History of group theory

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The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Joseph Louis Lagrange, Niels Henrik Abel and Évariste Galois were early researchers in the field of group theory.

Early 19th century

The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Augustin Louis Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so three important threads in its pre-history are developed here.

Development of permutation groups

One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4.

An early source occurs in the problem of forming an equation of degree m having as its roots m of the roots of a given equation of degree ${\displaystyle n>m}$. For simple cases, the problem goes back to Johann van Waveren Hudde (1659). Nicholas Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Thomas Le Seur (1703–1770) (1748) and Edward Waring (1762 to 1782) still further elaborated the idea. Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic.

Lagrange's goal (1770, 1771) was to understand why equations of third and fourth degree admit formulas for solutions, and a key object was the group of permutations of the roots. On this was built the theory of substitutions. He discovered that the roots of all Lagrange resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions, he invented a Calcul des Combinaisons. The contemporary work of Alexandre-Théophile Vandermonde (1770) developed the theory of symmetric functions and solution of cyclotomic polynomials. Leopold Kronecker has been quoted as saying that a new boom in algebra began with Vandermonde's first paper. Similarly Cauchy gave credit to both Lagrange and Vandermonde for studying symmetric functions and permutations of variables.

Paolo Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini was the first person to explore ideas in the theory of permutation groups such as the order of an element of a group, conjugacy, and the cycle decomposition of elements of permutation groups. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Pietro Abbati to himself, in which the group idea is prominent. However, he never formalized the concept of a group, or even of a permutation group.

Galois age fifteen, drawn by a classmate.

Évariste Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the posthumous publication of his collected papers in 1846 (Liouville, Vol. XI). He considered for the first time what is now called the closure property of a group of permutations, which he expressed as

if in such a group one has the substitutions S and T then one has the substitution ST.

Galois found that if ${\displaystyle r_1,r_2,\dots ,r_n}$ are the n roots of an equation, there is always a group of permutations of the r's such that

• every function of the roots invariable by the substitutions of the group is rationally known, and
• conversely, every rationally determinable function of the roots is invariant under the substitutions of the group.

In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with radicals.

Galois was the first to use the words group (groupe in French) and primitive in their modern meanings. He did not use primitive group but called equation primitive an equation whose Galois group is primitive. He discovered the notion of normal subgroups and found that a solvable primitive group may be identified to a subgroup of the affine group of an affine space over a finite field of prime order.

Groups similar to Galois groups are (today) called permutation groups. The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of isomorphism, although limited to the context of permutation groups. It was also Jordan who put the term group in wide use.

An abstract notion of a (finite) group appeared for the first time in Arthur Cayley's 1854 paper On the theory of groups, as depending on the symbolic equation ${\displaystyle {\theta }^n=1}$. Cayley proposed that any finite group is isomorphic to a subgroup of a permutation group, a result known today as Cayley's theorem. In succeeding years, Cayley systematically investigated infinite groups and the algebraic properties of matrices, such as the associativity of multiplication, existence of inverses, and characteristic polynomials.

Groups related to geometry

Felix Klein

Sophus Lie

Secondly, the systematic use of groups in geometry, mainly in the guise of symmetry groups, was initiated by Felix Klein's 1872 Erlangen program. The study of what are now called Lie groups started systematically in 1884 with Sophus Lie, followed by work of Wilhelm Killing, Eduard Study, Issai Schur, Ludwig Maurer, and Élie Cartan. The discontinuous (discrete group) theory was built up by Klein, Lie, Henri Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.

Appearance of groups in number theory

Ernst Kummer

The third root of group theory was number theory. Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work. Ernst Kummer's attempts to prove Fermat's Last Theorem resulted in work introducing groups describing factorization into prime numbers. In 1882, Heinrich M. Weber realized the connection between permutation groups and abelian groups and gave a definition that included a two-sided cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite groups).

Convergence

Camille Jordan

Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des substitutions et des équations algébriques (1870) is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the 19th century were Joseph Louis François Bertrand, Charles Hermite, Ferdinand Georg Frobenius, Leopold Kronecker, and Émile Mathieu; as well as William Burnside, Leonard Eugene Dickson, Otto Hölder, E. H. Moore, Ludwig Sylow, and Heinrich Martin Weber.

The convergence of the above three sources into a uniform theory started with Jordan's Traité and Walther von Dyck (1882) who first defined a group in the full modern sense. The textbooks of Weber and Burnside helped establish group theory as a discipline. The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms of Lie algebras.

