**Diophantus of Alexandria** (Ancient Greek : Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called * Arithmetica *, many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and of Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality.^{ [1] } This term was rendered as *adaequalitas* in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, during the Roman era, probably from between AD 200 and 214 to 284 or 298. Diophantus has variously been described by historians as either Greek,^{ [2] }^{ [3] }^{ [4] } or possibly Hellenized Egyptian,^{ [5] } or Hellenized Babylonian,^{ [6] } Many of these identifications may stem from confusion with the 4th-century rhetorician Diophantus the Arab.^{ [7] } Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) states:

- 'Here lies Diophantus,' the wonder behold.
- Through art algebraic, the stone tells how old:
- 'God gave him his boyhood one-sixth of his life,
- One twelfth more as youth while whiskers grew rife;
- And then yet one-seventh ere marriage begun;
- In five years there came a bouncing new son.
- Alas, the dear child of master and sage
- After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'

This puzzle implies that Diophantus' age *x* can be expressed as

*x*=*x*/6 +*x*/12 +*x*/7 + 5 +*x*/2 + 4

which gives *x* a value of 84 years. However, the accuracy of the information cannot be independently confirmed.

In popular culture, this puzzle was the Puzzle No.142 in * Professor Layton and Pandora's Box * as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.

*Arithmetica* is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which *Arithmetica* consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are also by Diophantus.^{ [8] } Some Diophantine problems from *Arithmetica* have been found in Arabic sources.

It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus.

“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”.^{ [9] }

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the Dark Ages, since the study of ancient Greek, and literacy in general, had greatly declined. The portion of the Greek *Arithmetica* that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370–1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 – 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.^{ [10] } In addition, some portion of the *Arithmetica* probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote:

- “No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .”

*Arithmetica* was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book *Algebra*. The * editio princeps * of *Arithmetica* was published in 1575 by Xylander. The best known Latin translation of *Arithmetica* was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.

The 1621 edition of *Arithmetica* by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy:

- “If an integer
*n*is greater than 2, then*a*^{n}+*b*^{n}=*c*^{n}has no solutions in non-zero integers*a*,*b*, and*c*. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”

Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.^{ [10] }

Diophantus wrote several other books besides *Arithmetica*, but very few of them have survived.

Diophantus himself refers^{[ citation needed ]} to a work which consists of a collection of lemmas called *The Porisms* (or *Porismata*), but this book is entirely lost.

Although *The Porisms* is lost, we know three lemmas contained there, since Diophantus refers to them in the *Arithmetica*. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any *a* and *b*, with *a* > *b*, there exist *c* and *d*, all positive and rational, such that

*a*^{3}−*b*^{3}=*c*^{3}+*d*^{3}.

Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.^{ [11] }

A book called *Preliminaries to the Geometric Elements* has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.^{ [12] }

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. How much he affected India is a matter of debate.

Diophantus is often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation.^{ [13] }

Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: *ax*^{2} + *bx* = *c*, *ax*^{2} = *bx* + *c*, and *ax*^{2} + *c* = *bx*. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers *a*, *b*, *c* to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 4 = 4*x* + 20 'absurd' because it would lead to a negative value for *x*. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.

Diophantus made important advances in mathematical notation, becoming the first person known to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:^{ [14] }

“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number *n*. Where we would write 12 + 6*n*/*n*^{2} − 3, Diophantus has to resort to constructions like: "... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three". Algebra still had a long way to go before very general problems could be written down and solved succinctly.

- ↑ Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond",
*Perspectives on Science*,**21**(3): 283–324, arXiv: 1210.7750 , Bibcode:2012arXiv1210.7750K, doi:10.1162/POSC_a_00101, S2CID 57569974 - ↑ Research Machines plc. (2004).
*The Hutchinson dictionary of scientific biography*. Abingdon, Oxon: Helicon Publishing. p. 312.**Diophantus (lived**Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.*c.*A.D. 270-280) - ↑ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics".
*A History of Mathematics*(Second ed.). John Wiley & Sons, Inc. p. 178. ISBN 0-471-54397-7.At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.

- ↑ Cooke, Roger (1997). "The Nature of Mathematics".
*The History of Mathematics: A Brief Course*. Wiley-Interscience. p. 7. ISBN 0-471-18082-3.Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.

- ↑ Victor J. Katz (1998).
*A History of Mathematics: An Introduction*, p. 184. Addison Wesley, ISBN 0-321-01618-1."But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."

- ↑ D. M. Burton (1991, 1995).
*History of Mathematics*, Dubuque, IA (Wm.C. Brown Publishers)."Diophantos was most likely a Hellenized Babylonian."

- ↑ Ad Meskens,
*Travelling Mathematics: The Fate of Diophantos' Arithmetic*(Springer, 2010), p. 48 n28. - ↑ J. Sesiano (1982).
*Books IV to VII of Diophantus'*Arithmetica*in the Arabic Translation Attributed to Qusta ibn Luqa*. New York/Heidelberg/Berlin: Springer-Verlag. p. 502. - ↑ Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.
- 1 2 Herrin, Judith (2013-03-18).
*Margins and Metropolis: Authority across the Byzantine Empire*. Princeton University Press. p. 322. ISBN 978-1400845224. - ↑ "Diophantus biography".
*www-history.mcs.st-and.ac.uk*. Retrieved 10 April 2018. - ↑ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
- ↑ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
- ↑ Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Scientific Biography, Encyclopedia.com, 2008.

