Introduction:

“You can learn more from solving one problem in many different ways than you can from

solving many different problems, each in only one way.”

Islamic civilization in the middle ages, like all of Europe, had a dichotomy between

theoretical and practical mathematics. Practical mathematics was the common subject,

“whereas theoretical and argumentative mathematics were reserved for specialists”

(Abedljaouad, 2006, p. 629). Between the eighth and the fifteenth centuries, Islamic

civilization produced a series of remarkable mathematicians. Among them was Ghiyath al-

Din Jamshid Mas’ud al-Kashi. Following this dichotomy, al-Kashi designed his book for use

by students who were looking to apply mathematics in their professions. The book does not

contain any theoretical proof for any problem, but it does contain methods for solution and

correctness verification, such as performing the opposite operation to check a result, and the

method of casting out nines to check whether the product, quotient, or root is correct.

Objective of the study:

1. Life History of Ghiyath Ai-Din Jamshid Mas'ud Al-Kashi

2. Contribution in Mathematics

3. Multiples Algorithm and Multiple Solutions

4. Law of Cosines

5. Fixed Point Iteration Method

6. Calculation of PI

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1. Life History of Jamshid al-Kashi

Al-Kashi was one of the best mathematicians in the Islamic world. He was born in

1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as

Timur, who was more interested in invading other areas than taking care of what he had. Due

to this, al-Kashi lived in poverty during his childhood and the beginning years of his

adulthood. He was born in Kashan which lies in a desert at the eastern foot of the Central

Iranian Range. At the time that al-Kashi was growing up Timur (often known as

Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and

restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests

in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between

his two sons, one of whom was Shah Rokh.

While Timur was undertaking his military campaigns, conditions were very difficult

with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and

devoted himself to astronomy and mathematics while moving from town to town. Conditions

improved markedly when Shah Rokh took over after his father's death. He brought economic

prosperity to the region and strongly supported artistic and intellectual life. With the changing

atmosphere, al-Kashi's life also improved markedly. The first event in al-Kashi's life which

we can date accurately is his observation of an eclipse of the moon which he made in Kashan

on 2 June 1406.

It is reasonable to assume that al-Kashi remained in Kashan where he worked on

astronomical texts. He was certainly in his home town on 1 March 1407 when he completed

Sullam Al-sama the text of which has survived. The full title of the work means The Stairway

of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of

Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to

obtain patronage from their kings, princes or rulers. Al-Kashi played this card to his

advantage and brought himself into favour in the new era where patronage of the arts and

sciences became popular. His Compendium of the Science of Astronomy written during 1410-

11 was dedicated to one of the descendants of the ruling Timurid dynasty.

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Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became

the capital of Timur's empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city.

Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was

to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij

which was based on the tables of Nasir al-Tusi. In the introduction al-Kashi says that without

the support of Ulugh Beg he could not have been able to complete it. In this work there are

trigonometric tables giving values of the sine function to four sexagesimal digits for each

degree of argument with differences to be added for each minute. There are also tables which

give transformations between different coordinate systems on the celestial sphere, in

particular allowing ecliptic coordinates to be transformed into equatorial coordinates.

The Khaqani Zij also contains :-

... detailed tables of the longitudinal motion of the sun, the moon, and the planets. Al-Kashi

also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical

latitudes, tables of eclipses, and tables of the visibility of the moon.

Al-Kashi had certainly found the right patron in Ulugh Beg since he founded a

university for the study of theology and science at Samarkand in about 1420 and he sought

out the best scientists to help with his project. Ulugh Beg invited Al-Kashi to join him at this

school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada.

There is little doubt that al-Kashi was the leading astronomer and mathematician at

Samarkand and he was called the second Ptolemy by an historian writing later in the same

century.

Letters which al-Kashi wrote in Persian to his father, who lived in Kashan, have

survived. These were written from Samarkand and give a wonderful description of the

scientific life there. In 1424 Ulugh Beg began the construction of an observatory in

Samarkand and, although the letters by al-Kashi are undated they were written at a time when

construction of the observatory had begun. The contents of one of these letters has only

recently been published.