Late 19th century

Groups in the 1870-1900 period were described as the continuous groups of Lie, the discontinuous groups, finite groups of substitutions of roots (gradually being called permutations), and finite groups of linear substitutions (usually of finite fields). During the 1880-1920 period, groups described by presentations came into a life of their own through the work of Cayley, Walther von Dyck, Max Dehn, Jakob Nielsen, Otto Schreier, and continued in the 1920-1940 period with the work of H. S. M. Coxeter, Wilhelm Magnus, and others to form the field of combinatorial group theory.

Finite groups in the 1870-1900 period saw such highlights as the Sylow theorems, Hölder's classification of groups of square-free order, and the early beginnings of the character theory of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu), and in the 1870s Klein's group-theoretic vision of geometry was being realized in his Erlangen program. The automorphism groups of higher dimensional projective spaces were studied by Jordan in his Traité and included composition series for most of the so-called classical groups, though he avoided non-prime fields and omitted the unitary groups. The study was continued by Moore and Burnside, and brought into comprehensive textbook form by Leonard Dickson in 1901. The role of simple groups was emphasized by Jordan, and criteria for non-simplicity were developed by Hölder until he was able to classify the simple groups of order less than 200. The study was continued by Frank Nelson Cole (up to 660) and Burnside (up to 1092), and finally in an early "millennium project", up to 2001 by Miller and Ling in 1900.

Continuous groups in the 1870-1900 period developed rapidly. Killing and Lie's foundational papers were published, Hilbert's theorem in invariant theory 1882, etc.

Early 20th century

In the period 1900–1940, infinite "discontinuous" (now called discrete groups) groups gained life of their own. Burnside's famous problem ushered in the study of arbitrary subgroups of finite-dimensional linear groups over arbitrary fields, and indeed arbitrary groups. Fundamental groups and reflection groups encouraged the developments of J. A. Todd and Coxeter, such as the Todd–Coxeter algorithm in combinatorial group theory. Algebraic groups, defined as solutions of polynomial equations (rather than acting on them, as in the earlier century), benefited heavily from the continuous theory of Lie. Bernard Neumann and Hanna Neumann produced their study of varieties of groups, groups defined by group theoretic equations rather than polynomial ones.

Continuous groups also had explosive growth in the 1900-1940 period. Topological groups began to be studied as such. There were many great achievements in continuous groups: Cartan's classification of semisimple Lie algebras, Hermann Weyl's theory of representations of compact groups, Alfréd Haar's work in the locally compact case.

Finite groups in the 1900-1940 grew immensely. This period witnessed the birth of character theory by Frobenius, Burnside, and Schur which helped answer many of the 19th century questions in permutation groups, and opened the way to entirely new techniques in abstract finite groups. This period saw the work of Philip Hall: on a generalization of Sylow's theorem to arbitrary sets of primes which revolutionized the study of finite soluble groups, and on the power-commutator structure of p-groups, including the ideas of regular p-groups and isoclinism of groups, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Hans Zassenhaus's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress on Frobenius groups, and a near classification of Zassenhaus groups.

Mid-20th century

Both depth, breadth and also the impact of group theory subsequently grew. The domain started branching out into areas such as algebraic groups, group extensions, and representation theory. Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify all finite simple groups in 1982. Completing and simplifying the proof of the classification are areas of active research.

Anatoly Maltsev also made important contributions to group theory during this time; his early work was in logic in the 1930s, but in the 1940s he proved important embedding properties of semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s return to logic proving various theories within the study of groups to be undecidable. Earlier, Alfred Tarski proved elementary group theory undecidable.

The period of 1960-1980 was one of excitement in many areas of group theory.

In finite groups, there were many independent milestones. One had the discovery of 22 new sporadic groups, and the completion of the first generation of the classification of finite simple groups. One had the influential idea of the Carter subgroup, and the subsequent creation of formation theory and the theory of classes of groups. One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification.

In discrete groups, the geometric methods of Jacques Tits and the availability the surjectivity of Serge Lang's map allowed a revolution in algebraic groups. The Burnside problem had tremendous progress, with better counterexamples constructed in the 1960s and early 1980s, but the finishing touches "for all but finitely many" were not completed until the 1990s. The work on the Burnside problem increased interest in Lie algebras in exponent p, and the methods of Michel Lazard began to see a wider impact, especially in the study of p-groups.

Continuous groups broadened considerably, with p-adic analytic questions becoming important. Many conjectures were made during this time, including the coclass conjectures.

Late 20th century

The last twenty years of the 20th century enjoyed the successes of over one hundred years of study in group theory.

In finite groups, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the classification of multiply transitive finite groups, the determination of the maximal subgroups of the simple groups and the corresponding classifications of primitive groups. In finite geometry and combinatorics, many problems could now be settled. The modular representation theory entered a new era as the techniques of the classification were axiomatized, including fusion systems, Luis Puig's theory of pairs and nilpotent blocks. The theory of finite soluble groups was likewise transformed by the influential book of Klaus Doerk and Trevor Hawkes which brought the theory of projectors and injectors to a wider audience.

