Vignettes of Ancient Mathematics

General Contents

The purpose of this site is to illustrate various mathematical techniques and strategies, mostly in ancient Greek mathematics, but other related examples will be included. This will not be a history of Greek mathematics but will contain examples designed to bring out a few interesting features. For the perspective of evidence, the techniques included will be of four sorts:

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.

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: Babylonian Mathematics, Hippocrates of Chios, Eudoxus, Autolycus, Euclid, Archimedes, Theodosius, Hero, Ptolemy, Diophantus, Pappus,
Return to General Contents

Autolycus of Pitane

On Moving Spheres (definitions and props. 1, 2)

Euclid

Elements, Book I (complete)
Elements, Book II (complete)
Elements, Book III (definitions and props. 1-6)
Elements, Book V (complete)
Elements, Book VI (complete)
Selections from Elements, Book VII (definitions and props. 1-22, 33)
Selections from Elements, Books VIII and IX (prop. VIII 22, VIII 23, prop. IX 8, prop. IX 12)
Selections from Elements, Book X (definitions, props. 1-9, Appendix 27)
Selections from Elements, Book XI (definitions, props. 1-8, 18)
Selections from Elements, Book XII (props. 1-2)

Selections from Euclid's Catoptrics (introduction, props. 1, 19, 30)

Selections from Euclid's Optics (introduction, props. 1-5, 8, 47)

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 with the comments of Eutocius (complete translation)

On Floating Bodies (Book I and selections from Book II)

Archimedes mechanical method with indivisibles: a very trivial illustration
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 of Bithynia, 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

Metrica I Preamble, 1-9: This page includes procedures for finding the area of a triangle.

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

Problems on the construction of a double mean proportion (Book III 1-3, 19-27): Pappus begins with a pseudo-diagram and then proceeds to explain how the "ancients" classified problems. He then gives four solutions, those of Eratosthenes, Nicomedes, Heron, and himself.
Theorems on the Archimedes spiral (Book IV §§21-25): Construction of the spiral, statement of the basic property (§21), proof that the figure bounded by the spiral of one rotation and a straight-line is 1/3 the circle generated by the straight-line (§22), generalization to area bounded by a straight-line and spiral from the center (of no more than one rotation) (§23), ratio of areas bounded by a straight-line and spiral from the center (of no more than one rotation) as the cubes of the bounding lines (§24), and ratios of quadrants (§25). Note that the proof of the ratio of the spiral area to the circle is very different from the proof in Archimedes in many important and interesting respects.
Theorems and claims on Nicomedes' 1st cochloid (Book IV §§26-29): Construction of the cochloid and its basic property (§26), claim that the cochloid monotonically approaches an asymptote and, given an angle, point outside the angle, and a length, construction of a line between the legs of the angle whose extension intersects the point(§27), proof that with the cochloid it is possible to find a double mean proportional between two lines (§28), and claim that the solution to finding a double mean proportion provides a solution to finding a cube in a given ratio to a given cube (§29).
Quadratrix (Book 1V §§30-34): Construction, discussion of use of the curve, rectification of the circle, geometrical construction from cylindrical spiral and construction from Archimedes Spiral
Division of angle by a given ratio (Book IV §§45-47): Division of angle or circular-arc in any given ratio by a quadratrix (§45) and by spiral (§46), and given two unequal circles constructing equal circular-arcs (§47)
Four propositions, three explicitly using quadratices (Book IV §§48-51): Construction of an isosceles triangle with a given ratio of the base angles to the vertex angle and construction of a regular polygon (48-49), construction of a circumference equal to a given straight-line (§49), construction of a circular-arc on a chord in a given ratio to the chord (§50) and constuction of incommensurable angles (§51)
Four lemmas for isoperimetric theorems (Book V §§11-14, pp. 234.22-242.12): The circular-arcs of circles are to one another as their diameters (§11); a circle has the same ratio to a section as the circumference of the circle to the circular arc of the section (§§12), part I: Similar segments of circles are to one another as the squares of their bases are to one another; part II: as their circular-arcs are to one another, so are their bases (§13), if two radii in one triangle form equal angles with radii in another, then the triangles formed by a tangent from one radius meeting the extension of the other and the perpendicular from the tangent to the other radius (half-chords) will be as the squares of the half chords.
Introductory lemmas on spherics and theorems relating to Theodosius, Sphaerica, iii 5 (Book VI §§1-11, pp. 474.1-488.25): Props. 1-4 introduce the Menelaus trilateral (spherical triangle), and then uses them to prove Sphaerica iii 5 (§5) and three variations, two proofs (§6 and §§7-9, which uses two-step proportion proof) of the case where the arcs on the oblique (see above) are not adjacent, and where the arcs on the initial are equal (§10), and where the arc nearer the equator on the oblique is larger than the one further away (§11).

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)

Diophantus

Arithmetica i: introduction, probs. 1-6
Arithmetica ii: probs. 8-10

Special Topics

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

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:

This is a brief discussion of arguments which use either an infinite number of steps or treat magnitudes as composed of an infinity of parts.

