you're living in the world God created, God can create real things. So you're living in a real world."

The membership of Descartes to the Rosicrucians is debated.

The initials of his name have been linked to the R.C. acronym widely used by Rosicrucians. Furthermore, in 1619 Descartes moved to Ulm which was a well renowned international center of the Rosicrucian movement. During his journey in Germany, he met Johannes Faulhaber who had previously expressed his personal commitment to join the brotherhood.

Descartes dedicated the work titled The Mathematical Treasure Trove of Polybius, Citizen of the World to "learned men throughout the world and especially to the distinguished B.R.C. (Brothers of the Rosy Cross) in Germany". The work was not completed and its publication is uncertain.

1618. Musicae Compendium. A treatise on music theory and the aesthetics of music, which Descartes dedicated to early collaborator Isaac Beeckman (written in 1618, first published-posthumously-in 1650).

1626-1628. Regulae ad directionem ingenii (Rules for the Direction of the Mind). Incomplete. First published posthumously in Dutch translation in 1684 and in the original Latin at Amsterdam in 1701 (R. Des-Cartes Opuscula Posthuma Physica et Mathematica). The best critical edition, which includes the Dutch translation of 1684, is edited by Giovanni Crapulli (The Hague: Martinus Nijhoff, 1966).

c. 1630. De solidorum elementis. Concerns the classification of Platonic solids and three-dimensional figurate numbers. Said by some scholars to prefigure Euler's polyhedral formula. Unpublished; discovered in Descartes' estate in Stockholm 1650, soaked for three days in the Seine in a shipwreck while being shipped back to Paris, copied in 1676 by Leibniz, and lost. Leibniz's copy, also lost, was rediscovered circa 1860 in Hannover.

1630-1631. La recherche de la vérité par la lumière naturelle (The Search for Truth by Natural Light) unfinished dialogue published in 1701.

1630-1633. Le Monde () and (Man). Descartes' first systematic presentation of his natural philosophy. Man was published posthumously in Latin translation in 1662; and The World posthumously in 1664.

1637. Discours de la méthode (Discourse on the Method). An introduction to the Essais, which include the Dioptrique, the Météores and the Géométrie.

1637. La Géométrie (Geometry). Descartes' major work in mathematics. There is an English translation by Michael Mahoney (New York: Dover, 1979).

1641. Meditationes de prima philosophia (Meditations on First Philosophy), also known as Metaphysical Meditations. In Latin; a second edition, published the following year, included an additional objection and reply, and a Letter to Dinet. A French translation by the Duke of Luynes, probably done without Descartes' supervision, was published in 1647. Includes six Objections and Replies.

1644. Principia philosophiae (Principles of Philosophy), a Latin textbook at first intended by Descartes to replace the Aristotelian textbooks then used in universities. A French translation, Principes de philosophie by Claude Picot, under the supervision of Descartes, appeared in 1647 with a letter-preface to Princess Elisabeth of Bohemia.

1647. Notae in programma (Comments on a Certain Broadsheet). A reply to Descartes' one-time disciple Henricus Regius.

1648. La description du corps humain (The Description of the Human Body). Published posthumously by Clerselier in 1667.

1648. Responsiones Renati Des Cartes... (Conversation with Burman). Notes on a Q&A session between Descartes and Frans Burman on 16 April 1648. Rediscovered in 1895 and published for the first time in 1896. An annotated bilingual edition (Latin with French translation), edited by Jean-Marie Beyssade, was published in 1981 (Paris: PUF).

1649. Les passions de l'âme (Passions of the Soul). Dedicated to Princess Elisabeth of the Palatinate.

1657. Correspondance (three volumes: 1657, 1659, 1667). Published by Descartes' literary executor Claude Clerselier. The third edition, in 1667, was the most complete; Clerselier omitted, however, much of the material pertaining to mathematics.

In January 2010, a previously unknown letter from Descartes, dated 27 May 1641, was found by the Dutch philosopher Erik-Jan Bos when browsing through Google. Bos found the letter mentioned in a summary of autographs kept by Haverford College in Haverford, Pennsylvania. The college was unaware that the letter had never been published. This was the third letter by Descartes found in the last 25 years.

Handwritten letter by Descartes, December 1638

Principia philosophiae, 1644

Oeuvres de Descartes edited by Charles Adam and Paul Tannery, Paris: Léopold Cerf, 1897-1913, 13 volumes; new revised edition, Paris: Vrin-CNRS, 1964-1974, 11 volumes (the first 5 volumes contains the correspondence). [This edition is traditionally cited with the initials AT (for Adam and Tannery) followed by a volume number in Roman numerals; thus AT VII refers to Oeuvres de Descartes volume 7.]

Étude du bon sens, La recherche de la vérité et autres écrits de jeunesse (1616-1631) edited by Vincent Carraud and Gilles Olivo, Paris: PUF, 2013.

Descartes, Œuvres complètes, new edition by Jean-Marie Beyssade and Denis Kambouchner, Paris: Gallimard, published volumes:

I: Premiers écrits. Règles pour la direction de l'esprit, 2016.

III: Discours de la Méthode et Essais, 2009.

VIII.1: Correspondance, 1 edited by Jean-Robert Armogathe, 2013.

VIII.2: Correspondance, 2 edited by Jean-Robert Armogathe, 2013.

1955. The Philosophical Works, E.S. Haldane and G.R.T. Ross, trans. Dover Publications. This work is traditionally cited with the initials HR (for Haldane and Ross) followed by a volume number in Roman numerals; thus HR II refers to volume 2 of this edition.

1988. The Philosophical Writings of Descartes in 3 vols. Cottingham, J., Stoothoff, R., Kenny, A., and Murdoch, D., trans. Cambridge University Press. This work is traditionally cited with the initials CSM (for Cottingham, Stoothoff, and Murdoch) or CSMK (for Cottingham, Stoothoff, Murdoch, and Kenny) followed by a volume number in Roman numeral; thus CSM II refers to volume 2 of this edition.

1998. René Descartes: The World and Other Writings. Translated and edited by Stephen Gaukroger. Cambridge University Press. (This consists mainly of scientific writings, on physics, biology, astronomy, optics, etc., which were very influential in the 17th and 18th centuries, but which are routinely omitted or much abridged in modern collections of Descartes' philosophical works.)

1633. The World, or Treatise on Light Archived 21 June 2021 at the Wayback Machine, tr. by Michael S. Mahoney.

1633. Treatise of Man, tr. by T. S. Hall. Cambridge, MA: Harvard University Press, 1972.

1637. The Geometry of René Descartes Archived 16 August 2021 at the Wayback Machine, trans. D. E. Smith & Marcia Latham (Chicago: Open Court, 1925).

1641. Meditations on First Philosophy, tr. by J. Cottingham, Cambridge: Cambridge University Press, 1996. Latin original. Alternative English title: Metaphysical Meditations. Includes six Objections and Replies. A second edition published the following year, includes an additional Objection and Reply and a Letter to Dinet. HTML Online Latin-French-English Edition Archived 27 August 2006 at the Wayback Machine.

1644. Principles of Philosophy Archived 30 September 2020 at the Wayback Machine, trans. V. R. Miller & R. P. Miller: (Dordrecht/Boston/London: Kluwer Academic Publishers, 1982).

1648. Descartes' Conversation with Burman, tr. by J. Cottingham, Oxford: Clarendon Press, 1989.

1649. Passions of the Soul Archived 16 August 2021 at the Wayback Machine, trans. S. H. Voss (Indianapolis: Hackett, 1989). Dedicated to Elisabeth of the Palatinate.

1. ^ Étienne Gilson argued in La Liberté chez Descartes et la Théologie (Alcan, 1913, pp. 132-147) that Duns Scotus was not the source of Descartes' Voluntarism. Although there exist doctrinal differences between Descartes and Scotus "it is still possible to view Descartes as borrowing from a Scotist Voluntarist tradition".

2. ^ Although the uncertain authorship of this most iconic portrait of Descartes was traditionally attributed to Frans Hals, there is no record of their meeting. During the 20th century the assumption was widely challenged.

3. This idea had already been proposed by ^ Spanish philosopher Gómez Pereira a hundred years ago in the form: "I know that I know something, anyone who knows exists, then I exist" (nosco me aliquid noscere, & quidquid noscit, est, ergo ego sum).

Pereira, Gómez. 1749 [1554]. "De Immortalitate Animae." Antoniana Margarita. p. 277.

Santos López, Modesto. 1986. "Gómez Pereira, médico y filósofo medinense." In Historia de Medina del Campo y su Tierra, volumen I: Nacimiento y expansión, edited by E. L. Sanz.