In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other

scientists in Samarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific

meetings where problems in astronomy were freely discussed. Usually these problems were

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too difficult for all except al-Kashi and Qadi Zada and on a couple of occasions only al-Kashi

succeeded. It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh

Beg at Samarkand and, despite al-Kashi's ignorance of the correct court behaviour and lack of

polished manners, he was highly respected by Ulugh Beg. After Al-Kashi's death, Ulugh Beg

described him as (see for example :-

... a remarkable scientist, one of the most famous in the world, who had a perfect command of

the science of the ancients, who contributed to its development, and who could solve the most

difficult problems.

Although al-Kashi had done some fine work before joining Ulugh Beg at Samarkand,

his best work was done while in that city. He produced his Treatise on the Circumference in

July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this

into sixteen decimal places. This was an achievement far beyond anything which had been

obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal

places in the 5th century). It would be almost 200 years before van Ceulen surpassed Al-

Kashi's accuracy with 20 decimal places.

Al-Kashi's most impressive mathematical work was, however, The Key to Arithmetic

which he completed on 2 March 1427. The work is a major text intended to be used in

teaching students in Samarkand, in particular al-Kashi tries to give the necessary mathematics

for those studying astronomy, surveying, architecture, accounting and trading. The authors of

describe the work as follows:-

In the richness of its contents and in the application of arithmetical and algebraic methods to

the solution of various problems, including several geometric ones, and in the clarity and

elegance of exposition, this voluminous textbook is one of the best in the whole of medieval

literature; it attests to both the author's erudition and his pedagogical ability.

Dold-Samplonius discussed several aspects of al-Kashi's Key to Arithmetic. For

example the measurement of the muqarnas refers to a type of decoration used to hide the

edges and joints in buildings such as mosques and palaces. The decoration resembles a

stalactite and consists of three-dimensional polygons, some with plane surfaces, and some

with curved surfaces. Al-Kashi uses decimal fractions in calculating the total surface area of

types of muqarnas. The qubba is the dome of a funerary monument for a famous person. Al-

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Kashi finds good methods to approximate the surface area and the volume of the shell

forming the dome of the qubba.

Rashed puts al-Kashi's important contribution into perspective. He shows that the main

advances brought in by al-Kashi are:-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.

(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for

real numbers such as π.

The last work by al-Kashi was The Treatise on the Chord and Sine which may have

been unfinished at the time of his death and then completed by Qadi Zada. In this work al-

Kashi computed sin 1° to the same accuracy as he had computed π in his earlier work. He

also considered the equation associated with the problem of trisecting an angle, namely a

cubic equation. He was not the first to look at approximate solutions to this equation since al-

Biruni had worked on it earlier. However, the iterative method proposed by al-Kashi was :-

... one of the best achievements in medieval algebra. ... But all these discoveries of al-Kashi's

were long unknown in Europe and were studied only in the nineteenth and twentieth centuries

by ... historians of science....

Let us end with one final comment on the al-Kashi's work in astronomy. We

mentioned earlier the astronomical tables Khaqani Zij produced by al-Kashi. It is worth

noting that Ulugh Beg also produced astronomical tables and sine tables, and it is almost

certain that these tables were based on al-Kashi's tables and almost certainly produced with

al-Kashi's help.

2. Contribution in Mathematics:

Al-Kāshī’s best-known work is Miftāḥ al-ḥisāb (1427), a veritable encyclopedia of

elementary mathematics intended for an extensive range of students; it also considers the

requirements of calculators—astronomers. land surveyors, architects, clerks, and merchants.

In the richness of its contents and in the application of arithmetical and algebraic methods to

the solution of various problems, including several geometric ones, and in the clarity and

elegance of exposition, this voluminous textbook is one of the best in the whole of medieval

literature; it attests to both the author’s erudition and his pedagogic ability.20 Because of its

high quality the Miftāḥ al-ḥisāb was often recopied and served as a manual for hundreds of

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years; a compendium of it was also used. The book’s title indicates that arithmetic was

viewed as the key to the solution of every kind of problem which can be reduced to

calculation, and al-Kāshī defined arithmetic as the “science of rules of finding numerical

unknowns with the aid of corresponding known quantities.”21 The Miftāḥ al-ḥisāb is

divided into five books preceded by an introduction: “On the Arithmetic of Integers,” “On the

Arithmetic of Fractions,” “On the ‘Computation of the Astronomers’”(on sexagesimal

arithmetic), “On the Measurement of Plane Figures and Bodies,” and “On the Solution of

Problems by Means of Algebra [linear and quadratic equations] and of the Rule of Two False

Assumptions, etc.” The work comprises many interesting problems and carefully analyzed

numerical examples.