In discrete groups, several areas of geometry came together to produce exciting new fields. Work on knot theory, orbifolds, hyperbolic manifolds, and groups acting on trees (the Bass–Serre theory), much enlivened the study of hyperbolic groups, automatic groups. Questions such as William Thurston's 1982 geometrization conjecture, inspired entirely new techniques in geometric group theory and low-dimensional topology, and was involved in the solution of one of the Millennium Prize Problems, the Poincaré conjecture.

Continuous groups saw the solution of the problem of hearing the shape of a drum in 1992 using symmetry groups of the laplacian operator. Continuous techniques were applied to many aspects of group theory using function spaces and quantum groups. Many 18th and 19th century problems are now revisited in this more general setting, and many questions in the theory of the representations of groups have answers.

Today

Group theory continues to be an intensely studied matter. Its importance to contemporary mathematics as a whole may be seen from the 2008 Abel Prize, awarded to John Griggs Thompson and Jacques Tits for their contributions to group theory.

Giambattista della Porta

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Giambattista della Porta
Giambattista della Porta in 1589
Born	late 1535 Vico Equense, Kingdom of Naples
Died	4 February 1615 (aged 79) Naples, Kingdom of Naples
Nationality	Italian
Scientific career
Fields	Occultism, astrology, alchemy, mathematics, meteorology, and natural philosophy

Giambattista della Porta (Italian pronunciation: [dʒambatˈtista della ˈpɔrta]; 1535 – 4 February 1615), also known as Giovanni Battista Della Porta, was an Italian scholar, polymath and playwright who lived in Naples at the time of the Renaissance, Scientific Revolution and Counter-Reformation.

Giambattista della Porta spent the majority of his life on scientific endeavours. He benefited from an informal education of tutors and visits from renowned scholars. His most famous work, first published in 1558, is entitled Magia Naturalis (Natural Magic). In this book he covered a variety of the subjects he had investigated, including occult philosophy, astrology, alchemy, mathematics, meteorology, and natural philosophy. He was also referred to as "professor of secrets".

Childhood

Giambattista della Porta was born at Vico Equense, near Naples, to the nobleman Nardo Antonio della Porta. He was the third of four sons and the second to survive childhood, having an older brother Gian Vincenzo and a younger brother Gian Ferrante. Della Porta had a privileged childhood including his education. His father had a thirst for learning, a trait he would pass on to all of his children. He surrounded himself with distinguished people and entertained the likes of philosophers, mathematicians, poets, and musicians. The atmosphere of the house resembled an academy for his sons. The members of the learned circle of friends stimulated the boys, tutoring and mentoring them, under the strict guidance of their father.

In addition to having talents for the sciences and mathematics, all the brothers were also extremely interested in the arts, music in particular. Despite their interest, none of them possessed any sort of talent for it, but they did not allow that to stifle their progress in learning theory. They were all accepted into the Scuola di Pitagora, a highly exclusive academy of musicians.

More aware of their social position than the idea that his sons could have professions in science, Nardo Antonio raised the boys more as gentlemen than as scholars. Therefore, the boys struggled to learn to sing, as that was considered a courtly accomplishment of gentlemen. They were taught to dance, ride, perform well in tournaments and games, and dress well. The training gave della Porta, at least earlier in his life, a taste for the finer aspects of privileged living.

Scientific disciplines

From De humana physiognomonia, 1586

In 1563, della Porta published De Furtivis Literarum Notis, a work about cryptography. In it, he described the first known digraphic substitution cipher. Charles J. Mendelsohn commented:

He was, in my opinion, the outstanding cryptographer of the Renaissance. Some unknown who worked in a hidden room behind closed doors may possibly have surpassed him in general grasp of the subject, but among those whose work can be studied he towers like a giant.

Della Porta invented a method which allowed him to write secret messages on the inside of eggs. Some of his friends were imprisoned by the Inquisition. At the gate of the prison, everything was checked except for eggs. Della Porta wrote messages on the eggshell using a mixture made of plant pigments and alum. The ink penetrated the eggshell which is semi-porous. When the eggshell was dry, he boiled the egg in hot water and the ink on the outside of the egg was washed away. When the recipient in prison peeled off the shell, the message was revealed once again on the egg white. De Furtivis Literarum Notis also contains one of the earliest known examples of music substitution ciphers.

