II. A Comparative Examination of the Euclidean and Heronian Demonstrations of Elem. II.5

Book II, comprising two definitions, 14 propositions (12 theorems and 2 problems), and a porism (after proposition II.4), is the smallest book of the Elements.Footnote ¹⁰ It is nevertheless considered one of the most important books of the whole treatise vis-à-vis the interpretation of the Elements, as the character of its propositions troubled historians for decades. According to the traditional interpretation of “geometrical algebra,” book II should be conceived as a kind of algebra textbook, in which Euclid exposed the method of “geometrical algebra.” It is within this framework that Heath (Heath 1908, i 372–3) interpreted the first ten propositions of book II as follows:

1. 1) a(b + c + d + ⋅⋅⋅) = ab + ac + ad + ⋅⋅⋅

2. 2) (a + b)a + (a + b)b = (a + b)²

3. 3) (a + b)a = ab + a ²

4. 4) (a + b)² = a ² + b ² + 2ab

5. 5) $\begin{array}{*{20}{c}} {ab + {{\left( {\frac{{a + b}}{2} - b} \right)}^2} = {{\left( {\frac{{a + b}}{2}} \right)}^2}} \end{array}$, or (a + b)(a − b) + b ² = a ²

6. 6) (2a + b)b + a ² = (a + b)², or (a + b)(b − a) + a ² = b ²

7. 7) a ² + b ² = 2ab + (a − b)²

8. 8) 4(a + b)a + b ² = [(a + b) + a]², or 4ab + (a − b)² = (a + b)²

9. 9) ${a^2} + {b^2} = 2\left[ {{{\left( {\frac{{a + b}}{2}} \right)}^2} + {{\left( {\frac{{a + b}}{2} - b} \right)}^2}} \right]$, or (a + b)² + (a − b)² = 2 (a ² + b ²)

10. 10) (2a + b)² + b ² = 2[a ² + (a + b)²], or (a + b)² + (b − a)² = 2 (a ² + b ²)

Now, a thorough account of what takes place in book II requires a detailed examination of each of its propositions. However, for the purposes of our argument, it is enough to examine in detail only one of these propositions, i.e. II.5, and refer to the rest on occasion:Footnote ¹¹

The last column of the table above provides the deductive connection of each (particular) argument with other, previously established, propositions of the Elements. So, for example, one is “justified” in constructing the square CEFB on the straight line segment CB because this kind of construction was demonstrated in I.46; similarly, one is “allowed” to claim that the complement CH is equal to the complement HF based on I.43, and so on. In addition, proposition II.5 itself is a lemma for proposition II.14: “To construct a square equal to the given rectilinear <figure>” (cf. Szabó Reference Szabó1978, 334). Within this framework, there can be little doubt that the character of Euclid's proof of II.5 is geometrical, and also, to use Fried and Unguru's words, “constructive and manipulatory at the diagrammatic level” (Fried and Unguru Reference Fried and Unguru2001, 21).Footnote ¹²

Now, Heron provides a different proof of the same proposition (II.5):Footnote ¹³

A careful examination of this text together with Heron's remarks leads us to the following observations regarding the characteristics of his approach:

(a) Heron offers alternative demonstrations “with the drawing of a sole line” (Lo Bello Reference Lo Bello2009, 22) only for theorems II.2-10. Two preliminary questions need to be asked regarding his choice: first, why does he not provide a similar demonstration for the first proposition of book II? And why does he stop after II.10? Fortunately, both questions are answered by Heron himself: he argues that his approach cannot be applied to II.1, without first “drawing a total of two lines” (ibid.). In the same line of thought, he says for II.11: “as for this figure, it is not possible for one to prove it without an illustration” (ibid., 47–8). From these statements it is evident that Heron's methodology cannot be applied when some sort of construction is involved. Constructions, however, are an essential characteristic of problem solving in geometry.Footnote ¹⁵ Thus, a first remark is that Heron's method – at least according to his own remarks – is only applicable to (geometrical) theorems. Let us now proceed beyond Heron's description and examine his actual mathematical practice.