4. While in the Netherlands he changed his address frequently, living among other places in Dordrecht (1628), ^ Franeker (1629), Amsterdam (1629-1630), Leiden (1630), Amsterdam (1630-1632), Deventer (1632-1634), Amsterdam (1634-1635), Utrecht (1635-1636), Leiden (1636), Egmond (1636-1638), Santpoort (1638-1640), Leiden (1640-1641), Endegeest (a castle near Oegstgeest) (1641-1643), and finally for an extended time in Egmond-Binnen (1643-1649).

5. He had lived with ^ Henricus Reneri in Deventer and Amsterdam, and had met with Constantijn Huygens and Vopiscus Fortunatus Plempius; Descartes was interviewed by Frans Burman at Egmond-Binnen in 1648. Henricus Regius, Jan Stampioen, Frans van Schooten, Comenius and Gisbertus Voetius were his main opponents.

6. The remains are, two centuries later, still resting between two other graves-those of the scholarly monks ^ Jean Mabillon and Bernard de Montfaucon-in a chapel of the abbey.

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2. ^ Bostock, D., Philosophy of Mathematics: An Introduction, Wiley-Blackwell, 2009, p. 43 Archived 1 August 2020 at the Wayback Machine: "All of Descartes, Locke, Berkeley, and Hume supposed that mathematics is a theory of our ideas, but none of them offered any argument for this conceptualist claim, and apparently took it to be uncontroversial."

3. ^ Gutting, Gary (1999). Pragmatic Liberalism and the Critique of Modernity. Cambridge University Press. p. 116. ISBN 978-0521649735. Modernity begins with Descartes' mutation of Augustinianism. Taylor emphasizes that "Descartes is in many ways profoundly Augustinian".

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5. ^ Gillespie, Michael Allen (1994). "Chapter One: Descartes and the Deceiver God". Nihilism Before Nietzsche. Chicago: University of Chicago Press. pp. 1-32, 263-64. ISBN 978-0226293479. Caton argues persuasively that Descartes uses the phrase genius malignus for deus deceptor to avoid the charge of blasphemy.

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7. ^ ""I think therefore I am" or "I think. Therefore I am"? René Descartes' saying "Cogito ergo sum" has been widely spread all over the world. In English it is commonly known as "I think therefore I am"". italki.

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40. Spencer, J., "'Love and Hatred are Common to the Whole Sensitive Creation': Animal Feeling in the Century before Darwin," in A. Richardson, ed., ^ After Darwin: Animals, Emotions, and the Mind (Amsterdam and New York: Rodopi, 2013),p. 37 Archived 16 August 2021 at the Wayback Machine.

41. ^ Workman, L. (2013). Charles Darwin: The Shaping of Evolutionary Thinking. Palgrave Macmillan. p. 177. ISBN 978-1-137-31323-2. Archived from the original on 16 August 2021.

42. Blom, John J., Descartes. His moral philosophy and psychology. New York University Press. 1978. ISBN 0-8147-0999-0

43. ^ Descartes, Rene "Meditations on First Philosophy, 3rd Ed., Translated from Latin by: Donald A. Cress

44. René Descartes, ^ Discourse de la Méthode (Leiden, Netherlands): Jan Maire, 1637, appended book: La Géométrie, book one, p. 299. Archived 8 October 2017 at the Wayback Machine From p. 299: " ... Et aa, ou a², pour multiplier a par soy mesme; Et a³, pour le multiplier encore une fois par a, & ainsi a l'infini; ... " ( ... and aa, or a ², in order to multiply a by itself; and a ³, in order to multiply it once more by a, and thus to infinity; ... )

45. ^ Gaukroger, S., "The nature of abstract reasoning: philosophical aspects of Descartes' work in algebra", in J. Cottingham, ed., The Cambridge Companion to Descartes (Cambridge: Cambridge University Press, 1992), pp. 91-114 Archived 16 August 2021 at the Wayback Machine.

46. ^ Grosholz, Emily (1991). Cartesian method and the problem of reduction. Oxford University Press. ISBN 978-0-19-824250-5. Archived from the original on 16 August 2021. But contemporary debate has tended to...understand [Cartesian method] merely as the 'method of doubt'...I want to define Descartes' method in broader terms...to trace its impact on the domains of mathematics and physics as well as metaphysics.

47. Harold J. Cook, in The Scientific Revolution in National Context, Roy Porter, Mikuláš Teich, (eds.), Cambridge University Press, 1992, pages 127-129

48. ^ Mendelson, Kenneth S. (13 February 2003). "Physical and colloquial meanings of the term "work"". American Journal of Physics. 71 (3): 279-281. doi:10.1119/1.1522707. ISSN 0002-9505.

49. ^ Descartes, R. (2013) [Letter to Huygens, Oct 5, 1637]. Bennett, J. (ed.). Selected correspondence of Descartes (PDF). p. 50.

50. Descartes, R. (2008) [1644]. Bennett, J. (ed.). Principles of philosophy (PDF). Part II, § 36.

51. Alexander Afriat, "Cartesian and Lagrangian Momentum" Archived 9 March 2017 at the Wayback Machine (2004).

52. ^ Rothman, Milton A. (1989). Discovering the natural laws: the experimental basis of physics (2nd ed.). New York: Dover. pp. 83-88. ISBN 9780486261782.

53. ^ Slowik, Edward (Fall 2017). "Descartes' Physics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.

54. ^ Frisinger, H. Howard (1977). The history of meteorology to 1800. New York: Science History Publications. ISBN 0-88202-036-6. OCLC 1694190.

55. ^ Martin, Craig (2011). Renaissance meteorology : Pomponazzi to Descartes. Baltimore: Johns Hopkins University Press. ISBN 978-1-4214-0244-4. OCLC 794700393.

56. ^ Heidegger [1938] (2002), p. 76 "Descartes... that which he himself founded... modern (and that means, at the same time, Western) metaphysics."

57. ^ Schmaltz, Tad M. Radical Cartesianism: The French Reception of Descartes Archived 16 August 2021 at the Wayback Machine p. 27 quotation:

The Descartes most familiar to twentieth-century philosophers is the Descartes of the first two Meditations, someone preoccupied with hyperbolic doubt of the material world and the certainty of knowledge of the self that emerges from the famous cogito argument.

58. ^ Roy Wood Sellars (1949) Philosophy for the future: the quest of modern materialism "Husserl has taken Descartes very seriously in a historical as well as in a systematic sense [...] [in The Crisis of the European Sciences and Transcendental Phenomenology, Husserl] finds in the first two Meditations of Descartes a depth which it is difficult to fathom, and which Descartes himself was so little able to appreciate that he let go "the great discovery" he had in his hands."

59. ^ Martin Heidegger [1938] (2002) The Age of the World Picture quotation:

For up to Descartes...a particular sub-iectum...lies at the foundation of its own fixed qualities and changing circumstances. The superiority of a sub-iectum...arises out of the claim of man to a...self-supported, unshakeable foundation of truth, in the sense of certainty. Why and how does this claim acquire its decisive authority? The claim originates in that emancipation of man in which he frees himself from obligation to Christian revelational truth and Church doctrine to a legislating for himself that takes its stand upon itself.

60. Briton, Derek ^ The modern practice of adult education: a postmodern critique p. 76

61. Heidegger [1938] (2002), p. 75 quotation: ^

With the interpretation of man as subiectum, Descartes creates the metaphysical presupposition for future anthropology of every kind and tendency.

62. ^ Schwartz, B. I., China and Other Matters (Cambridge, MA: Harvard University Press, 1996), p. 95 Archived 16 August 2021 at the Wayback Machine, quotation:

... the kind of anthropocentric subjectivism which has emerged from the Cartesian revolution.

63. ^ Charles B. Guignon Heidegger and the problem of knowledge Archived 16 August 2021 at the Wayback Machine p. 23

64. ^ Husserl, Edmund (1931) Cartesian Meditations: An Introduction to Phenomenology quotation:

When, with the beginning of modern times, religious belief was becoming more and more externalized as a lifeless convention, men of intellect were lifted by a new belief: their great belief in an autonomous philosophy and science. [...] in philosophy, the Meditations were epoch-making in a quite unique sense, and precisely because of their going back to the pure ego cogito. Descartes work has been used, in fact to inaugurates an entirely new kind of philosophy. Changing its total style, philosophy takes a radical turn: from naïve objectivism to transcendental subjectivism.

65. ^ Cottingham, John; Murdoch, Dugald; Stoothof, Robert (1984). "Comments on a Certain Broadsheet". The Philosophical Writings of Descartes. Cambridge University Press. p. 293. ISBN 978-0-521-28807-1. Archived from the original on 1 August 2020.

66. Bond, Steven (24 June 2011). ""R.C.": Rosicrucianism and Cartesianism in Joyce and Beckett". Miranda (4-2011). doi:10.4000/miranda.1939. ISSN 2108-6559. OCLC 5497224736. Archived from the original on 2 June 2018.

67. ^ Henri, John; Nolan, Lawrence (26 January 2015). Nolan, Lawrence (ed.). "Rosicrucian". The Cambridge Descartes Lexicon. Cambridge University Press: 659-60. doi:10.1017/CBO9780511894695.224. ISBN 978-0511894695. Archived from the original on 10 June 2018.

