XIX

PAPPUS OF ALEXANDRIA

WE have seen that the Golden Age of Greek geometry ended with the time of Apollonius of Perga. But the influence of Euclid, Archimedes and Apollonius continued, and for some time there was a succession of quite competent mathematicians who, although not originating anything of capital importance, kept up the tradition. Besides those who were known for particular investigations, e.g. of new curves or surfaces, there were such men as Geminus who, it cannot be doubted, were thoroughly familiar with the great classics. Geminus, as we have seen, wrote a comprehensive work of almost encyclopaedic character on the classification and content of mathematics, including the history of the development of each subject. But the beginning of the Christian era sees quite a different state of things. Except in sphaeric and astronomy (Menelaus and Ptolemy), production was limited to elementary textbooks of decidedly feeble quality. In the meantime it would seem that the study of higher geometry languished or was completely in abeyance, until Pappus arose to revive interest in the subject. From the way in which he thinks it necessary to describe the contents of the classical works belonging to the Treasury of Analysis, for example, one would suppose that by his time many of them were, if not lost, completely forgotten, and that the great task which he set himself was the re-establishment of geometry on its former high plane of achievement. Presumably such interest as he was able to arouse soon flickered out, but for us his work has an inestimable value as constituting, after the works of the great mathematicians which have actually survived, the most important of all our sources.

Date of Pappus.

Pappus lived at the end of the third century A.D. The authority for this date is a marginal note in a Leyden manuscript of chronological tables by Theon of Alexandria, where, opposite to the name of Diocletian, a scholium says, ‘In his time Pappus wrote’, Diocletian reigned from 284 to 305, and this must therefore be the period of Pappus’s literary activity. It is true that Suidas makes him a contemporary of Theon of Alexandria, adding that they both lived under Theodosius I (379-395). But Suidas was evidently not well acquainted with the works of Pappus; though he mentions a description of the earth by him and a commentary on four Books of Ptolemy’s Syntaxis, he has no word about his greatest work, the Synagoge. As Theon also wrote a commentary on Ptolemy and incorporated a great deal of the commentary of Pappus, it is probable that Suidas had Theon’s commentary before him and from the association of the two names wrongly inferred that they were contemporaries.

Works (commentaries) other than the Collection.

Besides the Synagoge, which is the main subject of this chapter, Pappus wrote several commentaries, now lost except for fragments which have survived in Greek or Arabic. One was a commentary on the Elements of Euclid. This must presumably have been pretty complete, for, while Proclus (on Eucl. I) quotes certain things from Pappus which may be assumed to have come in the notes on Book I, fragments of his commentary on Book X actually survive in the Arabic (see above, vol. i, pp. 154–5, 209), and again Eutocius in his note on Archimedes, On the Sphere and Cylinder, I. 13, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle, which problem would no doubt be solved by Pappus, as it is by a scholiast, in a note on XII. 1. Some of the references by Proclus deserve passing mention. (1) Pappus said that the converse of Post. 4 (equality of all right angles) is not true, i.e. it is not true that all angles equal to a right angle are themselves right, since the ‘angle’ between the conterminous arcs of two semicircles which are equal and have their diameters at right angles and terminating at one point is equal to, but is not, a right angle.¹ (2) Pappus said that, in addition to the genuine axioms of Euclid, there were others on record about unequals added to equals and equals added to unequals. Others given by Pappus are (says Proclus) involved by the definitions, e.g. that ‘all parts of the plane and of the straight line coincide with one another’, that ‘a point divides a line, a line a surface, and a surface a solid’, and that ‘the infinite is (obtained) in magnitudes both by addition and diminution‘.² (3) Pappus gave a pretty proof of Eucl. I. 5, which modern editors have spoiled when introducing it into text-books. If AB, AC are the equal sides in an isosceles triangle, Pappus compares the triangles ABC and ACB (i.e. as if he were comparing the triangle ABC seen from the front with the same triangle seen from the back), and shows that they satisfy the conditions of I. 4, so that they are equal in all respects, whence the result follows.³

Marinus at the end of his commentary on Euclid’s Data refers to a commentary by Pappus on that book.

Pappus's commentary on Ptolemy’s Syntaxis has already been mentioned (p. 274); it seems to have extended to six Books, if not to the whole of Ptolemy's work. The Fihrist says that he also wrote a commentary on Ptolemy's Planisphaerium, which was translated into Arabic by Thābit b. Qurra. Pappus himself alludes to his own commentary on the Analemma of Diodorus, in the course of which he used the conchoid of Nicomedes for the purpose of trisecting an angle.

We come now to Pappus's great work.

The Synagoge or Collection.

(α) Character of the work; wide range.

Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field. It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works (where still extant) rather than to enable them to be dispensed with. Thus in the case of the treatises included in the Treasury of Analysis there is a general introduction, followed by a general account of the contents, with lemmas, &c., designed to facilitate the reading of the treatises themselves. On the other hand, where the history of a subject is given, e.g. that of the problem of the duplication of the cube or the finding of the two mean proportionals, the various solutions themselves are reproduced, presumably because they were not easily accessible, but had to be collected from various sources. Even when it is some accessible classic which is being described, the opportunity is taken to give alternative methods, or to make improvements in proofs, extensions, and so on. Without pretending to great originality, the whole work shows, on the part of the author, a thorough grasp of all the subjects treated, independence of judgement, mastery of technique; the style is terse and clear; in short, Pappus stands out as an accomplished and versatile mathematician, a worthy representative of the classical Greek geometry.

(β) List of authors mentioned.

The immense range of the Collection can be gathered from a mere enumeration of the names of the various mathematicians quoted or referred to in the course of it. The greatest of them, Euclid, Archimedes and Apollonius, are of course continually cited, others are mentioned for some particular achievement, and in a few cases the mention of a name by Pappus is the whole of the information we possess about the person mentioned. In giving the list of the names occurring in the book, it will, I think, be convenient and may economize future references if I note in brackets the particular occasion of the reference to the writers who are mentioned for one achievement or as the authors of a particular book or investigation. The list in alphabetical order is: Apollonius of Perga, Archimedes, Aristaeus the elder (author of a treatise in five Books on the Elements of Conics or of ‘five Books on Solid Loci connected with the conics’), Aristarchus of Samos ( On the sizes and distaiices of the sun and moon), Autolycus (On the moving sphere), Carpus of Antioch (who is quoted as having said that Archimedes wrote only one mechanical book, that on sphere-making, since he held the mechanical appliances which made him famous to be nevertheless unworthy of written description: Carpus himself, who was known as mechanicus, applied geometry to other arts of this practical kind), Charmandrus (who added three simple and obvious loci to those which formed the beginning of the Plane Loci of Apollonius), Conon of Samos, the friend of Archimedes (cited as the propounder of a theorem about the spiral in a plane which Archimedes proved: this would, however, seem to be a mistake, as Archimedes says at the beginning of his treatise that he sent certain theorems, without proofs, to Conon, who would certainly have proved them had he lived), Demetrius of Alexandria (mentioned as the author of a work called ‘Linear considerations’, γραμμικα πιστάσєις, i.e. considerations on curves, as to which nothing more is known), Dinostratus, the brother of Menaechmus (cited, with Nicomedes, as, having used the curve of Hippias, to which they gave the name of quadratrix, τєτραγωνίζουσα, for the squaring of the circle), Diodorus (mentioned as the author of an Analemma), Eratosthenes (whose mean-finder, an appliance for finding two or any number of geometric means, is described, and who is further mentioned as the author of two Books ‘On means’ and of a work entitled ‘Loci with reference to means’), Erycinus (from whose Paradoxa are quoted various problems seeming at first sight to be inconsistent with Eucl. I. 21, it being shown that straight lines can be drawn from two points on the base of a triangle to a point within the triangle which are together greater than the other two sides, provided that the points in the base may be points other than the extremities), Euclid, Geminus the mathematician (from whom is cited a remark on Archimedes contained in his book ‘On the classification of the mathematical sciences’, see above, p. 223), Heraclitus (from whom Pappus quotes an elegant solution of a νєσις with reference to a square), Hermodorus (Pappus’s son, to whom he dedicated Books VII, VIII of his Collection), Heron of Alexandria (whose mechanical works are extensively quoted from), Hierius the philosopher (a contemporary of Pappus, who is mentioned as having asked Pappus's opinion on the attempted solution by ‘plane’ methods of the problem of the two means, which actually gives a method of approximating to a solution⁴), Hipparchus (quoted as practically adopting three of the hypotheses of Aristarchus of Samos), Megethion (to whom Pappus dedicated Book V of his Collection), Menelaus of Alexandria (quoted as the author of Sphaerica and as having applied the name παράδοξος to a certain curve), Nicomachus (on three means additional to the first three), Nicomedes, Pandrosion (to whom Book III of the Collection is dedicated), Pericles (editor of Euclid's Data), Philon of Byzantium (mentioned along with Heron), Philon of Tyana (mentioned as the discoverer of certain complicated curves derived from the interweaving of plectoid and other surfaces), Plato (with reference to the five regular solids), Ptolemy, Theodosius (author of the Sphaerica and On Days and Nights).

(γ) Translations and editions.

The first published edition of the Collection was the Latin translation by Commandinus (Venice 1589, but dated at the end ‘Pisauri apud Hieronymum Concordiam 1588’; reissued with only the title-page changed ‘Pisauri…1602’). Up to 1876 portions only of the Greek text had appeared, namely Books VII, VIII in Greek and German, by C. J. Gerhardt, 1871, chaps. 33–105 of Book V, by Eisenmann, Paris 1824, chaps. 45-52 of Book IV in Iosephi Torelli Veronensis Geometrica, 1769, the remains of Book II, by John Wallis (in Opera mathematica, III, Oxford 1699); in addition, the restorers of works of Euclid and Apollonius from the indications furnished by Pappus give extracts from the Greek text relating to the particular works. Breton de Champ on Euclid’s Porisms, Hailey in his edition of the Conics of Apollonius (1710) and in his translation from the Arabic and restoration respectively of the De sectione rationis and De sectione spatii of Apollonius (1706), Camerer on Apollonius's Tactiones (1795), Simson and Horsley in their restorations of Apollonius's Plane Loci and Inclinationes published in the years 1749 and 1770 respectively. In the years 1876–8 appeared the only complete Greek text, with apparatus, Latin translation, commentary, appendices and indices, by Friedrich Hultsch; this great edition is one of the first monuments of the revived study of the history of Greek mathematics in the last half of the nineteenth century, and has properly formed the model for other definitive editions of the Greek text of the other classical Greek mathematicians, e.g. the editions of Euclid, Archimedes, Apollonius, &c., by Heiberg and others. The Greek index in this edition of Pappus deserves special mention because it largely serves as a dictionary of mathematical terms used not only in Pappus but by the Greek mathematicians generally.

(δ) Summary of contents.

At the beginning of the work, Book I and the first 13 propositions (out of 26) of Book II are missing. The first 13 propositions of Book II evidently, like the rest of the Book, dealt with Apollonius’s method of working with very large numbers expressed in successive powers of the myriad, 10000. This system has already been described (vol. i, pp. 40, 54–7). The work of Apollonius seems to have contained 26 propositions (25 leading up to, and the 26th containing, the final continued multiplication).

Book III consists of four sections. Section (1) is a sort of history of the problem of finding two mean proportionals, in continued proportion, between two given straight lines.

It begins with some general remarks about the distinction between theorems and problems. Pappus observes that, whereas the ancients called them all alike by one name, some regarding them all as problems and others this theorems, a clear distinction was drawn by those who favoured more exact terminology. According to the latter a problem is that in which it is proposed to do or construct something, a theorem that in which, given certain hypotheses, we investigate that which follows from and is necessarily implied by them. Therefore he who propounds a theorem, no matter how he has become aware of the fact which is a necessary consequence of the premisses, must state, as the object of inquiry, the right result and no other. On the other hand, he who propounds a problem may bid us do something which is in fact impossible, and that without necessarily laying himself open to blame or criticism. For it is part of the solver’s duty to determine the conditions under which the problem is possible or impossible, and, ‘if possible, when, how, and in how many ways it is possible’, When, however, a man professes to know mathematics and yet commits some elementary blunder, he cannot escape censure. Pappus gives, as an example, the case of an-unnamed person ‘who was thought to be a great geometer’ but who showed ignorance in that he claimed to know how to solve the problem of the two mean proportionals by ‘plane’ methods (i.e. by using the straight line and circle only). He then reproduces the argument of the anonymous person, for the purpose of showing that it does not solve the problem as its author claims. We have seen (vol. i, pp. 269–70) how the method, though not actually solving the problem, does furnish a series of successive approximations to the real solution. Pappus adds a few simple lemmas assumed in the exposition.

Next comes the passage⁵, already referred to, on the distinction drawn by the ancients between (1) plane problems or problems which can be solved by means of the straight line and circle, (2) solid problems, or those which require for their solution one or more conic sections, (3) linear problems, or those which necessitate recourse to higher curves still, curves with a more complicated and indeed a forced or unnatural origin (βєβιασμνην) such as spirals, quadratrices, cochloids and cissoids, which have many surprising properties of their own. The problem of the two mean proportionals, being a solid problem, required for its solution either conics or some equivalent, and, as conics could not be constructed by purely geometrical means, various mechanical devices were invented such as that of Eratosthenes (the mean-finder), those described in the Mechanics of Philon and Heron, and that of Nicomedes (who used the ‘cochloidal’ curve). Pappus proceeds to give the solutions of Eratosthenes, Nicomedes and Heron, and then adds a fourth which he claims as his own, but which is practically the same as that attributed by Eutocius to Sporus. All these solutions have been given above (vol. i, pp. 258–64, 266–8).

