• Vignettes of Ancient Mathematics

by Henry Mendell, Cal. State U., L.A.

Textual Notes

Copyright

• Texts with explanatory diagrams.  The diagrams will be 'modern' in the sense that they will walk the reader through the proof.
• Paraphrases or summaries with explanatory diagrams.  Here the argument does occur in our sources, but a simpler paraphrase was used to facilitate understanding.  For example, the mathematician may have needed the elaborate original text to explain matters easily understood by a series of well constructed diagrams.
• Simple illustrations of techniques:  these illustrations are simpler than the examples which occur in ancient texts and so are useful for learning the techniques.
• Reconstructions of arguments which are lost, but which seem plausible.  Some of these are standard in the modern literature; others express the personal tastes of the author.

Contents:

Philosophical Discussions on Mathematical Issues

Authors: Presocratic Discussions, Peculiar Circle Squaring, Interesting curves discussed by philosophers, Aristotle, Texts on Astronomy
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Presocratic discussions of interest to mathematics:

Anaximander's Cosmos
Zeno's paradoxes on Motion and Size
 

Peculiar Circle Squaring:

Antiphon, a sophist of the 5th cent. B.C.E.
Bryson (mid-4th cent. B.C.E.)

Interesting curves, as discussed by philosophers

Some curves mentioned by Proclus and Iamblichus, as quoted by Simplicius

Aristotle and ps.-Aristotle

Physics 8 215a24-216a21: travel through media and the void
Physics Z 2.232a23-b29: the definition of 'faster' and the argument for it
Physics Z 7.237b28-238b22: arguments against finite traversals in infinite time, infinite traversals in finite time, and infinite bodies traversing
De caelo A 6 273a21-b27: an infinite body cannot have finite weight
Mechanica 1: the composition of changes, ordinary circular motion, and why longer balances are more precise

Philosophical texts pertaining to Astronomy

Simplicius and Geminus on early Greek Astronomy and testimonia (including Proclus) for Sosigenes (2nd cent. C.E.) on astronomy (this is a PDF file and must be viewed with Acrobat): Simplcius, In de Caelo Aristotelis 32.12-27, 474.7-28 (ad 291a29), 422.1-28 (ad 288a13-27), 488.3-24 (ad II 12 292b10), Geminus, The Elements of Astronomy I §§18-21, Simplicius, In Physica Aristotelis, 291.3-292.31 (quoting Alexander quoting Geminus quoting Posidonius), Simplicius, In de caelo Aristotelis, 491.13-510.35 (ad II 12 293a4-12), Proclus, Hypotyposis astron. posit., Ch. 4. 97.1-99.4, Proclus, In Rem Publicam ii 23.1-24.5.

Mathematical Texts

Authors: Hippocrates of Chios, Eudoxus, Euclid, Archimedes, Theodosius, Hero, Pappus, Ptolemy
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Hippocrates of Chios

Hippocrates of Chios
Introduction to the mathematics of lunules
Analysis of the quadrature of lunules as reported by Alexander
Analysis of the quadrature of lunules as reported by Eudemus
Comparison of the methods reported by Eudemus and Alexander

Eudoxus of Cnidus

Introduction to the astronomical models

Euclid

Selections from Euclid's Catoptrics

Selections from Euclid's Optics

Archimedes

The Sand-Reckoner (complete translation) or go directly to Ch. 1 or Ch. 2, or Ch. 3, or Ch. 4

Basic Lemmata for pre-Archimedean theory of orthotomes (a.k.a. parabolas) based on Quadrature of the Parabola 1-5

Quadrature of the Parabola (complete translation).