68. ^ Mansikka, Tomas (21 April 2018). "Esotericism in reverse. Descartes and the poetic imagination in the seventeenth century" (pdf). Approaching Religion. 8 (1): 84. doi:10.30664/ar.66731. ISSN 1799-3121. OCLC 8081521487.

69. ^ Cook, H. J., in Porter, R., & Teich, M., eds., The Scientific Revolution in National Context (Cambridge: Cambridge University Press, 1992), pp. 127-29 Archived 16 December 2021 at the Wayback Machine.

70. ^ Federico, Pasquale Joseph (1982). Descartes on Polyhedra: A Study of the "De solidorum elementis". Sources in the History of Mathematics and Physical Sciences. Vol. 4. Springer.

71. ^ Vlasblom, Dirk (25 February 2010). "Unknown letter from Descartes found". (in Dutch). Archived from the original on 8 November 2016.

72. ^ Vlasblom, Dirk. "Hoe Descartes in 1641 op andere gedachten kwam". (in Dutch). Archived from the original on 27 October 2016.

Clarke, Desmond (2006). Descartes: A Biography. Cambridge, England: Cambridge University Press. ISBN 978-0-521-82301-2.

Cottingham, John; Williams, B., eds. (1996). "Sixth Meditation: The existence of material things, and the real distinction between mind and body". Descartes: Meditations on First Philosophy: With Selections from the Objections and Replies. Cambridge Texts in the History of Philosophy. Cambridge: Cambridge University Press. pp. 50-62. doi:10.1017/CBO9780511805028.012. ISBN 978-0521558181.

Tyler Cowen (host), Agnes Callard (guest) (11 April 2018). "Agnes Callard on the Theory of Everything". Medium via Mercatus Center (Podcast). Conversations with Tyler Cowen. No. 38.

Farrell, John. "Demons of Descartes and Hobbes." Paranoia and Modernity: Cervantes to Rousseau (Cornell UP, 2006), chapter 7.

Gaukroger, Stephen (1995). Descartes: An Intellectual Biography. Oxford, England: Oxford University Press. ISBN 978-0-19-823994-9.

Gillespie, A. (2006). Descartes' Demon: A Dialogical Analysis of 'Meditations on First Philosophy.' Theory & Psychology, 16, 761-781.

Gottlieb, Anthony (2016). The Dream of Enlightenment: The Rise of Modern Philosophy. Allen Lane. p. 301. ISBN 978-0871404435.

Heidegger, Martin [1938] (2002) The Age of the World Picture in Off the Beaten Track Archived 21 November 2020 at the Wayback Machine pp. 57-85

Moreno Romo, Juan Carlos, Vindicación del cartesianismo radical, Anthropos, Barcelona, 2010.

Moreno Romo, Juan Carlos (Coord.), Descartes vivo. Ejercicios de hermenéutica cartesiana, Anthropos, Barcelona, 2007.

Negri, Antonio (2007) The Political Descartes, Verso.

Serfati, Michel, 2005, "Géometrie" in Ivor Grattan-Guinness, ed., Landmark Writings in Western Mathematics. Elsevier: 1-22.

Watson, Richard A. (2007). Cogito, Ergo Sum: a life of René Descartes. David R Godine. 2002, reprint 2007. ISBN 978-1-56792-335-3. Was chosen by the New York Public library as one of "25 Books to Remember from 2002"

Frisinger, H. Howard. "Chapter 3 ." The History of Meteorology: To 1800, American Meteorological Society, Boston, MA, 1983, pp. 37-40. ISBN 978-0882020365

Martin, Craig. "Chapter 6." Renaissance Meteorology: Pomponazzi to Descartes, Johns Hopkins University Press, Baltimore, MD, 2011. ISBN 978-1421401874

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René Descartes ( day-KART or DAY-kart ; French: [ʁənedekaʁt] ; Renatus Cartesius; 31 March 1596 - 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry.

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In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane ( two dimensions) and Euclidean space ( three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor-Dedekind axiom.

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.

Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.

The 11th-century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.

Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. La Geometrie, written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:

The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair ( x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z).

In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ, with θ normally measured counterclockwise from the positive x-axis. Using this notation, points are typically written as an ordered pair ( r, θ). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: . This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates.

In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis.

In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics.

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x ² + y ² = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x ² + y ² = r ² is the equation for any circle centered at the origin (0, 0) with a radius of r.

Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

where:

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let be the position vector of some point , and let be a nonzero vector. The plane determined by this point and vector consists of those points , with position vector , such that the vector drawn from to is perpendicular to . Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points such that

(The dot here means a dot product, not scalar multiplication.) Expanded this becomes

which is the point-normal form of the equation of a plane. This is just a linear equation:

Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation

is a plane having the vector as a normal.[20] This familiar equation for a plane is called the general form of the equation of the plane.[21]

In three dimensions, lines can not be described by a single linear equation, so they are frequently described by parametric equations:

where:

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section - though it may be degenerate, and all conic sections arise in this way. The equation will be of the form

As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space

The conic sections described by this equation can be classified using the discriminant

If the conic is non-degenerate, then:

if , the equation represents an ellipse;
if and , the equation represents a circle, which is a special case of an ellipse;

if , the equation represents a parabola;

if , the equation represents a hyperbola;

A quadric, or quadric surface, is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates , the general quadric is defined by the algebraic equation

Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and planes.

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points ( x ₁, y ₁) and ( x ₂, y ₂) is defined by the formula

which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

where m is the slope of the line.

In three dimensions, distance is given by the generalization of the Pythagorean theorem:

while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

The graph of is changed by standard transformations as follows:

Changing to moves the graph to the right units.

Changing to moves the graph up units.

Changing to stretches the graph horizontally by a factor of . (think of the as being dilated)

Changing to stretches the graph vertically.

Changing to and changing to rotates the graph by an angle .

There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.

For example, the parent function has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if , then it can be transformed into . In the new transformed function, is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative values, the function is reflected in the -axis. The value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like , reflects the function in the -axis when it is negative. The and values introduce translations, , vertical, and horizontal. Positive and values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.

is the relation that describes the unit circle.

For two geometric objects P and Q represented by the relations and the intersection is the collection of all points which are in both relations.[25]

For example, might be the circle with radius 1 and center : and might be the circle with radius 1 and center . The intersection of these two circles is the collection of points which make both equations true. Does the point make both equations true? Using for , the equation for becomes or which is true, so is in the relation . On the other hand, still using for the equation for becomes or which is false. is not in so it is not in the intersection.

Traditional methods for finding intersections include substitution and elimination.

Substitution: Solve the first equation for in terms of and then substitute the expression for into the second equation:

So our intersection has two points:

Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get . The in the first equation is subtracted from the in the second equation leaving no term. The variable has been eliminated. We then solve the remaining equation for , in the same way as in the substitution method:

So our intersection has two points:

For conic sections, as many as 4 points might be in the intersection.

The intersection of a geometric object and the -axis is called the -intercept of the object. The intersection of a geometric object and the -axis is called the -intercept of the object.

For the line , the parameter specifies the point where the line crosses the axis. Depending on the context, either or the point is called the -intercept.

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover Publications, ISBN 978-0486438320

Cajori, Florian (1999), A History of Mathematics, AMS, ISBN 978-0821821022

John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive.

Katz, Victor J. (1998), A History of Mathematics: An Introduction (2nd Ed.), Reading: Addison Wesley Longman, ISBN 0-321-01618-1

Struik, D. J. (1969), A Source Book in Mathematics, 1200-1800, Harvard University Press, ISBN 978-0674823556

Bissell, C. C., Cartesian geometry: The Dutch contribution

Boyer, Carl B. (1944), "Analytic Geometry: The Discovery of Fermat and Descartes", Mathematics Teacher, 37 (3): 99-105

Boyer, Carl B., Johann Hudde and space coordinates

Coolidge, J. L. (1948), "The Beginnings of Analytic Geometry in Three Dimensions", American Mathematical Monthly, 55 (2): 76-86, doi:10.2307/2305740, JSTOR 2305740

Pecl, J., Newton and analytic geometry

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In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight.

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In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies, in particular, that if the number of sign changes is zero or one, then there are exactly zero or one positive roots, respectively.

By a homographic transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan-Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoints intervals containing together all real roots of the polynomial, and containing each exactly one real root. Descartes rule of signs and homographic transformations of the variable are, nowadays, the basis of the fastest algorithms for computer computation of real roots of polynomials (see Real-root isolation).

Descartes himself used the transformation for using his rule for getting information of the number of negative roots.

Descartes' rule of signs

The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number. A root of multiplicity k is counted as k roots.