Section (2). The theory of means.

Next follows a section (pp. 69–105) on the theory of the different kinds of means. The discussion takes its origin from the statement of the ‘second problem’, which was that of ‘exhibiting the three means’ (i.e. the arithmetic, geometric and harmonic) ‘in a semicircle’. Pappus first gives a construction by which another geometer (άλλος τις) claimed to have solved this problem, but he does not seem to have understood it, and returns to the same problem later (pp. 80–2).

In the meantime he begins with the definitions of the three means and then shows how, given any two of three terms a, b, c in arithmetical, geometrical or harmonical progression, the third can be found. The definition of the mean (b) of three terms a, b, c in harmonic progression being that it satisfies the relation a : c = a— b : b — c, Pappus gives alternative definitions for the arithmetic and geometric means in corresponding form, namely for the arithmetic mean a : a = a — b : b — c and for the geometric a : b = a — b : b — c.

The construction for the harmonic mean is perhaps worth giving. Let AB, BG be two given straight lines. At A draw DAE perpendicular to AB, and make DA, AE equal. Join DB, BE. From G draw GF at right angles to AB meeting DB in F. D Join EF meeting AB in C. Then BC is the required harmonic mean.

For

Similarly, by means of a like figure, we can find BG when AB, BC are given, and AB when BC, BG are given (in the latter case the perpendicular DE is drawn through G instead of A).

Then follows a proposition that, if the three means and the several extremes are represented in one set of lines, there must be five of them at least, and, after a set of five such lines have been found in the smallest possible integers, Pappus passes to the problem of representing the three means with the respective extremes by dx lines drawn in a semicircle.

Given a semicircle on the diameter AC, and B any point on the diameter, draw BD at right angles to AC. Let the tangent at D meet AC produced in G, and measure DH along the tangent equal to DG. Join HB meeting the radius OD in K. Let BF be perpendicular to OD.

Then, exactly as above, it is shown that OK is a harmonic mean between OF and OD. Also BD is the geometric mean between AB, BC, while OC (= OD) is the arithmetic mean between AB, BC.

Therefore the six lines DO (= OC), OK, OF, AB, BC, BD supply the three means with the respective extremes.

But Pappus seems to have failed to observe that the ‘certain other geometer who has the same figure excluding the dotted lines, supplied the same in five lines. For he said that DF is ‘a harmonic mean’. It is in fact the harmonic mean between AB, BC, as is easily seen thus.

Since ODB is a right-angled triangle, and BF perpendicular to OD,

or

But

therefore

Therefore

that is,

and DF is the harmonic mean between AB, BC.

‘Consequently the five lines DO ( = OC), DF, AB, BC, BD exhibit all the three means with the extremes.

Pappus does not seem to have seen this, for he observes that the geometer in question, though saying that DF is a harmonic mean, does not say how it is a harmonic mean or between what straight lines.

In the next chapters (pp. 84–104) Pappus, following Nicomachus and others, defines seven more means, three of which were ancient and the last four more modern, and shows how we can form all ten means as linear functions of α, β, γ, where α, β, γ are in geometrical progression. The exposition has already been described (vol. i, pp. 86–9).

Section (3). The ‘Paradoxes’ of Erycinus.

The third section of Book III (pp. 104–30) contains a series of propositions, all of the same sort, which are curious rather than geometrically important. They appear to have been taken direct from a collection of Paradoxes by one Erycinus.⁶ The first set of these propositions (props. 28–34) are connected with Eucl. I. 21, which says that, if from the extremities of the base of any triangle two straight lines be drawn meeting at any point within the triangle, the straight lines are together less than the two sides of the triangle other than the base, but contain a greater angle. It is pointed out that, if the straight lines are allowed to be drawn from points in the base other than the extremities, their sum may be greater than the other two sides of the triangle.

The first case taken is that of a right-angled triangle ABC right-angled at B. Draw AD to any point D on BC. Measure on it DE equal to AB, bisect AE in F_f and join FC. Then shall DF + FC be > BA + AC.

For

Add DE and AB respectively, and we have

More elaborate propositions are next proved, such as the following.

1. In any triangle, except an equilateral triangle or an isosceles triangle with base less than one of the other sides, it is possible to construct on the base and within the triangle two straight lines meeting at a point, the sum of which is equal to the sum of the other two sides of the triangle (props. 29, 30).

2. In any triangle in which it is possible to construct two straight lines from the base to one internal point the sum of which is equal to the sum of the two sides of the triangle, it is also possible to construct two other such straight lines the sum of which is greater than that sum (Prop. 31).

3. Under the same conditions, if the base is greater than either of the other two sides, two straight lines can be so constructed from the base to an internal point which are respectively greater than the other two sides of the triangle; and the lines may be constructed so as to be respectively equal to the two sides, if one of those two sides is less than the other and each of them is less than the base (props. 32, 33).

4. The lines may be so constructed that their sum will bear to the sum of the two sides of the triangle any ratio less than 2 : 1 (Prop. 34).

As examples of the proofs, we will take the case of the scalene triangle, and prove the first and Part 1 of the third of the above propositions for such a triangle.

In the triangle ABC with base BC let AB be greater than AC.

Take D on BA such that BD = (BA + AC).

On DA between D and A take any point E, and draw EF parallel to BC. Let G be any point on EF; draw GH parallel to AB and join GC.

Now

Produce GC to K so that GK = EA + AC, and with G as centre and GK as radius describe a circle. This circle will meet HC and HG, because GH = EB > BD or DA + AC and > GK, a fortiori.

Then

To obtain two straight lines HG′, G′L such that HG′ + G′L > BA + AC, we have only to choose G′ so that HG′, G′L enclose the straight lines HG, GL completely.

Next suppose that, given a triangle ABC in which BC > BA > AC, we are required to draw from two points on BC to an internal point two straight lines greater respectively than BA, AC.

With B as centre and BA as radius describe the arc AEF. Take any point E on it, and any point D on BE produced but within the triangle. Join DC, and produce it to G so that DG = AC. Then with D as centre and DG as radius describe a circle. This will meet both BC and BD because BA > AC, and a fortiori DB > DG.

Then, if L be any point on BH, it is clear that BD, DL are two straight lines satisfying the conditions.

A point L′ on BH can be found such that DL′ is equal to AB by marking off DN on DB equal to AB and drawing with D as centre and DN as radius a circle meeting BH in L′. Also, if DH be joined, DH = AC.

Propositions follow (35–9) having a similar relation to the Postulate in Archimedes, On the Sphere and Cylinder, I, about conterminous broken lines one of which wholly encloses

the other, i.e. it is shown that broken lines, consisting of several straight lines, can be drawn with two points on the base of a triangle or parallelogram as extremities, and of greater total length than the remaining two sides of the triangle or three sides of the parallelogram.

props. 40–2 show that triangles or parallelograms can be constructed with sides respectively greater than those of a given triangle or parallelogram but having a less area.

Section (4). The inscribing of the five regular solids in a sphere.

The fourth section of Book III (pp. 132–62) solves the problems of inscribing each of the five regular solids in a given sphere. After some preliminary lemmas (props. 43–53), Pappus attacks the substantive problems (props. 54–8), using the method of analysis followed by synthesis in the case of each solid.

(a) In order to inscribe a regular pyramid or tetrahedron in the sphere, he finds two circular sections equal and parallel to one another, each of which contains one of two opposite edges as its diameter. If d be the diameter of the sphere, the parallel circular sections have d′ as diameter, where d² = d′².

(b) In the case of the cube Pappus again finds two parallel circular sections with diameter d′ such that d² = d′²; a square inscribed in one of these circles is one face of the cube and the square with sides parallel to those of the first square inscribed in the second circle is the opposite face.

(c) In the case of the octahedron the same two parallel circular sections with diameter d′ such that d² = d′² are used; an equilateral triangle inscribed in one circle is one face, and the opposite face is an equilateral triangle inscribed in the other circle but placed in exactly the opposite way.

(d) In the case of the icosahedron Pappus finds four parallel circular sections each passing through three of the vertices of the icosahedron; two of these are small circles circumscribing two opposite triangular faces respectively, and the other two circles are between these two circles, parallel to them, and equal to one another. The pairs of circles are determined in this way. If d be the diameter of the sphere, set out two straight lines x, y such that d, x, y are in the ratio of the sides of the regular pentagon, hexagon and decagon respectively described in one and the same circle. The smaller pair of circles have r as radius where r² = y², and the larger pair have r′ as radius where r′² = x².

(e) In the case of the dodecahedron the same four parallel circular sections are drawn as in the case of the icosahedron. Inscribed pentagons set the opposite way are inscribed in the two smaller circles; these pentagons form opposite faces. Regular pentagons inscribed in the larger circles with vertices at the proper points (and again set the opposite way) determine ten more vertices of the inscribed dodecahedron.

The constructions are quite different from those in Euclid XIII. 13, 15, 14, 16, 17 respectively, where the problem is first to construct the particular regular solid and then to ‘comprehend it in a sphere’, i. e. to determine the circumscribing sphere in each case. I have set out Pappus’s propositions in detail elsewhere.⁷

Book IV.

At the beginning of Book IV the title and preface are missing, and the first section of the Book begins immediately with an enunciation. The first section (pp. 176–208) contains Propositions 1–12 which, with the exception of props. 8–10, seem to be isolated propositions given for their own sakes and not connected by any general plan.

Section (1). Extension of the theorem of Pythagoras.

The first proposition is of great interest, being the generalization of Eucl. I. 47, as Pappus himself calls it, which is by this time pretty widely known to mathematicians. The enunciation is as follows.

‘If ABC be a triangle and on AB, AC any parallelograms whatever be described, as ABDE, ACFG, and if DE, FG produced meet in H and HA be joined, then the parallelograms ABDE' ACFG are together equal to the parallelogram contained by BC, HA in an angle which is equal to the sum of the angles ABC, DHA’,

Produce HA to meet BG in K, draw BL, CM parallel to KH meeting DE in L and FG in M, and join LNM.

Then BLHA is a parallelogram, and HA is equal and parallel to BL.

Similarly HA, CM are equal and parallel; therefore BL, CM are equal and parallel.

Therefore BLMC is a parallelogram; and its angle LBK is equal to the sum of the angles ABC, DHA.

Now

Similarly

Therefore, by addition,

It has been observed (by Professor Cook Wilson⁸) that the parallelograms on AB, AC need not necessarily be erected outwards from AB, AC. If one of them, e.g. that on AC, be drawn inwards, as in the second figure above, and Pappus's construction be made, we have a similar result with a negative sign, namely,

Again, if both ABDE and ACFG were drawn inwards, their sum would be equal to BLMC drawn outwards. Generally, if the areas of the parallelograms described outwards are regarded as of opposite sign to those of parallelograms drawn inwards,

we may say that the algebraic sum of the three parallelograms is equal to zero.

Though Pappus only takes one case, as was the Greek habit, I see no reason to doubt that he was aware of the results in the other possible cases.

props. 2, 3 are noteworthy in that they use the method and phraseology of Eucl. X, proving that a certain line in one figure is the irrational called minor (see Eucl. X. 76), and a certain line in another figure is ‘the excess by which the binomial exceeds the straight line which produces with a rational area a medial whole’ (Eucl. X. 77). The propositions 4–7 and 11–12 are quite interesting as geometrical exercises, but their bearing is not obvious : props. 4 and 12 are remarkable in that they are cases of analysis followed by synthesis applied to the proof of theorems. props. 8–10 belong to the subject of tangencies, being the sort of propositions that would come as particular cases in a book such as that of Apollonius On Contacts; Prop. 8 shows that, if there are two equal circles and a given point outside both, the diameter of the circle passing through the point and touching both circles is ‘given’; the proof is in many places obscure and assumes lemmas of the same kind as those given later à propos of Apollonius’s treatise; Prop. 10 purports to show how, given three unequal circles touching one another two and two, to find the diameter of the circle including them and touching all three.

Section (2). On circles inscribed in the ρβηλος ‘shoemaker’s knife’).

The next section (pp. 208–32), directed towards the demonstration of a theorem about the relative sizes of successive circles inscribed in the ρβηλος (shoemaker’s knife), is extremely interesting and clever, and I wish that I had space to reproduce it completely. The ρβηλος, which we have already met with in Archimedes’s ‘Book of Lemmas’, is formed thus. BC is the diameter of a semicircle BGC and BC is divided into two parts (in general unequal) at D; semicircles are described on BD, DC as diameters on the same side of BC as BGC is; the figure included between the three semicircles is the ρβηλος.

There is, says Pappus, on record an ancient proposition to the following effect. Let successive circles be inscribed in the ρβηλος touching the semicircles and one another as shown in the figure on p. 376, their centres being A, P, O …. Then, if p₁, p₂, p₃… be the perpendiculars from the centres A, P, O… on BC and d_l, d₂, d₃… the diameters of the corresponding circles,

He begins by some lemmas, the course of which I shall reproduce as shortly as I can.