In 1586 della Porta published a work on physiognomy, De humana physiognomonia libri IIII (1586). This influenced the Swiss eighteenth-century pastor Johann Kaspar Lavater as well as the 19th-century criminologist Cesare Lombroso. Della Porta wrote extensively on a wide spectrum of subjects throughout his life – for instance, an agricultural encyclopedia entitled "Villa" as well as works on meteorology, optics, and astronomy.

Phytognomonica, 1588

In 1589, on the eve of the early modern Scientific Revolution, della Porta became the first person to attack in print, on experimental grounds, the ancient assertion that garlic could disempower magnets. This was an early example of the authority of early authors being replaced by experiment as the backing for a scientific assertion. Della Porta's conclusion was confirmed experimentally by Thomas Browne, among others.

In later life, della Porta collected rare specimens and grew exotic plants. His work Phytognomonica lists plants according to their geographical location. In Phytognomonica the first observation of fungal spores is recorded, making him a pioneer of mycology.

His private museum was visited by travellers and was one of the earliest examples of natural history museums. It inspired the Jesuit Athanasius Kircher to begin a similar, even more renowned, collection in Rome.

Pioneering scientific society

Della Porta was the founder of a scientific society called the Academia Secretorum Naturae (Accademia dei Segreti). This group was more commonly known as the Otiosi, (Men of Leisure). Founded sometime before 1580, the Otiosi were one of the first scientific societies in Europe and their aim was to study the "secrets of nature." Any person applying for membership had to demonstrate they had made a new discovery in the natural sciences.

The Academia Secretorum Naturae was compelled to disband when its members were suspected of dealing with the occult. Della Porta was summoned to Rome by Pope Gregory XIII. Though he personally emerged from the meeting unscathed, the Academia Secretorum Naturae disbanded. Despite this incident, della Porta remained religiously devout and became a lay Jesuit brother.

Della Porta joined The Academy of the Lynxes in 1610.

Technological contributions

Chemical apparatus for a still from De distillatione, 1608

His interest in a variety of disciplines resulted in the technological advances of the following: agriculture, hydraulics, Military Engineering, instruments, and pharmacology. He published a book in 1606 on raising water by the force of the air. In 1608 he published a book on military engineering, and another on distillation.

Additionally, della Porta perfected the camera obscura. In a later edition of his Natural Magic, della Porta described this device as having a convex lens. Though he was not the inventor, the popularity of this work helped spread knowledge of it. He compared the shape of the human eye to the lens in his camera obscura, and provided an easily understandable example of how light could bring images into the eye.

Della Porta also claimed to have invented the first telescope, but died while preparing the treatise (De telescopiis) in support of his claim. His efforts were also overshadowed by Galileo Galilei's improvement of the telescope in 1609, following its introduction by Lippershey in the Netherlands in 1608.

In the book, della Porta also mentioned an imaginary device known as a sympathetic telegraph. The device consisted of two circular boxes, similar to compasses, each with a magnetic needle, supposed to be magnetized by the same lodestone. Each box was to be labelled with the 26 letters, instead of the usual directions. Della Porta assumed that this would coordinate the needles such that when a letter was dialled in one box, the needle in the other box would swing to point to the same letter, thereby helping in communicating.

Religious complications

A Catholic, della Porta was examined by the Inquisition in the years prior to 1578. He was forced to disband his Academia Secretorum Naturae, and in 1592 his philosophical works were prohibited from further publication by the Church; the ban was lifted in 1598. Porta's involvement with the Inquisition puzzles historians due to his active participation in charitable Jesuit works by 1585. A possible explanation for this lies in Porta's personal relations with Fra Paolo Sarpi after 1579.

Playwright

The 17 theatrical works that have survived from a total of perhaps 21 or 23 works comprise 14 comedies, one tragicomedy, one tragedy and one liturgical drama.

Comedies

• Lo Astrologo;
• La Carbonaria;
• La Chiappinaria;
• La Cintia;
• Gli Duoi fratelli rivali;
• La Fantesca;
• La Furiosa;
• Il Moro;
• L'Olimpia;
• I Simili;
• La Sorella;
• La Tabernaria;
• La Trappolaria;
• La Turca

Others

• La Penelope (tragicomedy);
• L'Ulisse (tragedy);
• Il Giorgio (liturgical drama)

Although they belong to the lesser-known tradition of the commedia erudita rather than the commedia dell'arte - which means they were written out as entire scripts instead of being improvised from a scenario - della Porta's comedies are eminently performable. While there are obvious similarities between some of the characters in della Porta's comedies and the masks of the commedia dell'arte, it should be borne in mind that the characters of the commedia erudita are uniquely created by the text in which they appear, unlike the masks, which remain constant from one scenario to another. Indeed, the masks of the improvised theatre evolved as stylised versions of recurring character types in written comedies. One of Della Porta's most notable stock characters was the parasito or parassita, a gluttonous trickster whose lack of moral scruples enabled him to pull off stunts that initially might risk bringing the plot crashing down, but ended up winning the day in unexpected ways. The term parasito was translated by John Florio in his Italian to English Dictionary first published in 1598 as a smell-feast, a flatterer, a parasite, a trencherd or bellie friend, one that saieth and doeth all things to please the humor of another, and agreeth unto him in all things to have his repast scotfree. Perhaps the best example of this type is Morfeo in the comedy La Fantesca.

