March 3, 1997

Pappus of Alexandria
(fl. c. 300-c. 350)

Very little is known of Pappus' life. Moreover, very little is known of what his actual contributions were. We do know that he recorded in one of his commentaries on the Almagest that he observed a solar eclipse on October 18, 320. He is regarded, though, as the last great mathematician of the Helenistic Age.

He wrote The Collection or The Synagogue, a treatise on geometry which we discuss here and several commentaries, now all lost except for some fragments in Greek or Arabic. One of the commentaries, we note from Proclus, was on The Elements.

Note that higher geometry was in complete abeyance until Pappus. From his descriptions, we may surmise that either the classical works were lost or forgotten.

His task is to `restore' geometry to a place of significance.

Basically, The Collection or The Synagogue is a treatise on Geometry, which included everything of interest to him. Whatever explanations or supplements to the works of the great geometers seemed to him necessary, he formulated them as lemmas.

Features:

• It is very broad, designed to revive classical geometry.
• It is a guide or handbook to be read with the Elements and other original works.
• Alternative methods of proof are often given.
• The work shows a thorough grasp of all the subjects treated, independence of judgment, mastery of technique; the style is terse and clear. Pappus is an accomplished and versatile mathematician.
• The range of names of predecessors is immense. In some cases, our only knowledge of some mathematicians is due to his citation.

Summary of Contents:

• Book I and first 13 propositions of Book II are missing. Book II was concerned with very large numbers - powers of myriads.
• Book III begins with a summary of finding two mean proportionals (a:x = x:y = a:y) between two straight lines. He also defines plane problems, solid problems, and linear problems. Pappus
  • distinguishes (1)  plane problems, solvable with straight edge and compass
  • distinguishes (2)  solid problems, requiring the conics for solution, e.g. solving certain cubics.
  • distinguishes (3)  linear problems, problems invoking spirals, quadratrices, and other higher curves
  • gives a constructive theory of means. That is, given any two of the numbers and the type of mean (arithmetic, geometric, or harmonic), he constructs the third.
  • describes the solution of the three famous problems of antiquity, asserts these are not plane problems  19 century.
  • treats the trisection problem, giving another solution involving a hyperbola and a circle.
  • inscribes the five regular solids in the sphere.

• Book IV covers an extention of theorem of Pythagorus for parallelograms constructed on the legs of any triangle. Also in Book IV is material on the Archimedian spiral, including methods of finding area of one turn -- differs from Archimedes. He also constructs the conchoid of Nicomedes. In addition, he constructs the quadratix in two different ways, (1) using a cylindrical helix, and (2) using a right cylinder, the base of which is an Archimedian spiral.
• Book V reproduces the work of Zeodorus on isoperimetric figures. Here we see in the introduction his comments on the sagacity of bees.
• Book VI determines the center of an ellipse as a perspective of a circle. It is also astronomical in nature. It has been called the ``Little Astronomy''. It covers optics - reflection and refraction.
• Book VII, the `Treasury of Analysis' is very important because it surveys a great number of works on geometric analysis of loci, nearly all of which are lost. Features:
  • The Book begins with a definition of aanalysis and synthesis.
    • Analysis, then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of systhesis. Unconditional controvertability required.
    • In Synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what before were antecedents, and successively connecting them one with the other, we arrive finally at the construction of what was sought.
  • A list of the books forming the `treasury' is included, together with a short description of their contents.
  • As an independent contribution Pappus formulated the volume of a solid of revolution, the result we now call the The Pappus - Guldin Theorem. P. Guldin (1577-1643)
  • Most of the remaining of the treatise is collections of lemmas that will assist the reader's understanding of the original works.

Theorem. If ABC is a triangle and on AB,AC any parallelograms are drawn as ABDE and ACFG, and if DE and FG are extended to H and HA be joined to K. Then BCNL is a parallelogram and

.

Proof. The proof is similar to the original proof of the Pythagorean  theorem as found in The Elements. First, BL and NC are defined to be parallel to HK. BLHA is a parallelogram and CAHM is a parallelogram. Hence BCNL is a parallelogram.

