Euclid’s Algorithm in the Mathematic Textbooks

(1852–1907)

K. Pazourek

Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

Abstract. The Euclid’s algorithm is a traditional method of determining the

greatest common divisor. It was commonly presented in the matematic textbooks

between 1852 and 1907 for (natural) numbers as well as for polynomials.

We focused on the arithmetics and algebra textbooks for three types of

secondary school: gymnasium, realka, realne gymnasium. We described the position

and different presentations of the Euclid’s algorithm in the textbooks and mentioned

various schemes for the algorithm and their development. Finaly, we showed that

some authors studied the generalization on this algorithm or its property like

finitness and computational complexity.

Introduction

The Euclid’s algorithm was presented in the mathematic textbooks for Czech secondary

schools since their origin, i. e. 1850’s. It was used to calculate the greatest common divisor

of the (natural) numbers or the polynomials. However, the term Euclid’s algorithm was never

used, the method was called postupne delenı [succesive division], [Bydzovsky, 1910] or retezove

delenı [chain division], [Tuma, 1887].

We focus on the mathematic textbooks for types of contemporary secondary schools which

were called gymnasium, realka and realne gymnasium. They offered general curriculum (gymna-

sium was based on teaching of Latin and Greek, realka was oriented on natural science, realne

gymnasium joined both approaches) and prepared students for universities (gymnasium, realne

gymnasium) and polytechnics (realka, realne gymnasium). The textbooks were commonly of

three types: arithmetics textbooks, algebra textbooks or geometry textbooks. We are inter-

ested in the first two types. The Euclid’s algorithm was introduced in the textbooks for the

first grade of both lower and higher level of the secondary schools, in the chapters devoted to

the divisibility.

Position of the Divisibility in the Textbooks

The chapters about divisibility are located among the revision of arithmetic operations,

lectures on vıcejmenne veliciny (i. e. counting with units) and the chapter about fractions.

The divisibility is shown in the terms of the positive integers, even though the positivity is

not mentioned (in later books, like [Bydzovsky, 1910], there is claimed that the positive numbers

are considered; [Machovec 1886] actually worked with all integers).

The chapter on the divisibility is often structured in the following way: First, there is a

definition of the divisibility. Then there are introduced the terms of prime numbers (prvocet,

prvocıslo na prosto,. . . ), relative primes (prvocty na pospol,. . . ), composite numbers, odd and

even numbers. The text continues with the divisibility rules for 2, 3, 4, 5, 8, 9, and 11. In some

textbooks, there are mentioned the divisibility rules for the powers of 2, 5 and 10. The rule for

composite numbers is often demonstrated by the example of number 6.

In the algebra textbooks, the divisibility rules are mainly prooved, as well as the basic

statement of the Euclid’s algorithm, i. e. the common divisor of the divident and the divisor

also divides the reminder:

∀a, b, c, d ∈ N : (a|b & a|c & bmod c = d) ⇒ a|d.

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WDS'08 Proceedings of Contributed Papers, Part I, 23–26, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS

PAZOUREK: EUCLID’S ALGORITHM IN MATHEMATIC TEXTBOOKS (1852–1907)

The text continues with the integer factorisation, determining the greatest common divisor

(including the Euclid’s algorithm) and the least common multiple. However, [Mocnık, 1852] did

not set the term of the common divisor and omit this topic.

[Soldat, 1901] and [Hoza, 1892] added some historical notes on the topic.

Euclid’s Algorithm

Euclid’s Algorithm for (Natural) Numbers

The Euclid’s algorithm was often presented as the second method of determining the great-

est common divisor, the method based on the integer factorization is prefered.

Some authors of the arithmetics textbooks did not formulate the algorithm properly: They

just solved several exercises without an explanation of the algorithm ([Jarolımek, 1863], [Smolık,

1863], [Stary, 1882]), or they only commented these exercises without presenting general notion

([Fischer, 1870]).

On the other hand, the algebra textbooks presented the algorithm in general way, it is

usually formulated as a theorem with a proof ([Hora, 1875]) or a partial explanation. Few

authors mentioned the finiteness of the algorithm: . . . Such division must finish once; because

every reminder is lower then the corresponding divisor, the numbers diminish and they can not

become negative. [Simerka, 1863].

During the considered period 1852–1907, the algorithm was written in the different schemes.

Ther are some examples on the Figure 1.

Figure 1. Schemes: a) Fleischer, 1862; b) Jarolımek, 1863; c) Simerka, 1868; d) Fischer, 1870;

e) Studnicka, 1877; e) Taftl, 1883.

It seems that the most popular schemes were of the shape aproaching Fleischer’s and

Fischer’s ones. The possible reason is its similarity with the schemes used to describe the

integer factorization or finding all divisors of the given number.

Deeper Insights in the Theory

[Machovec, 1886] presented the Euclid’s algorithm in very exact way: The necessary theo-

rems and the algorithm itself were formulated and proved rigorously. Moreover, the divisibility

is discussed on the set of integers. That allows to accelerate the algorithm by using not only

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PAZOUREK: EUCLID’S ALGORITHM IN MATHEMATIC TEXTBOOKS (1852–1907)

the positive reminders, but the reminders of the smallest absolute value. That is why Machovec

could write the scheme in the Figure 2.