In the first book of the Miftāḥ, al-Kāshī describes in detail a general method of

extracting roots of integers. The integer part of the root is obtained by means of what is now

called the Ruffini—Horner method. If the root is irrational, (a and r are integers), the

fractional part of the root is calculated according to the approximate formula 22Al-Kāshī

himself expressed all rules of computation in words, and his algebra is always purely

“rhetorical.” In this connection he gives the general rule for raising a binomial to any natural

power and the additive rule for the successive determination of binomial coefficients; and he

constructs the so-called Pascal’s triangle (for n = 9). The same methods were presented

earlier in the Jāmiʿal-ḥisāb biʾl takht waʾl-tuzāb (“Arithmetic by Means of Board and

Dust”) of Nasīr al-Din al-Tũsī (1265). The origin of these methods is unknown. It is possible

that they were at least partly developed by al-Khayyāmī the influence of Chinese algebra is

also quite plausible.23

Noteworthy in the second and the third book is the doctrine of decimal fractions, used

previously by al-Kāshī in his Risāla al-muhītīyya. It was not the first time that decimal

fractions appeared in an Arabic mathematical work; they are in the Kitāb al-fusūl fiʾl-

hisāb al-Hindi (“Treatise of Arithmetic”) of al Ulīdisī (mid-tenth century) and were used

occasionally also by Chinese scientists.24 But only al-Kāshī introduced the decimal fractions

methodically, with a view to establishing a system of fractions in which (as in the

sexagesimal system) all operations would be carried out in the same manner as with integers.

It was based on the commonly used decimal numeration, however, and therefore accessible to

those who were not familiar with the sexagesimal arithmetic of the astronomers. Operations

with finite decimal fractions are explained in detail, but al-Kāshī does not mention the

phenomenon of periodicity. To denote decimal fractions, written on the same line with the

integer, he sometimes separated the integer by a vertical line or wrote in the orders above the

figures; but generally he named only the lowest power that determined all the others. In the

second half of the fifteenth century and in the sixteenth century al-Kāshī’s decimal fractions

found a certain circulation in Turkey, possibly through ʿAlī Qūshjī, who had worked with

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him at Samarkand and who sometime after the assassination of Ulugh Bēg and the fall of the

Byzantine empire settled in Constantinople. They also appear occasionally in an anonymous

Byzantine collection of problems from the fifteenth century which was brought to Vienna in

1562. It is also possible that al-Kāshī’s ideas had some influence on the propagation of

decimal fractions in Europe.

In the fifth book al-Kāshī mentions in passing that for the fourth-degree equations he

had discovered “the method for the determination of unknowns in. . . seventy problems which

had not been touched upon by either ancients or contemporaries.” He also expressed his

intention to devote a separate work to this subject, but it seems that he did not complete this

research. Al-Kāshī’s theory should be analogous to the geometrical theory of cubic equations

developed much earlier by Abu’l-Jũd Muhammad ibn Laith, al-Khayyāmī (eleventh century),

and their followers: the positive roots of fourth-degree equations were constructed and

investigated as coordinates of points of intersection of the suitable pairs of conics. It must be

added that actually there are only sixty-five (not seventy) types of fourth-degree equations

reducible to the forms considered by Muslim mathematicians, that is, the forms having terms

with positive coefficients on both sides of the equation. Only a few cases of fourth-degree

equations were studied before al-Kāshī.

Al-Kāshī’s greatest mathematical achievements are Risāla al-muhitiyya and Risāla al-

watar waʾl-jaib, both written in direct connection with astronomical researches and

especially in connection with the increased demands for more precise trigonometrical tables.

At the beginning of the Risāla al-muḥītīyya al-Kāshī points out that all approximate

values of the ratio of the circumference of a circle to its diameter, that is, of π, calculated by

his predecessors gave a very great (absolute) error in the circumference and even greater

errors in the computation of the areas of large circles, Al-Kāshī tackled the problem of a more

accurate computation of this ratio, which he considered to be irrational, with an accuracy

surpassing the practical needs of astronomy, in terms of the then-usual standard of the size of

the visible universe or of the “sphere of fixed stars.”27 For that purpose he assumed, as had

the Iranian astronomer Qutb al-Din al-Shīrāzī (thirteenth-fourteenth centuries), that the radius

of this sphere is 70,073.5 times the diameter of the earth. Concretely, al-KĀshī posed the

problem of calculating the said ratio with such precision that the error in the circumference

whose diameter is equal to 600,000 diameters of the earth will be smaller than the thickness

of a horse’s hair. Al-Kāshī used the following old Iranian units of measurement: I parasang

(about 6 kilometers) = 12,000 cubits, 1 cubit = 24 inches (or fingers), 1 inch = 6 widths of a

medium-size grain of barley, and I width of a barley grain = 6 thicknesses of a horse’s hair.