By ``sliding" DE to HL, it is easy to see that

and by sliding HA to MK it follows that

Thus

Similarly,

Thus

Note. Both parallelograms need not be drawn outside ABC.

The statement on the bees celebrates the hexagonal shape of their honeycombs.

[ The bees], believing themselves, no doubt, to be entrusted with the task of bringing from the gods to the more cultured part of mankind a share of ambrosia in this form, do not think it proper to pour it carelessly into earth or wood or any other unseemly and irregular material, but, collecting the fairest parts of the sweetest flowers growing on the earth, from them they prepare for the reception of the honey the vessels called honeycombs, [with cells] all equal, similar and adjacent, and hexagonal in form.

That they have contrived this in accordance with a certain geometrical forethought we may thus infer. They would necessarily think that the figures must all be adjacent one to another and have their sides common, in order that nothing else might fall into the interstices and so defile their work. Now there are only three rectilineal figures which would satisfy the condition, I mean regular figures which are equilateral and equiangular, inasmuch as irregular figures would be displeasing to the bees  . [These being] the triangle, the square and the hexagon, the bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others.

Bees, then, know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each.

Pappus also discusses the three and four lines theorem of Apollonius.

Succinctly, given three lines: Find the locus of points for which the product of the distances from two lines is the square of the distance of the third.

Given four lines: Find the locus of points for which the product of the distances from two lines is the product of the distance of the other two.

Pappus' Theorem. Volume of revolution = (area bounded by the curve)
(distance traveled by the center of gravity)

Volume of revolution

Area bounded by the curve:

The center of gravity:

Pappus:

Pappus', on the Pappus-Guldin Thm

`Figures generated by a complete revolution of a plane figure about an axis are in a ratio compounded (1) of the ratio of the areas of the figures, and (2) of the ratio of the straight lines similarly drawn to (i.e. drawn to meet at the same angles) the axes of rotation from the respective centres of gravity. Figures generated by incomplete revolutions are in the ratio compounded (1) of the ratio of the areas of the figures and (2) of the ratio of the arcs described by the centres of gravity of the respective figures, the latter ratio being itself compounded (a) of the ratio of the straight lines similarly drawn (from the respective centres of gravity to the axes of rotation) and (b) of the ratio of the angles contained (i.e. described) about the axes of revolution by the extremities of the said straight lines (i.e. the centres of gravity).'

Pappus' theorem surface area.

Hypatia (c. 370-418)

• daughter of Theon of Alexandria
• by 397 Christianity became the state religion of the Roman empire ``pagamni'' was banned.
• In about 400 Hypatia became a leader of a neoplatonic school in Alexandria.
• She was said to be ``mistress'' of the whole of Pagon science, especially philosophy and medicine.
• was a speaker of great enough eloquence to be considered a threat by Christians. She was slain in 418 by a fanatical mob when she refused to repudiate her beliefs. (This may have resulted because of a dispute between the prefect Orestus and the bishop Cyrillus.)
• She wrote commentaries on Diophantus and on Ptolemy and Apollonius.

After Hypatia, Alexandrian mathematics came to an end - perhaps even before.

Back to Athens: The Academia of Plato, which had access to it own ample financial means, maintained itself for a longer time. There was here the commentator Proclus whom we have already discussed.

After came Isidore of Alexandria and Damascius of Damascus. They were heads of the school. There was also Simplicius, who wrote commentaries on Aristotle. But in 529, on the order of the Emperor Justinian, the school of Athens, the last rampart of the pagan world, was closed.

The last center of Greek culture was Constantinople. Here lived Isidore of Milete and Anthemius of Tralles, both architects and mathematicians.

It was probably Isidore that added the so-called 15th Book of the Elements, which contains propositions on regular polyhedra. At least, the propositions were probably his.

After these last flutterings, the history of Greek mathematics dies like a snuffed candle.

The Decline of Greek Mathematics

Why did mathematics decline so dramatically from the Golden Age?

• There were always only a few that could afford to spend their lives pursuing mathematics.
• The teaching tradition diminished partly due to the political strife around the eastern Mediterranean.
• Roman influence was important.
• Arab hegemony - destruction of the library of Alexandria - and of the seat of learning.
• Christian intolerance.

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