On the Equilibria of Planes, Book I and comments of Eutocius (complete translation)
On the Equilibria of Planes I Assumptions, Props. 1-5: preliminary matters (with comments of Eutocius).
On the Equilibria of Planes I 6-7: weights balance in inverse proportion to the distances from the fulcrum (with comments of Eutocius).
On the Equilibria of Planes I 8: the center of weight of one of two parts of a body, given the centers of the weights of the whole and the remaining part.
On the Equilibria of Planes I 9-10: the center of weight of a parallelogram.
On the Equilibria of Planes I 11-12: the center of weight of similar triangles.
On the Equilibria of Planes I 13-14: the center of weight of a triangle (with comments of Eutocius).
On the Equilibria of Planes I 15: the center of weight of a trapezoid (with comments of Eutocius).

On the Equilibria of Planes, Book I and comments of Eutocius, propositions 1-8
On the Equilibria of Planes II 1: parabolas that can be applied to a line balance in inverse proportion of their centers of weight. (with comments of Eutocius)
On the Equilibria of Planes II 2: after definition of figure inscribed familiarly in a parabolic segment and statement of two lemmas, the center of weight of a figure inscribed familiarly in a parabolic segment lies on the diameter of the parabolic segment. (with comments of Eutocius)
On the Equilibria of Planes II 3: familiarly inscribed figures in similar parabolic segments with the same number of sides have their centers of weight similarly positioned (note, for Archimedes, all parabolic segments will be similar). (with comments of Eutocius)
On the Equilibria of Planes II 4: The center of the weight of every segment enclosed by a straight-line and the section of a right angled cone is on the diameter of the segment. (with comments of Eutocius)
On the Equilibria of Planes II 5: If a rectilinear-figure is familiarly inscribed in a segment enclosed by a straight-line and the section of a right angled cone, the center of the weight of the whole segment is nearer to the vertex of the segment than the center of the inscribed rectilinear-figure. (with comments of Eutocius)
On the Equilibria of Planes II 6: Given a segment enclosed by a straight-line and the section of a right angled cone it is possible to inscribe familiarly into the segment a rectilinear-figure so that the straight-line between the centers of weight of the segment and the inscribed rectilinear-figure is less than any proposed straight-line. (with comments of Eutocius)
On the Equilibria of Planes II 7: The centers of the weights of two similar segments enclosed by a straight-line and the section of a right angled cone cut the diameters in the same ratio. (with comments of Eutocius)
On the Equilibria of Planes II 8: The center of the weight of every segment enclosed by a straight-line and the section of a right angled cone divides the diameter of the segment to that the part of it at the vertex of the segment is half-again that at the base. (with comments of Eutocius)

Method 1: using the principle of the balance and treating a plane as slivers of lines, to square a parabola
Method 2: using the principle of the balance and treating a solid as slivers of planes, to compare the volumes of a sphere, circumscribed cylinder, and inscribed cone (with a great circle of the sphere as base)
Method 14 (summary): using the principle of the balance and treating a solid as slivers of planes, to cube a 'hoof'

On Conoids and Spheroids 1: a basic proportion theorem

Thabit Ibn Qurra's translation of the construction of the regular heptagon (prop. 17, 18 of the treatise), which he attributes to Archimedes