In particular, if the number of sign changes is zero or one, the number of positive roots equals the number of sign changes

As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself. For example, the negative roots of are the positive roots of

Thus, applying Descartes' rule of signs to this polynomial gives the number of negative roots of the original polynomial.

The polynomial

has one sign change between the second and third terms (the sequence of signs is (+, +, -, -). Therefore it has exactly one positive root. To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial , to obtain the polynomial

This polynomial has two sign changes (the sequence signs is (-, +, +, -)), meaning that this second polynomial has two or zero positive roots; thus the original polynomial has two or zero negative roots.

In fact, the factorization of the first polynomial is

so the roots are -1 (twice) and +1 (once).

The factorization of the second polynomial is

So here, the roots are +1 (twice) and -1 (once), the negation of the roots of the original polynomial.

Any nth degree polynomial has exactly n roots in the complex plane, if counted according to multiplicity. So if f( x) is a polynomial which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the minimum number of nonreal roots is equal to

where p denotes the maximum number of positive roots, q denotes the maximum number of negative roots (both of which can be found using Descartes' rule of signs), and n denotes the degree of the equation.

The polynomial

has one sign change; so the number of positive real roots is one. As

has no sign change, the original polynomial has no negative real roots. So the number of nonreal roots is

Since nonreal roots of a polynomial with real coefficients must occur in conjugate pairs, it results can that has exactly two nonreal roots and one real root, which is positive.

The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case.

If the real polynomial P has k real positive roots counted with multiplicity, then for every a > 0 there are at least k changes of sign in the sequence of coefficients of the Taylor series of the function e ax P( x). For a sufficiently large, there are exactly k such changes of sign.

In the 1970s Askold Khovanskii developed the theory of fewnomials that generalises Descartes' rule. The rule of signs can be thought of as stating that the number of real roots of a polynomial is dependent on the polynomial's complexity, and that this complexity is proportional to the number of monomials it has, not its degree. Khovanskiǐ showed that this holds true not just for polynomials but for algebraic combinations of many transcendental functions, the so-called Pfaffian functions.

This article incorporates material from Descartes' rule of signs on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie , is a technique for getting information on the number of positive real roots of a polynomial.

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Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

Most optical phenomena can be accounted for using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation.

Some phenomena depend on the fact that light has both wave-like and particle-like properties. Explanation of these effects requires quantum mechanics. When considering light's particle-like properties, the light is modelled as a collection of particles called " photons". Quantum optics deals with the application of quantum mechanics to optical systems.

Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, and medicine (particularly ophthalmology and optometry). Practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics.

Optics began with the development of lenses by the ancient Egyptians and Mesopotamians. The earliest known lenses, made from polished crystal, often quartz, date from as early as 2000 BC from Crete (Archaeological Museum of Heraclion, Greece). Lenses from Rhodes date around 700 BC, as do Assyrian lenses such as the Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world. The word optics comes from the ancient Greek word ὀπτική ( optikē), meaning "appearance, look".

Greek philosophy on optics broke down into two opposing theories on how vision worked, the intromission theory and the emission theory. The intromission approach saw vision as coming from objects casting off copies of themselves (called eidola) that were captured by the eye. With many propagators including Democritus, Epicurus, Aristotle and their followers, this theory seems to have some contact with modern theories of what vision really is, but it remained only speculation lacking any experimental foundation.

Plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in . Some hundred years later, Euclid (4th-3rd century BC) wrote a treatise entitled where he linked vision to geometry, creating geometrical optics. He based his work on Plato's emission theory wherein he described the mathematical rules of perspective and described the effects of refraction qualitatively, although he questioned that a beam of light from the eye could instantaneously light up the stars every time someone blinked. Euclid stated the principle of shortest trajectory of light, and considered multiple reflections on flat and spherical mirrors. Ptolemy, in his treatise , held an extramission-intromission theory of vision: the rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarized much of Euclid and went on to describe a way to measure the angle of refraction, though he failed to notice the empirical relationship between it and the angle of incidence. Plutarch (1st-2nd century AD) described multiple reflections on spherical mirrors and discussed the creation of magnified and reduced images, both real and imaginary, including the case of chirality of the images.

During the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world. One of the earliest of these was Al-Kindi (c. 801-873) who wrote on the merits of Aristotelian and Euclidean ideas of optics, favouring the emission theory since it could better quantify optical phenomena. In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for lenses and curved mirrors. In the early 11th century, Alhazen (Ibn al-Haytham) wrote the Book of Optics ( Kitab al-manazir) in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment. He rejected the "emission theory" of Ptolemaic optics with its rays being emitted by the eye, and instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and then entered the eye, although he was unable to correctly explain how the eye captured the rays. Alhazen's work was largely ignored in the Arabic world but it was anonymously translated into Latin around 1200 A.D. and further summarised and expanded on by the Polish monk Witelo making it a standard text on optics in Europe for the next 400 years.

In the 13th century in medieval Europe, English bishop Robert Grosseteste wrote on a wide range of scientific topics, and discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light, basing it on the works Aristotle and Platonism. Grosseteste's most famous disciple, Roger Bacon, wrote works citing a wide range of recently translated optical and philosophical works, including those of Alhazen, Aristotle, Avicenna, Averroes, Euclid, al-Kindi, Ptolemy, Tideus, and Constantine the African. Bacon was able to use parts of glass spheres as magnifying glasses to demonstrate that light reflects from objects rather than being released from them.

The first wearable eyeglasses were invented in Italy around 1286. This was the start of the optical industry of grinding and polishing lenses for these "spectacles", first in Venice and Florence in the thirteenth century, and later in the spectacle making centres in both the Netherlands and Germany. Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses rather than using the rudimentary optical theory of the day (theory which for the most part could not even adequately explain how spectacles worked). This practical development, mastery, and experimentation with lenses led directly to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle making centres in the Netherlands.

In the early 17th century, Johannes Kepler expanded on geometric optics in his writings, covering lenses, reflection by flat and curved mirrors, the principles of pinhole cameras, inverse-square law governing the intensity of light, and the optical explanations of astronomical phenomena such as lunar and solar eclipses and astronomical parallax. He was also able to correctly deduce the role of the retina as the actual organ that recorded images, finally being able to scientifically quantify the effects of different types of lenses that spectacle makers had been observing over the previous 300 years. After the invention of the telescope, Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope, using two convex lenses to produce higher magnification.

Optical theory progressed in the mid-17th century with treatises written by philosopher René Descartes, which explained a variety of optical phenomena including reflection and refraction by assuming that light was emitted by objects which produced it. This differed substantively from the ancient Greek emission theory. In the late 1660s and early 1670s, Isaac Newton expanded Descartes' ideas into a corpuscle theory of light, famously determining that white light was a mix of colours which can be separated into its component parts with a prism. In 1690, Christiaan Huygens proposed a wave theory for light based on suggestions that had been made by Robert Hooke in 1664. Hooke himself publicly criticised Newton's theories of light and the feud between the two lasted until Hooke's death. In 1704, Newton published and, at the time, partly because of his success in other areas of physics, he was generally considered to be the victor in the debate over the nature of light.

Newtonian optics was generally accepted until the early 19th century when Thomas Young and Augustin-Jean Fresnel conducted experiments on the interference of light that firmly established light's wave nature. Young's famous double slit experiment showed that light followed the law of superposition, which is a wave-like property not predicted by Newton's corpuscle theory. This work led to a theory of diffraction for light and opened an entire area of study in physical optics. Wave optics was successfully unified with electromagnetic theory by James Clerk Maxwell in the 1860s.

The next development in optical theory came in 1899 when Max Planck correctly modelled blackbody radiation by assuming that the exchange of energy between light and matter only occurred in discrete amounts he called quanta. In 1905, Albert Einstein published the theory of the photoelectric effect that firmly established the quantization of light itself. In 1913, Niels Bohr showed that atoms could only emit discrete amounts of energy, thus explaining the discrete lines seen in emission and absorption spectra. The understanding of the interaction between light and matter that followed from these developments not only formed the basis of quantum optics but also was crucial for the development of quantum mechanics as a whole. The ultimate culmination, the theory of quantum electrodynamics, explains all optics and electromagnetic processes in general as the result of the exchange of real and virtual photons. Quantum optics gained practical importance with the inventions of the maser in 1953 and of the laser in 1960.

Following the work of Paul Dirac in quantum field theory, George Sudarshan, Roy J. Glauber, and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light.

Classical optics is divided into two main branches: geometrical (or ray) optics and physical (or wave) optics. In geometrical optics, light is considered to travel in straight lines, while in physical optics, light is considered as an electromagnetic wave.

Geometrical optics can be viewed as an approximation of physical optics that applies when the wavelength of the light used is much smaller than the size of the optical elements in the system being modelled.

Geometrical optics, or ray optics, describes the propagation of light in terms of "rays" which travel in straight lines, and whose paths are governed by the laws of reflection and refraction at interfaces between different media. These laws were discovered empirically as far back as 984 AD and have been used in the design of optical components and instruments from then until the present day. They can be summarised as follows:

When a ray of light hits the boundary between two transparent materials, it is divided into a reflected and a refracted ray.