I. If (Fig. 1) two circles with centres A, C of which the former is the greater touch externally at B, and another circle with centre G touches the two circles at K, L respectively, then KL produced cuts the circle BL again in D and meets AC produced in a point E such that AB : BC = AE : EC. This is easily proved, because the circular segments DL, LK are similar, and CD is parallel to AG. Therefore

Also

For

FIG. 1.

But

Therefore

And

II. Let (Fig. 2) BC, BD, being in one straight line, be the diameters of two semicircles BGC, BED, and let any circle as FGH touch both semicircles, A being the centre of the circle. Let M be the foot of the perpendicular from A on BC, r the radius of the circle FGH. There are two cases according as BD lies along BC or B lies between D and C; i.e. in the first case the two semicircles are the outer and one of the inner semicircles of the ρβηλος, while in the second case they are the two inner semicircles; in the latter case the circle FGH may either include the two semicircles or be entirely external to them. Now, says Pappus, it is to be proved that

in case (1)

and in case (2)

FIG. 2.

We will confine ourselves to the first case, represented in the figure (Fig. 2).

Draw through A the diameter HF parallel to BC. Then, since the circles BGC, HGF touch at G, and BC, HF are parallel diameters, GHB, GFC are both straight lines.

Let E be the point of contact of the circles FGH and BED; then, similarly, BEF, HED are straight lines.

Let HK, FL be drawn perpendicular to BC.

By the similar triangles BGC, BKH we have

and by the similar triangles BLF, BED

But

therefore

or

Therefore

And

therefore

It is next proved that BK . LC = AM².

For, by similar triangles BKH, FLC,

Lastly, since BC : BD = BL : BK, from above,

Also

III. We now (Fig. 3) take any two circles touching the semicircles BGC, BED and one another. Let their centres be A and P, H their point of contact, d, d′ their diameters respectively. Then, if AM, PN are drawn perpendicular to BC, Pappus proves that

Draw BF perpendicular to BC and therefore touching the semicircles BGC, BED at B. Join AP, and produce it to meet BF in F.

Now, by II. (a) above,

and for the same reason = BN : PH;

it follows that

Therefore (Lemma I), if the two circles touch the semicircle BED in R, E respectively, FRE is a straight line and EF. FR = FH².

But

If now BH meets PN in 0 and MA produced in S, we have, by similar triangles, FH: FB = PH: PO = AH: AS, whence PH = PO and SA = AH, so that O, S are the intersections of PN, AM with the respective circles.

FIG. 3.

Join BP, and produce it to meet MA in K.

Now

And

Therefore KS = AS, and KA = d, the diameter of the circle EHG.

Lastly,

that is,

or

IV. We now come to the substantive theorem.

Let FGH be the circle touching all three semicircles (Fig. 4).

We have then, as in Lemma II,

and for the same reason (regarding FGH as touching the semicircles BGC, DUC)

From the first relation we have

FIG. 4.

whence DC: BD = KL : BK, and inversely BD : DC = BK : KL,

while, from the second relation, BC : CD = CK : CL,

whence

Consequently

or

But we saw in Lemma II (6) that BK . LC = AM².

Therefore

For the second circle Lemma III gives us

whence, since p₁ = d₁, p₂ = 2d₂.

For the third circle

whence p₃ = 3d₃.

And so on ad infinitum.

The same proposition holds when the successive circles, instead of being placed between the large and one of the small semicircles, come down between the two small semicircles.

Pappus next deals with special cases (1) where the two smaller semicircles become straight lines perpendicular to the diameter of the other semicircle at its extremities, (2) where we replace one of the smaller semicircles by a straight line through D at right angles to BC, and lastly (3) where instead of the semicircle DUC we simply have the straight line DC and make the first circle touch it and the two other semicircles.

Pappus’s propositions of course include as particular cases the partial propositions of the same kind included in the ‘Book of Lemmas’ attributed to Archimedes (props. 5, 6); cf. p. 102.

Sections (3) and (4). Methods of squaring the circle, and of
trisecting (or dividing in any ratio) any given angle.

The last sections of Book IV (pp. 234–302) are mainly devoted to the solutions of the problems (1) of squaring or rectifying the circle and (2) of trisecting any given angle or dividing it into two parts in any ratio. To this end Pappus gives a short account of certain curves which were used for the purpose.

(α) The Archimedean spiral.

He begins with the spiral of Archimedes, proving some of the fundamental properties. His method of finding the area included (1) between the first turn and the initial line, (2) between any radius vector on the first turn and the curve, is worth giving because it differs from the method of Archimedes. It is the area of the whole first turn which Pappus works out in detail. We will take the area up to the radius vector OB, say.

With centre O and radius OB draw the circle A′BCD.

Let BC be a certain fraction, say 1 /nth, of the arc BCDA′, and CD the same fraction, OC, OD meeting the spiral in F, E respectively. Let KS, SV be the same fraction of a straight line KR, the side of a square KNLR. Draw ST, VW parallel to KN meeting the diagonal KL of the square in U, Q respectively, and draw MU, PQ parallel to KR.

With O as centre and OE, OF as radii draw arcs of circles meeting OF, OB in H, G respectively.

For brevity we will now denote a cylinder in which r is the radius of the base and h the height by (cyl. r, h) and the cone with the same base and height by (cone r, h).

By the property of the spiral,

whence

Now

Similarly

and so on.

The sectors OBC, OCD … form the sector OA′DB, and the sectors OFG, OEH … form a figure inscribed to the spiral.

In like manner the cylinders (KN, TN), (ST, TW) … form the cylinder (KN, NL), while the cylinders (MN, NT), (PT, TW) … form a figure inscribed to the cone (KN, NL).

Consequently

We have a similar proportion connecting a figure circumscribed to the spiral and a figure circumscribed to the cone.

By increasing n the inscribed and circumscribed figures can be compressed together, and by the usual method of exhaustion we have ultimately

The ratio of the sector OA′DB to the complete circle is that of the angle which the radius vector describes in passing from the position OA to the position OB to four right angles, that is, by the property of the spiral, r : a, where r = OB, a = OA.

Therefore

Similarly the area of the spiral cut off by any other radius vector

Therefore (as Pappus proves in his next proposition) the first area is to the second as r³ to r′³.

Considering the areas cut off by the radii vectores at the points where the revolving line has passed through angles of π, π, π and 2π respectively, we see that the areas are in the ratio of ()³, ()³, ()³, 1 or 1, 8, 27, 64, so that the areas of the spiral included in the four quadrants are in the ratio of 1, 7, 19, 37 (Prop. 22).

(β) The conchoid of Nicomedes.

The conchoid of Nicomedes is next described (chaps. 26–7), and it is shown (chaps. 28, 29) how it can be used to find two geometric means between two straight lines, and consequently to find a cube having a given ratio to a given cube (see vol. i, pp. 260–2 and pp. 238–40, where I have also mentioned Pappus’s remark that the conchoid which he describes is the first conchoid, while there also exist a second, a third and a fourth which are of use for other theorems).

(γ) The quadratrix.

The quadratrix is taken next (chaps. 30–2), with Sporus’s criticism questioning the construction as involving a petitio principii. Its use for squaring the circle is attributed to Dinostratus and Nicomedes. The whole substance of this subsection is given above (vol. i, pp. 226–30).

Two constructions for the quadratrix by means of
‘surface-loci’

In the next chapters (chaps. 33, 34, props. 28, 29) Pappus gives two alternative ways of producing the quadratrix ‘by means of surface-loci’, for which he claims the merit that they are geometrical rather than ‘too mechanical’ as the traditional method (of Hippias) was.

(1) The first method uses a cylindrical helix thus.

Let ABC be a quadrant of a circle with centre B, and let BD be any radius. Suppose that EF, drawn from a point E on the radius BD perpendicular to BC, is (for all such radii) in a given ratio to the arc DC.

‘I say’, says Pappus, ‘that the locus of E is a certain curve.’

Suppose a right cylinder erected from the quadrant and a cylindrical helix CGH drawn upon its surface. Let DH be the generator of this cylinder through D, meeting the helix in H. Draw BL, EI at right angles to the plane of the quadrant, and draw HIL parallel to BD.

Now, by the property of the helix, EI(= DH) is to the arc CD in a given ratio. Also EF : (arc CD) = a given ratio.

Therefore the ratio EF : EI is given. And since EF, EI are given in position, FI is given in position. But FI is perpendicular to BC. Therefore FI is in a plane given in position, and so therefore is I.

But I is also on a certain surface described by the line LH, which moves always parallel to the plane ABC, with one extremity L on BL and the other extremity H on the helix. Therefore I lies on the intersection of this surface with the plane through FI.

Hence I lies on a certain curve. Therefore E, its projection on the plane ABC, also lies on a curve.

In the particular case where the given ratio of EF to the arc CD is equal to the ratio of BA to the arc CA, the locus of E is a quadratrix.

[The surface described by the straight line LH is a plectoid. The shape of it is perhaps best realized as a continuous spiral staircase, i.e. a spiral staircase with infinitely small steps. The quadratrix is thus produced as the orthogonal projection of the curve in which the plectoid is intersected by a plane through BC inclined at a given angle to the plane ABC. It is not difficult to verify the result analytically.]

(2) The second method uses a right cylinder the base of which is an Archimedean spiral.

Let ABC be a quadrant of a circle, as before, and EF, perpendicular at F to BC, a straight line of such length that EF is to the arc DC as AB is to the arc ADC.

Let a point on AB move uniformly from A to B while, in the same time, AB itself revolves uniformly about B from the position BA to the position BC. The point thus describes the spiral AGB. If the spiral cuts BD in G,

or

Therefore

Draw GK at right angles to the plane ABC and equal to BG. Then GK, and therefore K, lies on a right cylinder with the spiral as base.

But BK also lies on a conical surface with vertex B such that its generators all make an angle of π with the plane ABC.

Consequently K lies on the intersection of two surfaces, and therefore on a curve.

Through K draw LKI parallel to BD, and let BL, EI be at right angles to the plane ABC.

Then LKI, moving always parallel to the plane ABC, with one extremity on BL and passing through K on a certain curve, describes a certain plectoid, which therefore contains the point I.

Also IE = EF, IF is perpendicular to BC, and hence IF, and therefore I, lies on a fixed plane through BC inclined to ABCat an angle of π.

Therefore I, lying on the intersection of the plectoid and the said plane, lies on a certain curve. So therefore does the projection of I on ABC, i.e. the point E.

The locus of E is clearly the quadratrix.

[This result can also be verified analytically.]

(δ) Digression: a spiral on a sphere.

Prop. 30 (chap. 35) is a digression on the subject of a certain spiral described on a sphere, suggested by the discussion of a spiral in a plane.

Take a hemisphere bounded by the great circle KLM, with H as pole. Suppose that the quadrant of a great circle HNK revolves uniformly about the radius HO so that K describes the circle KLM and returns to its original position at K, and suppose that a point moves uniformly at the same time from H to K at such speed that the point arrives at K at the same time that HK resumes its original position. The point will thus describe a spiral on the surface of the sphere between the points H and K as shown in the figure.

Pappus then sets himself to prove that the portion of the surface of the sphere cut off towards the pole between the spiral and the arc HNK is to the surface of the hemisphere in a certain ratio shown in the second figure where ABC is a quadrant of a circle equal to a great circle in the sphere, namely the ratio of the segment ABC to the sector DABC.

Draw the tangent CF to the quadrant at C. With C as centre and radius CA draw the circle AEF meeting CF in F.

Then the sector CAF is equal to the sector ADC (since CA² = 2AD² while ).

It is required, therefore, to prove that, if S be the area cut off by the spiral as above described,

Let KL be a (small) fraction, say 1 /nth, of the circumference of the circle KLM, and let HPL be the quadrant of the great circle through H, L meeting the spiral in P. Then, by the property of the spiral,

Let the small circle NPQ passing through P be described about the pole H.

Next let FE be the same fraction, 1/nth, of the arc FA that KL is of the circumference of the circle KLM, and join EC meeting the arc ABC in B. With C as centre and CB as radius describe the arc BG meeting CF in G.

Then the arc CB is the same fraction, 1/nth, of the arc CBA that the arc FE is of FA (for it is easily seen that while ). Therefore, since (arc CBA) = (arc HPL), (arc CB) = (arc HP), and chord CB = chord HP.

Now

(a consequence of Archimedes, On Sphere and Cylinder, I. 42).

And

Therefore

(sector HPN) : (sector HKL) = (sector CBG) : (sector CEF).

Similarly, if the arc LL′ be taken equal to the arc KL and the great circle through H, L′ cuts the spiral in P′, and a small circle described about H and through P′ meets the arc HPL in p; and if likewise the arc BB′ is made equal to the arc BC, and CB’ is produced to meet AF in E′, while again a circular arc with C as centre and CB′ as radius meets CE in b,

And so on.

Ultimately then we shall get a figure consisting of sectors on the sphere circumscribed about the area S of the spiral and a figure consisting of sectors of circles circumscribed about the segment CBA; and in like manner we shall have inscribed figures in each case similarly made up.

The method of exhaustion will then give

[We may, as an illustration, give the analytical equivalent of this proposition. If ρ, ω be the spherical coordinates of P with reference to H as pole and the arc HNK as polar axis, the equation of Pappus’s curve is obviously ω = 4 ρ.

