6

planations. We have one bit of information. In discussing Heron of Alexandria (c. 200 A. D.), who lived four centuries later, Eutocius tells us that he found square and cube roots by the same method as Archimedes. That remark, together with testing for certain reconstruction formulas, seems to indicate that at times he used methods of Double False Position, as in evaluating the expression √2, and

7

at other times he used the formula Ä

ing the square root of 3.

=

(a + N),

as in find.

Now, we have Heron's processes described in his own words. A tenth century manuscript of Heron's Metrica was recently discovered by Dr. R. Schöne in the Serailo Library in Constantinople. In describing how to find the square root of 720, the area of a certain triangle, Heron says:

"Now, since 720 has no rational square root, we shall find the root differing from it by a very small error. Since the next larger square to 720 is 729 and its side is 27, we divide 720 by 27. The result is 26. To this add 27; that gives 53%. One-half of this is 26 1⁄21⁄2. Therefore the next root of 720 is 26 1⁄2 1⁄2. Multiplying 26 1⁄2 3 by itself gives 720%, so that the error amounts to only % of unity. Should we wish, however, to obtain an error smaller than 6, we may put in place of 729, the value we have just found, 720 86, and if we do this, we shall find an error much smaller than

For want of a name we shall call this the "method of averages." If a is a rational root of N, then N÷a = a. But if it is an approximation, and a little too large, as in Heron's example, then N÷ a = a1, a little smaller than the exact root. Heron takes as

the final approximation ¦(a + a1) = (a + 1).

2

2

For approximating the cube root Heron used the method of Double False Position. He explains this process as follows: "We shall now explain how to find the cube root of 100 units. Take the two cubes nearest to 100, one larger, the other smaller. Note how

• Cantor, I (1907), pp. 315-17; consult S. Günther, "Quadratische Irrationaliteten der Alten," in Abhandl. zur Geschichte der Math., Vol. IV.

7 G. Wertheim, "Heron's Ausziehung der irrationalen Kubikwurzeln," in Zeitschrift für Math. u. Phys., XLIX (1899), Hist. Lit. Abt., pp. 1-3.

8 Curtze, in Zeitschrift für Math. u. Phys., XLII (1897), p. 117-119; Wertheim, in XLIV (1899), pp. 1-3; Bibliotheca Mathematica (hereafter called Bibl. Math.), third series, VIII (1907–1908), p. 412; Cantor, I (1907), p. 374.

much larger the first one is, namely 25, and how much smaller the second one is, namely 36. Then multiply 36 by 5; the result is 180. To this add 100, which gives 280 [and divide 180 by 280]; the result is 14. Add this to the root of the smaller cube, namely 4; the result is 414. Which is the value of the cube root of 100 units as nearly exact as possible."

The ambiguity of the numbers in this example (5 may be √125 and also √125-100, etc.) excludes a precise knowledge of Heron's process; but it seems to differ from those used by the medieval writers. Curtze, Wertheim, Cantor, and Eneström have all worked out different reconstruction formulas for Heron's process for approximating cube roots.

4. THEON'S METHOD OF EXHAUSTION

The first instance in extant writing of a method of finding square root similar to our present method is found in the writings of Theon of Alexandria (c. 365 A. D.). He gives a very full description of how to evaluate irrational roots in sexagesimal fractions. It is a geometrical process referred to a figure. In it he applies a process of exhaustion to Euclid's geometrical counterpart of the expression (a + x)2. = a2 + 2ax + x2. This is followed by a generalizing rule. Expressed in modern analytical form, if a2 + b = (a + x)2 b a2 + 2ax + x2 and a is a first approximation, then x = approxi

20

=

mately, and va2 + b = a + approximately. In his illustrative

b 20

example he finds the square root of 4500° to be 67° 4' 45".

Diophantus did brilliant work in approximation, but this was chiefly in connection with inequalities and indeterminate equations, both of which are outside of our topic.

5. SUMMARY

1. We find the first concept of roots in Egypt or in Babylon; in Egypt we also find the solution of quadratic equations by single false position.

2. The concepts of irrationality and of higher roots are first found among the Greeks. Theirs are also the earliest records of a scientific notion of limit and of scientific approximation.

'Gow, History of Greek Mathematics, Cambridge, 1884, p. 55.