In mathematics, a **Diophantine equation** is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A **linear Diophantine equation** equates to a constant the sum of two or more monomials, each of degree one. An **exponential Diophantine equation** is one in which unknowns can appear in exponents.

The area of study known as the **history of mathematics** is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

**Pell's equation**, also called the **Pell–Fermat equation**, is any Diophantine equation of the form where *n* is a given positive nonsquare integer and integer solutions are sought for *x* and *y*. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose *x* and *y* coordinates are both integers, such as the trivial solution with *x* = 1 and *y* = 0. Joseph Louis Lagrange proved that, as long as *n* is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of *n* by rational numbers of the form *x*/*y*.

**Algebraic number theory** is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

**Claude Gaspard Bachet de Méziriac** was a French mathematician, linguist, poet and classics scholar born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy.

* The Compendious Book on Calculation by Completion and Balancing*, also known as

* Arithmetica* is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.

**Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ** was an Egyptian mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.

The **history of mathematical notation** includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

The **eighth problem of the second book of Arithmetica** by Diophantus is to divide a square into a sum of two squares.

**Algebra** is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

In number theory, **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2. The cases *n* = 1 and *n* = 2 have been known since antiquity to have infinitely many solutions.

A timeline of **number theory**.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

**Fermat's right triangle theorem** is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:

**Isabella Grigoryevna Bashmakova** was a Russian historian of mathematics.

* Diophantus and Diophantine Equations* is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by Nauka in 1972 under the title

- Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304"
*Byzantion*53. Brussels, 1983: 682-710. - Bachet de Méziriac, C.G.
*Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus*. Paris: Lutetiae, 1621. - Bashmakova, Izabella G.
*Diophantos. Arithmetica and the Book of Polygonal Numbers. Introduction and Commentary*Translation by I.N. Veselovsky. Moscow: Nauka [in Russian]. - Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"",
*Historia Scientiarum*, 6 (1996)37-41. - Christianidis, J. "Une interpretation byzantine de Diophante",
*Historia Mathematica*, 25 (1998) 22-28. - Czwalina, Arthur.
*Arithmetik des Diophantos von Alexandria*. Göttingen, 1952. - Heath, Sir Thomas,
*Diophantos of Alexandria: A Study in the History of Greek Algebra*, Cambridge: Cambridge University Press, 1885, 1910. - Robinson, D. C. and Luke Hodgkin.
*History of Mathematics*, King's College London, 2003. - Rashed, Roshdi.
*L’Art de l’Algèbre de Diophante*. éd. arabe. Le Caire : Bibliothèque Nationale, 1975. - Rashed, Roshdi.
*Diophante. Les Arithmétiques*. Volume III: Book IV; Volume IV: Books V–VII, app., index. Collection des Universités de France. Paris (Société d’Édition “Les Belles Lettres”), 1984. - Sesiano, Jacques.
*The Arabic text of Books IV to VII of Diophantus’ translation and commentary*. Thesis. Providence: Brown University, 1975. - Sesiano, Jacques.
*Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā*, Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8, doi : 10.1007/978-1-4613-8174-7. - Σταμάτης, Ευάγγελος Σ.
*Διοφάντου Αριθμητικά. Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις*. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963. - Tannery, P. L.
*Diophanti Alexandrini Opera omnia: cum Graecis commentariis*, Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: vol. 1, vol. 2) - Ver Eecke, P.
*Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones*, Bruges: Desclée, De Brouwer, 1921. - Wertheim, G.
*Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria*. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.

- Bashmakova, Izabella G. "Diophante et Fermat,"
*Revue d'Histoire des Sciences*19 (1966), pp. 289-306 - Bashmakova, Izabella G.
*Diophantus and Diophantine Equations*. Moscow: Nauka 1972 [in Russian]. German translation:*Diophant und diophantische Gleichungen*. Birkhauser, Basel/ Stuttgart, 1974. English translation:*Diophantus and Diophantine Equations*. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997. - Bashmakova, Izabella G. “Arithmetic of Algebraic Curves from Diophantus to Poincaré,”
*Historia Mathematica*8 (1981), 393-416. - Bashmakova, Izabella G., Slavutin, E.I.
*History of Diophantine Analysis from Diophantus to Fermat*. Moscow: Nauka 1984 [in Russian]. - Heath, Sir Thomas (1981).
*A history of Greek mathematics*.**2**. Cambridge University Press: Cambridge. - Rashed, Roshdi, Houzel, Christian.
*Les Arithmétiques de Diophante : Lecture historique et mathématique*, Berlin, New York : Walter de Gruyter, 2013. - Rashed, Roshdi,
*Histoire de l’analyse diophantienne classique : D’Abū Kāmil à Fermat*, Berlin, New York : Walter de Gruyter. - Vogel, Kurt (1970). "Diophantus of Alexandria".
*Dictionary of Scientific Biography*.**4**. New York: Scribner.

- Media related to Diophantus at Wikimedia Commons

Wikiquote has quotations related to: Diophantus |

Wikisource has the text of the 1911 Encyclopædia Britannica article Diophantus . |

- O'Connor, John J.; Robertson, Edmund F., "Diophantus",
*MacTutor History of Mathematics archive*, University of St Andrews - Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
- Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.
- Review of Sesiano's Diophantus Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
- Latin translation from 1575 by Wilhelm Xylander

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.