If now the radius of the sphere is taken as unity, we have as the element of area

Therefore

Therefore

The second part of the last section of Book IV (chaps. 36–41, pp. 270–302) is mainly concerned with the problem of trisecting any given angle or dividing it into parts in any given ratio. Pappus begins with another account of the distinction between plane, solid and linear problems (cf. Book III, chaps. 20–2) according as they require for their solution (1) the straight line and circle only, (2) conics or their equivalent, (3) higher curves still, ‘which have a more complicated and forced (or unnatural) origin, being produced from more irregular surfaces and involved motions. Such are the curves which are discovered in the so-called loci on surfaces, as well as others more complicated still and many in number discovered by Demetrius of Alexandria in his Linear considerations and by Philon of Tyana by means of the interlacing of plectoids and other surfaces of all sorts, all of which curves possess many remarkable properties peculiar to them. Some of these curves have been thought by the more recent writers to be worthy of considerable discussion; one of them is that which also received from Menelaus the name of the paradoxical curve. Others of the same class are spirals, quadratrices, cochloids and cissoids’. He adds the often-quoted reflection on the error committed by geometers when they solve a problem by means of an ‘inappropriate class’ (of curve or its equivalent), illustrating this by the use in Apollonius, Book V, of a rectangular hyperbola for finding the feet of normals to a parabola passing through one point (where a circle would serve the purpose), and by the assumption by Archimedes of a solid νєσις in his book On Spirals (see above, pp. 65–8).

Trisection (or division in any ratio) of any angle.

The method of trisecting any angle based on a certain νєσις is next described, with the solution of the νєσις itself by means of a hyperbola which has to be constructed from certain data, namely the asymptotes and a certain point through which the curve must pass (this easy construction is given in Prop. 33, chap. 41–2). Then the problem is directly solved (chaps. 43, 44) by means of a hyperbola in two ways practically equivalent, the hyperbola being determined in the one case by the ordinary Apollonian property, but in the other by means of the focus-directrix property. Solutions follow of the problem of dividing any angle in a given ratio by means (1) of the quadratrix, (2) of the spiral of Archimedes (chaps. 45, 46). All these solutions have been sufficiently described above (vol. i, pp. 235–7, 241–3, 225–7).

Some problems follow (chaps. 47–51) depending on these results, namely those of constructing an isosceles triangle in which either of the base angles has a given ratio to the vertical angle (Prop. 37), inscribing in a circle a regular polygon of any number of sides (Prop. 38), drawing a circle the circumference of which shall be equal to a given straight line (Prop. 39), constructing on a given straight line AB a segment of a circle such that the arc of the segment may have a given ratio to the base (Prop. 40), and constructing an angle incommensurable with a given angle (Prop. 41).

Section (5). Solution of the νєσις of Archimedes, ‘On Spirals’,
Prop. 8, by means of conics.

Book IV concludes with the solution of the νєσις which, according to Pappus, Archimedes unnecessarily assumed in On Spirals, Prop. 8. Archimedes’s assumption is this. Given a circle, a chord (BC) in it less than the diameter, and a point A on the circle the perpendicular from which to BC cuts BC in a point D such that BD > DC and meets the circle again in E, it is possible to draw through A a straight line ARP cutting BC in R and the circle in P in such a way that RP shall be equal to DE (or, in the phraseology of νєσєις, to place between the straight line BC and the circumference of the circle a straight line equal to DE and verging towards A).

Pappus makes the problem rather more general by not requiring PR to be equal to DE, but making it of any given length (consistent with a real solution). The problem is best exhibited by means of analytical geometry.

If BD = a, DC = b, AD = c (so that DE = ab/c), we have to find the point R on BC such that AR produced solves the problem by making PR equal to k, say.

Let DR = x. Then, since BR . RC = PR . RA, we have

An obvious expedient is to put y for √(c² + x²), when we have

and

These equations represent a parabola and a hyperbola respectively, and Pappus does in fact solve the problem by means of the intersection of a parabola and a hyperbola; one of his preliminary lemmas is, however, again a little more general. In the above figure y is represented by RQ.

The first lemma of Pappus (Prop. 42, p. 298) states that, if from a given point A any straight line be drawn meeting a straight line BC given in position in R, and if RQ be drawn at right angles to BC and of length bearing a given ratio to AR, the locus of Q is a hyperbola.

For draw AD perpendicular to BC and produce it to A′ so that

Measure DA″ along DA equal to DA′.

Then, if QN be perpendicular to AD,

that is,

and the locus of Q is a hyperbola.

The equation of the hyperbola is clearly

where µ is a constant. In the particular case taken by Archimedes QR = RA, or µ = 1, and the hyperbola becomes the rectangular hyperbola (2) above.

The second lemma (Prop. 43, p. 300) proves that, if BC is given in length, and Q is such a point that, when QR is drawn perpendicular to BC, BR . RC = k . QR, where k is a given length, the locus of Q is a parabola.

Let O be the middle point of BC, and let OK be drawn at right angles to BC and of length such that

Let QN′ be drawn perpendicular to OK.

Then

Therefore the locus of Q is a parabola.

The equation of the parabola referred to DB, DE as axes of x and y is obviously

which easily reduces to

In Archimedes’s particular case k = ab/c.

To solve the problem then we have only to draw the parabola and hyperbola in question, and their intersection then gives Q, whence R, and therefore ARP, is determined.

Book V. Preface on the Sagacity of Bees.

It is characteristic of the great Greek mathematicians that, whenever they were free from the restraint of the technical language of mathematics, as when for instance they had occasion to write a preface, they were able to write in language of the highest literary quality, comparable with that of the philosophers, historians, and poets. We have only to recall the introductions to Archimedes’s treatises and the prefaces to the different Books of Apollonius’s Conics. Heron, though severely practical, is no exception when he has any general explanation, historical or other, to give. We have now to note a like case in Pappus, namely the preface to Book V of the Collection. The editor, Hultsch, draws attention to the elegance and purity of the language and the careful writing; the latter is illustrated by the studied avoidance of hiatus.⁹ The subject is one which a writer of taste and imagination would naturally find attractive, namely the practical intelligence shown by bees in selecting the hexagonal form for the cells in the honeycomb. Pappus does not disappoint us; the passage is as attractive as the subject, and deserves to be reproduced.

‘It is of course to men that God has given the best and most perfect notion of wisdom in general and of mathematical science in particular, but a partial share in these things he allotted to some of the unreasoning animals as well. To men, as being endowed with reason, he vouchsafed that they should do everything in the light of reason and demonstration, but to the other animals, while denying them reason, he granted that each of them should, by virtue of a certain natural instinct, obtain just so much as is needful to support life. This instinct may be observed to exist in very many other species of living creatures, but most of all in bees. In the first place their orderliness and their submission to the queens who rule in their state are truly admirable, but much more admirable still is their emulation, the cleanliness they observe in the gathering of honey, and the forethought and housewifely care they devote to its custody. Presumably because they know themselves to be entrusted with the task of bringing from the gods to the accomplished portion of mankind a share of ambrosia in this form, they do not think it proper to pour it carelessly on ground or wood or any other ugly and irregular material; but, first collecting the sweets of the most beautiful flowers which grow on the earth, they make from them, for the reception of the honey, the vessels which we call honeycombs, (with cells) all equal, similar and contiguous to one another, and hexagonal in form. And that they have contrived this by virtue of a certain geometrical forethought we may infer in this way. They would necessarily think that the figures must be such as to be contiguous to one another, that is to say, to have their sides common, in order that no foreign matter could enter the interstices between them and so defile the purity of their produce. Now only three rectilineal figures would satisfy the condition, I mean regular figures which are equilateral and equiangular; for the bees would have none of the figures which are not uniform…. There being then three figures capable by themselves of exactly filling up the space about the same point, the bees by reason of their instinctive wisdom chose for the construction of the honeycomb the figure which has the most angles, because they conceived that it would contain more honey than either of the two others.

‘Bees, then, know just this fact which is of service to themselves, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material used in constructing the different figures. We, however, claiming as we do a greater share in wisdom than bees, will investigate a problem of still wider extent, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest plane figure of all those which have a perimeter equal to that of the polygons is the circle’

Book V then is devoted to what we may call isoperimetry, including in the term not only the comparison of the areas of different plane figures with the same perimeter, but that of the contents of different solid figures with equal surfaces.

Section (1). Isoperimetry after Zenodorus.

The first section of the Book relating to plane figures (chaps. 1–10, pp. 308–34) evidently followed very closely the exposition of Zenodorus πєρ σομτρων σχημάτων (see pp. 207–13, above); but before passing to solid figures Pappus inserts the proposition that of all circular segments having the same circumference the semicircle is the greatest, with some preliminary lemmas which deserve notice (chaps. 15, 16).

(1) ABC is a triangle right-angled at B. With C as centre and radius CA describe the arc AD cutting CB produced in D. To prove that (R denoting a right angle)

Draw AF at right angles to CA meeting CD produced in F, and draw BH perpendicular to AF. With A as centre and AB as radius describe the arc GBE.

Now

and, componendo,

But (by an easy lemma which has just preceded)

whence

and

Therefore

Therefore

whence, inversely,

and, componendo,

[If α be the circular measure of BCA, this gives (if AC = b)

or

that is,

(2) ABC is again a triangle right-angled at B. With C as centre and CA as radius draw a circle AD meeting BC produced in D. To prove that

Draw AE at right angles to AC. With A as centre and AC as radius describe the circle FCE meeting AB produced in F and AE in E.

Then,

Therefore

Inversely,

and, componendo,

Inversely,

We come now to the application of these lemmas to the proposition comparing the area of a semicircle with that of other segments of equal circumference (chaps. 17, 18).

A semicircle is the greatest of all segments of circles which
have the same circumference.

Let ABC be a semicircle with centre G, and DEF another segment of a circle such that the circumference DEF is equal to the circumference ABC. I say that the area of ABC is greater than the area of DEF.

Let H be the centre of the circle DEF. Draw EHK, BG at right angles to DF, AC respectively. Join DH, and draw LHM parallel to DF.

Then

Also

Therefore the sector LHE is to the sector AGB in the ratio duplicate of that which the sector LHE has to the sector DHE.

Therefore

Now (1) in the case of the segment less than a semicircle and (2) in the case of the segment greater than a semicircle

by the lemmas (1) and (2) respectively.

That is,

Therefore the half segment EDK is less than the half semicircle AGB, whence the semicircle ABC is greater than the segment DEF.

We have already described the content of Zenodorus’s treatise (pp. 207–13, above) to which, so far as plane figures are concerned, Pappus added nothing except the above proposition relating to segments of circles.

Section (2). Comparison of volumes of solids having their
surfaces equal. Case of the sphere.

The portion of Book V dealing with solid figures begins (p. 350. 20) with the statement that the philosophers who considered that the creator gave the universe the form of a sphere because that was the most beautiful of all shapes also asserted that the sphere is the greatest of all solid figures which have their surfaces equal; this, however, they had not proved, nor could it be proved without a long investigation. Pappus himself does not attempt to prove that the sphere is greater than all solids with the same surface, but only that the sphere is greater than any of the five regular solids having the same surface (chap. 19) and also greater than either a cone or a cylinder of equal surface (chap. 20).

Section (3). Digression on the semi-regular solids
of Archimedes.

He begins (chap. 19) with an account of the thirteen semi-regular solids discovered by Archimedes, which are contained by polygons all equilateral and all equiangular but not all similar (see pp. 98–101, above), and he shows how to determine the number of solid angles and the number of edges which they have respectively; he then gives them the go-by for his present purpose because they are not completely regular; still less does he compare the sphere with any irregular solid having an equal surface.

The sphere is greater than any of the regular solids which
has its surface equal to that of the sphere.

The proof that the sphere is greater than any of the regular solids with surface equal to that of the sphere is the same as that given by Zenodorus. Let P be any one of the regular solids, S the sphere with surface equal to that of P. To prove that S > P. Inscribe in the solid a sphere s and suppose that r is its radius. Then the surface of P is greater than the surface of s, and accordingly, if R is the radius of S, R > r. But the volume of S is equal to the cone with base equal to the surface of S, and therefore of P, and height equal to R; and the volume of P is equal to the cone with base equal to the surface of P and height equal to r. Therefore, since R > r, volume of S > volume of P.

Section (4). Propositions on the lines of Archimedes,
‘On the Sphere and Cylinder’

For the fact that the volume of a sphere is equal to the cone with base equal to the surface, and height equal to the radius, of the sphere, Pappus quotes Archimedes, On the Sphere and Cylinder, but thinks proper to add a series of propositions (chaps. 20–43, pp. 362–410) on much the same lines as those of Archimedes and leading to the same results as Archimedes obtains for the surface of a segment of a sphere and of the whole sphere (Prop. 28), and for the volume of a sphere (Prop. 35). Prop. 36 (chap. 42) shows how to divide a sphere into two segments such that their surfaces are in a given ratio and Prop. 37 (chap. 43) proves that the volume as well as the surface of the cylinder circumscribing a sphere is 1 times that of the sphere itself.