History of optics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/History_of_optics

Modern ophthalmic lens making machine

Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, followed by theories on light and vision developed by ancient Greek philosophers, and the development of geometrical optics in the Greco-Roman world. The word optics is derived from the Greek term τα ὀπτικά meaning 'appearance, look'. Optics was significantly reformed by the developments in the medieval Islamic world, such as the beginnings of physical and physiological optics, and then significantly advanced in early modern Europe, where diffractive optics began. These earlier studies on optics are now known as "classical optics". The term "modern optics" refers to areas of optical research that largely developed in the 20th century, such as wave optics and quantum optics.

Early history

In the fifth century BCE, Empedocles postulated that everything was composed of four elements; fire, air, earth and water. He believed that Aphrodite made the human eye out of the four elements and that she lit the fire in the eye which shone out from the eye making sight possible. If this were true, then one could see during the night just as well as during the day, so Empedocles postulated an interaction between rays from the eyes and rays from a source such as the sun. He stated that light has a finite speed.

In the 4th century BC Chinese text, credited to the philosopher Mozi, it is described how light passing through a pinhole creates an inverted image in a "collecting-point" or "treasure house".

In his Optics Greek mathematician Euclid observed that "things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal". In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Ptolemy.

In 55 BC, Lucretius, a Roman atomist, wrote:

For from whatsoever distances fires can throw us their light and breathe their warm heat upon our limbs, they lose nothing of the body of their flames because of the interspaces, their fire is no whit shrunken to the sight.

In his Catoptrica, Hero of Alexandria showed by a geometrical method that the actual path taken by a ray of light reflected from a plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.

The Indian Buddhists, such as Dignāga in the 5th century and Dharmakirti in the 7th century, developed a type of atomism which defined the atoms which make up the world as momentary flashes of light or energy. They viewed light as being an atomic entity equivalent to energy, though they also viewed all matter as being composed of these light/energy particles.

Geometrical optics

See also: Geometrical optics and Ray (optics)

The early writers discussed here treated vision more as a geometrical than as a physical, physiological, or psychological problem. The first known author of a treatise on geometrical optics was the geometer Euclid (c. 325 BC–265 BC). Euclid began his study of optics as he began his study of geometry, with a set of self-evident axioms.

1. Lines (or visual rays) can be drawn in a straight line to the object.
2. Those lines falling upon an object form a cone.
3. Those things upon which the lines fall are seen.
4. Those things seen under a larger angle appear larger.
5. Those things seen by a higher ray, appear higher.
6. Right and left rays appear right and left.
7. Things seen within several angles appear clearer.

Euclid did not define the physical nature of these visual rays but, using the principles of geometry, he discussed the effects of perspective and the rounding of things seen at a distance.

Where Euclid had limited his analysis to simple direct vision, Hero of Alexandria (c. AD 10–70) extended the principles of geometrical optics to consider problems of reflection (catoptrics). Unlike Euclid, Hero occasionally commented on the physical nature of visual rays, indicating that they proceeded at great speed from the eye to the object seen and were reflected from smooth surfaces but could become trapped in the porosities of unpolished surfaces. This has come to be known as emission theory.

Hero demonstrated the equality of the angle of incidence and reflection on the grounds that this is the shortest path from the object to the observer. On this basis, he was able to define the fixed relation between an object and its image in a plane mirror. Specifically, the image appears to be as far behind the mirror as the object really is in front of the mirror.

Like Hero, Claudius Ptolemy in his second-century Optics considered the visual rays as proceeding from the eye to the object seen, but, unlike Hero, considered that the visual rays were not discrete lines, but formed a continuous cone.

Optics documents Ptolemy's studies of reflection and refraction. He measured the angles of refraction between air, water, and glass, but his published results indicate that he adjusted his measurements to fit his (incorrect) assumption that the angle of refraction is proportional to the angle of incidence.

In the Islamic world

Reproduction of a page of Ibn Sahl's manuscript showing his discovery of the law of refraction, now known as Snell's law

Al-Kindi (c. 801–873) was one of the earliest important optical writers in the Islamic world. In a work known in the west as De radiis stellarum, al-Kindi developed a theory "that everything in the world ... emits rays in every direction, which fill the whole world."