Theodosius, Sphaerica

Sphaerica iii 1:  Let a segment less then a semicircle be erected perpendicular to a circle on a chord which is less than a diameter and let the segment be divided unequally at some point.  Then the line from the point to the larger segment of the initial circles is smaller than the line to any other point on the circular-arc of the larger segment of the circle.  The theorem considers other conditions, including the case where the segment is erected on a diameter.
Sphaerica iii 2:  The same as theorem iii 1, except that the segment is inclined towards the smaller part of the original circle.  The theorem considers the same cases as in iii 1.
Sphaerica iii 3:  If two great circles intersect each other and equal arcs are taken on each side of the intersection point on each great circle, then the opposite straight-lines connecting the end points are equal.
Sphaerica iii 4:  If two great circles intersect each other and equal arcs are taken on each side of the intersection point on one of them (the first), and planes parallel planes intersect the sphere at the end points of the two arcs so that the section of the two intersecting circles intersects one of the planes, where the two planes cut off arcs of the second circle smaller than the equal arcs, then the arc on the second circle cut off by intersection point and the plane that doesn't intersect one section of the two great circles is larger than the arc cut off by the one that does and the intersection point.
The following theorems use this set-up: Two great circles (we shall identify as the oblique and the latitude) intersect another at right angles (the initial).  The poles of the latitude and the oblique are on the initial.
Sphaerica iii 5:  If two equal successive arcs are marked off in the same quadrant of the oblique and latitudes are drawn through their end-points to the original circle, thus marking off two arcs on it, the arc nearer to the latitude is larger than the one further away.
Sphaerica iii 9: Let great circles be drawn through the end-points of the marked out arcs and the pole of the latitude.  Then, each pair of great circles marks out arcs on the latitude.  Those nearer the initial are larger.
Sphaerica iii 10:  Let great circles be drawn through the oblique between the two other circles from the pole of the latitude to the latitude (i.e. two longitudes).  This marks out two arcs on each of the oblique and latitude from the initial.  The ratio of the arc nearer to the initial on the latitude to the corresponding arc on the oblique is the same as the ratio of the next arc on the latitude to an arc smaller than the corresponding arc on the oblique.

Hero of Alexandria

Mechanica i 32-34:  Three theorems concerning the balance, the first from Archimedes, Books on Levers, the second from Archimedes (presumably the same book), and the third likely to be from the same book.

Mechanica ii 35-41:  Six theorems on center of weight, at least some attributed to Archimedes.

Pappus of Alexandria, Collectio Mathematica

Ptolemy, Almagest

Derivation of the position of the center of the solar deferent on the eccentric model (iii 4).  This also illustrates the claim that only three points, the time intervals between them and the mean motion are needed to construct an eccentric model.
Almagest i 10, H43-45: Let , be arcs of a circle. Then > => Chord() : Chord() < :
Almagest xii 1, Trigonometric Lemma of Apollonius (xii 1)

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Special Topics

List of Topics: Quadratrix (a circle squaring and angle dividing curve), Infinitary Arguments, The Method of Exhaustion, Trigonometry, Early Modern Mathematics
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Quadratrix (a circle squaring and angle dividing curve)

Philosophical texts referring to the quadratrix and its history: Proclus or Iamblichus, as quoted by Simplicius

Discussions in Pappus, Mathematical Collection
Generation of the curve and its history (iv 30)
Discussion of the objections of Sporus to the curve (iv 31)
Rectification of the circumference of a circle and quadrature of a circle by the curve (iv 31-32)
Geometrical construction of the curve from a cylindrical spiral (iv 33)
Construction from an Archimedean spiral (iv 34)
Division of an angle or circular-arc in any given ratio by the curve (iv 45)
Given two unequal circles construction of equal circular-arcs (mentions quadratrix and Archimedes spiral) (iv 47)
Construction of a circumference equal to a given straight-line by the curve (iv 50)
Construction of incommensurable angles by the curve (iv 51)

To be added: the two other mentions of the quadratrix in Pappus

Infinitary Arguments:

The Method of Exhaustion:

Approximation:
1.  Euclid, Elements XII 2:  Circles are as the squares of their diameters.
2.  Archimedes: Quadrature of the Parabola 24: geometrical quadrature of the parabola

Compression:  Archimedes, Quadrature of the Parabola, 16. the mechanical quadrature of the parabola.

