Years Ago David E. Rowe, Editor

Was Uncle TomRight That QuadraticProblems Can’t BeSolved with the Ruleof False Position?ALBRECHT HEEFFER

Years Ago features essays by historians and

mathematicians that take us back in time. Whether

addressing special topics or general trends, individual

mathematicians or ‘‘schools’’ (as in schools of fish), the

idea is always the same: to shed new light on the

mathematics of the past. Submissions are welcome.

� Submissions should be uploaded to http://tmin.edmgr.com

or sent directly to David E. Rowe,

rowe@mathematik.uni-mainz.de

False Position in Mesopotamia and Egypt

TThe rule of false position was—without much doubt—the oldest method for solving linear problems in thehistory of mathematics. One distinguishes between

two variations, single and double false position. Single falseposition was practiced within the two oldest mathematicalcultures, that is, Mesopotamia and Egypt (c. 1800 BCE). Thishad been largely unknown since the script and language ofboth these civilizations was lost between antiquity and theearly nineteenth century. Egyptian hieroglyphs and thehieratic script became comprehensible during the 1820s,whereas the cuneiform script for the Akkadian language wasdeciphered in1857, followinga contest between four scholars.The second half of the nineteenth century was mostly devotedto linguistic and cultural studies of these ancient civilizations;their traditions of mathematical practice became known onlyat the beginning of the twentieth century. A first edition of anEgyptian mathematical text was published by Peet in 1923;Otto Neugebauer’s pioneering work on Old-Babylonianmathematics began soon thereafter.

False position is a method for solving simple linear prob-lems or aha-problems (quantity problems), like problem 24from the Rhind Mathematical Papyrus (Clagett 1999, III, 140):‘‘A quantity with 1/7th of it added, becomes 19. [What is thequantity?].’’ The method requires one to make a guess for theunknown quantity and calculate the alternative result. Aconvenient guess in this case is 7 so that 7 with 1/7th addedbecomes 8. The solution then proceeds with ‘‘As many timesas 8 must be multiplied to give 19, so many times 7 must bemultiplied togive the required number.’’ The result ofwhich isexpressed as the sum 16þ 1

2þ 18 (in Egyptian mathematics

fractional parts were always expressed as unit fractions). Totranslate this procedure in modern symbolism, false positiondeals with problems of the form ax = b, begins with a guessfor the unknown x, and calculates the corresponding b0. Thereal value of x is then determined by correcting the guess bythe proportion b:b0. The modern reader might wonder whyone would not just calculate b/a to determine x. The reason isthat in most problems a consists of a complex form of arith-metical operations and fractions. This is illustrated by a well-documented problem from the Old-Babylonian tablet YBC4652 about the weight of a stone (see Fig. 1): ‘‘I found a stone(but) it was not marked. A 7th part I tore off, an 11th part Ijoined. A 13th part I tore off. I weighed it: 1 mina. The originalstone was what?’’ (Friberg 2005, 32). In modern-day symbol-ism the problem can be represented by the equation

x� 1

7x þ 1

11x � 1

7x

� �� 1

13x � 1

7x þ 1

11x � 1

7x

� �� �¼ 1

whichmakes it immediately evidentwhy calculatingb/a is notanoptionhere. It also illustrateswhyweshouldavoid thinking

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DOI 10.1007/s00283-013-9404-6

about ancient mathematics in terms of equations. A conve-nient choice for the falsepositionherewouldbe1001, the LCMof 7, 11, and 13, facilitating the calculation of b0.

Double False PositionThe method of double false position emerged much later andwas used to solve more general linear problems. It turns up inArabic mathematics from the ninth century under the termhisab al-khata’ayn or its Latinization elchataym in Fibonacci’sLiber Abbaci (1228). It has been conjectured that the termrefers to Khitan or Cathai, an old designation for ‘‘Chinese’’(Martzloff 2006: 90). However, this interpretation is highlyunlikely as the phrase is a literal Arabic rendering of ‘‘methodof the two errors.’’ As an example we consider a linear prob-lem from Fibonacci (Sigler 2002: 460):

A certain man went on business to Lucca, next to Flor-ence, and thenback toPisa, andhemadedouble in eachcity, and in each city he spent 12 denari, and in the endnothing was left to him. It is sought how much he had atthe beginning.

As a first guess, also called position, Fibonacci takes 12.Doubling it in the first city gives 24, spending 12 leaves 12.So after the last city he is left with 12, which gives an errorof 12. As a second position he takes 11, doubled is 22,minus 12 leaves 10, doubled again is 20, minus 12 is 8,doubled a last time makes 16, minus 12 leaves 4 as the endresult and as the second error. To calculate the correctvalue of the unknown, the following procedure is applied:multiply the first error with the second position (12 9

11 = 132) and subtract from this the product of the seconderror with the first position (4 9 12 = 48), leaving 84.When this value is divided by the difference between thetwo errors (12 - 4 = 8) the result is the unknown (10 1

2).By translating this procedure into modern-day symbolism

we can verify the reasoning. The general format of such linearproblems isax + b = c. Twoguessesarebeingmade, letus sayx0 and x1. The first guess leads to a value ax0 + b from whichwe calculate the difference with c, and denote it as c0. We alsodo this for the second guess, which leads to c1 = ax1 + b – c.