Among the lemmatic propositions in this section of the Book props. 21, 22 may be mentioned. Prop. 21 proves that, if C, E be two points on the tangent at H to a semicircle such that CH = HE, and if CD, EF be drawn perpendicular to the diameter AB, then (CD + EF) CE = AB . DF; Prop. 22 proves a like result where C, E are points on the semicircle, CD, EF are as before perpendicular to AB, and EH is the chord of the circle subtending the arc which with CE makes up a semicircle ; in this case (CD + EF) CE = EH . DF. Both results are easily seen to be the equivalent of the trigonometrical formula

or, if certain different angles be taken as x, y,

Section (5). Of regular solids with surfaces equal, that is greater which has more faces.

Returning to the main problem of the Book, Pappus shows that, of the five regular solid figures assumed to have their surfaces equal, that is greater which has the more faces, so that the pyramid, the cube, the octahedron, the dodecahedron and the icosahedron of equal surface are, as regards solid content, in ascending order of magnitude (props. 38–56). Pappus indicates (p. 410. 27) that ‘some of the ancients’ had worked out the proofs of these propositions by the analytical method; for himself, he will give a method of his own by synthetical deduction, for which he claims that it is clearer and shorter. We have first propositions (with auxiliary lemmas) about the perpendiculars from the centre of the circumscribing sphere to a face of (a) the octahedron, (b) the icosahedron (props. 39, 43), then the proposition that, if a dodecahedron and an icosahedron be inscribed in the same sphere, the same small circle in the sphere circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron (Prop. 48); this last is the proposition proved by Hypsicles in the so-called ‘Book XIV of Euclid’, Prop. 2, and Pappus gives two methods of proof, the second of which (chap. 56) corresponds to that of Hypsicles. Prop. 49 proves that twelve of the regular pentagons inscribed in a circle are together greater than twenty of the equilateral triangles inscribed in the same circle. The final propositions proving that the cube is greater than the pyramid with the same surface, the octahedron greater than the cube, and so on, are props. 52–6 (chaps. 60–4). Of Pappus’s auxiliary propositions, Prop. 41 is practically contained in Hypsicles’s Prop. 1, and Prop. 44 in Hypsicles’s last lemma; but otherwise the exposition is different.

Book VI.

On the contents of Book VI we can be brief. It is mainly astronomical, dealing with the treatises included in the so-called Little Astronomy, that is, the smaller astronomical treatises which were studied as an introduction to the great Syntaxis of Ptolemy. The preface says that many of those who taught the Treasury of Astronomy, through a careless understanding of the propositions, added some things as being necessary and omitted others as unnecessary. Pappus mentions at this point an incorrect addition to Theodosius, Sphaerica, III. 6, an omission from Euclid’s Phaenomena, Prop. 2, an inaccurate representation of Theodosius, On Days and Nights, Prop. 4, and the omission later of certain other things as being unnecessary. His object is to put these mistakes right. Allusions are also found in the Book to Menelaus’s Sphaerica, e.g. the statement (p. 476. 16) that Menelaus in his Sphaerica called a spherical triangle τρίπλєυρον, three-side.

The Sphaerica of Theodosius is dealt with at some length (chaps. 1–26, props. 1–27), and so are the theorems of Autolycus On the moving Sphere (chaps. 27–9), Theodosius On Days and Nights (chaps. 30–6, props. 29–38), Aristarchus On the sizes and distances of the Sun and Moon (chaps. 37–40, including a proposition, Prop. 39 with two lemmas, which is corrupt at the end and is not really proved), Euclid’s Optics (chaps. 41–52, props. 42–54), and Euclid’s Phaenomena (chaps. 53–60, props. 55–61).

Problem arising out of Euclid’s ‘Optics’

There is little in the Book of general mathematical interest except the following propositions which occur in the section on Euclid’s Optics.

Two propositions are fundamental in solid geometry, namely:

(a) If from a point A above a plane AB be drawn perpendicular to the plane, and if from B a straight line BD be drawn perpendicular to any straight line EF in the plane, then will AD also be perpendicular to EF (Prop. 43).

(b) If from a point A above a plane AB be drawn to the plane but not at right angles to it, and AM be drawn perpendicular to the plane (i.e. if BM be the orthogonal projection of BA on the plane), the angle ABM is the least of all the angles which AB makes with any straight lines through B, as BP, in the plane; the angle ABP increases as BP moves away from BM on either side; and, given any straight line BP making a certain angle with BA, only one other straight line in the plane will make the same angle with BA, namely a straight line BP′ on the other side of BM making the same angle with it that BP does (Prop. 44).

These are the first of a series of lemmas leading up to the main problem, the investigation of the apparent form of a circle as seen from a point outside its plane. In Prop. 50 (= Euclid, Optics, 34) Pappus proves the fact that all the diameters of the circle will appear equal if the straight line drawn from the point representing the eye to the centre of the circle is either (a) at right angles to the plane of the circle or (b), if not at right angles to the plane of the circle, is equal in length to the radius of the circle. In all other cases (Prop. 51 = Eucl. Optics, 35) the diameters will appear unequal. Pappus’s other propositions carry farther Euclid’s remark that the circle seen under these conditions will appear deformed or distorted (παρєσπασμνος), proving (Prop. 53, pp. 588–92) that the apparent form will be an ellipse with its centre not, ‘as some think’, at the centre of the circle but at another point in it, determined in this way. Given a circle ABDE with centre O, let the eye be at a point F above the plane of the circle such that FO is neither perpendicular to that plane nor equal to the radius of the circle. Draw FG perpendicular to the plane of the circle and let ADG be the diameter through G. Join AF, DF, and bisect the angle AFD by the straight line FC meeting AD in C. Through C draw BE perpendicular to AD, and let the tangents at B, E meet AG produced in K. Then Pappus proves that C (not O) is the centre of the apparent ellipse, that AD, BE are its major and minor axes respectively, that the ordinates to AD are parallel to BE both really and apparently, and that the ordinates to BE will pass through K but will appear to be parallel to AD. Thus in the figure, C being the centre of the apparent ellipse, it is proved that, if LCM is any straight line through C, LC is apparently equal to CM (it is practically assumed—a proposition proved later in Book VII, Prop. 156—that, if LK meet the circle again in P, and if PM be drawn perpendicular to AD to meet the circle again in M, LM passes through C).

The test of apparent equality is of course that the two straight lines should subtend equal angles at F.

The main points in the proof are these. The plane through CF, CK is perpendicular to the planes BFE, PFM and LFR ; hence CF is perpendicular to BE, QF to PM and HF to LR, whence BC and CE subtend equal angles at F : so do LH, HR, and PQ, QM.

Since FC bisects the angle AFD, and AC : CD = AK : KD (by the polar property), CFK is a right angle. And CF is the intersection of two planes at right angles, namely AFK and BFE, in the former of which FK lies; therefore KF is perpendicular to the plane BFE, and therefore to FN. Since therefore (by the polar property) LN : NP = LK : KP, it follows that the angle LFP is bisected by FN ; hence LN, NP are apparently equal.

Again

Therefore the angles LFC, CFM are equal, and LC, CM are apparently equal.

Lastly LR : PM = LK : KP = LN : NP = LF : FP; therefore the isosceles triangles FLR, FPM are equiangular; therefore the angles PFM, LFR, and consequently PFQ, LFH, are equal. Hence LP, RM will appear to be parallel to AD.

We have, based on this proposition, an easy method of solving Pappus’s final problem (Prop. 54). ‘Given a circle ABDE and any point within it, to find outside the plane of the circle a point from which the circle will have the appearance of an ellipse with centre C.’

We have only to produce the diameter AD through C to the pole K of the chord BE perpendicular to AD and then, in the plane through AK perpendicular to the plane of the circle, to describe a semicircle on CK as diameter. Any point F on this semicircle satisfies the condition.

Book VII. On the ‘Treasury of Analysis’.

Book VII is of much greater importance, since it gives an account of the books forming what was called the Treasury of Analysis (άναλυόμєνος τόπος) and, as regards those of the books which are now lost, Pappus’s account, with the hints derivable from the large collection of lemmas supplied by him to each book, practically constitutes our only source of information. The Book begins (p. 634) with a definition of analysis and synthesis which, as being the most elaborate Greek utterance on the subject, deserves to be quoted in full.

‘The so-called ’Aναλυόμєνος is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga and Aristaeus the elder, and proceeds by way of analysis and synthesis.’

Definition of Analysis 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 synthesis : for in analysis we assume that which is sought as if it were already done (γєγονός), and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (νάπαλιν λύσιν).

‘But 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 another, we arrive finally at the construction of what was sought; and this we call synthesis.

‘Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical.

(1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false.

(2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in the reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.’

This statement could hardly be improved upon except that it ought to be added that each step in the chain of inference in the analysis must be unconditionally convertible; that is, when in the analysis we say that, if A is true, B is true, we must be sure that each statement is a necessary consequence of the other, so that the truth of A equally follows from the truth of B. This, however, is almost implied by Pappus when he says that we inquire, not what it is (namely B) which follows from A, but what it is (B) from which A follows, and so on.

List of works in the ‘Treasury of Analysis’.

Pappus adds a list, in order, of the books forming the ’Aναλυόμєνος, namely:

‘Euclid’s Data, one Book, Apollonius’s Cutting-off of a ratio, two Books, Cutting-off of an area, two Books, Determinate Section, two Books, Contacts, two Books, Euclid’s Porisms, three Books, Apollonius’s Inclinations or Vergings (νєύσєις), two Books, the same author’s Plane Loci, two Books, and Conics, eight Books, Aristaeus’s Solid Loci, five Books, Euclid’s Surface-Loci, two Books, Eratosthenes’s On means, two Books. There are in all thirty-three Books, the contents of which up to the Conics of Apollonius I have set out for your consideration, including not only the number of the propositions, the diorismi and the cases dealt with in each Book, but also the lemmas which are required; indeed I have not, to the best of my belief, omitted any question arising in the study of the Books in question.’

Description of the treatises.

Then follows the short description of the contents of the various Books down to Apollonius’s Conics; no account is given of Aristaeus’s Solid Loci, Euclid’s Surface-Loci and Eratosthenes’s On means, nor are there any lemmas to these works except two on the surface-Loci at the end of the Book.

The contents of the various works, including those of the lost treatises so far as they can be gathered from Pappus, have been described in the chapters devoted to their authors, and need not be further referred to here, except for an addendum to the account of Apollonius’s Conics which is remarkable. Pappus has been speaking of the ‘locus with respect to three or four lines’ (which is a conic), and proceeds to say (p. 678. 26) that we may in like manner have loci with reference to five or six or even more lines; these had not up to his time become generally known, though the synthesis of one of them, not by any means the most obvious, had been worked out and its utility shown. Suppose that there are five or six lines, and that p₁, p₂, p₃, p₄, p₅ or p_l, p₂, p₃, p₄, p₅, p₆ are the lengths of straight lines drawn from a point to meet the five or six at given angles, then, if in the first case p₁ p₂ p₃ = λ p₄ p₅ a (where λ is a constant ratio and a a given length), and in the second case p₁ p₂ p₃ = λ p₄ p₅ p₆, the locus of the point is in each case a certain curve given in position. The relation could not be expressed in the same form if there were more lines than six, because there are only three dimensions in geometry, although certain recent writers had allowed themselves to speak of a rectangle multiplied by a square or a rectangle without giving any intelligible idea of what they meant by such a thing (is Pappus here alluding to Heron’s proof of the formula for the area of a triangle in terms of its sides given on pp. 322–3, above ?). But the system of compounded ratios enables it to be expressed for any number of lines thus,Pappus proceeds in language not very clear (p. 680. 30); but the gist seems to be that the investigation of these curves had not attracted men of light and leading, as, for instance, the old geometers and the best writers. Yet there were other important discoveries still remaining to be made. For himself, he noticed that every one in his day was occupied with the elements, the first principles and the natural origin of the subject-matter of investigation; ashamed to pursue such topics, he had himself proved propositions of much more importance and utility. In justification of this statement and ‘in order that he may not appear empty-handed when leaving the subject’, he will present his readers with the following.

(Anticipation of Guldin’s Theorem.)

The enunciations are not very clearly worded, but there is no doubt as to the sense.

‘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).’

Here, obviously, we have the essence of the celebrated theorem commonly attributed to P. Guldin (1577–1643), ‘quantitas rotunda in viam rotationis ducta producit Potestatem Rotundam uno grado altiorem Potestate sive Quantitate Rotata’.¹⁰

Pappus adds that

‘these propositions, which are practically one, include any number of theorems of all sorts about curves, surfaces, and solids, all of which are proved at once by one demonstration, and include propositions both old and new, and in particular those proved in the twelfth Book of these Elements.’

Hultsch attributes the whole passage (pp. 680. 30–682. 20) to an interpolator, I do not know for what reason; but it seems to me that the propositions are quite beyond what could be expected from an interpolator, indeed I know of no Greek mathematician from Pappus’s day onward except Pappus himself who was capable of discovering such a proposition.’

If the passage is genuine, it seems to indicate, what is not elsewhere confirmed, that the Collection originally contained, or was intended to contain, twelve Books.

Lemmas to the different treatises.