Figure 2. Scheme in Machovec’s textbook (1886).

Furthermore, the computational complexity of the Euclid’s algorithm is considered in fol-

lowing way (We remark that the theory of algorithms just developed at that time): . . . By

this method, we must reach the reminder, which equals 0. That follows from the fact that all

reminders are integers and the following inequality holds for their absolute values:

B > R1 > R2 · · · > Rn−1 > Rn.

(B stands for the divisor, Ri for the reminders, i = 1, . . . n.) We claimed that the process

could be accelerate, if we take the smallest reminders (precisely, their absolute values) R1, R2,

R3. . . Then

2R1 < B,

2R2 < R1,

2R3 < R2,

2R4 < R3,

...

2Rn < Rn−1,

By multiplying (the inequalities), we obtain 2nR1R2 . . . Rn < BR1R2R3 . . . Rn−1 and the divi-

sion by the equation R1R2 . . . Rn−1 = R1R2 . . . Rn−1 gives 2nRn < B. Because Rn is greater

or equal to 1, it rather holds 2n < B. Considering the number of used divisions is (n + 1), the

previous inequality implies:

If we consider the list of the smallest reminders (the reminders with the smallest absolute

values) during determining the greatest common divisor by succesive division, it requires to

divide at least one more time than the highest power of 2 which is still smaller than B, i. e.

the smaller number of given two ones. For example, if the smaller number of given two ones is

125, we must divide at least 7-times, because 26

= 64 and 27

= 128. ([Machovec, 1886], 70–71.)

Euclid’s Algorithm for the Polynomials

The Euclid’s algorithm for the polynomials is presented in the algebra textbooks, not in

the arithmetics textbooks. The polynomials are considered as a general kind of quantity. That

is why this topic is not provide with a large introduction, it generally consists only from few

remarks. The text usually refers to the previous algorithm for (natural) numbers. See Figure 3

for example.

Furthermore, the following statement is often presented: We do not change the greatest

common divisor of the quantities A and B, if we multiply or divide A by a number, which is

not a factor of B; the same is true for B;. . . ([Fleischer, 1862], 49.) This rule is applied to

simplify the calculations, there is calculated with polynomials with integral coefficients.

Both examples and exercises consist of polynomials of degrees 5 or lower.

General Euclid’s Algorithm

[Hora, 1875] remarked: This method for determining the greatest common measure (di-

visor) without the integer factorization of the given numbers corresponds with the method for

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PAZOUREK: EUCLID’S ALGORITHM IN MATHEMATIC TEXTBOOKS (1852–1907)

Figure 3. Euclid’s algorithm for polynomials in Machovec’s textbook (1886).

determining the greatest common measure of two homogenous quantities (e. g. two lines (it

means the line segments)). We take the smaller quantity from the greater much as possible,

than we take the reminder from the smaller quantity, the new reminder from it etc. If nothing

remains in any step, the last remainder, which not vanishes, is the greatest common measure of

the given quantities.

However, this statement is unique throughout all studied textbooks, no further author

presented this general setting.

Conclusion

The Euclid’s algorithm is commonly presented method in the arithmetics and algebra

textbooks published between 1852 and 1907. The authors presented the algorithm in different

ways, from the slight suggestions in the exercises to the complete formulation of exact theorems

and its proofs. It can be noted the tendency to improve the formulations and the schemes and

to approach the method to the students in more comfortable way.

References

Demuth, O., Kryl, R., Kucera, A., Teorie algoritmu, SPN, Prague, 1984.

Bydzovsky, B., Arithemtika pro IV. trıdu skol realnych, JCM, Prague, 1910.

Fischer, F., X., Arithmetika pro prvnı a druhou trıdu nizsıho gymnasia, F. X. Fischer, Prague, 1870.

Hora, F., A., Dra Frant. Ryt. Mocnıka aritmetika i algebra pro vyssı trıdy skol strednıch, B. Tempsky,

Prague, 1875.

Hoza, F., Algebra pro vyssı realky, I. L. Kober, Prague, 1892.

Jarolımek, C., Poctarstvı pro prvnı a druhou trıdu nizsı realne skoly, Ant. Augusta, Prague, 1863.

Machovec, F., Algebra pro vyssı trıdy skol strednıch, F. Tempsky, Prague, 1886.

Mocnık, F., Kniha pocetnı pro prvnı trıdu nızsı realnı skoly, Wien, 1852.

Smolık, J., Pocetnı kniha pro nizsı gymnasium, J. G. Calve, Prague, 1863.

Soldat, H., Dra Em. Taftla algebra pro vyssı trıdy strednıch skol ceskych, JCM, Prague, 1901.

Stary, V., Arithmetika pro prvnı, druhou a tretı trıdu skol realnych F. Tempsky, Prague, 1882.

Studnicka, F., J., Algebra pro vyssı trıdy skol strednıch, F. J. Studnicka, Prague, 1877.

Simerka, V., Algebra cili poctarstvı obecne pro vyssı gymnasia, Edv. Gregr, Prague, 1863.

Taftl, E., Algebra, Max. Cermak, Klatovy, 1883.

Tuma, F., Arithmetika pro prvou a druhou trıdu skol gymnasijnıch, F. Tuma, Prague, 1887.

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