The law of reflection says that the reflected ray lies in the plane of incidence, and the angle of reflection equals the angle of incidence.

The law of refraction says that the refracted ray lies in the plane of incidence, and the sine of the angle of refraction divided by the sine of the angle of incidence is a constant:

,

where n is a constant for any two materials and a given colour of light. If the first material is air or vacuum, n is the refractive index of the second material.

The laws of reflection and refraction can be derived from Fermat's principle which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Geometric optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.

Reflections can be divided into two types: specular reflection and diffuse reflection. Specular reflection describes the gloss of surfaces such as mirrors, which reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual ( real) or extrapolated ( virtual) location in space. Diffuse reflection describes non-glossy materials, such as paper or rock. The reflections from these surfaces can only be described statistically, with the exact distribution of the reflected light depending on the microscopic structure of the material. Many diffuse reflectors are described or can be approximated by Lambert's cosine law, which describes surfaces that have equal luminance when viewed from any angle. Glossy surfaces can give both specular and diffuse reflection.

In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with the surface normal, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays and the normal lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection.

For flat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. The law also implies that mirror images are parity inverted, which we perceive as a left-right inversion. Images formed from reflection in two (or any even number of) mirrors are not parity inverted. Corner reflectors produce reflected rays that travel back in the direction from which the incident rays came. This is called retroreflection.

Refraction occurs when light travels through an area of space that has a changing index of refraction; this principle allows for lenses and the focusing of light. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction and another medium with index of refraction . In such situations, Snell's Law describes the resulting deflection of the light ray:

where and are the angles between the normal (to the interface) and the incident and refracted waves, respectively.[42]

The index of refraction of a medium is related to the speed, v, of light in that medium by

where c is the speed of light in vacuum.

Snell's Law can be used to predict the deflection of light rays as they pass through linear media as long as the indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. In most materials, the index of refraction varies with the frequency of the light. Taking this into account, Snell's Law can be used to predict how a prism will disperse light into a spectrum. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton.

Some media have an index of refraction which varies gradually with position and, therefore, light rays in the medium are curved. This effect is responsible for mirages seen on hot days: a change in index of refraction air with height causes light rays to bend, creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). Optical materials with varying index of refraction are called gradient-index (GRIN) materials. Such materials are used to make gradient-index optics.

For light rays travelling from a material with a high index of refraction to a material with a low index of refraction, Snell's law predicts that there is no when is large. In this case, no transmission occurs; all the light is reflected. This phenomenon is called total internal reflection and allows for fibre optics technology. As light travels down an optical fibre, it undergoes total internal reflection allowing for essentially no light to be lost over the length of the cable.[42]

A device which produces converging or diverging light rays due to refraction is known as a lens. Lenses are characterized by their focal length: a converging lens has positive focal length, while a diverging lens has negative focal length. Smaller focal length indicates that the lens has a stronger converging or diverging effect. The focal length of a simple lens in air is given by the lensmaker's equation.

Ray tracing can be used to show how images are formed by a lens. For a thin lens in air, the location of the image is given by the simple equation

where is the distance from the object to the lens, is the distance from the lens to the image, and is the focal length of the lens. In the sign convention used here, the object and image distances are positive if the object and image are on opposite sides of the lens.[44]

Incoming parallel rays are focused by a converging lens onto a spot one focal length from the lens, on the far side of the lens. This is called the rear focal point of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens.

With diverging lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at a spot one focal length in front of the lens. This is the lens's front focal point. Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal point, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens. As with mirrors, upright images produced by a single lens are virtual, while inverted images are real.

Lenses suffer from aberrations that distort images. Monochromatic aberrations occur because the geometry of the lens does not perfectly direct rays from each object point to a single point on the image, while chromatic aberration occurs because the index of refraction of the lens varies with the wavelength of the light.

In physical optics, light is considered to propagate as a wave. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The speed of light waves in air is approximately 3.0×10 ⁸ m/s (exactly 299,792,458 m/s in vacuum). The wavelength of visible light waves varies between 400 and 700 nm, but the term "light" is also often applied to infrared (0.7-300 μm) and ultraviolet radiation (10-400 nm).

The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated. The existence of electromagnetic waves was predicted in 1865 by Maxwell's equations. These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves. Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered.

Many simplified approximations are available for analysing and designing optical systems. Most of these use a single scalar quantity to represent the electric field of the light wave, rather than using a vector model with orthogonal electric and magnetic vectors. The Huygens-Fresnel equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of superposition of waves. The Kirchhoff diffraction equation, which is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens-Fresnel principle can be found in the articles on diffraction and Fraunhofer diffraction.

More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect the interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light.

All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing.

Gaussian beam propagation is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics.

In the absence of nonlinear effects, the superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances. This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are , both the wave crests and wave troughs align. This results in constructive interference and an increase in the amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in destructive interference and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect.

combined
waveform

wave 1

wave 2

Two waves in phase

Two waves 180° out
of phase

Since the Huygens-Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns. Interferometry is the science of measuring these patterns, usually as a means of making precise determinations of distances or angular resolutions. The Michelson interferometer was a famous instrument which used interference effects to accurately measure the speed of light.

The appearance of thin films and coatings is directly affected by interference effects. Antireflective coatings use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with thickness one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550 nm. More complex designs using multiple layers can achieve low reflectivity over a broad band, or extremely low reflectivity at a single wavelength.

Diffraction is the process by which light interference is most commonly observed. The effect was first described in 1665 by Francesco Maria Grimaldi, who also coined the term from the Latin diffringere, 'to break into pieces'. Later that century, Robert Hooke and Isaac Newton also described phenomena now known to be diffraction in Newton's rings while James Gregory recorded his observations of diffraction patterns from bird feathers.

The first physical optics model of diffraction that relied on the Huygens-Fresnel principle was developed in 1803 by Thomas Young in his interference experiments with the interference patterns of two closely spaced slits. Young showed that his results could only be explained if the two slits acted as two unique sources of waves rather than corpuscles. In 1815 and 1818, Augustin-Jean Fresnel firmly established the mathematics of how wave interference can account for diffraction.

The simplest physical models of diffraction use equations that describe the angular separation of light and dark fringes due to light of a particular wavelength (λ). In general, the equation takes the form

where is the separation between two wavefront sources (in the case of Young's experiments, it was two slits), is the angular separation between the central fringe and the th order fringe, where the central maximum is .[58]

This equation is modified slightly to take into account a variety of situations such as diffraction through a single gap, diffraction through multiple slits, or diffraction through a diffraction grating that contains a large number of slits at equal spacing. More complicated models of diffraction require working with the mathematics of Fresnel or Fraunhofer diffraction.

X-ray diffraction makes use of the fact that atoms in a crystal have regular spacing at distances that are on the order of one angstrom. To see diffraction patterns, x-rays with similar wavelengths to that spacing are passed through the crystal. Since crystals are three-dimensional objects rather than two-dimensional gratings, the associated diffraction pattern varies in two directions according to Bragg reflection, with the associated bright spots occurring in unique patterns and being twice the spacing between atoms.[58]

Diffraction effects limit the ability for an optical detector to optically resolve separate light sources. In general, light that is passing through an aperture will experience diffraction and the best images that can be created (as described in diffraction-limited optics) appear as a central spot with surrounding bright rings, separated by dark nulls; this pattern is known as an Airy pattern, and the central bright lobe as an Airy disk. The size of such a disk is given by

where θ is the angular resolution, λ is the wavelength of the light, and D is the diameter of the lens aperture. If the angular separation of the two points is significantly less than the Airy disk angular radius, then the two points cannot be resolved in the image, but if their angular separation is much greater than this, distinct images of the two points are formed and they can therefore be resolved. Rayleigh defined the somewhat arbitrary " Rayleigh criterion" that two points whose angular separation is equal to the Airy disk radius (measured to first null, that is, to the first place where no light is seen) can be considered to be resolved. It can be seen that the greater the diameter of the lens or its aperture, the finer the resolution. Interferometry, with its ability to mimic extremely large baseline apertures, allows for the greatest angular resolution possible.

Refractive processes take place in the physical optics limit, where the wavelength of light is similar to other distances, as a kind of scattering. The simplest type of scattering is Thomson scattering which occurs when electromagnetic waves are deflected by single particles. In the limit of Thomson scattering, in which the wavelike nature of light is evident, light is dispersed independent of the frequency, in contrast to Compton scattering which is frequency-dependent and strictly a quantum mechanical process, involving the nature of light as particles. In a statistical sense, elastic scattering of light by numerous particles much smaller than the wavelength of the light is a process known as Rayleigh scattering while the similar process for scattering by particles that are similar or larger in wavelength is known as Mie scattering with the Tyndall effect being a commonly observed result. A small proportion of light scattering from atoms or molecules may undergo Raman scattering, wherein the frequency changes due to excitation of the atoms and molecules. Brillouin scattering occurs when the frequency of light changes due to local changes with time and movements of a dense material.