The theorem of Ibn Haytham

This theory of the active power of rays had an influence on later scholars such as Ibn al-Haytham, Robert Grosseteste and Roger Bacon.

Ibn Sahl, a mathematician active in Baghdad during the 980s, is the first Islamic scholar known to have compiled a commentary on Ptolemy's Optics. His treatise Fī al-'āla al-muḥriqa "On the burning instruments" was reconstructed from fragmentary manuscripts by Rashed (1993). The work is concerned with how curved mirrors and lenses bend and focus light. Ibn Sahl also describes a law of refraction mathematically equivalent to Snell's law. He used his law of refraction to compute the shapes of lenses and mirrors that focus light at a single point on the axis.

Alhazen (Ibn al-Haytham), "the father of Optics"

Ibn al-Haytham (known in as Alhacen or Alhazen in Western Europe), writing in the 1010s, received both Ibn Sahl's treatise and a partial Arabic translation of Ptolemy's Optics. He produced a comprehensive and systematic analysis of Greek optical theories. Ibn al-Haytham's key achievement was twofold: first, to insist, against the opinion of Ptolemy, that vision occurred because of rays entering the eye; the second was to define the physical nature of the rays discussed by earlier geometrical optical writers, considering them as the forms of light and color. He then analyzed these physical rays according to the principles of geometrical optics. He wrote many books on optics, most significantly the Book of Optics (Kitab al Manazir in Arabic), translated into Latin as the De aspectibus or Perspectiva, which disseminated his ideas to Western Europe and had great influence on the later developments of optics. Ibn al-Haytham was called "the father of modern optics".

Avicenna (980–1037) agreed with Alhazen that the speed of light is finite, as he "observed that if the perception of light is due to the emission of some sort of particles by a luminous source, the speed of light must be finite." Abū Rayhān al-Bīrūnī (973-1048) also agreed that light has a finite speed, and stated that the speed of light is much faster than the speed of sound.

Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in Al-Andalus during the second half of the 11th century, wrote a work on optics later translated into Latin as Liber de crepisculis, which was mistakenly attributed to Alhazen. This was a "short work containing an estimation of the angle of depression of the sun at the beginning of the morning twilight and at the end of the evening twilight, and an attempt to calculate on the basis of this and other data the height of the atmospheric moisture responsible for the refraction of the sun's rays." Through his experiments, he obtained the value of 18°, which comes close to the modern value.

In the late 13th and early 14th centuries, Qutb al-Din al-Shirazi (1236–1311) and his student Kamāl al-Dīn al-Fārisī (1260–1320) continued the work of Ibn al-Haytham, and they were among the first to give the correct explanations for the rainbow phenomenon. Al-Fārisī published his findings in his Kitab Tanqih al-Manazir (The Revision of [Ibn al-Haytham's] Optics).

In medieval Europe

The English bishop Robert Grosseteste (c. 1175–1253) wrote on a wide range of scientific topics at the time of the origin of the medieval university and the recovery of the works of Aristotle. Grosseteste reflected a period of transition between the Platonism of early medieval learning and the new Aristotelianism, hence he tended to apply mathematics and the Platonic metaphor of light in many of his writings. He has been credited with discussing 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.

Setting aside the issues of epistemology and theology, Grosseteste's cosmogony of light describes the origin of the universe in what may loosely be described as a medieval "big bang" theory. Both his biblical commentary, the Hexaemeron (1230 x 35), and his scientific On Light (1235 x 40), took their inspiration from Genesis 1:3, "God said, let there be light", and described the subsequent process of creation as a natural physical process arising from the generative power of an expanding (and contracting) sphere of light.

Optical diagram showing light being refracted by a spherical glass container full of water. (from Roger Bacon, De multiplicatione specierum)

His more general consideration of light as a primary agent of physical causation appears in his On Lines, Angles, and Figures where he asserts that "a natural agent propagates its power from itself to the recipient" and in On the Nature of Places where he notes that "every natural action is varied in strength and weakness through variation of lines, angles and figures."

The English Franciscan, Roger Bacon (c. 1214–1294) was strongly influenced by Grosseteste's writings on the importance of light. In his optical writings (the Perspectiva, the De multiplicatione specierum, and the De speculis comburentibus) he cited a wide range of recently translated optical and philosophical works, including those of Alhacen, Aristotle, Avicenna, Averroes, Euclid, al-Kindi, Ptolemy, Tideus, and Constantine the African. Although he was not a slavish imitator, he drew his mathematical analysis of light and vision from the writings of the Arabic writer, Alhacen. But he added to this the Neoplatonic concept, perhaps drawn from Grosseteste, that every object radiates a power (species) by which it acts upon nearby objects suited to receive those species. Note that Bacon's optical use of the term species differs significantly from the genus/species categories found in Aristotelian philosophy.

