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:
Summary of Contents:
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)
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?