Dispersion occurs when different frequencies of light have different phase velocities, due either to material properties ( material dispersion) or to the geometry of an optical waveguide ( waveguide dispersion). The most familiar form of dispersion is a decrease in index of refraction with increasing wavelength, which is seen in most transparent materials. This is called "normal dispersion". It occurs in all dielectric materials, in wavelength ranges where the material does not absorb light. In wavelength ranges where a medium has significant absorption, the index of refraction can increase with wavelength. This is called "anomalous dispersion".

The separation of colours by a prism is an example of normal dispersion. At the surfaces of the prism, Snell's law predicts that light incident at an angle θ to the normal will be refracted at an angle arcsin(sin (θ) / n). Thus, blue light, with its higher refractive index, is bent more strongly than red light, resulting in the well-known rainbow pattern.

Material dispersion is often characterised by the Abbe number, which gives a simple measure of dispersion based on the index of refraction at three specific wavelengths. Waveguide dispersion is dependent on the propagation constant. Both kinds of dispersion cause changes in the group characteristics of the wave, the features of the wave packet that change with the same frequency as the amplitude of the electromagnetic wave. "Group velocity dispersion" manifests as a spreading-out of the signal "envelope" of the radiation and can be quantified with a group dispersion delay parameter:

where is the group velocity.[63] For a uniform medium, the group velocity is

where n is the index of refraction and c is the speed of light in a vacuum. This gives a simpler form for the dispersion delay parameter:

If D is less than zero, the medium is said to have positive dispersion or normal dispersion. If D is greater than zero, the medium has negative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components slow down more than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. This causes the spectrum coming out of a prism to appear with red light the least refracted and blue/violet light the most refracted. Conversely, if a pulse travels through an anomalously (negatively) dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.

The result of group velocity dispersion, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fibres, since if dispersion is too high, a group of pulses representing information will each spread in time and merge, making it impossible to extract the signal.

Polarization is a general property of waves that describes the orientation of their oscillations. For transverse waves such as many electromagnetic waves, it describes the orientation of the oscillations in the plane perpendicular to the wave's direction of travel. The oscillations may be oriented in a single direction ( linear polarization), or the oscillation direction may rotate as the wave travels ( circular or elliptical polarization). Circularly polarised waves can rotate rightward or leftward in the direction of travel, and which of those two rotations is present in a wave is called the wave's chirality.

The typical way to consider polarization is to keep track of the orientation of the electric field vector as the electromagnetic wave propagates. The electric field vector of a plane wave may be arbitrarily divided into two perpendicular components labeled x and y (with z indicating the direction of travel). The shape traced out in the x-y plane by the electric field vector is a Lissajous figure that describes the polarization state. The following figures show some examples of the evolution of the electric field vector (blue), with time (the vertical axes), at a particular point in space, along with its x and y components (red/left and green/right), and the path traced by the vector in the plane (purple): The same evolution would occur when looking at the electric field at a particular time while evolving the point in space, along the direction opposite to propagation.

Elliptical polarization

In the leftmost figure above, the x and y components of the light wave are in phase. In this case, the ratio of their strengths is constant, so the direction of the electric vector (the vector sum of these two components) is constant. Since the tip of the vector traces out a single line in the plane, this special case is called linear polarization. The direction of this line depends on the relative amplitudes of the two components.

In the middle figure, the two orthogonal components have the same amplitudes and are 90° out of phase. In this case, one component is zero when the other component is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this requirement: the x component can be 90° ahead of the y component or it can be 90° behind the y component. In this special case, the electric vector traces out a circle in the plane, so this polarization is called circular polarization. The rotation direction in the circle depends on which of the two phase relationships exists and corresponds to right-hand circular polarization and left-hand circular polarization.

In all other cases, where the two components either do not have the same amplitudes and/or their phase difference is neither zero nor a multiple of 90°, the polarization is called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse). This is shown in the above figure on the right. Detailed mathematics of polarization is done using Jones calculus and is characterised by the Stokes parameters.

Changing polarization

Media that have different indexes of refraction for different polarization modes are called birefringent. Well known manifestations of this effect appear in optical wave plates/retarders (linear modes) and in Faraday rotation/ optical rotation (circular modes). If the path length in the birefringent medium is sufficient, plane waves will exit the material with a significantly different propagation direction, due to refraction. For example, this is the case with macroscopic crystals of calcite, which present the viewer with two offset, orthogonally polarised images of whatever is viewed through them. It was this effect that provided the first discovery of polarization, by Erasmus Bartholinus in 1669. In addition, the phase shift, and thus the change in polarization state, is usually frequency dependent, which, in combination with dichroism, often gives rise to bright colours and rainbow-like effects. In mineralogy, such properties, known as pleochroism, are frequently exploited for the purpose of identifying minerals using polarization microscopes. Additionally, many plastics that are not normally birefringent will become so when subject to mechanical stress, a phenomenon which is the basis of photoelasticity. Non-birefringent methods, to rotate the linear polarization of light beams, include the use of prismatic polarization rotators which use total internal reflection in a prism set designed for efficient collinear transmission.

Media that reduce the amplitude of certain polarization modes are called dichroic, with devices that block nearly all of the radiation in one mode known as polarizing filters or simply " polarisers". Malus' law, which is named after Étienne-Louis Malus, says that when a perfect polariser is placed in a linear polarised beam of light, the intensity, I, of the light that passes through is given by

where

I ₀ is the initial intensity,

and is the angle between the light's initial polarization direction and the axis of the polariser.

A beam of unpolarised light can be thought of as containing a uniform mixture of linear polarizations at all possible angles. Since the average value of is 1/2, the transmission coefficient becomes

In practice, some light is lost in the polariser and the actual transmission of unpolarised light will be somewhat lower than this, around 38% for Polaroid-type polarisers but considerably higher (>49.9%) for some birefringent prism types.

In addition to birefringence and dichroism in extended media, polarization effects can also occur at the (reflective) interface between two materials of different refractive index. These effects are treated by the Fresnel equations. Part of the wave is transmitted and part is reflected, with the ratio depending on angle of incidence and the angle of refraction. In this way, physical optics recovers Brewster's angle. When light reflects from a thin film on a surface, interference between the reflections from the film's surfaces can produce polarization in the reflected and transmitted light.

Natural light

Most sources of electromagnetic radiation contain a large number of atoms or molecules that emit light. The orientation of the electric fields produced by these emitters may not be correlated, in which case the light is said to be unpolarised. If there is partial correlation between the emitters, the light is partially polarised. If the polarization is consistent across the spectrum of the source, partially polarised light can be described as a superposition of a completely unpolarised component, and a completely polarised one. One may then describe the light in terms of the degree of polarization, and the parameters of the polarization ellipse.

Light reflected by shiny transparent materials is partly or fully polarised, except when the light is normal (perpendicular) to the surface. It was this effect that allowed the mathematician Étienne-Louis Malus to make the measurements that allowed for his development of the first mathematical models for polarised light. Polarization occurs when light is scattered in the atmosphere. The scattered light produces the brightness and colour in clear skies. This partial polarization of scattered light can be taken advantage of using polarizing filters to darken the sky in photographs. Optical polarization is principally of importance in chemistry due to circular dichroism and optical rotation (" circular birefringence") exhibited by optically active ( chiral) molecules.

Modern optics encompasses the areas of optical science and engineering that became popular in the 20th century. These areas of optical science typically relate to the electromagnetic or quantum properties of light but do include other topics. A major subfield of modern optics, quantum optics, deals with specifically quantum mechanical properties of light. Quantum optics is not just theoretical; some modern devices, such as lasers, have principles of operation that depend on quantum mechanics. Light detectors, such as photomultipliers and channeltrons, respond to individual photons. Electronic image sensors, such as CCDs, exhibit shot noise corresponding to the statistics of individual photon events. Light-emitting diodes and photovoltaic cells, too, cannot be understood without quantum mechanics. In the study of these devices, quantum optics often overlaps with quantum electronics.

Specialty areas of optics research include the study of how light interacts with specific materials as in crystal optics and metamaterials. Other research focuses on the phenomenology of electromagnetic waves as in singular optics, non-imaging optics, non-linear optics, statistical optics, and radiometry. Additionally, computer engineers have taken an interest in integrated optics, machine vision, and photonic computing as possible components of the "next generation" of computers.

Today, the pure science of optics is called optical science or optical physics to distinguish it from applied optical sciences, which are referred to as optical engineering. Prominent subfields of optical engineering include illumination engineering, photonics, and optoelectronics with practical applications like lens design, fabrication and testing of optical components, and image processing. Some of these fields overlap, with nebulous boundaries between the subjects terms that mean slightly different things in different parts of the world and in different areas of industry. A professional community of researchers in nonlinear optics has developed in the last several decades due to advances in laser technology.