The value of the unknown can then be found by calculating

x ¼ x1c0 � x0c1

c0 � c1ð1Þ

Thevalidity of the procedure can beproved by substitutingthe expressions above for c0 and c1 into (1) and working outthe algebra, which gives x ¼ c�b

a :

On the Origins of Double False PositionBefore the advent of algebra, double false position was themost widely used method for solving linear problems acrossdifferent cultures. Themethod probably emerged from the so-called excess-and-deficit problems in ancient Chinese math-ematics. Although these problems are formulated differently,they share the same solution recipe. The most importantclassic of Chinese mathematics, the Nine Chapters, includes achapter with twenty excess-and-deficit problems or Ying-buzu (chap. 7). The first problem reads as follows (Chemla2004, 559, translated from the French):

Suppose a certain number of people are going topurchase goods and when they each pay 8, there is anexcess of 3. If they pay 7, there is a deficit of 4. It isasked, respectively, the number of people and howmuch the goods cost.

Commentaries on this text explain how to perform thecalculation on a counting board. The prices are set on top(x0, x1), the excess (c0) and deficit (c1) below (with thedeficit expressed as a negative number). Then a crosswisemultiplication is performed and the sum of these productsis divided by the difference between the two prices, thus((8 9 4) + (7 9 3))/(8 - 7). The result, 53, is the price ofthe goods. As can be seen, this procedure corresponds withexpression (1). The number of people is found by addingthe excess and deficit and dividing by the differencebetween the two prices, thus (3 + 4)/(8 - 7) = 7. If theArabic version of double false position indeed dependedon earlier excess-and-deficit problems from China, then it isa strange irony that the method spread to Europe, to beimported again in China four centuries later through Jesuit

Figure 1. Old-Babylonian tablet using false position (plate 13 from Neugebauer and Sachs 1945).

THE MATHEMATICAL INTELLIGENCER

translations of Western arithmetic books. Karine Chemladescribed this as ‘‘closing the loop’’ (Chemla 1997).

Double false positionwas alsoused to solve linearproblemsin multiple unknowns. We find many such solutions in Arabicand medieval European arithmetic. A typical example is anepigram problem from Ancient Greek identified as part of theEuclid corpus (Heiberg 1883, VIII, 286). A mule and ass arewalking together. Themule says to the ass, ‘‘If you gavemeoneof your sacks, I would have as many as you.’’ The ass replies, ‘‘Ifyou gave me one of your sacks, I would have twice as many asyou.’’ The question is, ‘‘How many sacks does each have?’’

The general form of this problem can be represented bytwo linear equations in two unknowns:

x þ a ¼ c y � að Þ

y þ b ¼ d x � bð Þ

A solution by double false position, as found in medievalarithmetic, would for example take 4 as the first position forthe number of sacks of the mule. The ass then should have 6(since 6 + 2 = 2 9 4). The error thus is 6 - 5 = 1. If wetake 8 for the second position, the ass should have 10 sacks(8 + 2 = 2 9 5) and the error is -3. Multiplying the firsterror with the second position and subtracting the multi-plication of the second error with the first position yields1 9 8 -(-3) 9 4 = 20. Dividing this by the differencebetween the two errors (1-(-3) = 4) gives 5 for the num-ber of sacks of the mule. Hence, the ass carries 7 sacks.

Extending the Method to Quadratic Problems

Asystematic solutionof systemsof linear equations inmultipleunknowns appears only from the sixteenth century. Aboutthat time some authors of arithmetic books began wonderingif double false position could be applied to quadratic prob-lems. Christoff Rudolff, who published the first book almostcompletely devoted to algebra (Die Coss, 1525), listed eightpossible equation types, an extension and generalization ofthe six types known from Arabic algebra. In modern sym-bolism, these were:

1Þaxnþ1 ¼ bxn; 2Þaxnþ2 ¼ bxn; 3Þaxnþ3 ¼ bxn;

4Þaxnþ4 ¼ bxn;

5Þaxnþ2 þ bxnþ1 ¼ cxn; 6Þaxnþ2 þ bxn ¼ cxnþ1;

7Þaxnþ1 þ bxn ¼ cxnþ2; and 8Þaxnþ4 þ bxnþ2 ¼ cxn:

