Editing Ancient Greek mathematics (section)

==Mathematics in late antiquity==
The mathematicians in the later Roman era from the 4th century onward generally had few notable original works, however, they are distinguished for their commentaries and expositions on the works of earlier mathematicians. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity.{{sfn|Mansfeld|2016}}

=== Pappus' ''Collection''===
[[Pappus of Alexandria]] compiled a canon of results of earlier mathematics in the ''Collection'' in eight books, of which part of book II and books III–VII are extant in Greek and book VIII is extant in Arabic. The collection attempts to sum up the whole of Ancient Greek mathematics up to that time as interpreted by Pappus: Book III is framed as a letter to [[Pandrosion]], a mathematician in Athens, and discusses three construction problems and attempts to solve them: [[doubling the cube]], [[angle trisection]], and [[squaring the circle]]. Book IV discusses classical geometry, which Pappus divides into plane geometry, Line geometry, and Solid geometry, and includes a discussion of Archimedes' construction of the [[Arbelos]], otherwise only known via a pseudo-Archimedean work, [[Book of Lemmas]]. Book V discusses isoperimetric figures, summarizing otherwise lost works by [[Zenodotus]] and [[Archimedes]] on isoperimetric plane figures and solid figures, respectively. Book VI deals with astronomy, providing commentary on some of the works of the Little Astronomy corpus. Book VII deals with analysis, providing epitomes and lemmas from otherwise lost works. Book VIII deals with mechanics. The Greek version breaks off in the middle of a sentence discussing [[Hero of Alexandria]], but a complete edition of the book survives in Arabic.<ref>{{cite book |last1=Pappus |first1=of Alexandria |title=Book 7 of the Collection |date=1986 |publisher=Springer-Verlag |isbn=978-0-387-96257-3 |url=https://archive.org/details/book7ofcollectio0000papp |access-date=4 May 2025}}</ref>

=== Commentaries ===
The commentary tradition, which had begun during the Hellenistic period, continued into late antiquity. The first known commentary on the ''Elements'' was written by [[Hero of Alexandria]], who likely set the format for future commentaries. [[Serenus of Antinoöpolis]] wrote a lost commentary on the ''Conics'' of Apollonius, along with two works that survive, ''Section of a Cylinder'' and ''Section of a Cone'', expanding on specific subjects in the ''Conics''.{{sfn|Acerbi|2018|p=274}} Pappus wrote a commentary on Book X of the elements, dealing with incommensurable magnitudes. [[Heliodorus of Larissa]] wrote a summary of the Optics.{{sfn|Netz|2022}}

Many of the late antique commentators were associated with Neoplatonist philosophy; [[Porphyry of Tyre]], a student of Plotinus, the founder of [[Neoplatonism]], wrote a commentary on Ptolemy's ''Harmonics''. [[Iamblichus]], who was himself a student of Porphyry, wrote a commentary on Nicomachus' Introduction to Arithmetic. In Alexandria in the 4th century, [[Theon of Alexandria]] wrote commentaries on the writings of [[Ptolemy]], including a commentary on the ''Almagest'' and two commentaries on the ''Handy Tables'', one of which is more of an instruction manual ("Little Commentary"), and another with a much more detailed exposition and derivations ("Great Commentary"). [[Hypatia]], Theon's daughter, also wrote a commentary on Diophantus' ''Arithmetica'' and a commentary on the ''Conics'' of Apollonius, which have not survived.{{sfn|Cameron|1990}} 

In the 5th century, in Athens, [[Proclus]] wrote a commentary on Euclid's elements, which the first book survives. Proclus' contemporary, [[Domninus of Larissa]], wrote a summary of Nicomachus' Introduction to Arithmetic, while [[Marinus of Neapolis]], Proclus' successor, wrote an ''Introduction to Euclid's Data''. Meanwhile in Alexandria, [[Ammonius Hermiae]], [[John Philoponus]] and [[Simplicius of Cilicia]] wrote commentaries on the works of Aristotle that preserve information on earlier mathematicians and philosophers. [[Eutocius of Ascalon]] (c.&nbsp;480–540), another student of Ammonius, wrote commentaries that are extant on Apollonius' ''Conics'' along with some treatises of Archimedes: ''On the Sphere and Cylinder'', ''Measurement of a Circle'', and ''On Balancing Planes'' (though the authorship of the last one is disputed).{{sfn|Netz|2022|pp=427-428}} In Rome, Boethius, seeking to preserve Ancient Greek philosophical, translated works on the [[quadrivium]] into Latin, deriving much of his work on Arithmetic and Harmonics from Nicomachus.{{sfn|Netz|2022|pp=429-430}}

After the closure of the Neoplatonic schools by the emperor [[Justinian]] in 529 AD, the institution of mathematics as a formal enterprise entered a decline. However, two mathematicians connected to the Neoplatonic tradition were commissioned to build the [[Hagia Sophia]]: [[Anthemius of Tralles]] and [[Isidore of Miletus]]. Anthemius constructed many advanced mechanisms and wrote a work ''On Surprising Mechanisms'' which treats "burning mirrors" and skeptically attempts to explain the function of [[Archimedes' heat ray]]. Isidore, who continued the project of the Hagia Sophia after Anthemius' death, also supervised the revision of Eutocius' commentaries of Archimedes. From someone in Isidore's circle we also have a work on polyhedra that is transmitted pseudepigraphically as ''Book XV'' of Euclid's ''Elements.''{{sfn|Netz|2022|pp=432-433}}