A laser is a device that emits light (electromagnetic radiation) through a process called stimulated emission. The term laser is an acronym for Light Amplification by Stimulated Emission of Radiation. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses. Because the microwave equivalent of the laser, the maser, was developed first, devices that emit microwave and radio frequencies are usually called masers.

The first working laser was demonstrated on 16 May 1960 by Theodore Maiman at Hughes Research Laboratories. When first invented, they were called "a solution looking for a problem". Since then, lasers have become a multibillion-dollar industry, finding utility in thousands of highly varied applications. The first application of lasers visible in the daily lives of the general population was the supermarket barcode scanner, introduced in 1974. The laserdisc player, introduced in 1978, was the first successful consumer product to include a laser, but the compact disc player was the first laser-equipped device to become truly common in consumers' homes, beginning in 1982. These optical storage devices use a semiconductor laser less than a millimetre wide to scan the surface of the disc for data retrieval. Fibre-optic communication relies on lasers to transmit large amounts of information at the speed of light. Other common applications of lasers include laser printers and laser pointers. Lasers are used in medicine in areas such as bloodless surgery, laser eye surgery, and laser capture microdissection and in military applications such as missile defence systems, electro-optical countermeasures (EOCM), and lidar. Lasers are also used in holograms, bubblegrams, laser light shows, and laser hair removal.

Kapitsa-Dirac effect

The Kapitsa-Dirac effect causes beams of particles to diffract as the result of meeting a standing wave of light. Light can be used to position matter using various phenomena (see optical tweezers).

Optics is part of everyday life. The ubiquity of visual systems in biology indicates the central role optics plays as the science of one of the five senses. Many people benefit from eyeglasses or contact lenses, and optics are integral to the functioning of many consumer goods including cameras. Rainbows and mirages are examples of optical phenomena. Optical communication provides the backbone for both the Internet and modern telephony.

The human eye functions by focusing light onto a layer of photoreceptor cells called the retina, which forms the inner lining of the back of the eye. The focusing is accomplished by a series of transparent media. Light entering the eye passes first through the cornea, which provides much of the eye's optical power. The light then continues through the fluid just behind the cornea-the anterior chamber, then passes through the pupil. The light then passes through the lens, which focuses the light further and allows adjustment of focus. The light then passes through the main body of fluid in the eye-the vitreous humour, and reaches the retina. The cells in the retina line the back of the eye, except for where the optic nerve exits; this results in a blind spot.

There are two types of photoreceptor cells, rods and cones, which are sensitive to different aspects of light. Rod cells are sensitive to the intensity of light over a wide frequency range, thus are responsible for black-and-white vision. Rod cells are not present on the fovea, the area of the retina responsible for central vision, and are not as responsive as cone cells to spatial and temporal changes in light. There are, however, twenty times more rod cells than cone cells in the retina because the rod cells are present across a wider area. Because of their wider distribution, rods are responsible for peripheral vision.

Ciliary muscles around the lens allow the eye's focus to be adjusted. This process is known as accommodation. The near point and far point define the nearest and farthest distances from the eye at which an object can be brought into sharp focus. For a person with normal vision, the far point is located at infinity. The near point's location depends on how much the muscles can increase the curvature of the lens, and how inflexible the lens has become with age. Optometrists, ophthalmologists, and opticians usually consider an appropriate near point to be closer than normal reading distance-approximately 25 cm.

The optical power of corrective lenses is measured in diopters, a value equal to the reciprocal of the focal length measured in metres; with a positive focal length corresponding to a converging lens and a negative focal length corresponding to a diverging lens. For lenses that correct for astigmatism as well, three numbers are given: one for the spherical power, one for the cylindrical power, and one for the angle of orientation of the astigmatism.

Optical illusions (also called visual illusions) are characterized by visually perceived images that differ from objective reality. The information gathered by the eye is processed in the brain to give a percept that differs from the object being imaged. Optical illusions can be the result of a variety of phenomena including physical effects that create images that are different from the objects that make them, the physiological effects on the eyes and brain of excessive stimulation (e.g. brightness, tilt, colour, movement), and cognitive illusions where the eye and brain make unconscious inferences.

Cognitive illusions include some which result from the unconscious misapplication of certain optical principles. For example, the Ames room, Hering, Müller-Lyer, Orbison, Ponzo, Sander, and Wundt illusions all rely on the suggestion of the appearance of distance by using converging and diverging lines, in the same way that parallel light rays (or indeed any set of parallel lines) appear to converge at a vanishing point at infinity in two-dimensionally rendered images with artistic perspective. This suggestion is also responsible for the famous moon illusion where the moon, despite having essentially the same angular size, appears much larger near the horizon than it does at zenith. This illusion so confounded Ptolemy that he incorrectly attributed it to atmospheric refraction when he described it in his treatise, .

Another type of optical illusion exploits broken patterns to trick the mind into perceiving symmetries or asymmetries that are not present. Examples include the café wall, Ehrenstein, Fraser spiral, Poggendorff, and Zöllner illusions. Related, but not strictly illusions, are patterns that occur due to the superimposition of periodic structures. For example, transparent tissues with a grid structure produce shapes known as moiré patterns, while the superimposition of periodic transparent patterns comprising parallel opaque lines or curves produces line moiré patterns.

Microscopes were first developed with just two lenses: an objective lens and an eyepiece. The objective lens is essentially a magnifying glass and was designed with a very small focal length while the eyepiece generally has a longer focal length. This has the effect of producing magnified images of close objects. Generally, an additional source of illumination is used since magnified images are dimmer due to the conservation of energy and the spreading of light rays over a larger surface area. Modern microscopes, known as compound microscopes have many lenses in them (typically four) to optimize the functionality and enhance image stability. A slightly different variety of microscope, the comparison microscope, looks at side-by-side images to produce a stereoscopic binocular view that appears three dimensional when used by humans.

The first telescopes, called refracting telescopes, were also developed with a single objective and eyepiece lens. In contrast to the microscope, the objective lens of the telescope was designed with a large focal length to avoid optical aberrations. The objective focuses an image of a distant object at its focal point which is adjusted to be at the focal point of an eyepiece of a much smaller focal length. The main goal of a telescope is not necessarily magnification, but rather collection of light which is determined by the physical size of the objective lens. Thus, telescopes are normally indicated by the diameters of their objectives rather than by the magnification which can be changed by switching eyepieces. Because the magnification of a telescope is equal to the focal length of the objective divided by the focal length of the eyepiece, smaller focal-length eyepieces cause greater magnification.

Since crafting large lenses is much more difficult than crafting large mirrors, most modern telescopes are reflecting telescopes, that is, telescopes that use a primary mirror rather than an objective lens. The same general optical considerations apply to reflecting telescopes that applied to refracting telescopes, namely, the larger the primary mirror, the more light collected, and the magnification is still equal to the focal length of the primary mirror divided by the focal length of the eyepiece. Professional telescopes generally do not have eyepieces and instead place an instrument (often a charge-coupled device) at the focal point instead.

The optics of photography involves both lenses and the medium in which the electromagnetic radiation is recorded, whether it be a plate, film, or charge-coupled device. Photographers must consider the reciprocity of the camera and the shot which is summarized by the relation

Exposure ∝ ApertureArea × ExposureTime × SceneLuminance

In other words, the smaller the aperture (giving greater depth of focus), the less light coming in, so the length of time has to be increased (leading to possible blurriness if motion occurs). An example of the use of the law of reciprocity is the Sunny 16 rule which gives a rough estimate for the settings needed to estimate the proper exposure in daylight.

A camera's aperture is measured by a unitless number called the f-number or f-stop, f/#, often notated as , and given by

where is the focal length, and is the diameter of the entrance pupil. By convention, "f/#" is treated as a single symbol, and specific values of f/# are written by replacing the number sign with the value. The two ways to increase the f-stop are to either decrease the diameter of the entrance pupil or change to a longer focal length (in the case of a zoom lens, this can be done by simply adjusting the lens). Higher f-numbers also have a larger depth of field due to the lens approaching the limit of a pinhole camera which is able to focus all images perfectly, regardless of distance, but requires very long exposure times.[90]

The field of view that the lens will provide changes with the focal length of the lens. There are three basic classifications based on the relationship to the diagonal size of the film or sensor size of the camera to the focal length of the lens:

Normal lens: angle of view of about 50° (called normal because this angle considered roughly equivalent to human vision) and a focal length approximately equal to the diagonal of the film or sensor.

Wide-angle lens: angle of view wider than 60° and focal length shorter than a normal lens.

Long focus lens: angle of view narrower than a normal lens. This is any lens with a focal length longer than the diagonal measure of the film or sensor. The most common type of long focus lens is the telephoto lens, a design that uses a special telephoto group to be physically shorter than its focal length.