The Method of Exhaustion:

A way of avoiding infinitary arguments invented by Eudoxus. Eventually, included will be the major categories of this type of argument (by approximation or compression and by one step or two steps)

Approximation:
1. Euclid, XII Elements 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, .

Translations:

The translations will occasionally employ non-conventional ways of expressing the Greek text that is intended to capture the wording of the Greek better (this will develop as new translations appear).

Words of construction: there are two words in Greek that are commonly translated with the same word, 'draw'. 'Draw' is reserved for the verb 'agô', while 'inscribe' is used for 'graphô'.

Numbers: the Greeks use two words for numbers, plêthos (a collection of things) and arithmos (a plethos of units). It is usual to translate plethos as 'multitude' and arithmos as 'number'. This can be deceptive, and the usual practice here will be to leave the two terms transliterated and not translated. The plurals are plêthê and arithmoi.

Squares, rectangles, other figures, and angles:
Squares: the full expression for a square ABGD is something like: 'the square inscribed up from line/arithmos AB', but this is commonly abbreviated to 'the from AB', which is awkward in English. Hence, 'that from AB' will be commonly used.

Rectangles: the full expression for a rectangle ABGD is something like 'the rectangular area/arithmos enclosed by lines/arithmoi AB, BG'. This and the expression for arithmoi can be abbreviated to 'the rectangle enclosed by lines/arithmoi AB, BG', and with other variations, to 'the by AB, BG', which is awkward English. 'that by AB, BG' will be commonly used. Occasionally, the lettering will be abbreviated from 'AB, BG' to 'ABG'.

Other figures: the language works the same as for rectangles.

Angles: the full, but rarely used expression for an angle is something like: 'the angle enclosed by lines AB, BG', which is abbreviated to 'the by ABG', which is awkward English. 'That by ABG' will be commonly used, but because this can easily be confused with the definite description of a rectangle, 'the angle by ABG' will often occur. Following other conventions, 'that under ABG' will also occur.

Ratios, proportion
Proportion: In Greek, there are three words for proportion: ana logon, analogon, and analogia. The last is a noun and may happily be translated as 'proportion'. The first is a prepositional phrase, 'in ratio', while the second is an adverb formed from the first, 'proportionally'. Unfortunately, the adverb needs to function as an adjective, so that this translation makes for awkward English. Moreover, the distinction between ana logon and analogon would not have been evident in most writing in the Hellenistic Age. So it works out better to translate ana logon as 'in ratio' and analogon as 'in-ratio', which preserves the prepositional phrasing.

Ratios: there are two common ways of expressing ratio in Greek, a language where word order is very free:
General: In Greek there is a strong tendency not to break up the expressions 'A to B' and "G to D'. This allows the reader to do what we do with our symbolic representation, to see these as relational units.
Has A to B a/the (same) ratio that/which G to D (where 'has' can go before 'a ratio' or before 'that/which', but occasionally in the relative clause). Obviously, the natureal rendering in English would be, "A has a/the (same) ratio to B that G has to D." Since the expression is less common, and mostly occurs in statements of theorems or restatements of theorems, it seems easiest to let word order in English have sway, "A has to B a/the (same) ratio that/which G to D." This decision may be changed in future translations to: "A to B has a/the (same) ratio that/which G to D."

(Is) as A to B so G to D: the two most common forms are to have 'is' in front, along with any conjunctions, adverbs, etc. or to omit it altogether, where there might be a conjunction after 'as'. 'Is' also occurs before (very rarely, twice in Archimedes) or after 'so'. To avoid breaking up the proportion, where 'is' occurs in front, the translation will be "It is: as A to B so G to D," and where the verb is left out, there will be no verb, "as A to B so G to D." Keep in mind that the verb 'to be' is unnecessary when universal or eternal truths are expressed, while English not only needs a verb, but requires a dummy subject for the verb.

Since Greek is a highly gendered language (masculine, feminine, and neuter), ambiguities between these expressions will be rarer than in English. For example, the word for 'angle' is feminine and that for 'rectangle' neuter. Hence, 'the(f) by ABG' would almost never be confused for 'the(n) by ABG' (the exception being genitve plurals). Keep this terseness of Greek in mind when you read 'the angle by ABG' or 'the rectangle by ABG'. This somewhat makes up for the fact that, except for the use of numerals and letters to name parts of diagrams and arithmoi, the language of Greek mathematics is non-symbolic. In Byzantine manuscripts there are further abbreviations, for square or rectangle (the expected) and even for expressing ratios that come close, but not quite to symbolic representation.

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.

The copyright for all the sites of Vignettes of Ancient Mathematics belongs to Henry Mendell. Permission to use any translations, diagrams, or other texts by him is only granted for personal use and for use in a classroom using links to this site. For all other uses, including publication or commercial use, please inquire of the author.

Contents:

Philosophical Discussions on Mathematical Issues

Presocratic discussions of interest to mathematics:

Peculiar Circle Squaring:

Interesting curves, as discussed by philosophers

Aristotle and ps.-Aristotle