Rudolff considered it impossible to apply false position toany equation beyond type 1.1 However, Gemma Frisius tookup the challenge in his Arithmeticae practicae methodusfacilis (1540), a work reprinted in more than a hundrededitions during the sixteenth century.2 Frisius (1540, f. XXXr)stated that he could also solve Rudolff’s second, third, and

fourth type by double false position, and he did so in severalexamples. One such problem involves a rectangular field of200 square ells of which the length is a half larger than itsbreadth. Taking x for the breadth and (3/2)x for the length,this leads to the pure quadratic equation, 3

2 x2 ¼ 200 or Ru-dolff’s type 2. Now since equations of types 2, 3, and 4 allreduce to a form with only one quadratic or one cubic term,the square or cubic termcanbe treated linearly and a solutionbecomes possible by single false position. The calculation ofthe value of theunknown then requires only the extractionofsquare or cube roots. However, this is not how Frisius pro-ceeds, because he wanted to show how to solve suchproblems by double false position. He poses 4 as a firstposition for the breadth. Hence, the length is 6, the area 24,and the error 176. He then takes 20 for the second position,the length being 30, the area 600, and the second error 400.He squares the two positions and multiplies them crosswisewith the two errors. The sum of these products is divided bythe sum of the errors, resulting in the value 133 1

3 for thesquare of the unknown. This procedure in modern symbol-ism leads to the expression:

x2 ¼ x21c0 þ x2

0c1

c0 þ c1:

After Frisius, several others tried to solve full quadraticequations using double false position. The GermanRechenmeister Simon Jacob from Coburg published anarithmetic book, Ein new und wolgegrundt Rechenbuch(1565, a revised and expanded version of his 1557 edition), inwhich he devoted a large section to the regel falsi, as he callsfalse position. He showed that he was acquainted with Ru-dolff’s Coss as well as with Frisius’s work, and remarked thatFrisius remained silent on the application of false position tothe other types of equations (f. 273v). Jacob solved severalproblems of type 2, 3, and 4 following Frisius. He thenremarked that problems of these types can also be solved in adifferent way, not noted by Frisius, and shows how for aproblem of type 2. This problem reduces to the equation16 x2 ¼ 54 in which he uses the false positions for the squareterm and solves the problem as a linear one. He used thesame linearization technique to solve problems of types 5 to8. An example of his innovative approach to the full qua-dratic is problem 37, an equation of type 5 (f. 276v, seeFig. 2): ‘‘Find me a number, when 6 added to it, and 2 sub-tracted from that samenumber, andmultiplying the sum withthe difference, 84 results from it.’’ As a first position he takes 4resulting in 20, an error of 64. The second position is 6, givingthe value 48 or an error of 36.

The calculation of the solution follows the known proce-dure according to Jacob (‘‘procedier nach allgemeiner Falsi’’),but there is a curious difference.He adds 2 to the value of eachposition:

ð6þ 2Þ2 � 64� ð4þ 2Þ2 � 36

64� 36¼ 100

1Noted byFrisius (1540, f.XXIIIv) anddiscussedbyseveral authorswithout identifying thequotebyRudolff, 1525, fol. Hvjv: ‘‘Durch diseerst equationoder reglwerdenentricht vnd

auffgelost all exempl vnnd fragen so durch die regl de tre vnd durch die regl falsi practicirt werden. Hie wil ich dich erinnert haben das sich die falsi allein streckt auff die erst Coss

dann es ist nit muglich ein exempl der andern, drittn, vierden etc. regln durch sie zu machen.’’2This was first noted by Cantor, 1894, vol. II, pp. 411–413.

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and then takes the root of the result and subtracts 2, giving 8as the solution. So what trick is involved here? To understandwhat Simon Jacob is hiding we have to look at the equations.The problem can be represented by the type 5 equation x2 þ4x � 12 ¼ 84: Intriguingly, all his examples are normalizedequations in which the coefficient of the higher-order term is1. Such equations can be written as

x2 þ bx þ c ¼ 0 or x þ b

2

� �2

þc� b2

4¼ 0

As Jacob is linearizing the higher-order term in equationtypes 2 to 4 he is now linearizing ðx þ b

2Þ2:

This shows that the value he calculated as 2 is simply b/2.This approach was pursued further by Elcius Edouardus

Leon Mellema (1544–1622) of northern Holland, author ofbooks on arithmetic, double-entry bookkeeping, and aFrench-Flemishdictionary.HisArithmetique waspublished inAntwerp, a first part in 1582 when he was a reckoning masterin this city, and a second in 1586. His method is explained inthe second part, of which only one extant copy is known(Antwerp City Library).3 Mellema, who was aware of theprevious work of both Frisius and Jakob, follows the eightequation types of Rudolff, presenting solutions for each ofthem first by algebra and then by double false position. Hismethod for types 2, 3, and 4 is the same as Frisius’s, so let uslook at an example of a type 5 equation (Mellema 1586,336-337): ‘‘Find me a number to which 25 is added and [fromthe number] 15 subtracted and multiplying the sum and thedifference, 225 results from it.’’ For a first position Mellematakes 35 and calculates the error as c0 = (35 + 25)(35 – 15) –225 = 975. For the second he chooses 25 and calculatesc1 = (25 + 25)(25 – 15) – 225 = 275. Next he calculates a

value by dividing 10 by 2, which is 5, and adds this to the twopositions. These modified values are used to calculate ourfamiliar procedure as

