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Historical Topics: Elementary Algebra Author(s): D. R. Green Source: Mathematics in School, Vol. 7, No. 3 (May, 1978), pp. 12-15 Published by: The Mathematical Association Stable URL: http://www.jstor.org/stable/30214226 . Accessed: 22/04/2014 11:13 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to Mathematics in School. http://www.jstor.org This content downloaded from 130.239.116.185 on Tue, 22 Apr 2014 11:13:06 AM All use subject to JSTOR Terms and Conditions

Historical Topics: Elementary Algebra

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Page 1: Historical Topics: Elementary Algebra

Historical Topics: Elementary AlgebraAuthor(s): D. R. GreenSource: Mathematics in School, Vol. 7, No. 3 (May, 1978), pp. 12-15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214226 .

Accessed: 22/04/2014 11:13

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

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Page 2: Historical Topics: Elementary Algebra

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Illustration from the title page of Cardan's Ars Magna, 1545 (from the Turner Collection, Univer- sity of Keele)

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ELEMENTARY

ALGEBRA by D. R. Green CAMET Loughborough University

As with many branches of mathematics, the origins of algebra are somewhat obscure. The stages through which algebra has passed have been named by G. Nesselmann (1 842) as: rhetorical, syncopated, and symbolic. Originally the problems and solutions were set out in long-hand form using words, without any abbreviations or symbols (other than those for the numbers themselves) - this was the rhetorical form. Later on abbreviations were introduced for the various words which kept occurring, e.g. for the operations and unknowns - this was the syncopated form. Gradually there were developed special symbols e.g. +, x, x2, -x - this being the symbolic form. The above three stages were by no means clearcut, and much overlapping and variation can be discerned. Nevertheless the overall trend from the lengthy and clumsy rhetorical algebra through to the terse and efficient symbolic algebra of today has been a development of major significance in the history of mathematics.

The Babylonians It is thought that algebra originated in Babylonia and quite sophisticated problems are to be found on clay tablets dating back to King Hammurabi's reign around 1700 BC. A typical problem and solution (translated into English and decimal notation!) is as follows:

1. Length, width. I have multiplied length and width to obtain the area 192. I have added length and width to obtain 28. Required to find: length and width

2. 28 is the sum; 192 is the area

3. 16 is the length; 12 is the width 4. The method: (i) Take half of 28, to give 14 (ii) 14x14=196 (iii) 196- 192= 4 (iv) Square root of 4 is 2 (v) 14+2=16, length (vi) 14- 2= 12, width

5. I multiply length and width 16x 12= 192, area

Note the five stages in the working:

1. Statement of the problem 2. Assembly of the data 3. Statement of the answer 4. Outline of the method 5. Checking the answer

This rhetorical approach would be repeated with a large variety of problems so that the nature of the method of solution would become apparent. A general example (using unknowns like x and y) was not possible and only specific examples with particular numbers could be handled. The reader might care to repeat the above problem using f for length and w for width and so check the validity of the method in general.'

The above problem essentially requires solving a quadratic equation, and there are many such problems found on ancient clay tablets from Mesopotamia. The Babylonians were also able to solve certain kinds of cubic and quartic equations, with numerical coefficients. A clay tablet containing values of n3+n2 has been found, for n from 1 to 30. This table could be directly used to find a solution to equations such as

n3 + n2= 12, of course,

and even equations like

3x3+ 2x2= 224

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Page 3: Historical Topics: Elementary Algebra

could be dealt with by making certain transformations.

e.g. Multiply through by 9:

27x3+ 18X2= 201 6

Let 3x=p:

p3+ 2p2= 201 6 Let p= 2n:

8n3+ 8n2 = 201 6

Divide by 8:

n3 + n2= 252

Now by consulting the table it is seen that n = 6, so x= 4. Despite a notational system very inferior to our own the Babylonians showed themselves as very fine algebraists.

As Carl Boyer2 has said "The solution of quadratic and cubic equations in Mesopotamia is a remarkable achievement to be admired not so much for the high level of technical skill as for the maturity and flexibility of the algebraic concepts that are involved". The reader might care to construct a table of values of n3+n2 for n= 1 to 10 (say) and use it to solve the following problems: (i) 2x3+3x2= 540 (ii) p3+4p2= 1053 (For solutions, see note 3.)

The Egyptians Many of the problems of the Ahmes or Rhind Papyrus (referred to in a previous article4) are arithmetical but some can be more accurately classified as algebraic, requiring the solving of linear equations.

The unknown, which we usually designate as x, was called "aha" or heap. Problem 24 is typical:

"Heap and one seventh of heap is 19. What is heap?"

We could solve as follows:

x+7 x=19 = 7 Ax=19 = x=Ux 19 = x= 161

However the Egyptian method was quite different, and based on making a guess and then correcting it, known as the "rule of false", a method still in use until modern times. (i) Assume heap= 7 (ii) Then heap and

- heap is 8

(iii) However it should be 19 Thus we must increase our guess (7) by a factor F. (The Egyptians worked with unit fractions so they in fact would take not 9 but its equivalent 2+y+ 1.)

(iv) Thus the answer is 7(2+ + ) which gives 16 +y+ 1/8

The reader might like to solve by the method of false, working with unit fractions, the similar problem:

"Heap and ( heap is 13. What is heap?" (For solution, seu note 5.)

The Greeks The Greek approach to algebra was geometric. To take a typical example which is familiar to many of us consider the following theorem (Theorem 41 in Hall and Stevens" based on Euclid 11.4):

"If a straight line is divided internally at any point, the square on the given line is equal to the sum of the squares on the two segments together with twice the rectangle contained by the segments."

This is made much clearer by the diagram in Fig. 1, where the "straight line" is represented by AB

A a b B

aI

bI

Fig. 1

The Euclidean proof (given in Hall and Stevens6, page 222) is based on geometrical construction. In terms of algebra, however, we really have the statement

(a + b)2 = a2 +b2 + 2ab

For the earlier Greeks there was considerable difficulty in dealing with a2 other than in terms of an actual square, and their algebra was thus most cumbersome and so could not be developed fully.

Later on Diophantus made a fresh approach to algebra, introducing his syncopated style with abbreviated words. Quite different words were originally used in algebra to express what we now would call x, x2, x3 and so on. Diophantus, in his major work on higher arithmetic, called the Arithmetica, used the following system.

modern Symbol abbreviation for meaning equivalent

S MONAAEE (MONADES) units 1 s alQCPos (ARITHMOS) number x AT ArNAMIC (DUNAMIS) power x2 KT KrBOE (KUBOS) cube x3 ATA ArNAMODrNAMIE power power X4

No symbol for addition was used, and the minus sign was A, derived from AEIlE (LEIPIS) meaning v "lacking". Equality was shown by Eanr, meaning "is equal to", or iaos, meaning "equal". The Greeks did not use our modern Hindu-Arabic system of numeration but had their own less sophisticated decimal system. The first nine Greek letters stood for the integers 1 to 9:

C1 0 -Y E S 0 1 2 3 4 5 6 7 8 9

and later Greek letters represented 10, 20 to 90: L x X A o 7r e

10 20 30 40 50 60 70 80 90

Thus the expression

syA KT? q EaTL X-q

represents x3- X38 1 x2=38

or 3x- (8X3 + 2) = 38

The reader might like to interpret the expression below found in a copy of one of Diophantus' manuscripts:

100" A A OP C~7 C1 (See note 7 for the solution.)

Diophantus' importance, however, does not so much lie in his first steps toward modern abbreviated algebraic notation as in the mathematical problems which he solved. His major work, the Arithmerica, consisted of 13 books (only the first six, alas, surviving) dealing with indeterminate equations -

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Page 4: Historical Topics: Elementary Algebra

those with more than one solution, typically several equations in several unknowns and solution restricted to the positive rationals or integers. This is a work of great ingenuity but belonging more to number theory than to algebra as we know it.

The Hindus and Arabs The Hindus Brahmagupta (c. 630) and Bhaskara (c. 11 50) were two of the most important algebraists of the middle ages. The Hindus used a syncopated style and solved the quadratic equations by completing the square, and attempted to find all possible integer solutions to Diophantine equations, which Diophantus himself had not done, including the famous, and misnamed, Pell equation:

y2= ax2+ 1 where a is a non-square integer.

When the Arabs conquered India to the east and North Africa and Spain to the west, at the turn of the eighth century, their scholars acquired the great scientific and mathematical works of both Greek and Hindu traditions. Thus these important works were translated into Arabic and so preserved, and the highly efficient Hindu system of numeration, which we use today, was made available to the world at large. When the European Christians pushed westwards (reconquering Spain for example) beginning in the late eleventh century, many Arabic documents were translated into Latin and greatly influenced subsequent European mathematics.

The word "algebra" The term "algebra" is a Latinised version of the Arabic word al-jabr which occurred in the title of a book written around AD 825 by the Arab Mohammed ibn-Musa al-Khowarizmi. The full title of the book was Hisab al-jabr w'al-muqabalah which possibly means "science of transposition (of negative terms) and cancellation (of like terms)"

For the example, with the equation x3- 4x2+ 5= x2+ X + 5

al-jabr gives x3+ 5= 5x2+ X + 5

al-muqabalah gives X3= 5X2+ X

Al-Khowarizmi's book was an exhaustive treatise on solving quadratic equations by the method of "completing the square". Only positive answers are given, in keeping with the understanding of the time. Carl Boyer2 says "So systematic and exhaustive was al-Khowarizmi's exposition that his readers must have had little difficulty in mastering the solutions. In this sense, then, al-Khowarizmi is entitled to be known as 'the father of algebra"'

The Italian Algebraists The book on arithmetic and elementary algebra Liber Abaci by the Italian Leonardo of Pisa (known as Fibonacci) can be said to have been the first significant European work, written in 1202. However, little real progress was made until the sixteenth century when a number of Italian algebraists made advances in solving cubic and high degree equations.

These were not the first mathematicians to solve cubics but the earlier efforts (Babylonian, Persian, Greek, etc.) had been directed towards specific equations whereas the sixteenth century Italians were able to deal with general classes of cubic and quartic equations.

Scipione del Ferro (1465-1 526), Tartaglia (1 500-57) and Cardano (1501-76) all contributed to the solving of cubic equations of various kinds and Cardano's pupil

Ferrari (1522-65) went on to extend Cardano's work to solve the general quartic. The two most influential algebra books published were Cadano's Ars Magna (1545) which described a comprehensive set of methods for solving cubic and biquadratic (quartic) equations and Bombelli's Algebra which summarised the existing knowledge and went beyond in dealing to some extent with complex numbers, a subject which had intruded into Cardano's work in solving equations. (See note 8 for a discussion of the history of complex numbers.)

One of the most important books in the whole history of mathematics was, undoubtedly, the Ars Magna of Gerolamo Cardano, published in 1545 in Nirnberg. At this time negative numbers and even zero were treated with suspicion so all cubic equations had to be formulated to have only positive coefficients. Thus x3 + 2x = 5 would be treated quite differently from x3= 2x + 3 for example rather than both being of the type x3+ ax= b. This meant that there were many different forms which a cubic could take. Cardano, with help from Tartaglia (and from del Ferro's manuscripts which he later examined) knew how to solve case of the types x3+ ax= N ("cube and first power equal to number" as Cardano expressed it) and x3+ ax2= N and from these he developed techniques for similarly dealing with the other cases.

We take as an example on solving a cubic, the case "On the Cube Equal to the First Power and Number", adapted from Witmer's translation of the Ars Magna9.

Rule When the cube of one-third the coefficient of x is not greater than the square of one-half the constant of the equation, subtract the former from the latter and add the square root of the remainder to one- half the constant of the equation and, again, subtract it from the same half, giving two terms, the sum of whose cube roots gives the value of x. (See'o for the modern algebraic expression.)

Example x3= 6x+ 40 Raise 2, one-third the coefficient of x, to the cube, which makes 8; subtract this from 400, the square of 20, one-half the constant, making 392; the square root of this added to 20 makes 20+ /392 and subtracted from 20 makes 20- /392; and the sum of the cube roots of these, 3 20+/352 + 3 20--/392, is the value of x.

The reader might like to solve x3= 6x+ 6 by this method (for solution, see note 11).

Many such rules and examples are given in what amounts to a very long and rather tedious book. With the poor notation also employed in the original Latin version the Ars Magna is even more difficult to follow and the "geometric proofs" (in deference to Euclid) do nothing to add to its readability. To illustrate the notation, the expression above:

3 /20+ /392+ 3 /20- e/392

appeared in the Ars Magna as

R v.cubicam 20.p.R .392.p.R .v.cubica 20.i. R .392.

where R means "radix" or "root", i.e. / R .v. means "radix universalis" i.e. J-e, but the

extent is unspecified. cubica(m) means "cube", so "R.cubicam" means 3/ Smeans "plus"

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Page 5: Historical Topics: Elementary Algebra

However, it was a major step forward at that time, and a remarkable achievement. The intrigues and scandals associated with Cardano's obtaining Tartaglia's secret makes interesting reading (see any standard history of mathematics text) but should not mislead the reader into believing the Cardano's own contribution was not outstanding. The Europeans There was a great surge of interest in algebra through- out Western Europe in the fifteenth and sixteenth centuries. A French work of great importance, comparable to Fibonacci's Liber Abaci of 1 202, was Triparty en la science des nombres by Chuquet (c. 1 500). The third part of this three part work concerned algebra and Chuquet invented an exponential notation, the forerunner to our modern system. For example, the terms which we would express as 5x3, 4X2, 6x, 8, 3x-2 he designated by .5.3, .4.2,.6.1, .8.0, .3.2.m. It was, however, some two centuries before an exponent notation was to become widely used.

Another great French mathematician, Francois Vi'te (1540-1603), known as Vieta, made great advances in algebra. All down the ages algebra had been a "bag of tricks" dealing largely with numerical examples and hosts of special cases. There seemed to be no way to talk about the general quadratic or cubic, for example, nothing equivalent to having a figure of a general triangle, which the geometer could use. A basic problem was distinguishing between unspecified coefficients and unknown variables. For the first time Vieta made this distinction: vowels represented the unknown variables, consonants represented the constants. Had Vieta been completely modern he might have adopted a system such that he could write all quadratic equations as BA2 + CA+ D, where A is the

unknown. However, he still had one foot in the past, and used the same letter with a differing word following to indicate the power: A, A quadratum, A cubum ... for A, A2, A3 for example. (Others abbreviated this to A, Aq, Ac, etc.)

The advances being made in arithmetic notation, referred to in a previous articlel2 were also of influence being naturally relevant to the generalised arithmetic which is algebra. Vieta, for example did adopt the Germanic symbols + and -. Various mathematicians improved the symbolism and by the time of Newton (1642-1727) the notation used in algebra had become much as we use it today and the growth of modern algebra was to follow.

References and notes 1. Stage 1 e+ w= Sum

ew = Area Stage 4 (i) (e+w)/2

(ii) (e+ w)2/4 (iii) (e+ w)2/4-tw= (V- w)2/4 (iv) /I(f- w)2141 = (V- w)1/2 (v) (+ w)/2+ (V- w)/2 =f (vi) (V+ w)/2- (V- w)/2 = w

2. Boyer, C. B. A History of Mathematics. Wiley, 1968 3. (i) x=6 (ii) p=9 4. Green, D. R. Historical Topics: Plane Trigonometry

Mathematics in School, Vol. 6, No. 5, November 1977, p. 7 5. 101=10+1+1 6. Hall, H. S. and Stevens, F. H. A School Geometry Parts I-VI

(First Edition) (1903) Macmillan, 1951 7. 2x3+8x- (5x2+4)=44 8. Green, D. R. The Historical Development of Complex Numbers

The Mathematical Gazette, Vol. 60, No. 412, June 1976, p. 99 9. Witmer, T. R. The Great Art (by G. Cardano). M.I.T. Press, 1 968.

10. Given x3= aX+ N and provided (a/3)3 < (N/2)2 then: x = 3N/2+(/2 2)2- "W+3V N/2- -(/)

11. 3/4+3N/2 12. Green, D. R. Historical Topics: Arithmetic symbols and algebraic

notation. Mathematics in School, Vol. 6, No. 1, January 1977, p. 2

THIE RHOMBIC I)OIEC IlEDRON By Henry Lulli, Fremont High, Los Angeles, CA 90003

Fig. 1 Fig. 2

A

B s

DL ic

F G B

/i/ --/-- ------ -- L-

D~ H C

There exist two dodecahedra: the regular Platonic polyhedron of 12 pentagonal faces; and one with 12 rhombic faces. The latter, a stellated cube, is the dual of the Archimedean cuboctahedron. It is also the rhombic dodecahedron. It can be made from four square sheets of paper without the aid of geometric instruments, scissors or glue.

The model is made in three sections: a top, a central ring and a bottom. The ring is made of two strips. Both strips, top and bottom are composed of three rhombi each. To avoid extra folds in the finished model, a fourth sheet is used as a template from which the measurements are transferred. The folds should be as accurate as possible as the errors are cumulative, causing the rhombi to buckle. Pencil in the lines which will be referred to again, particularly at the intersections.

Start with a square sheet ABCD. Crease both diagonals. Open the sheet and fold B to the centre. Now open the crease and fold B to the quarter point. Finally, fold B to the last crease. See Figure 1. Label the 3/16 point E, as shown. Open the creases.

Fold AD to E; and BC on top, creasing through E. Open the sheet. Label these lines FI and GH as in Figure 2. Locate point J as the intersection of two lines. First, fold C to F; then,

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