(b) Heron claims that he provides a proof of II.5 – in fact, of II.2-10 – without using any diagram. His intention, however, is not substantiated into mathematical practice: the truth is that Heron does use at least one single-line diagram in all propositions for which he provides alternative demonstrations. In fact, to be precise, Heron applies prima facie single-line diagrams. This is how he introduces his single-line diagrams for the case of II.2:

It is evident from this passage that the reader of Heron is asked to imagine a single-dimension line segment (AB) divided in a particular way (at the point G), as if it were a two-dimension area; i.e. a diagram (cf. Mueller Reference Mueller1981, 47). This methodology cannot be associated with algebra whatsoever: the objects involved are geometrical (line segments, rectangles, squares), no arithmetical operation is performed on these objects, all manipulations that take place are geometrical, the language applied is purely geometrical, e.g. “the surface that the two lines AD, DB enclose is equal to the sum of the two surfaces that the line BD and the two segments AG, GD enclose,” and each deductive step relies on previously established propositions in books I and II – the geometrical character of which (book I) has not been in question.Footnote ¹⁶

(c) Heron's idea to demonstrate II.2-10 without using a diagram is not as innovative as it usually appears in the literature. Traces of this practice appear even in Euclid's text. Let us illustrate this point by providing an example from II.9: “If a straight line is cut into equal and unequal <segments>, the squares <described> on the unequal segments of the whole <line> are double the <square described> on the half <of the line> and the square <described> on the <line> between the cuts.” The diagram provided in this proposition is:

One can easily notice here that what is requested in the specification of the proposition, i.e. “. . .that the squares <described> on the <lines> AD, DB are double the squares <described> on the <lines> AC, CD,” does not appear on the diagram – the same holds true with all relevant arguments involved in II.9 and in II.10. The omission of the explicit drawing of the diagram does not affect the geometric character of Euclid's arguments.

(d) Heron is doing geometrical analysis and synthesis, for which he provides descriptions:

In practice, what Heron does in II.5 is to decompose the square described on the line segment BG into smaller pieces, and subsequently reassemble these pieces into the rectangle contained by DA, DB and the smaller square described on DG. The deductive order in which these operations are performed is helped by the Euclidean diagram of II.5 (al-Nayrīzī’s version):

(e) Heron does not see – how could he? – the “algebraic” equivalence between the propositions II.5-6, or II.9-10 (cf. Szabó Reference Szabó1978, 332–3).

The conclusion of this brief discussion is that the available textual evidence does not support any algebraic interpretation of Heron's method – especially if by “algebraic” we denote algebra as it was understood before the seventeenth century.Footnote ¹⁸ To be more precise, there is as much algebra in Heron as there is in Euclid, i.e. none.

By virtue of the above, it is a historiographical mistake to consider Heron's alternative proofs to Elem.II.1-10 as the starting point of geometrical algebra. This will become even more evident in the next section, in which, by using three texts as a case study, we will illustrate some of the fundamental characteristics of premodern algebra; namely, that (a) it was a method of solving problems; (b) it was following a specific pattern; (c) it was employing a technical vocabulary; and (d) it was carried out within a conceptual framework which is remarkably different compared to modern algebra (especially regarding the key notions of “polynomial” and “equation”). Nothing similar happens in Heron's proofs.

III. Premodern Algebra: The Cases of Diophantus, al-Sulamī, and al-Khwārizmī

We have already noted that, according to the traditional interpretation of “geometrical algebra,” proposition II.5: “If a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole <line>, together with the square <described> on the <line> between the cuts, is equal to the square <described> on the half <of the whole line>,” is represented by the following algebraic equalities: $\begin{array}{*{20}{c}} {ab + {{( {\frac{{a + b}}{2} - b})}^2} = {{( {\frac{{a + b}}{2}} )}^2}} \end{array}$ or (a + b)(a − b) + b ² = a ².Footnote ¹⁹ These equalities are connected to the following problem: “It is requested to find two numbers, the addition and multiplication of which make given numbers.”Footnote ²⁰ We present below the English translations of the latter problem, as solved by Diophantus, al-Sulamī, and al-Khwārizmī, three mathematicians who belong to the premodern stage of the history of algebra.Footnote ²¹

The following table presents these solutions in concise format. We apply there a notation consistent with the conceptual framework of premodern algebra:

1. (a) For the sought-after numbers, we use the capital letters A and B.Footnote ²²

2. (b) The technical (algebraic) terms in our texts are: “arithmos” (in italics) and “dynamis,” in Diophantus, and “thing” and “māl” (in italics), in al-Sulamī and al-Khwārizmī.Footnote ²³ The terms “arithmos” and “thing” are denoted by x, and the terms “dynamis” and “māl” are denoted by x ².