Modern zoom lenses may have some or all of these attributes.

The absolute value for the exposure time required depends on how sensitive to light the medium being used is (measured by the film speed, or, for digital media, by the quantum efficiency). Early photography used media that had very low light sensitivity, and so exposure times had to be long even for very bright shots. As technology has improved, so has the sensitivity through film cameras and digital cameras.

Other results from physical and geometrical optics apply to camera optics. For example, the maximum resolution capability of a particular camera set-up is determined by the diffraction limit associated with the pupil size and given, roughly, by the Rayleigh criterion.

The unique optical properties of the atmosphere cause a wide range of spectacular optical phenomena. The blue colour of the sky is a direct result of Rayleigh scattering which redirects higher frequency (blue) sunlight back into the field of view of the observer. Because blue light is scattered more easily than red light, the sun takes on a reddish hue when it is observed through a thick atmosphere, as during a sunrise or sunset. Additional particulate matter in the sky can scatter different colours at different angles creating colourful glowing skies at dusk and dawn. Scattering off of ice crystals and other particles in the atmosphere are responsible for halos, afterglows, coronas, rays of sunlight, and sun dogs. The variation in these kinds of phenomena is due to different particle sizes and geometries.

Rainbows are the result of a combination of internal reflection and dispersive refraction of light in raindrops. A single reflection off the backs of an array of raindrops produces a rainbow with an angular size on the sky that ranges from 40° to 42° with red on the outside. Double rainbows are produced by two internal reflections with angular size of 50.5° to 54° with violet on the outside. Because rainbows are seen with the sun 180° away from the centre of the rainbow, rainbows are more prominent the closer the sun is to the horizon.

Further reading

Born, Max; Wolf, Emil (2002). Principles of Optics. Cambridge University Press. ISBN 978-1-139-64340-5.

Hecht, Eugene (2002). Optics (4 ed.). Addison-Wesley Longman, Incorporated. ISBN 978-0-8053-8566-3.

Serway, Raymond A.; Jewett, John W. (2004). Physics for scientists and engineers (6, illustrated ed.). Belmont, CA: Thomson-Brooks/Cole. ISBN 978-0-534-40842-8.

Tipler, Paul A.; Mosca, Gene (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics. 2. W.H. Freeman. ISBN 978-0-7167-0810-0.

Lipson, Stephen G.; Lipson, Henry; Tannhauser, David Stefan (1995). Optical Physics. Cambridge University Press. ISBN 978-0-521-43631-1.

Fowles, Grant R. (1975). Introduction to Modern Optics. Courier Dover Publications. ISBN 978-0-486-65957-2.

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Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light.

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Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.

The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays. This behavior was first described by Hero of Alexandria ( AD c. 10-70).

Specular reflection may be contrasted with diffuse reflection, in which light is scattered away from the surface in a range of directions.

When light encounters a boundary of a material, it is affected by the optical and electronic response functions of the material to electromagnetic waves. Optical processes, which comprise reflection and refraction, are expressed by the difference of the refractive index on both sides of the boundary, whereas reflectance and absorption are the real and imaginary parts of the response due to the electronic structure of the material. The degree of participation of each of these processes in the transmission is a function of the frequency, or wavelength, of the light, its polarization, and its angle of incidence. In general, reflection increases with increasing angle of incidence, and with increasing absorptivity at the boundary. The Fresnel equations describe the physics at the optical boundary.

Reflection may occur as specular, or mirror-like, reflection and diffuse reflection. Specular reflection reflects all light which arrives from a given direction at the same angle, whereas diffuse reflection reflects light in a broad range of directions. The distinction may be illustrated with surfaces coated with glossy paint and matte paint. Matte paints exhibit essentially complete diffuse reflection, while glossy paints show a larger component of specular behavior. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

Light propagates in space as a wave front of electromagnetic fields. A ray of light is characterized by the direction normal to the wave front ( wave normal). When a ray encounters a surface, the angle that the wave normal makes with respect to the surface normal is called the angle of incidence and the plane defined by both directions is the plane of incidence. Reflection of the incident ray also occurs in the plane of incidence.

The law of reflection states that the angle of reflection of a ray equals the angle of incidence, and that the incident direction, the surface normal, and the reflected direction are coplanar.

When the light impinges perpendicularly to the surface, it is reflected straight back in the source direction.

The phenomenon of reflection arises from the diffraction of a plane wave on a flat boundary. When the boundary size is much larger than the wavelength, then the electromagnetic fields at the boundary are oscillating exactly in phase only for the specular direction.

The law of reflection can also be equivalently expressed using linear algebra. The direction of a reflected ray is determined by the vector of incidence and the surface normal vector. Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction (all unit vectors) is:[4][5]

where is a scalar obtained with the dot product. Different authors may define the incident and reflection directions with different signs. Assuming these Euclidean vectors are represented in column form, the equation can be equivalently expressed as a matrix-vector multiplication:

where is the so-called Householder transformation matrix, defined as:

in terms of the identity matrix and twice the outer product of .

Reflectivity is the ratio of the power of the reflected wave to that of the incident wave. It is a function of the wavelength of radiation, and is related to the refractive index of the material as expressed by Fresnel's equations. In regions of the electromagnetic spectrum in which absorption by the material is significant, it is related to the electronic absorption spectrum through the imaginary component of the complex refractive index. The electronic absorption spectrum of an opaque material, which is difficult or impossible to measure directly, may therefore be indirectly determined from the reflection spectrum by a Kramers-Kronig transform. The polarization of the reflected light depends on the symmetry of the arrangement of the incident probing light with respect to the absorbing transitions dipole moments in the material.

Measurement of specular reflection is performed with normal or varying incidence reflection spectrophotometers ( reflectometer) using a scanning variable-wavelength light source. Lower quality measurements using a glossmeter quantify the glossy appearance of a surface in gloss units.

When light is propagating in a material and strikes an interface with a material of lower index of refraction, some of the light is reflected. If the angle of incidence is greater than the critical angle, total internal reflection occurs: all of the light is reflected. The critical angle can be shown to be given by

When light strikes an interface between two materials, the reflected light is generally partially polarized. However, if the light strikes the interface at Brewster's angle, the reflected light is completely linearly polarized parallel to the interface. Brewster's angle is given by

The image in a flat mirror has these features:

It is the same distance behind the mirror as the object is in front.

It is the same size as the object.

It is the right way up (erect).

It is reversed.

It is virtual, meaning that the image appears to be behind the mirror, and cannot be projected onto a screen.

The reversal of images by a plane mirror is perceived differently depending on the circumstances. In many cases, the image in a mirror appears to be reversed from left to right. If a flat mirror is mounted on the ceiling it can appear to reverse up and down if a person stands under it and looks up at it. Similarly a car turning left will still appear to be turning left in the rear view mirror for the driver of a car in front of it. The reversal of directions, or lack thereof, depends on how the directions are defined. More specifically a mirror changes the handedness of the coordinate system, one axis of the coordinate system appears to be reversed, and the chirality of the image may change. For example, the image of a right shoe will look like a left shoe.

A classic example of specular reflection is a mirror, which is specifically designed for specular reflection.

In addition to visible light, specular reflection can be observed in the ionospheric reflection of radiowaves and the reflection of radio- or microwave radar signals by flying objects. The measurement technique of x-ray reflectivity exploits specular reflectivity to study thin films and interfaces with sub-nanometer resolution, using either modern laboratory sources or synchrotron x-rays.

Non-electromagnetic waves can also exhibit specular reflection, as in acoustic mirrors which reflect sound, and atomic mirrors, which reflect neutral atoms. For the efficient reflection of atoms from a solid-state mirror, very cold atoms and/or grazing incidence are used in order to provide significant quantum reflection; ridged mirrors are used to enhance the specular reflection of atoms. Neutron reflectometry uses specular reflection to study material surfaces and thin film interfaces in an analogous fashion to x-ray reflectivity.

1. ^ Tan, R.T. (2013), Specularity, Specular Reflectance. In: Ikeuchi K. (eds) Computer Vision, Springer, Boston, MA, doi:10.1007/978-0-387-31439-6, ISBN 978-0-387-31439-6

2. ^ Sir Thomas Little Heath (1981). A history of Greek mathematics. Volume II: From Aristarchus to Diophantus. ISBN 978-0-486-24074-9.

3. ^ Farin, Gerald; Hansford, Dianne (2005). Practical linear algebra: a geometry toolbox. A K Peters. pp. 191-192. ISBN 978-1-56881-234-2. Archived from the original on 2010-03-07. Practical linear algebra: a geometry toolbox at Google Books

4. ^ Comninos, Peter (2006). Mathematical and computer programming techniques for computer graphics. Springer. p. 361. ISBN 978-1-85233-902-9. Archived from the original on 2018-01-14.

# data[].attributes.attachment_link.description
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays.