# Early History

The Beginning

The history of Euclid’s Elements is quite complicated. It survived 2,300 years of wars, famine, and the general upheaval of civilizations as they rose and fell. A complete picture of how Euclid’s Elements reached us is, therefore, still an active area of research.

Surprisingly, given how important the book is, very little is known about the author. Certainly, Euclid was Greek and lived around 300 BC in Alexandria, Egypt. In writing Euclid’s Elements, he brought together work from a variety of authors spanning the previous 200 years or more, as well as adding his own material.

After Euclid, we know of various commentaries, such as Geminus (∼1st century BC), Heron (∼60 A.D), Porphyry (∼233-309 A.D), Pappus (∼290-350 A.D), Proclus (412-485 A.D), and Simplicius (490-560 A.D.). This reveals that interest in Euclid was still very active during the period between 300 B.C and 500 A.D. Fragments of these commentaries have survived, including the complete commentaries from Proclus on Book I. The work by Proclus is particularly important because he refers back to earlier authors, whose original work has been lost. The commentaries offer an important glimpse into the issues that were of interest to these early mathematicians. Heath (1925, volume 1) describes in some detail the work of these authors.

Our earliest copies of Euclid's Elements still in existence include two 9th-century manuscripts.

The first is a manuscript written in Greek, dated to 888 AD, and written in Constantinople on parchment by the scribe (clerk) Stephanos. It is a copy of Euclid's Elements edited by Theon of Alexandria sometime in the 4th century AD. This copy is held at the Bodleian in Oxford and is known as manuscript D'Orville 301. A facsimile can be found at this location, but a simple Google search will easily locate it.

The first 14 leaves are in two different hands from the rest of the manuscript. Leaves 2 to 4 appear to be notes from Arethas the original owner; leaves up to leaf fourteen, written somewhat untidily, are considered to have been written in the 13th century, presumably because the original leaves were damaged or lost. Leaves beyond leaf 14 are from the original scribe of the 9th century and are made with a very elegant hand. These two written styles are shown below:

Part of leaves 7 and 8. Leaf 8 (right) shows the first proposition to construct an equilateral triangle. The style of writing is somewhat untidy and written in the 13th century presumably to replace damaged leaves.

Pages 58 and 59 show the superior hand of the scribe from Constantinople and the neatness of the diagrams. This page shows proposition 30 from book 3 "To bisect a given circumference". Also, notice the notes made by subsequent readers in the margins.

The second is a copy from the 9th/10th century that was discovered by Peyrard in 1808 in the Vatican library. The exact date for the manuscript is not know, but a 9/10th century date is assumed. What is significant about this copy is that it contains a version of Euclid that is earlier than Theon's and thus closer to the original. There was considerable excitement when the manuscript was first discovered. Current editions of Euclid such as ones from Heath or from Dana Densmore are based on Heiberg' rendering of Euclid. Heiberg based his work on Peyrard's copy as well as many other sources. in order to recreate an authentic version of Euclid.

A page from the Vatican Peyrard manuscript (Vat.gr.190.pt.1) showing Book I, Pythagoras' theorem.

Page from the Vatican manuscript (Peyrard, Vat.gr.190.pt.1) showing Proposition I, constructing an equilateral triangle.

The following chart shows a timeline of Euclid down the ages, which I have put together. The period we know least about is from 500AD to 1300 AD. The Arabs were most active in the period 800 to 1100AD during their Golden Era. translating the Greek and possibly even Latin editions of Euclid to Arabic. The Euclid's Elements was definitely in Constantinople in 888 AD. However, the period 1100 to 1200 is of most interest since this is the time when Euclid was rediscovered in Europe. A number of editions emerged at this time (a topic still under active research) due to contact between Northern Europe and the Arab civilization in Spain and the brief Norman occupation of Sicily between 999 and 1139AD.

Arabic Influence

With the fall of the western half of the Roman empire in 476 AD and the violent rejection of all pagan works, including philosophy and mathematics by the new Christian fundamentalists, Europe fell into what is known as the Dark Ages. However, it was not as dark as it may have seemed because, on the fringes of the old Roman empire, particularly in places like Ireland and Northern England, some semblance of intellectual activity continued though on a reduced scale at some of the new Celtic monasteries such as Lindisfarne and Jarrow.

However, the bulk of intellectual activity shifted to the Islamic world. The 9th and 12th centuries and even before, witnessed the Golden Age of Islam. Not only did Islam preserve much of the old Greek mathematics literature it also contributed significantly to the field.

Much of what we know comes from the "Fihrist of al-Nadim" (fihrist: an Arabic word that means list, a catalog, or table of contents), which is a remarkable compendium of the knowledge and literature from tenth-century Islam compiled by Ibn al-Nadim (c. 998). It references approx. 10,000 books and 2,000 authors. The book has 10 chapters, with the seventh chapter cataloging Philosophy and Ancient Sciences. Among this list is of course, Euclid's Elements. The author writes that missions were sent out by the Caliphs of Baghdad to gather books from the Byzantines in Constantinople. The first mission is reported to have been carried out between 754 and 775, by Caliph Harun al-Rashid, who also started a private library. A second mission under the Caliph al-Ma'mun between 813-833 continued to collect books. It was Caliph al-Ma'mun who made the library public as well as founding a translation and research center that also included an astronomical observatory. The site was called the House of Wisdom or the Grand Library of Baghdad. Both Muslim and Christian scholars were employed to translate texts from the Greek and Syriac languages into Arabic. Sadly, the library was destroyed in 1258 during the Mongol siege of Baghdad. Wikipedia has an extensive page on the House of Wisdom.

Euclid's Elements was one of many Greek mathematical texts translated into Arabic by the library. The most well-known translator was al-Hajjaj. According to the Fihrist, al-Hajjaj translated the Elements twice. The second translation is considered the most trustworthy. A copy of the second version exists in the Leiden MS, (Codex Leidenssis 399, I). However, there is some recent dispute as to how much al-Hajjaj text actually remains in this manuscript as it appears to have been edited by other authors. The following is a ChatGPT translation of the first page of this manuscript:

"Praise be to God, the Lord of the world, and may God bestow His grace upon Muhammad and his entire family! This is the book of Euclid on the elements of geometry, a precursor to the study of dimensions, just as the study of letters of the alphabet, which are the elements of writing, precedes the art of writing. Jahja [Ibn] Khalid Ibn Barmak, at the command of Al-Rashid Harun Ibn Al-Mahdi, the faithful emperor and reigning caliph, ordered the translation of this book from the Greek language into Arabic.

Later, by the will of God, when Imam Al-Mamun Abd-Allah Ibn Harun became the faithful emperor and caliph, who had a deep appreciation for the study of letters and supported scholars of literature, Al-Hadschadsch Ibn Yusuf understood that he would gain his favor if he illuminated, explained, and condensed this book. He did not leave any gaps; he corrected and refined it until he had thoroughly worked on the book and condensed and corrected it, as it is in this copy, for the use of gifted individuals and scholars, without changing the original edition left in the hands of readers. Then, Abul-Abbas Al-Fadhl Ibn-Hatim Al-Narizi added commentaries, connected words correctly, and explained in all chapters Euclid added appropriate elements from other geometric works and from the writings of those who explained Euclid's book."

The al-Hajjaj second version was later to form the basis for the first Arabic to Latin translation by Adelard of Bath.

Two other important Arabic translations, which made their way to Europe, were by Ishaq (830-910) and Thabit (died 901). In addition to these authors, there are approximately fifty Arabian commentaries are known today (Sonja Brentjes, 2001). Many of these are listed in the Fihrist. The transmission of Euclid to Arabic is complex and still an active area of research. For those interested, two key researchers include Sonja Brentjes in Berlin and Gregg De Young in Cairo.

The image below is from a 1466 copy of the translation by Ishaq (See Gregg de Young for more information).

MS Ishaq 1466 from Mathematical Treasure: Early Translations of Euclid's Elements into Arabic

Middle Ages Europe

During the 13th and 12th centuries, there was a flurry of Latin translations of Euclid from Arabic and at least one from a Greek source (Palermo, Sicily).

Adelard of Bath

The first of these translators was the English philosopher by the name Adelard of Bath, born around 1080 and died possibly between 1142 and 1152. Legend has it (A short account of the history of mathematics, Rouse, 1915, pp165) that under the disguise of a Muslim student, he attended some lectures at Cordova in about 1120 and obtained an Arabic copy of Euclid’s Elements. However, this is not certain, and I was not able to trace the origins of this story. In fact, there doesn't seem to be any documented evidence that he ever traveled to Spain. I also note that the author Simon Webb, in his booklet "The Life and Times of Adelard of Bath", comes to the same conclusion.

However, this doesn't mean he never traveled to Spain. For example, his translation of the astronomical tables of al-Khwarizmi came from the revised edition of the Spanish Cordova astronomer Maslama ibn Ahmad al-Majriti. Of course, he may have obtained this elsewhere, for example, in Sicily or even in France.

In his travels, starting near the end of the 11th century, Adelard is known to have first traveled to France, where he studied in Tours and taught at Laon. From Laon, he traveled for about 7 years, visiting Salerno, Sicily, Cilicia, Syria, and possibly Palestine. He then appears again in the historical record in Bath in 1130. He wrote a book on the Astrolabe and dedicated it to the future king, Henry II. It is in this book that he states he translated the Elements.

At one point, Adelard traveled to Salerno and on to Sicily, where in Sicily it is not certain, though he dedicated his commentary De eodem et diverso (“On Sameness and Diversity”), to William, bishop of Syracuse. It is possible, for example, that he may have acquired Euclid's Elements in Sicily. At that time, Sicily was ruled by the Normans, where there was active translation going on in Palermo. By some mechanism he came into contact with Euclid, specifically an Arab edition (very close to the edition by al-Nayrizi) from which he made a Latin translation, now called Adelard I. This is considered the first complete Latin translation of Euclid that we know of. The oldest copy is the manuscript Trinity College 47 (fols. 139–80) held at Oxford (Note fols. 104–38 in the same MS are a copy of Adelard II). MS 47 has been dated to or before 1150, and Busard/Clagett even suggest that it might have been written by Adelard himself due to its age but also due to the fact that in a number of places, Arabic words have been crossed out and replaced by the Latin word in the same had as the original word. In a sense, a correction made by Adelard himself. Unfortunately, no online copy of this MS is available to view, which seems surprising given its historical importance. This purports to be an early 14th century copy of Adelard I (Note this link is sometimes bad).

You can find further details of Adelard I edition in the list of web links

Adelard was not the only person, however, to provide Arabic to Latin translations at this time. Other important individuals include Hermann of Carinthia (1105/1110-1154+), Gerard of Cremona (1114-1187), Robert of Chester (12th century, also known as Robert of Ketton), and John of Tynemouth (13th century). All of these editions were derived from Arabic sources. It wasn't until 1505 that a more widely available direct translation from the Greek was made by Bartolomeo Zamberti (from an unknown source) and published in Vienna in Latin. Note that a direct translation was made in the 12th century in Palermo from a Greek source. It is now believed (Busard, Campanus of Novara) that this source was used, among other sources, by Campanus of Novara to create this edition in the 13th century.

It is now also believed that Adelard II was written by Robert of Chester and Adelard III by John of Tynemouth.

It should be said that the chronology and development of Euclid editions in medieval Europe is complex. Many manuscripts were produced at this time (by many, we're talking probably in the 100s), and many of them contain slight variations. By comparison, once printing was developed in the 16/15th century, Euclid editions were printed in the 1000s if not 10,000s in a very short time.

Robert of Chester (Adelard II)

Adelard II (but written by Robert of Chester, also known as Robert of Ketton) is an unusual translation because it doesn't have the proofs and only has summaries of the propositions. Not only that, the figures, though present, are largely unlabeled and not very well done. It was this edition that became the most popular version in the medieval edges. It is said that Adelard II has more extant manuscripts than any other edition. Menso Folkerts lists more than 50 such manuscripts. The edition appears to be a mix of text from Boethius, Arabic sources, as well as unknown sources. There are, however some variations between the extant copies of Adelard II. Zbigniew Król has a nice summary, see references.

This edition was probably also used by Campanus as his base text, which became the standard medieval Euclid for many years. His and other editions in the thirteenth and fourteenth centuries used the enunciations and added new proofs. This is why, by the 16th century, work started to 'restore' Euclid to its original form.

Looking at Proposition 1 in Adelard II, the last sentence says: "Therefore, deduce an argument from the description of the circle." It's almost as if this edition was an exercise book which explains why it doesn't include the proofs.

For example, here is the Latin and Chatpgt translation to English for Proposition 1 (Latin text obtained from Busard's "Robert of Chester's Redaction of Euclid's Elements, the So-Called Adelard II Version: Volume I":

Latin:

"Triangulum equilaterum super datam lineam rectam collocare.

A duobus terminis date linee ipsam lineam occupando cum circino duos circulos sese invicem secantes describe at ab ipsa communi seccione circulorum ad duos terminos linee proposite duas rectas lineas dirige. Deinde ergo ex circuli descripcione argumentum elice."

English:

"To construct an equilateral triangle on a given straight line.

From the two given endpoints of the line, occupy the line itself with a compass and describe two circles intersecting each other. Then, from the common section of the circles, draw two straight lines to the two endpoints of the given line. Therefore, deduce an argument from the description of the circle."

Heath:

Compared to the full proposition 1 from Heath:

"On a given finite straight line to construct an equilateral triangle.

Let AB be the given finite straight line. Thus it is required to construct an equilateral triangle on a the straight line AB.