3. (c) For the assignment of a name to an unnamed sought-after number we use the symbol “:=”. Thus, for example, in Diophantus, the action of assigning the name “10 units and 1 arithmos” to the greater of the two sought-after numbers (say A), is denoted as: “A := 10 and 1x”. Likewise, in al-Khwārizmī, the assignment of the name “one thing” to the one of the two parts of 10 is denoted by “A := 1x”.Footnote ²⁴

4. (d) For the arithmetical operations involved, we apply the standard symbols. There is, however, a subtle point here: in modern algebra an expression like “a plus b” (denoted by a + b) could be seen as denoting both the operation of addition and also the outcome of it. In premodern algebra, instead, a verb meaning “to add” is used for the addition, while the conjunction “and” is used to denote the outcome of the addition. We reserve the symbol “+”, only to denote the operation of addition. Thus, for example, the sentence “add the māl to twenty-one; the result is twenty-one dirhams and a māl” is denoted as: 21 + 1x ² → 21 and 1x ².

5. (e) Finally, the symbol “=” is used only to denote equations. For the outcome of an operation we use the single arrow “→”, while we reserve the double arrow “⇉” for the prescriptions stated in the enunciations.

By virtue of the above, the three solutions appear as follows:

By virtue of the three solutions presented above, the essential characteristics of “premodern algebra” can be summarized as follows:

(1) To begin with, premodern algebra was a method of solving problems. This is explicitly stated in the texts: for example, in the introduction to the Arithmetica, Diophantus refers to arithmetical problems which, as he writes “are solved if you follow the way that I will show” (Tannery Reference Tannery1893, i 4.10-11). In this way, Diophantus reveals his agenda, i.e. to teach a method of solving problems. The same holds true for al-Khwārizmī, who writes in his preface that the caliph al-Ma'mūn

Thus, we can describe al-Khwārizmī’s project as an effort to provide a treatise, useful in teaching how to solve practical problems by algebra.

Besides descriptions such as the ones presented above, the fact that algebra was conceived as a method of problem solving is primarily confirmed by the structure of the premodern algebra textbooks: Arithmetica is a huge collection of problems, accompanied by a theoretical introduction in which Diophantus exposes the basic stages and the technical vocabulary of an algebraic solution (Christianidis and Oaks Reference Christianidis and Oaks2013, 130–134); similarly, most Arabic algebra books have, mutatis mutandis, a first part which is devoted to the rules of algebra (usually called “algebra proper” in the literature), followed by a second part which comprises a collection of worked-out problems.Footnote ²⁶ This general structure already appears in the very first book on algebra in Arabic, the book of al-Khwārizmī – which also contains a third part comprising practical problems like problems of mensuration and inheritance, most of which (but not all) are solved by algebra. Some Arabic algebra textbooks also provide in the first part, in which they deal with “algebra proper,” justification for several rules by means of either geometry or arithmetic.Footnote ²⁷ The later practice originates in late antiquity, where we find geometrical justifications of arithmetical rules, or arithmetical interpretations of geometrical propositions.

(2) Second, premodern algebra was a part of arithmetic. The arithmetical character of the calculations involved is explicitly stated by Diophantus and al-Khwārizmī themselves: Diophantus speaks about “problems in arithmetic” (Tannery Reference Tannery1893–5, i, 4.10), while al-Khwārizmī opens his book by stating: “When I considered what people generally want in calculating, I found that it always is a number” (Rosen Reference Rosen1831, 5).Footnote ²⁸ As Oaks aptly remarks: “What he is telling us is that whether one wishes to find an unknown line segment, weight, span of time, or amount of money, calculations are always numerical. In fact, we find problems about all kinds of quantifiable objects solved by algebra” (Oaks Reference Oaks2007, 559). Thus, even in cases in which the problems solved by the premodern algebraists are practical problems of mensuration, inheritance, etc., what is finally asked is to find one or more arithmetical values. Notice that all calculations involved in this procedure are always arithmetical.Footnote ²⁹

