The Synagogue, also known as the Collection, is a mathematical work in eight books by Pappus of Alexandria, written in the early fourth century CE. It gathers, explains and extends the geometry of earlier Greek mathematicians, Euclid, Archimedes and Apollonius among them. The first book and the start of the second are lost. The surviving books treat Apollonius's method for multiplying large numbers, the means and the finding of two mean proportionals, the curves used for problems beyond ruler and compass, plane figures and solids of equal perimeter or surface, the shorter astronomical works read before Ptolemy, and mechanics. Book VII, the longest, is a guide to the body of works known as the Treasury of Analysis, with the lemmas needed to read them.
Pappus addresses several books to named readers, Pandrosion, Megethion and Hermodorus. The work frequently offers alternative solutions to problems and corrects earlier writers and contemporary teachers. The text survives mainly through a single tenth-century Byzantine manuscript, from which the other copies descend. A Latin translation by Federico Commandino was published in 1588, and Descartes took up a locus problem from Book VII in his Geometry.
Pappus of Alexandria Synagogue: summary and contents (PDF)
Pappus of Alexandria Synagogue: summary and contents (Markdown)
Contents
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Βʹ Book II
What survives of Book II begins in the middle of Apollonius's method for multiplying large numbers without carrying out the multiplication. The numbers are split into their bases and powers of ten, and the products are counted in myriads of increasing order. The book ends by applying the method to the letters of a single hexameter line, taken as numerals.
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Γʹ Book III
Addressed to Pandrosion, the third book begins by distinguishing problem from theorem and correcting problems that self-styled geometers had sent through Pandrosion. It treats the finding of two mean proportionals, with Pappus's criticism of one geometer's supposed plane construction, and then sets out the arithmetic, geometric, harmonic and further means and how each is built from a proportion.
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Δʹ Book IV
Book IV opens with Pappus's extension of Pythagoras's theorem to parallelograms drawn on any two sides of a triangle. It goes on to the curves used for problems beyond ruler and compass: the spiral of Archimedes and the area it bounds, the quadratrix and the conchoid, and their use in dividing an angle. The book ends on a solid neusis that Archimedes was criticised for using.
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Εʹ Book V
Dedicated to Megethion, Book V begins with the bees, whose hexagonal cells hold the most honey for the wax spent, and then compares plane figures of equal perimeter with each other and with the circle. It turns to solids of equal surface and the sphere, proving theorems on the surfaces of spherical segments, and closes by showing that only five regular solids can exist.
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Ϛʹ Book VI
Solutions to difficulties in the little astronomy, the shorter works read before Ptolemy, make up Book VI. Pappus corrects teachers who misreport Theodosius's Spherics and On Days and Nights and Euclid's Phaenomena, and proves the propositions they skip on spherical triangles, the risings and settings of the zodiac, and the lengths of days and nights. He ends by sending the reader on to Ptolemy.
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Ζʹ Book VII
The treasury of analysis, the body of works by Euclid, Apollonius and Aristaeus written for solving geometrical problems, is the subject of Book VII, addressed to Hermodorus. Pappus defines analysis and synthesis, surveys each work in turn, and then supplies the lemmas needed to read them, on tangent circles, cyclic quadrilaterals and proportions in right-angled triangles.
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Ηʹ Book VIII
Mechanics is the subject of Book VIII, addressed to Hermodorus, which opens by setting out, after Heron's school, the theoretical and manual parts of the art. Problems follow on centres of gravity and on drawing an ellipse through five given points, used to recover a broken cylinder. It ends with a practical account of a crane with compound pulleys for lifting a stone onto a wall.