Two-step exhaustion:
1.  Fundamental Lemma for two step exhaustion: Scholion to Theodosius, Sphaerica iii 9: given three lines of the same kind, AB, G, DE, with AB > G, to find a line BZ, such that G < BZ < AB and BZ is commensurable with DE.
2. Archimedes, On the Equilibrium of Planes I 6-7: weights balance inverse proportion to the distances from the fulcrum. This is the oldest mathematical example of the method.
3.  Theodosius, Sphaerica iii 9:  This theorem contributes to the problem of measuring arcs on an equator given arcs marked out on an oblique great circle.
4.  Theodosius, Sphaerica iii 10: This theorem also contributes to the problem of measuring arcs on an equator given arcs marked out on an oblique great circle.
5.  Two-step Compression Argument: Pappus, Mathematical Collection v 12: a circle has the same ratio to a section as the circumference of the circle to the circular arc of the section.
6.  An expansion of Theodosius, Sphaerica iii 5, in Pappus, Mathematical Collection vi 7-9:  Latitudes through on-adjacent equal arcs on the oblique in the required configuration mark off unequal arcs on the initial with the larger nearer to the equator (see above or Pappus, Mathematical Collection vi 7-9).  Keep in mind that Pappus gives a direct and simple proof of the theorem, at prop. 6.
7.  Aristotle's argument, De caelo A 6 273a21-b27, that no infinite body can have finite weight may also be a trace of the method.

Trigonometry:

Variations on the tangent rule (anachronistically): > => Tan() : Tan() > :
1. Scholion to Theodosius, Sphaerica iii 11: Given two right triangles with one leg equal and the other unequal (with c adjacent to angle , and d adjacent to angle ), d > c => d : c > : or anachronistically: Cot() > Cot() => Cot() : Cot() > : .
2. Euclid, Optics 8: Equal and parallel magnitudes at an unequal distance from the eye are not seen proportionally to the distances, but the proposition actually proves the same, stronger proposition as the scholion to Theodosius, Sphaerica iii 11.
3. Ptolemy, Almagest xii 1 (lemma of Apollonius): in any triangle ABC, if BC > AC, then AC : BC-AC > ABC : ACB.

Variations on the sine rule (anachronistically): > => : > Sin() : Sin()
4. Scholion 16 to Aristarchus, On the Sizes and Distances of the Sun and the Moon, prop.4: in a triangle, let a be opposite and b be opposite . Then a > b => b : a > : or anachronistically: Csc() > Csc() => Csc() : Csc() > :
5. Ptolemy, Almagest i 10, H43-45: Let , be arcs of a circle. Then > => Chord() : Chord() < :

Early Modern Mathematics:

Isaac Barrow, Mathematical Lectures, pp. 30-31 of John Kirby's translation. Proof of the sum of an infinite non-standard arithmetical series using Cavalieri's method and using convergence.

To be added in the near future or under construction:
A discussion of horn angles and a neo-Platonic paradox
Hippocrates on lunules (translation of the text with elaborate explanation)
More neo-Platonic discussions of interesting curves
Mathematical discussions in Aristotle

• Textual Notes

Annotations:

If the text is a quotation or translation, blue text will indicate additions or annotations.

Ancient Greek texts often take a right angle to be a unit. It is a matter of debate when degrees were introduced. The earliest Greek text to use degrees (imported from Babylon) is Hypsicles, Anaphoricus (2nd cent. B.C.E.). It is convenient to use a symbol for a right angle. I use rho, .

Lettering:

If the text is a reconstruction or conjecture, the lettering is English.

If the text is a translation or a summary, then the lettering will be standard English equivalents of Greek letters with two exceptions:
A, B, G (gamma), D (delta), E, Z, H (eta), Q (theta), I, K, L (lambda), M, N, X (xi), O, P (pi), R (rho), S (sigma), T, U (upsilon), F (phi), C (chi), Y (psi), W (omega), J (waw or digamma).

If the list of letters is long and involves thinking of the letters as a sequences, where following Greek letter order would be onerous on the reader, English lettering is used, for example, Archimedes, On Conoids and Spheroids, prop. 1.

Of particular interest to historians is the way in which some early texts, Aristotle, Eudemus (quoted by Simplicius), Archimedes, sometimes refer to figures with the preposition: epi + genitive.  To indicate the use of such an expression in the translation, they will be underlined, e.g. EG will mean that the Greek has something like:  ef' hê EG. In some cases, however, this has been written out.

• Copyright