3. We find among them the first recorded use of three approximation methods: namely, the methods of averages and of double false position, by Archimedes and Heron; and the method of exhaustion, by Theon. Theon's method differs from Heron's methods in giving sharper relief to the limit idea; the method of averages and the method of double false position really have two variable limits. Though differing in principle and operation, Heron's method of averages and Theon's method of exhaustion give the same results at each step. For, if Na2+ b, then will N √a2 + b =

=

4. The existence of a root, and never more than one root, was taken for granted, among both Egyptians and Greeks.

III

ALGORITHMIC METHODS FOR APPROXIMATING

THE ROOTS OF PURE POWERS

I. THE HINDU METHOD OF EXHAUSTION

The superior notation and convenient symbolism of the Hindus enabled them to use processes of inversion to a degree impossible to Theon. In fact that became the favorite line of attack in much of their mathematics. Square and cube roots were found by Āryabhaṭa1 (b. 476 A. D.) from the inverse of the formulas a2+2ab+b2 and a3 + 3a2b + 3ab2 + b3.

When the Hindu decimal notation with its place values came into use, the work of finding roots was so arranged that an application of the inversion process would evolve the digits of the root in order (whence "evolution" in texts on arithmetic and algebra). To effect this the number was divided into periods and the work arranged in columns and lines. This "Hindu method," as it was often called in the Middle Ages, is described for us by Brahmagupta (b. 598 A. D.).2 Śrīdhara,3 in his Triśatikā (1040 A. D.) gives the rules in full, as also does Bhaskara (b. 1114 A. D.). The following is a free translation from Bhaskara, by Taylor:

"The first place on the right is called ghana or cube; the two next places aghana or not-cube.-Subtract the cube contained in the final period from the said period; put down the root of the cube in a separate line, and after multiplying its square by three, divide the antecedent figure by the result, and write down the quotient in the separate line: Then multiply the square of the quotient by the preceding number in that line and by three, and after subtracting the product from the next antecedent figure cube the said quotient, and subtract the result from the next antecedent figure. Thus repeat the process through all the figures. The separate line contains the Cube Root."

1 Cantor, I (1907), p. 625.

2 H. T. Colebrooke's translations of Brahmagupta and Bhaskara, London, 1817, pp. 279-81. Hereafter referred to as Colebrooke.

Kaye, on Tristika, in Bibl. Math. (3) XIII (1912–13), p. 209.

4 Taylor's translation of Lilawati, Bombay, 1816, p. 20.

5

The Hindu method was adopted by the Arabs, and Omar Khayyam (c. 1100 A. D.) wrote an exposition of it in a work that has been lost. From the Arabs it came to Christian Europe through the writings of Leonardo of Pisa (1202); and the arrangement of the work as given in Sacrobosco's Algorismus (c. 1240) remained in use until it was supplanted by that of Peurbach in the fifteenth century. Our modern arrangement of the work in using the Hindu method begins with Peurbach (1423-1461), whose prestige as an arithmetician and an astronomer caused his schema to be adopted throughout Europe. Slight modifications were added by Chuquet (1484), LaRoche (1520), and Gemma Frisius (1540). In the Oriental countries an abacus-like arrangement obtained.

2. FORMULAS FOR IRRATIONAL ROOTS IN THE MIDDLE AGES The Hindu view of irrationals differed somewhat from that of the Greeks. To the latter, irrational numbers represented incommensurable lines, and hence they were slow to evaluate them arithmetically. To the Hindu an irrational number was "one, the root of which is required, but cannot be found without residue.”7 As far back as before 200 A. D., in the Sulvasūtras, we find remarkably close approximations for √2, √3, and other surds, by the altar computers. Thus Baudhāyana states that √2

8

=

Their method is not known. It does not

seem to have been a method of inversion. But later writers' used the principle of inversion for irrationals almost entirely.

b

20

Among the Arabs Theon's rule, va2 + b = a + was used by Albategnius (c. 920), Abûl-Wefâ (940-998), al-Nasavi (c. 1025),

5 Consult the introduction to C. J. Gerhardt's Das Rechenbuch des Maximus Planudes, Halle, 1865.

"Treutlein, "Das Rechnen in 16. Jahrhundert," in Abhandl. Math. Wissenschaft (1877); Triparty in Boncompagni's Bulletino di bibliografia e di storia delle scienze matematiche e fisiche (hereafter referred to as Boncompagni), XIII (1880), p. 695; Cantor, II (1900), p. 411.

7 Colebrooke, p. 145.

Kaye, Indian Mathematics, Calcutta, 1915, pp. 2-8; S. Günther, Quadratische Irrationaliteten der Alten, pp. 39–42; p. 112.

Colebrooke, p. 155, Example 51.