With centre A and distance AB let the circle BCD be described; again with centre B and distance BA let the circle ACE be described;

and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined.

Now, since the point A is the centre of the circle CDB, AC is equal to AB.

Again, since the point B is the centre of the circle CAE, BC is equal to BA.

But CA was also proved equal to AB;

therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are also equal to one another;

therefore CA is also equal to CB.

Therefore the three straight lines CA, AB, BC are equal to one another."

In general the Robert of Chester editions, especially the early ones, were very poorly copied.

John of Tynemouth (Adelard III)

Adelard III is now believed to have been compiled by John of Tynemouth (Knorr, 1990). The Adelard III series was written after Adelard II and appears to have been used by Roger Bacon, as Bacon quotes material from this edition. Adelard III appears to have used Adelard II but with the proofs added back.

Gerard of Cremona (1114-1187)

Gerard of Cremona was born in Cremona in Lombardy, and was one of the most prolific translators of scientific books from Arabic into Latin. He did all is translation in Toledo, Spain after moving there sometime around 1144. In total, he translated 87 books from the Arabic language. Gerard's edition is based on Ishaq-Thabit Arabic text as well as some text from editions by al-Hajjaj. Gerard's edition is by far superior to Robert of Chester's edition (but aren't they all). See "The Commentary of Al-Nayrizi on Book I of Euclid's Elements in Geometry" (Anthony Lo Bello) or the recent article by Gregg de Young for more details. Gerard's editon doesn't appear to have amde much impact however on subsequent editions.

Campanus of Novara (1220-1296) - Italian mathematician, astronomer, astrologer, and physician

A variety number of editions of Euclid's Elements were written in the 12th and early 13th centuries but the most important edition was published by Campanus of Novara sometime between 1255 and 1259. This Latin edition formed the basis for all subsequent editions until new editions were printed from Greek sources in the 16th century. It was also the first edition of Euclid to be printed on paper by Erhard Ratdolt in Venice in 1482.

This edition was based on the limited Adelard II (Robert of Chester) edition but with the proofs from other sources added back in. It's not entirely sure why he didn't use Adelard III which was much more complete. It is now believed (Busard, Campanus of Novara) that Campanus likely obtained some of the new material from the Palermo Greek-Latin translation.

The Campanus edition was by far the most copied version and 131 manuscripts of Campanus' version in Euclid's Elements still exist. This is a 1482 printed version of Campanus' edition.

References:

On a medieval circle quadrature: De circulo quadrando, Wilbur R. Knorr, Historia Mathematica Volume 18, Issue 2, May 1991, Pages 107-128 (Discusses Adelard III) (https://www.sciencedirect.com/science/article/pii/031508609190495J)

The medieval manuscripts of Trinity College, Oxford: a descriptive catalogue, RICHARD GAMESON , 2018. Describes in detail the many books in the Trinity College 47 manuscript. https://medieval.bodleian.ox.ac.uk/pdfs/trinity/MS_47.pdf

Link to Trinity College 47 manuscript, but no download is available.

Euclid in Medieval Europe by Menso Folkerts, lists medieval manuscripts related to Euclid. https://math.berkeley.edu/~wodzicki/160/Euclid_in_Middle_Ages.pdf

MS. D'Orville 301, old Euclid manuscript (888 AD), https://medieval.bodleian.ox.ac.uk/catalog/manuscript_4146

MS. D'Orville 301, OCR version. https://www.claymath.org/euclid_index/

Harley MS 5266, 14th century copy of Adelard I. https://www.bl.uk/manuscripts/FullDisplay.aspx?ref=Harley_MS_5266

Mathematical Treasures Adelard

Zbigniew Król , SCIENTIFIC HERITAGE: THE RECEPTION AND TRANSMISSION OF EUCLIDIAN GEOMETRY IN WESTERN CIVILIZATION, DIALOGUE AND UNIVERSALISM No. 4/2012

Wilbur R. Knorr, John of Tynemouth alias John of London: Emerging Portrait of a Singular Medieval Mathematician, in The British Journal for the History of Science XXIII (1990) 293-330.

Busard, H. L. L. (2005). Campanus of Novara and Euclid's Elements. Germany: Steiner.

Gregg de Young, Historia Mathematica Volume 19, Issue 2, May 1992, Pages 188-199 Isāq ibn unayn, unayn ibn Isāq, and the third Arabic translation of Euclid's Elements,

Sonja Brentjes, Science in Context 14(1/2), 39–84 (2001), Observations on Hermann of Carinthia’s Version of the Elements and its Relation to the Arabic Transmission