After the description of the treatises forming the Treamry of Analysis come the collections of lemmas given by Pappus to assist the student of each of the books (except Euclid’s Data) down to Apollonius’s Conics, with two isolated lemmas to the Surface-Loci of Euclid. It is difficult to give any summary or any general idea of these lemmas, because they are very numerous, extremely various, and often quite difficult, requiring first-rate ability and full command of all the resources of pure geometry. Their number is also greatly increased by the addition of alternative proofs, often requiring lemmas of their own, and by the separate formulation of particular cases where by the use of algebra and conventions with regard to sign we can make one proposition cover all the cases. The style is admirably terse, often so condensed as to make the argument difficult to follow without some little filling-out; the hand is that of a master throughout. The only misfortune is that, the books elucidated being lost (except the Conics and the Cutting-off of a ratio of Apollonius), it is difficult, often impossible, to see the connexion of the lemmas with one another and the problems of the book to which they relate. In the circumstances, all that I can hope to do is to indicate the types of propositions included in the lemmas and, by way of illustration, now and then to give a proof where it is sufficiently out of the common.

(α) Pappus begins with Lemmas to the Sectio rationis and Sectio spatii of Apollonius (props. 1–21, pp. 684–704). The first two show how to divide a straight line in a given ratio, and how, given the first, second and fourth terms of a proportion between straight lines, to find the third term. The next section (props. 3–12 and 16) shows how to manipulate relations between greater and less ratios by transforming them, e.g. componendo, convertendo, &c., in the same way as Euclid transforms equal ratios in Book V ; Prop. 16 proves that, according as a : b > or < c : d, ad > or < bc. props. 17–20 deal with three straight lines a, b, c in geometrical progression, showing how to mark on a straight line containing a, b, c as segments (including the whole among ‘segments’), lengths equal to a + c ± 2 √(ac); the lengths are of course equal to a + c ± 2b respectively. These lemmas are preliminary to the problem (Prop. 21), Given two straight lines AB, BC (C lying between A and B), to find a point D on BA produced such that . This is, of course, equivalent to the quadratic equation (a + x) : x = (a – c + x) : (a + c – 2), and, after marking off AE along AD equal to the fourth term of this proportion, Pappus solves the equation in the usual way by application of areas.

(β) Lemmas to the ‘Determinate Section’ of Apollonius.

The next set of Lemmas (props. 22–64, pp. 704–70) belongs to the Determinate Section of Apollonius. As we have seen (pp; 180–1, above), this work seems to have amounted to a Theory of Involution. Whether the application of certain of Pappus’s lemmas corresponded to the conjecture of Zeuthen or not, we have at all events in this set of lemmas some remarkable applications of ‘geometrical algebra’. They may be divided into groups as follows

I. props. 22, 25, 29.

If in the figure AD . DC = BD . DE, then

The proofs by proportions are not difficult. Prop. 29 is an alternative proof by means of Prop. 26 (see below). The algebraic equivalent may be expressed thus: if ax = by, then

II. props. 30, 32, 34.

If in the same figure AD . DE = BD . DC, then

props. 32, 34 are alternative proofs based on other lemmas (props. 31, 33 respectively). The algebraic equivalent may be stated thus: if ax = by, then

III. props. 35, 36.

If AB . BE = CB . BD>, then AB : BE = DA . AC : CE . ED, and CB : BD = AC . GE : AD . DE, results equivalent to the following: if ax = by, then

IV. props. 23, 24, 31, 57, 58.

If AB = CD, and E is any point in CD,

and similar formulae hold for other positions of E. If E is between B and C, AC . CD = AE . ED - BE . EC; and if E is on AD produced, BC . EC = AE . ED + BD . DC.

V. A small group of propositions relate to a triangle ABC with two straight lines AD, AE drawn from the vertex A to points on the base BC in accordance with one or other of the conditions (a) that the angles BAC, DAE are supplementary, (b) that the angles BAE, DAC are both right angles or, as we

may add from Book VI, Prop. 12, (c) that the angles BAD, EAC are equal. The theorems are:

Two proofs are given of the first theorem. We will give the first (Prop. 26) because it is a case of theoretical analysis followed by synthesis. Describe a circle about ABD: produce EA, CA to meet the circle again in F, G, and join BF, FG.

Substituting GC . CA for BC . CD and FE . EA for BE . ED, we have to inquire whether GC . CA : CA² = FE . EA : AE²,

i.e whether

i.e whether

i.e. whether the triangles GAF, CAE are similar or, in other words, whether GF is parallel to BC.

But GF is parallel to BC, because, the angles BAC, DAE being supplementary, DAE = GAB = GFB, while at the same time DAE = suppt. of FAD = FBD.

The synthesis is obvious.

An alternative proof (Prop. 27) dispenses with the circle, and only requires EKH to be drawn parallel to CA to meet AB, AD in H, K.

Similarly (Prop. 28) for case (b) it is only necessary to draw FG through D parallel to AC meeting BA in F and AE produced in G.

Then, FAG, ADF (= DAC) being both right angles, FD . DG = DA².

Therefore

In case (c) a circle is circumscribed to ADE cutting AB in F and AC in G. Then, since FAD = GAE, the arcs DF, EG are equal and therefore FG is parallel to DE. The proof is like that of case (a).

VI. props. 37, 38.

If AB : BC = AD² : DC, whether AB be greater or less than AD, then

[E in the figure is a point such that ED = CD.]

The algebraical equivalent is: If then ac = b ².

These lemmas are subsidiary to the next (props. 39, 40), being used in the first proofs of them.

props. 39, 40 prove the following:

If ACDEB be a straight line, and if

then

if, again,

then

If AB = a, BC = b, BD = c, BE = d, the algebraic equivalents are the following.

VII. props. 41, 42, 43.

If AD . DC = BD . DE, suppose that in Figures (1) and (2)

k = AE + CB, and in Figure (3) k = AE — BC, then

The algebraical equivalents for Figures (1) and (2) respectively may be written (if a = AD, b = DC, c = BD, d = DE):

If ab = cd, then

Figure (3) gives other varieties of sign. Troubles about sign can be avoided by measuring all lengths in one direction from an origin O outside the line. Thus, if OA = a, OB = b, &c., the proposition may be as follows:

If

then

VIII. props. 45–56.

More generally, if AD . DC = BD . DE and k = AE ± BC, then, if F be any point on the line, we have, according to the position of F in relation to A, B, C, D, E,

Algebraically, if 0A = a, OB = b … OF = x, the equivalent is: If (d — a) (d — c) = (b — d) (e — d), and k = (e — a) + (b — c),

then

By making x = a, b, c, e successively in this equation, we obtain the results of props. 41–3 above.

IX. props. 59–64.

In this group props. 59, 60, 63 are lemmas required for the remarkable propositions (61, 62, 64) in which Pappus investigates ‘singular and minimum’ values of the ratio

where (A, D), (B, C) are point-pairs on a straight line and P is another point on the straight line. He finds, not only when the ratio has the ‘singular and minimum (or maximum)’ value, but also what the value is, for three different positions of P in relation to the four given points.

I will give, as an illustration, the first case, on account of its elegance. It depends on the following Lemma . AEB being a semicircle on AB as diameter, C, D any two points on AB, and CE, DF being perpendicular to AB, let EF be joined and

produced, and let BG be drawn perpendicular to EG. To

prove that

Join GC, GD, FB, EB, AF.

(1) Since the angles at G, D are right, F, G, B, D are concyclic.

Similarly E, G, B, C are concyclic.

Therefore

And the triangles GCB, DGB also have the angle CBG common; therefore they are similar, and CB : BG = BG : BD,

or

(2) We have AB . BD = BF ²;

therefore, by subtraction,

(3) Similarly AB . BC = BE ²;

therefore, by subtraction, from the same result (1),

Thus the lemma gives an extremely elegant construction for squares equal to each of the three rectangles.

Now suppose (A, D),(B, C) to be two point-pairs on a straight line, and let P, another point on it, be determined by the relation

then, says Pappus, the ratio AP . PD : BP . PC is singular and a minimum, and is equal to

On AD as diameter draw a circle, and draw BF, CG perpendicular to AD on opposite sides.

Then, by hypothesis, AB . BD : AC . CD = BP ² : CP ²;

therefore

or

whence the triangles FBP, GCP are similar and therefore equiangular, so that FPG is a straight line.

Produce GC to meet the circle in H, join FH, and draw DK perpendicular to FH produced. Draw the diameter FL and join LH.

Now, by the lemma, FK ² = AC . BD, and HK ² = AB . CD;

therefore

Since, in the triangles FHL, PCG, the angles at H, C are right and FLH = PGC, the triangles are similar, and

But

therefore

Therefore

The proofs of props. 62 and 64 are different, the former being long and involved. The results are:

Prop. 62. If P is between C and D, and

then the ratio AP . PB : CP . PD is singular and a minimum and is equal to { √(AC . BD) + √(AD . BC) } ² : DC ².

Prop. 64. If P is on AD produced, and

then the ratio AP . PD : BP . PC is singular and a maximum, and is equal to AD ² : { √(AC . BD) + √(AB. CD) } ².

(γ) Lemmas on the Nєύσєις of Apollonius.

After a few easy propositions (e.g. the equivalent of the proposition that, if ax + x ² = by + y ², then, according as a > or < b, a + x > or < b + y), Pappus gives (Prop. 70) the lemma leading to the solution of the νєσις with regard to the rhombus (see pp. 190–2, above), and after that the solution by one Heraclitus of the same problem with respect to a square (props. 71, 72, pp. 780–4). The problem is, Given a square ABCD, to draw through B a straight line, meeting CD in H and AD produced in E, such that HE is equal to a given length.

The solution depends on a lemma to the effect that, if any straight line BHE through B meets CD in H and AD produced in E, and if EF be drawn perpendicular to BE meeting BC produced in F, then

Draw EG perpendicular to BF.

Then the triangles BCH, EGF are similar and since BC = EG) equal in all respects: therefore EF = BH.

Now

or

But, the angles HCF, HEF being right, H, C, F, E are concyclic, and

Therefore, by subtraction,

Taking away the common part, BC. CF, we have

Now suppose that we have to draw BHE through B in such a way that HE = k. Since BC, EH are both given, we have only to determine a length x such that x ² = BC ² + k ², produce BC to F so that CF = x, draw a semicircle on BF as diameter, produce AD to meet the semicircle in E, and join BE . BE is thus the straight line required.

Prop. 73 (pp. 784–6) proves that, if D be the middle point of BC, the base of an isosceles triangle ABC, then BC is the shortest of all the straight lines through D terminated by the straight lines AB, AC, and the nearer to BC is shorter than the more remote.

There follows a considerable collection of lemmas mostly showing the equality of certain intercepts made on straight lines through one extremity of the diameter of one of two semicircles having their diameters in a straight line, either one including or partly including the other, or wholly external to one another, on the same or opposite sides of the diameter

I need only draw two figures by way of illustration.

In the first figure (Prop. 83), ABC, DEF being the semicircles, BEKC is any straight line through C cutting both; FG is made equal to AD; AB is joined; GH is drawn perpendicular to BK produced. It is required to prove that

FIG. 1.

BE = KH. (This is obvious when from L, the centre of the semicircle DEF, LM is drawn perpendicular to BK.) If E, K coincide in the point M′ of the semicircle so that B′CH′ is a tangent, then B′ M′ = M′ H′ (props. 83, 84).

In the second figure (Prop. 91) D is the centre of the semicircle ABC and is also the extremity of the diameter of the semicircle DEF. If BEGF be any straight line through

FIG. 2.

F cutting both semicircles, BE = EG. This is clear, since DE is perpendicular to BG.

The only problem of any difficulty in this section is Prop. 85 (p. 796). Given a semicircle ABC on the diameter AC and a point D on the diameter, to draw a semicircle passing through D and having its diameter along DC such that, if CEB be drawn touching it at E and meeting the semicircle ABC in B, BE shall be equal to AD.

The problem is reduced to a problem contained in Apollonius’s Determinate Section thus.

Suppose the problem solved by the semicircle DEF, BE being equal to AD. Join E to the centre G of the semicircle

DEF. Produce DA to H, making HA equal to AD. Let K be the middle point of DC.

Since the triangles ABC, GEC are similar,

Therefore

Take a straight line L such that AD ² = L . 2 DC;

therefore

or

Therefore, given the two straight lines HD, DK (or the three points H, D, K on a straight line), we have to find a point G between D and K such that

which is the second epitagma of the third Problem in the Determinate Section of Apollonius, and therefore may be taken as solved. (The problem is the equivalent of the solution of a certain quadratic equation.) Pappus observes that the problem is always possible (requires no διορισμός), and proves that it has only one solution.

(δ)Lemmas on the treatise ‘On contacts’ by Apollonius.

These lemmas are all pretty obvious except two, which are important, one belonging to Book I of the treatise, and the other to Book II. The two lemmas in question have already been set out a propos of the treatise of Apollonius (see pp. 182–5, above). As, however, there are several cases of the first (props. 105, 107, 108, 109), one case (Prop. 108, pp. 836–8), different from that before given, may be put down here: Given a circle and two points D, E within it, to draw straight lines through D, E to a point A on the circumference in such a way that, if they meet the circle again in B, C, BC shall be parallel to DE.

We proceed by analysis. Suppose the problem solved and DA, EA drawn (‘inflected’) to A in such a way that, if AD,

AE meet the circle again in B, C, BC is parallel to DE.

Draw the tangent at B meeting ED produced in F.

Then

therefore A, E, B, F are concyclic, and consequently

But the rectangle AD. DB is given, since it depends only on the position of D in relation to the circle, and the circle is given.

Therefore the rectangle FD. DE is given.

And DE is given; therefore FD is given, and therefore F.