The above claim is further confirmed by the fact that in Islam the art of algebra (that is, the art of solving problems by algebra) was conceived as belonging to arithmetic. In fact, several Islamic works which provide a classification of sciences either categorize algebra under arithmetic, that is “the science of numbers” (ᶜilm al-ᶜadad), or they do not mention algebra as a distinct discipline at all. Al-Muhaqqiq (Al-Muhaqqiq Reference Al-Muhaqqiq and al-Hassan2001) informs us that such works were produced by al-Kindī (d. 250/863), Qusṭā ibn Lūqā (d. 360/912), al-Fārābi (d. 339/950), Ikhwān al-Ṣafāᵓ (mid-fourth/tenth century), Ibn al-Nadīm (d. 385/995), al-Khuwārizmī (d. 387/997), Ibn Farīghūn (d. second half of the fourth/tenth century), Ibn Sīnā (d. 428/1037), al-Bīrūnī (d. 440/1048), al-Lūkarī (d. post-503/1109), al-Ghazālī (d. 505/1111), Fakhr al-Dīn al-Rāzī (d. 606/1209), Ibn Khaldūn (d. 808/1406), al-Suyūṭī (d. 911/1505), Ṭāshköprüzāde (d. 968/1560), and Ṣadr al-Dīn al-Shīrāzī (d. 1050/1640).Footnote ³⁰ Another later leading intellectual figure, Quṭb al-Dīn al-Shīrāzī (1236-1311), himself both philosopher and scientist, who wrote on the classification of sciences, also classified algebra as belonging to arithmetic. As Bakar points out, apparently for Quṭb al-Dīn “algebra and the science of calculation were subdivisions of arithmetic” (Bakar 1998, 254). Similar classifications also appear in the Byzantine period; thus, for example, the Treatise of the four mathematical sciences: arithmetic, music, geometry, and astronomy by Georgios Pachymeres (thirteenth century) includes, in the section on arithmetic, besides selected passages from Euclid's arithmetical books of the Elements and Nicomachus’ Arithmetical introduction, also material from Diophantus’ Arithmetica (cf. Megremi and Christianidis Reference Μegremi and Christianidis2015).

(3) By virtue of (1) and (2), it is now clear that premodern algebra should best be described as a numerical problem-solving method. In late antiquity and throughout the middle ages several other methods were also used in numerical problem solving. Among those there were the “rule of three,” the “single/double false position,” the “arithmetical reasoning,” the “epilogismos,” etc.Footnote ³¹ However, compared to the aforementioned methods, algebra is remarkably different, since it follows an entirely different pattern and has a specific technical vocabulary and its own syntax.Footnote ³² More importantly, what distinguishes algebra from other numerical methods of solving problems is, first and foremost, the action of “naming the unknown” and performing operations with it. To be more precise, algebra is a method of problem solving that involves: (a) naming the unknown(s); (b) working through the operations stated in the enunciation of the problem with the named term(s); (c) setting up an equation as the outcome of the previous processes; (d) manipulating and solving the equation; and, finally, (e) answering the problem, based on the solution of the equation. All the texts we discussed in this section (Diophantus, al-Sulamī, al-Khwārizmī) illustrate the structure described above.Footnote ³³ Additionally to the aforementioned characteristics, premodern algebra is paired with a conceptual background with regard to the key algebraic notions of “polynomial” and “equation” that differs profoundly from the corresponding modern understanding.Footnote ³⁴

To conclude this section, the introduction of the new historiographical concept of “premodern algebra” has much to offer. First, it allows us to distinguish algebra from other problem solving techniques that were used at the time (based on the aforementioned technical characteristics). Second – and perhaps more important – by introducing the notion of premodern algebra one can realize the conceptual distance that separates premodern from modern algebra. For, although algebra never ceased to serve as a problem solving method, from the seventeenth century onwards it underwent a profound transformation which led to an understanding of algebra as a science covering the theory of equations and the study of qualities in general.Footnote ³⁵