Several later works, including the influential A Moral Treatise on the Eye (Latin: Tractatus Moralis de Oculo) by Peter of Limoges (1240–1306), helped popularize and spread the ideas found in Bacon's writings.

Another English Franciscan, John Pecham (died 1292) built on the work of Bacon, Grosseteste, and a diverse range of earlier writers to produce what became the most widely used textbook on optics of the Middle Ages, the Perspectiva communis. His book centered on the question of vision, on how we see, rather than on the nature of light and color. Pecham followed the model set forth by Alhacen, but interpreted Alhacen's ideas in the manner of Roger Bacon.

Like his predecessors, Witelo (born circa 1230, died between 1280 and 1314) drew on the extensive body of optical works recently translated from Greek and Arabic to produce a massive presentation of the subject entitled the Perspectiva. His theory of vision follows Alhacen and he does not consider Bacon's concept of species, although passages in his work demonstrate that he was influenced by Bacon's ideas. Judging from the number of surviving manuscripts, his work was not as influential as those of Pecham and Bacon, yet his importance, and that of Pecham, grew with the invention of printing.

Theodoric of Freiberg (ca. 1250–ca. 1310) was among the first in Europe to provide the correct scientific explanation for the rainbow phenomenon, as well as Qutb al-Din al-Shirazi (1236–1311) and his student Kamāl al-Dīn al-Fārisī (1260–1320) mentioned above.

Renaissance and early modern period

Johannes Kepler (1571–1630) picked up the investigation of the laws of optics from his lunar essay of 1600. Both lunar and solar eclipses presented unexplained phenomena, such as unexpected shadow sizes, the red color of a total lunar eclipse, and the reportedly unusual light surrounding a total solar eclipse. Related issues of atmospheric refraction applied to all astronomical observations. Through most of 1603, Kepler paused his other work to focus on optical theory; the resulting manuscript, presented to the emperor on January 1, 1604, was published as Astronomiae Pars Optica (The Optical Part of Astronomy). In it, Kepler described the inverse-square law governing the intensity of light, reflection by flat and curved mirrors, and principles of pinhole cameras, as well as the astronomical implications of optics such as parallax and the apparent sizes of heavenly bodies. Astronomiae Pars Optica is generally recognized as the foundation of modern optics (though the law of refraction is conspicuously absent).

Willebrord Snellius (1580–1626) found the mathematical law of refraction, now known as Snell's law, in 1621. Subsequently, René Descartes (1596–1650) showed, by using geometric construction and the law of refraction (also known as Descartes' law), that the angular radius of a rainbow is 42° (i.e. the angle subtended at the eye by the edge of the rainbow and 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.

Christiaan Huygens (1629–1695) wrote several works in the area of optics. These included the Opera reliqua (also known as Christiani Hugenii Zuilichemii, dum viveret Zelhemii toparchae, opuscula posthuma) and the Traité de la lumière.

Isaac Newton (1643–1727) investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light. He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton's theory of colour. From this work he concluded that any refracting telescope would suffer from the dispersion of light into colours. He went on to invent a reflecting telescope (today known as a Newtonian telescope), which showed that using a mirror to form an image bypassed the problem. In 1671 the Royal Society asked for a demonstration of his reflecting telescope. Their interest encouraged him to publish his notes On Colour, which he later expanded into his Opticks. Newton argued that light is composed of particles or corpuscles and were refracted by accelerating toward the denser medium, but he had to associate them with waves to explain the diffraction of light (Opticks Bk. II, Props. XII-L). Later physicists instead favoured a purely wavelike explanation of light to account for diffraction. Today's quantum mechanics, photons and the idea of wave-particle duality bear only a minor resemblance to Newton's understanding of light.

In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. In 1704, Newton published Opticks, in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?"

Diffractive optics

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665. Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits in his double slit interferometer. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

Lenses and lensmaking

See also: Timeline of telescope technology

Although disputed, archeological evidence has been suggested of the use of lenses in ancient times over a period of several millennia. It has been proposed that glass eye covers in hieroglyphs from the Old Kingdom of Egypt (c. 2686–2181 BCE) were functional simple glass meniscus lenses. The so-called Nimrud lens, a rock crystal artifact dated to the 7th century BCE, might have been used as a magnifying glass, although it could have simply been a decoration.

The earliest written record of magnification dates back to the 1st century CE, when Seneca the Younger, a tutor of Emperor Nero, wrote: "Letters, however small and indistinct, are seen enlarged and more clearly through a globe or glass filled with water." Emperor Nero is also said to have watched the gladiatorial games using an emerald as a corrective lens.