If follows that the tangent FB is given in position, and therefore B is given. Therefore BDA is given and consequently AE also.

To solve the problem, therefore, we merely take F on ED produced such that FD . DE = the given rectangle made by the segments of any chord through D, draw the tangent FB, join BD and produce it to A, and lastly draw AE through to C; BC is then parallel to DE.

The other problem (Prop. 117, pp. 848–50) is, as we have seen, equivalent to the following: Given a circle and three points D, E, F in a straight line external to it, to inscribe in the circle a triangle ABC such that its sides pass severally through the three points D, E, F. For the solution, see pp. 182–4, above.

(ε) The Lemmas to the Plane Loci of Apollonius (props. 119–26, pp. 852–64) are mostly propositions in geometrical algebra worked out by the methods of Eucl., Books II and VI. We may mention the following:

Prop. 122 is the well-known proposition that, if D be the middle point of the side BC in a triangle ABC,

props. 123 and 124 are two cases of the same proposition, the enunciation being marked by an expression which is also found in Euclid’s Data. Let AB : BC be a given ratio, and

let the rectangle CA . AD be given; then, if BE is a mean proportional between DB, BC, ‘the square on AE is greater by the rectangle CA . AD than in the ratio of AB to BC to the square on EC’, by which is meant that

or

The algebraical equivalent may be expressed thus (if AB = a, BC = b, AD = c, BE = x):

If

Prop. 125 is remarkable: If C, D be two points on a straight line AB,

This is equivalent to the general relation between four points on a straight line discovered by Simson and therefore wrongly known as Stewart’s theorem:

(Simson discovered this theorem for the more general case where D is a point outside the line ABC.)

An algebraical equivalent is the identity

Pappus’s proof of the last-mentioned lemma is perhaps worth giving.

C, D being two points on the straight line AB, take the point F on it such that

Then

and

or

and therefore

From (1) we derive

and from (2)

We have now to prove that

or

i.e. (if DA . AC be subtracted from each side)

that

i.e. (if AF . CD be subtracted from each side)

that

or

which is true, since, by (1) above, FD : DB = AC : CB.

(ζ) Lemmas to the ‘Porisms’ of Euclid.

The 38 Lemmas to the Porisms of Euclid form an important collection which, of course, has been included in one form or other in the ‘restorations’ of the original treatise. Chasles¹¹ in particular gives a classification of them, and we cannot do better than use it in this place: ‘23 of the Lemmas relate to rectilineal figures, 7 refer to the harmonic ratio of four points, and 8 have reference to the circle.

‘Of the 23 relating to rectilineal figures, 6 deal with the quadrilateral cut by a transversal; 6 with the equality of the anharmonic ratios of two systems of four points arising from the intersections of four straight lines issuing from one point with two other straight lines; 4 may be regarded as expressing a property of the hexagon inscribed in two straight lines; 2 give the relation between the areas of two triangles which have two angles equal or supplementary; 4 others refer to certain systems of straight lines; and the last is a case of the problem of the Cutting-off of an area.’

The lemmas relating to the quadrilateral and the transversal are 1, 2, 4, 5, 6 and 7 (props. 127, 128, 130, 131, 132, 133). Prop. 130 is a general proposition about any transversal

whatever, and is equivalent to one of the equations by which we express the involution of six points. If A, A′; B, B′; C, C′ be the points in which the transversal meets the pairs of opposite sides and the two diagonals respectively, Pappus’s result is equivalent to

props. 127, 128 are particular cases in which the transversal is parallel to a side; in Prop. 131 the transversal passes through the points of concourse of opposite sides, and the result is equivalent to the fact that the two diagonals divide into proportional parts the straight line joining the points of concourse of opposite sides; Prop. 132 is the particular case of Prop. 131 in which the line joining the points of concourse of opposite sides is parallel to a diagonal; in Prop. 133 the transversal passes through one only of the points of concourse of opposite sides and is parallel to a diagonal, the result being CA² = CB . CB′.

props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16, 19) establish the equality of the anharmonic ratios which four straight lines issuing from a point determine on two transversals; but both transversals are supposed to be drawn from the same point on one of the four straight lines. Let

AB, AC, AD be cut by transversals HBCD, HEFG. It is required to prove that

Pappus gives (Prop. 129) two methods of proof which are practically equivalent. The following is the proof ‘by compound ratios’.

Draw HK parallel to AF meeting DA and AE produced in K, L; and draw LM parallel to AD meeting GH produced in M.

Then

In exactly the same way, if DH produced meets LM in M′

we prove that

Therefore

(The proposition is proved for HBCD and any other transversal not passing through H by applying our proposition twice, as usual.)

props. 136, 142 are the reciprocal; Prop. 137 is a particular case in which one of the transversals is parallel to one of the straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145 another case of Prop. 129.

The Lemmas 12, 13, 15, 17 (props. 138, 139, 141, 143) are equivalent to the property of the hexagon inscribed in two straight lines, viz. that, if the yertices of a hexagon are situate, three and three, on two straight lines, the points of concourse of opposite sides are in a straight line; in props. 138, 141 the straight lines are parallel, in props. 139, 143 not parallel.

Lemmas 20, 21 (props. 146, 147) prove that, when one angle of one triangle is equal or supplementary to one angle of another triangle, the areas of the triangles are in the ratios of the rectangles contained by the sides containing the equal or supplementary angles.

The seven Lemmas 22, 23, 24, 26, 26, 27, 34 (props. 148–53 and 160) are propositions relating to the segments of a straight line on which two intermediate points are marked. Thus;

props. 148, 150.

If C, D be two points on AB, then

(a) if

(b) if

props. 149, 151.

If

then

and

props. 152, 153.

If

Prop. 160.

If AB : BC = AD : DC, then, if E be the middle point of AC,

The Lemmas about the circle include the harmonic properties of the pole and polar, whether the pole is external to the circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a problem, Given a segment of a circle on AB as base, to inflect straight lines AC, BC to the segment in a given ratio to one another.

Prop. 156 is one which Pappus has already used earlier in the Collection. It proves that the straight lines drawn from the extremities of a chord (DE) to any point (F) of the circumference divide harmonically the diameter (AB) perpendicular to the chord. Or, if ED, FK be parallel chords, and EF, DK meet in G, and EK, DF in H, then

Since AB bisects DE perpendicularly, (arc AE) = (arc AD) and EFA = AFD, or AF bisects the angle EFD.

Since the angle AFB is right, FB bisects HFG, the supplement of EFD.

Therefore (Eucl. VI. 3) GB: BH = GF: FH = GA : AH, and, alternately and inversely, AH : HB = AG : GB.

Prop. 157 is remarkable in that (without any mention of a conic) it is practically identical with Apollonius’s Conics III. 45 about the foci of a central conic. Pappus’s theorem is as follows. Let AB be the diameter of a semicircle, and

from A, B let two straight lines AE, BD be drawn at right angles to AB. Let any straight line DE meet the two perpendiculars in D, E and the semicircle in F. Further, let FG be drawn at right angles to DE, meeting AB produced in G.

It is to be proved that

Since F, D, G, B are concyclic, BDG = BFG.

And, since AFB, EFG are both right angles, BFG = LAFE.

But, since A, E, G, F are concyclic, AFE = AGE.

Therefore

and the right-angled triangles DBG, are similar.

Therefore

or

In Apollonius G and the corresponding point G′ on BA produced which is obtained by drawing F′G′ perpendicular to ED (where DE meets the circle again in F′) are the foci of a central conic (in this case a hyperbola), and DE is any tangent to the conic; the rectangle AE . BD is of course equal to the square on half the conjugate axis.

(η) The Lemmas to the Conics of Apollonius (pp. 918–1004) do not call for any extended notice. There are a large number of propositions in geometrical algebra of the usual kind, relating to the segments of a straight line marked by a number of points on it; propositions about lines divided into proportional segments and about similar figures; two propositions relating to the construction of a hyperbola (props. 204, 206) and a proposition (208) proving that two hyperbolas with the same asymptotes do not meet one another. There are also two propositions (221, 222) equivalent to an obvious trigonometrical formula. Let ABCD be a rectangle, and let any straight line through A meet DC produced in E and BC (produced if necessary) in F.

Then

For

Also

Therefore

i.e

This is equivalent to

The algebraical equivalents of some of the results obtained by the usual geometrical algebra may be added.

props. 178, 179, 192–4.

Prop. 195.

Prop. 196.

props. 197, 199, 198.

props. 200, 201. If (a + b)x = b², then and (2b + a)a = (a + b) (a + b - x).

Prop. 207. If (a + b)b = 2a², then a = b.

(θ) The two Lemmas to the Surface-Loci of Euclid have already been mentioned as significant. The first has the appearance of being a general enunciation, such as Pappus is fond of giving, to cover a class of propositions. The enunciation may be translated as follows: ‘If AB be a straight line, and CD a straight line parallel to a straight line given in position, and if the ratio AD . DB : DC² be given, the point C lies on a conic section. If now AB be no longer given in position, and the points A, B are no longer given but lie (respectively) on straight lines AE, EB given in position, the point C raised above (the plane containing AE, EB) lies on a surface given in position. And this was proved.’ Tannery was the first to explain this intelligibly; and his interpretation only requires the very slight change in the text of substituting єθєίαις for єθєα in the phrase γνηται δπρòς θσєι єθєα τας AE, EB. It is not clear whether, when AB ceases to be given in position, it is still given in length. If it is given in length and A, B move on the lines AE, EB respectively, the surface which is the locus of C is a complicated one such as Euclid would hardly have been in a position to investigate. But two possible cases are indicated which he may have discussed, (1) that in which AB moves always parallel to itself and varies in length accordingly, (2) that in which the two lines on which A, B move are parallel instead of meeting at a point. The loci in these two cases would of course be a cone and a cylinder respectively.

The second Lemma is still more important, since it is the first statement on record of the focus-directrix property of the three conic sections. The proof, after Pappus, has been set out above (pp. 119–21).

(ι) An unallocated Lemma.

Book VII ends (pp. 1016–18) with a lemma which is not given under any particular treatise belonging to the Treasury of Analysis, but is simply called ‘Lemma to the’Avαλυόμєνος’. If ABC be a triangle right-angled at B, and AB, BC be divided at F, G so that AF : FB = BG : GC = AB : BC, and if AEG, CEF be joined and BE joined and produced to D, then shall BD be perpendicular to AC.

The text is unsatisfactory, for there is a long interpolation containing an attempt at a proof by reductio ad abdurdum; but the genuine proof is indicated, although it breaks off before it is quite complete.

Since

or

But, by hypothesis,

therefore

From this point the proof apparently proceeded by analysis. ‘Suppose it done’ (γєγοvτω), i.e. suppose the proposition true, and BED perpendicular to AC.

Then, by similarity of triangles, AD : DB = AB : BC; therefore AF : FB = AD : DB, and consequently the angle ADB is bisected by DF.

Similarly the angle BDC is bisected by DG.

Therefore each of the angles BDF, BDG is half a right angle, and consequently the angle FDG is a right angle.

Therefore B, G, D, F are concyclic; and, since the angles FDB, BDG are equal,

This is of course the result above proved.

Evidently the interpolator tried to clinch the argument by proving that the angle BDA could not be anything but a right angle.

Book VIII.

Book VIII of the Collection is mainly on mechanics, although it contains, in addition, some propositions of purely geometrical interest.

Historical preface.

It begins with an interesting preface on the claim of theoretical mechanics, as distinct from the merely practical or industrial, to be regarded as a mathematical subject. Archimedes, Philon, Heron of Alexandria are referred to as the principal exponents of the science, while Carpus of Antioch is also mentioned as having applied geometry to ‘certain (practical) arts’.

The date of Carpus is uncertain, though it is probable that he came after Geminus; the most likely date seems to be the first or second century A.D. Simplicius gives the authority of Iamblichus for the statement that Carpus squared the circle by means of a certain curve, which he simply called a curve generated by a double motion.¹² Proclus calls him ‘Carpus the writer on mechanics (ό μηχανικός)’, and quotes from a work of his on Astronomy some remarks about the relation between problems and theorems and the ‘priority in order’ of the former.¹³ Proclus also mentions him as having held that an angle belongs to the category of quantity (ποσόν), since it represents a sort of ‘distance’ between the two lines forming it, this distance being ‘extended one way’ (ϕ’ ν διєστώς) though in a different sense from that in which a line represents extension one way, so that Carpus′s view appeared to be ‘the greatest possible paradox’¹⁴; Carpus seems in reality to have been anticipating the modern view of an angle as representing divergence rather than distance, and to have meant by ϕ’ ν in one sense (rotationally), as distinct from one way or in one dimension (linearly).