ð30Þ2 � 975� ð40Þ2 � 475

975� 275¼ 625:

Taking the square root of 625 and subtracting the value 5,he arrives at 20 for the final value of the unknown.

The auxiliary value, which he calculates, corresponds tob2a ¼ 5:Thegeneralprocedureheand Jacobare following thuscorresponds to

x þ b

2a

� �2

¼ðx0 þ b

2aÞ2 � c1 � ðx1 þ b

2aÞ2 � c0

c1 � c0

The value of the unknown is then found by taking thesquare root of this result and subtracting b

2a. The search foran application to the full quadratic hereby came to an end.Double false position emerged in a period when algebrawas still in its infancy and the method was considered as aneasy way to solve linear problems without the use ofalgebra. To apply it to the full quadratic, both Jacob andMellema had to rely on knowledge of algebra (writingdown the equations and calculating the value of b

2a).Eventually they arrived at a procedure which became morecomplex than the calculation of the roots of the quadratic.

EpilogueWhen I posted Mellema’s method on the now extinctmailing list Historia Mathematica in November 2004, asan answer to a question on double false position, Ireceived a long message from a certain Uncle Tom,

Figure 2. A solution to the full quadratic in Simon Jacob’s Ein new und wolgegrundt Rechenbuch (from Google books).

3Smeur (1960) was the first to describe this book and to note the application of double false position to the full quadratic. He wrote about double false position in several later

publications.

THE MATHEMATICAL INTELLIGENCER

starting as follows: ‘‘In all unkindness, which may wellmirror my own failure to understand, I cannot be sure youare right’’ and then vehemently claiming that it is plainimpossible to apply false position to the full quadratic. Ianswered him, explaining Mellema’s method as shownabove. He wrote back: ‘‘Forgive, but I regret it is you whodo not see. As I detailed it in the generality in my pre-vious letter Mellema’s ‘method,’ if not a confidence trick,is just a blind.’’ After some further exchange of mathe-matical arguments, he eventually revealed his identity‘‘May I beg apology for unintentionally omitting my namefrom what I wrote to you,’’ signed by the acronym D.T. W. The writer Derek Thomas Whiteside, a highlyrespected historian of mathematics, best known for hismeticulous edition of eight huge volumes of The Mathe-matical Papers of Isaac Newton, passed away in 2008. Iwas never able to convince him.

REFERENCES

Cantor, Moritz (1880–1898) Vorlesungen uber Geschichte der Math-

ematik (4 vols.), Teubner, Leipzig.

Chemla, Karine (1997) ‘‘Reflections on the world-wide history of

double false position, or how the loop was closed,’’ Centaurus,

39, pp. 97–120.

Chemla, Karine, and Shuchun Guo (eds. tr.) (2004) Les neuf chapitres.

Le classique mathematique de la Chine ancienne et ses

commentaires. Edition critique bilingue traduite, Dunod, Paris.

Clagett, Marshall (1999) Ancient Egyptian Science (3 vols). American

Philosophical Society, Philadelphia.

Friberg, Joran (2005) Unexpected links between Egyptian and

Babylonian mathematics, World Scientific, Singapore.

Heiberg, Johan Ludvig (1883) Euclidis Opera Omnia Vol I–IX, Teubner,

Leipzig.

Martzloff, Jean-Claude (2006) A history of Chinese mathematics,

Springer, Heidelberg.

Mellema, Elcius Edouardus Leon (1586) Second volume de l’arithme-

tique, Antwerp.

Neugebauer, Otto, and Abraham Joseph Sachs (1945) Mathematical

Cuneiform Texts, The American Oriental Society, New Haven,

Connecticut.

Sigler, Laurence (2002) Fibonacci’s Liber Abaci: A Translation into

Modern English of Leonardo Pisano’s Book of Calculation

(Sources and Studies in the History of Mathematics and Physical

Sciences), Springer, Heidelberg.

Smeur, A.J.E.M. (1960) De Zestiende-eeuwse Nederlandse

Rekenboeken, Martinus Nijhoff, s’Gravenhage.

Department of Philosophy

and Moral Science (LW01)

Ghent University

Blandijnberg 2

9000 Ghent

Belgium

e-mail: albrecht.heeffer@ugent.be

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