Ibn al-Haytham (Alhacen) wrote about the effects of pinhole, concave lenses, and magnifying glasses in his 11th century Book of Optics (1021 CE). The English friar Roger Bacon, during the 1260s or 1270s, wrote works on optics, partly based on the works of Arab writers, that described the function of corrective lenses for vision and burning glasses. These volumes were outlines for a larger publication that was never produced, so his ideas never saw mass dissemination.

Between the 11th and 13th centuries, so-called "reading stones" were invented. Often used by monks to assist in illuminating manuscripts, these were primitive plano-convex lenses, initially made by cutting a glass sphere in half. As the stones were experimented with, it was slowly understood that shallower lenses magnified more effectively. Around 1286, possibly in Pisa, Italy, the first pair of eyeglasses was made, although it is unclear who the inventor was.

The earliest known working telescopes were the refracting telescopes that appeared in the Netherlands in 1608. Their inventor is unknown: Hans Lippershey applied for the first patent that year followed by a patent application by Jacob Metius of Alkmaar two weeks later (neither was granted since examples of the device seemed to be numerous at the time). Galileo greatly improved upon these designs the following year. Isaac Newton is credited with constructing the first functional reflecting telescope in 1668, his Newtonian reflector.

The earliest known examples of compound microscopes, which combine an objective lens near the specimen with an eyepiece to view a real image, appeared in Europe around 1620. The design is very similar to the telescope and, like that device, its inventor is unknown. Again claims revolve around the spectacle making centers in the Netherlands including claims it was invented in 1590 by Zacharias Janssen and/or his father, Hans Martens, claims it was invented by rival spectacle maker, Hans Lippershey, and claims it was invented by expatriate Cornelis Drebbel who was noted to have a version in London in 1619.

Galileo Galilei (also sometimes cited as a compound microscope inventor) seems to have found after 1609 that he could close focus his telescope to view small objects and, after seeing a compound microscope built by Drebbel exhibited in Rome in 1624, built his own improved version. The name microscope was coined by Giovanni Faber, who gave that name to Galileo Galilei's compound microscope in 1625.

Quantum optics

Main article: Quantum optics

Light is made up of particles called photons and hence inherently is quantized. Quantum optics is the study of the nature and effects of light as quantized photons. The first indication that light might be quantized came from Max Planck in 1899 when he correctly modelled blackbody radiation by assuming that the exchange of energy between light and matter only occurred in discrete amounts he called quanta. It was unknown whether the source of this discreteness was the matter or the light. In 1905, Albert Einstein published the theory of the photoelectric effect. It appeared that the only possible explanation for the effect was the quantization of light itself. Later, Niels Bohr showed that atoms could only emit discrete amounts of energy. The understanding of the interaction between light and matter following from these developments not only formed the basis of quantum optics but also were crucial for the development of quantum mechanics as a whole. However, the subfields of quantum mechanics dealing with matter-light interaction were principally regarded as research into matter rather than into light and hence, one rather spoke of atom physics and quantum electronics.

This changed with the invention of the maser in 1953 and the laser in 1960. Laser science—research into principles, design and application of these devices—became an important field, and the quantum mechanics underlying the laser's principles was studied now with more emphasis on the properties of light, and the name quantum optics became customary.

As laser science needed good theoretical foundations, and also because research into these soon proved very fruitful, interest in quantum optics rose. Following the work of 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 (see degree of coherence). This led to the introduction of the coherent state as a quantum description of laser light and the realization that some states of light could not be described with classical waves. In 1977, Kimble et al. demonstrated the first source of light which required a quantum description: a single atom that emitted one photon at a time. Another quantum state of light with certain advantages over any classical state, squeezed light, was soon proposed. At the same time, development of short and ultrashort laser pulses—created by Q-switching and mode-locking techniques—opened the way to the study of unimaginably fast ("ultrafast") processes. Applications for solid state research (e.g. Raman spectroscopy) were found, and mechanical forces of light on matter were studied. The latter led to levitating and positioning clouds of atoms or even small biological samples in an optical trap or optical tweezers by laser beam. This, along with Doppler cooling was the crucial technology needed to achieve the celebrated Bose–Einstein condensation.

Other remarkable results are the demonstration of quantum entanglement, quantum teleportation, and (recently, in 1995) quantum logic gates. The latter are of much interest in quantum information theory, a subject which partly emerged from quantum optics, partly from theoretical computer science.

Today's fields of interest among quantum optics researchers include parametric down-conversion, parametric oscillation, even shorter (attosecond) light pulses, use of quantum optics for quantum information, manipulation of single atoms and Bose–Einstein condensates, their application, and how to manipulate them (a sub-field often called atom optics).