Pappus tells us that Heron distinguished the logical, i.e. theoretical, part of mechanics from the practical or manual (χєιρονργικόν), the former being made up of geometry, arithmetic, astronomy and physics, the latter of work in metal, architecture, carpentering and painting; the man who had been trained from his youth up in the sciences aforesaid as well as practised in the said arts would naturally prove the best architect and inventor of mechanical devices, but, as it is difficult or impossible for the same person to do both the necessary mathematics and the practical work, he who has not the former must perforce use the resources which practical experience in his particular art or craft gives him. Other varieties of mechanical work included by the ancients under the general term mechanics were (1) the use of the mechanical powers, or devices for moving or lifting great weights by means of a small force, (2) the construction of engines of war for throwing projectiles a long distance, (3) the pumping of water from great depths, (4) the devices of ‘wonder-workers’ (θανμασιονργοί), some depending on pneumatics (like Heron in the Pneumatica), some using strings, &c., to produce movements like those of living things (like Heron in ‘Automata and Balancings’), some employing floating bodies (like Archimedes in ‘Floating Bodies’), others using water to measure time (like Heron in his ‘Water-clocks’), and lastly ‘sphere-making’, or the construction of mechanical imitations of the movements of the heavenly bodies with the uniform circular motion of water as the motive power. Archimedes, says Pappus, was held to be the one person who had understood the cause and the reason of all these various devices, and had applied his extraordinarily versatile genius and inventiveness to all the purposes of daily life, and yet, although this brought him unexampled fame the world over, so that his name was on every one′s lips, he disdained (according to Carpus) to write any mechanical work save a tract on sphere-making, but diligently wrote all that he could in a small compass of the most advanced parts of geometry and of subjects connected with arithmetic. Carpus himself, says Pappus, as well as others applied geometry to practical arts, and with reason: ‘for geometry is in no wise injured, nay it is by nature capable of giving substance to many arts by being associated with them, and, so far from being injured, it may be said, while itself advancing those arts, to be honoured and adorned by them in return.’

The object of the Book.

Pappus then describes the object of the Book, namely to set out the propositions which the ancients established by geometrical methods, besides certain useful theorems discovered by himself, but in a shorter and clearer form and in better logical sequence than his predecessors had attained. The sort of questions to be dealt with are (1) a comparison between the force required to move a given weight along a horizontal plane and that required to move the same weight upwards on an inclined plane, (2) the finding of two mean proportionals between two unequal straight lines, (3) given a toothed wheel with a certain number of teeth, to find the diameter of, and to construct, another wheel with a given number of teeth to work on the former. Each of these things, he says, will be clearly understood in its proper place if the principles on which the ‘centrobaric doctrine’ is built up are first set out. It is not necessary, he adds, to define what is meant by ‘heavy’ and ‘light’ or upward and downward motion, since these matters are discussed by Ptolemy in his Mathematica; but the notion of the centre of gravity is so fundamental in the whole theory of mechanics that it is essential in the first place to explain what is meant by the ‘centre of gravity’ of any body.

On the centre of gravity.

Pappus then defines the centre of gravity as ‘the point within a body which is such that, if the weight be conceived to be suspended from the point, it will remain at rest in any position in which it is put’.¹⁵ The method of determining the point by means of the intersection, first of planes, and then of straight lines, is next explained (chaps. 1, 2), and Pappus then proves (Prop. 2) a proposition of some difficulty, namely that, if D, E, F be points on the sides BC, CA, AB of a triangle ABC

such that

then the centre of gravity of the triangle ABC is also the centre of gravity of the triangle DEF.

Let H, K be the middle points of BC, CA respectively; join AH, BK. Join HK meeting DE in L.

Then AH, BK meet in G, the centre of gravity of the triangle ABC, and AG = 2 GH, BG = 2 GK, so that

Now, by hypothesis,

whence

and, if we halve the antecedents,

therefore

whence, componendo,

But

Now, ELD being a transversal cutting the sides of the triangle KHC, we have

[This is ‘Menelaus’s theorem’; Pappus does not, however, quote it, but proves the relation ad hoc in an added lemma by drawing CM parallel to DE to meet HK produced in M. The proof is easy, for

It follows from (2) and (3) that

and, since AB is parallel to HK, and AH, BK are straight lines meeting in G, FGL is a straight line.

[This is proved in another easy lemma by reductio ad absurdum.]

We have next to prove that EL = LD.

Now [again by ‘Menelaus’s theorem’, proved ad hoc by drawing CN parallel to HK to meet ED produced in N]

But, by (1) above, CE : EK = BD : DH;

therefore

so that (EK : KC) . (CH : HD) = 1, and therefore, from (4),

It remains to prove that FG = 2 GL, which is obvious by parallels, since

Two more propositions follow with reference to the centre of gravity. The first is, Given a rectangle with AB, BC as adjacent sides, to draw from C a straight line meeting the side opposite BC in a point D such that, if the trapezium ADCB is hung from the point D, it will rest with AD, BC horizontal.

In other words, the centre of gravity must be in DL drawn perpendicular to BG. Pappus proves by analysis that CL² = 3BL², so that the problem is reduced to that of dividing BC into parts BL, LC such that this relation holds. The latter problem is solved (Prop. 6) by taking a point, say X, in CB such that CX = 3XB, describing a semicircle on BC as diameter and drawing XY at right angles to BC to meet the semicircle in Y, so that XY² = BC², and then dividing CB at L so that

The second proposition is this (Prop. 7). Given two straight lines AB, AC, and B a fixed point on AB, if CD be drawn with its extremities on AC, AB and so that AC : BD is a given ratio, then the centre of gravity of the triangle ADC will lie on a straight line.

Take E, the middle point of AC, and F a point on DE such that DF = 2 FE. Also let H be a point on BA such that BH = 2HA. Draw FG parallel to AC. Then AG = AD, and AH = AB; A therefore HG = BD.

Also FG = AE = AC. Therefore, since the ratio AC . BD is given, the ratio GH : GF is given.

And the angle FGH (= A) is given; therefore the triangle FGH is given in species, and consequently the angle GHF is given. And if is a given point, Therefore HF is a given straight line, and it contains the centre of gravity of the triangle ADC.

The inclined plane.

Prop. 8 is on the construction of a plane at a given inclination to another plane parallel to the horizon, and with this Pappus leaves theory and proceeds to the practical part. Prop. 9 (p. 1054. 4 sq.) investigates the problem ‘Given a weight which can be drawn along a plane parallel to the horizon by a given force, and a plane inclined to the horizon at a given angle, to find the force required to draw the weight upwards on the inclined plane’. This seems to be the first or only attempt in ancient times to investigate motion on an inclined plane, and as such it is curious, though of no value.

Let A be the weight which can be moved by a force C along a horizontal plane. Conceive a sphere with weight equal to A placed in contact at L with the given inclined plane; the circle OGL represents a section of the sphere by a vertical plane passing through E its centre and LK the line of greatest slope drawn through the point L. Draw EGH horizontal and therefore parallel to MN in the plane of section, and draw LF perpendicular to EH. Pappus seems to regard the plane as rough, since he proceeds to make a system in equilibrium about FL as if L were the fulcrum of a lever. Now the weight A acts vertically downwards along a straight line through E. To balance it, Pappus supposes a weight B attached with its centre of gravity at G.

Then

and, since KMN is given, the ratio EF : EL, and therefore the ratio (EL — EF) : EF, is given; thus B is found.

Now, says Pappus, if D is the force which will move B along a horizontal plane, as C is the force which will move A along a horizontal plane, the sum of C and D will be the force required to move the sphere upwards on the inclined plane. He takes the particular case where θ = 60^°. Then sin θ is approximately (he evidently uses for ),

and

Suppose, for example, that A = 200 talents; then B is 1300 talents. Suppose further that C is 40 man-power; then, since D : G = B : A, D = 260 man-power; and it will take D + C, or 300 man-power, to move the weight up the plane!

Prop. 10 gives, from Heron’s Barulcus, the machine consisting of a pulley, interacting toothed wheels, and a spiral screw working on the last wheel and turned by a handle; Pappus merely alters the proportions of the weight to the force, and of the diameter of the wheels. At the end of the chapter (pp. 1070–2) he repeats his construction for the finding of two mean proportionals.

Construction of a conic through five points.

Chaps. 13–17 are more interesting, for they contain the solution of the problem of constructing a conic through five given points. The problem arises in this way. Suppose we are given a broken piece of the surface of a cylindrical column such that no portion of the circumference of either of its base is left intact, and let it be required to find the diameter of a circular section of the cylinder. We take any two points A, B on the surface of the fragment and by means of these we find five points on the surface all lying in one plane section, in general oblique. This is done by taking five different radii and drawing pairs of circles with A, B as centres and with each of the five radii successively. These pairs of circles with equal radii, intersecting at points on the surface, determine five points on the plane bisecting AB at right angles. The five points are then represented on any plane by triangulation.

Suppose the points are A, B, C, D, E and are such that no two of the lines connecting the different pairs are parallel.

This case can be reduced to the construction of a conic through the five points A, B, D, E, F where EF is parallel to AB. This is shown in a subsequent lemma (chap. 16).

For, if EF be drawn through E parallel to AB, and if CD meet AB in O and EF in O′, we have, by the well-known proposition about intersecting chords,

whence O′F is known, and F is determined.

We have then (Prop. 13) to construct a conic through A, B, D, E, F, where EF is parallel to AB.

Bisect AB, EF at V, W; then VW produced both ways is a diameter. Draw DR, the chord through D parallel to this diameter. Then R is determined by means of the relation

in this way.

Join DB, RA, meeting EF in K, L respectively.

Then, by similar triangles,

Therefore, by (1),

whence HL is determined, and therefore L. The intersection of AL, DH determines R.

Next, in order to find the extremities P, P′ of the diameter through V, W, we draw ED, RF meeting PP′ in M, N respectively.

Then, as before,

Therefore

and similarly we can find the value of P′V. VP.

Now, says Pappus, since P′W . WP and P′V . VP are given areas and the points V, W are given, P, P′ are given. His determination of P, P′ amounts (Prop. 14 following) to an elimination of one of the points and the finding of the other by means of an equation of the second degree.

Take two points Q, Q′ on the diameter such that

Q, Q′ are thus known, while P, P′ remain to be found.

By (α)

whence

Therefore, by means of (β),

so that

or

Thus P can be found, and similarly P′.

The conjugate diameter is found by virtue of the relation

where p is the latus rectum to PP′ determined by the property of the curve

Problem, Given two conjugate diameters of an ellipse, to find the axes.

Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are given two conjugate diameters, we can find the axes. The construction is as follows. Let AB, CD be conjugate diameters (CD being the greater), E the centre.

Produce EA to H so that

Through A draw FG parallel to CD. Bisect EH in K, and draw KL at right angles to EH meeting FG in L.

With L as centre, and LE as radius, describe a circle cutting GF in G, F.

Join EF, EG, and from A draw AM, AN parallel to EF, EG respectively.

Take points P, R on EG, EF such that

Then EP is half the major axis, and ER half the minor axis. Pappus omits the proof.

Problem of seven hexagons in a circle.

Prop. 19 (chap. 23) is a curious problem. To inscribe seven equal regular hexagons in a circle in such a way that one is about the centre of the circle, while six others stand on its sides and have the opposite sides in each case placed as chords in the circle.

Suppose GHKLNM to be the hexagon so described on HK, a side of the inner hexagon; OKL will then be a straight line. Produce OL to meet the circle in P.

Then OK = KL = LN. Therefore, in the triangle OLN, OL = 2LN, while the included angle OLN (= 120°) is also given. Therefore the triangle is given in species; therefore the ratio ON : NL is given, and, since ON is given, the side NL of each of the hexagons is given.

Pappus gives the auxiliary construction thus. Let AF be taken equal to the radius OP. Let AC = AF, and on AC as base describe a segment of a circle containing an angle of 60°. Take CE equal to AC, and draw EB to touch the circle at B.

Then he proves that, if we join AB, AB is equal to the length of the side of the hexagon required.

Produce BC to D so that BD = BA, and join DA. ABD is then equilateral.

Since EB is a tangent to the segment, AE . EC = EB² or AE : EB = EB : EC, and the triangles EAB, EBC are similar.

Therefore

and

But CF = 2CA ; therefore AC : CF = DC : CB, and AD, BF are parallel.

Therefore BF : AD = BC : CD = 2:1, so that

Also FBC = BDA = 60°, so that ABF = 120°, and the triangle ABF is therefore equal and similar to the required triangle NLO.

Construction of toothed wheels and indented screws.

The rest of the Book is devoted to the construction (1) of toothed wheels with a given number of teeth equal to those of a given wheel, (2) of a cylindrical helix, the cochlias, indented so as to work on a toothed wheel. The text is evidently defective, and at the end an interpolator has inserted extracts about the mechanical powers from Heron’s Mechanics.

---

¹ Proclus on Eucl. I, pp. 189–90.

²Ib., pp. 197. 6–198. 15.

³Ib., pp. 249. 20–250. 12.

⁴ See vol. i, pp. 268–70.

⁵ Pappus, iii, p. 54. 7–22.

⁶ Pappus, iii, p. 106. 5–9.

⁷ Vide notes to Euclid’s propositions in The Thirteen Books of Euclid's Elements, pp. 473, 480, 477, 489–91, 501–3.

⁸ Mathematical Gazette, vii, p. 107 (May 1913).

⁹ Pappus, vol. iii, p. 1233.

¹⁰ Centrobaryca, Lib. ii, chap. viii, Prop. 3. Viennae 1641.

¹¹ Chasles, Les trois livrs de Porismes d’Euclide, Paris, 1860, pp. 74 sq.

¹² Simplicius on Arist. Categ., p. 192, Kalbfleiseh.

¹³ Proclus on Eucl. I, pp. 241–3.

¹⁴ Ib., pp. 125. 25–126. 6.

¹⁵ Pappus, viii, p. 1030. 11–13.