The great-circle circumference of the earth is considered to be about 8,000 parasangs, so al-

Kāshī’s requirement is equivalent to the computation of π with an error no greater than 0.5

7

·10-17. This computation was accomplished by means of elementary operations, including the

extraction of square roots, and the technique of reckoning is elaborated with the greatest care.

Al-Kāshī’s measurement of the circumference is based on a computation of the

perimeters of regular inscribed and circumscribed polygons, as had been done by

Archimedes, but it follows a somewhat different procedure. All calculations are performed in

sexagesimal numeration for a circle with a radius of 60. Al-Kāshī’s fundamental theorem—in

modern notation—is as follows: In a circle with radius r,

where crd α° is the chord of the arc α° and α° < 180°. Thus al-Kāshī applied here the

“trigonometry of chords” and not the trigonometric lines themselves. If α = 2φ° and d = 2,

then al-Kāshī’s theorem may be written trigonometrically as which is found in the work of J.

H. Lambert (1770). The chord of 60° is equal to r, and so it is possible by means of this

theorem to calculate successively the chords c1, c2, c3. . . . of the arcs 120°, 150°, 165°, in

general the value of the chord cn of the arc will be . The chord cn being known, we may,

according to Pythagorean theorem, find the side of the regular inscribed 3 · 2n-sided polygon,

for this side an is also the chord of the supplement of the arc αn° up to 180°. The side bn of a

similar circumscribed polygon is determined by the proportion bn: an = r: h, where h is the

apothem of the inscribed polygon. In the third section of his treatise al-Kāshī ascertains that

the required accuracy will be attained in the case of the regular polygon with 3·228 = 805,

306, 368 sides.

He resumes the computation of the chords in twenty-eight extensive tables; he verifies

the extraction of the roots by squaring and also by checking by 59 (analogous to the checking

by 9 in decimal numeration); and he establishes the number of sexagesimal places to which

the values used must be taken. We can concisely express the chords cn and the sides an by

formulas and where the number of radicals is equal to the index n. In the sixth section, by

multiplying a28 by 3·228, one obtains the perimeter p28 of the inscribed 3·228-sided polygon

and then calculates the perimeter p28 of the corresponding similar circumscribed polygon.

Finally the best approximation for 2π r is accepted as the arithmetic mean whose sexagesimal

value for r = 1 is 6 16I 59II 28III 1IV 34V 51VI 46VIII 50IX, where all places are correct. In the

eighth section al-Kāshī translates this value into the decimal fraction 2π=

6.2831853071795865, correct to sixteen decimal places. This superb result far surpassed all

previous determinations of π. The decimal approximation π ≈ 3.14 corresponds to the famous

boundary values found by Archimedes, Ptolemy used the sexagesimal value 3 8I 30II (≈

3.14166), and the results of al-Kāshī’s predecessors in the Islamic countries were not much

better. The most accurate value of π obtained before al-Kāshī by the Chinese scholar Tsu

Chʾung-chih (fifth century) was correct to six decimal places. In Europe in 1597 A. van

8

Roomen approached al-Kāshī’s result by calculating π to fifteen decimal places; later Ludolf

van Ceulen calculated π to twenty and then to thirty-two places (published 1615).

In his Risāla al-walar waʾl-jaib al-Kāshī again calculates the value of sin 1° to ten

correct sexagesimal places; the best previous approximations, correct to four places, were

obtained in the tenth century by Abuʾl-Wafāʾ and Ibn Yũnus. Al-Kāshī derived the equation

for the trisection of an angle, which is a cubic equation of the type px = q + x3—or, as the

Arabic mathematicians would say, “Things are equal to the cube and the number.” The

trisection equation had been known in the Islamic countries since the eleventh century; one

equation of this type was solved approximately by al-Bīūnī to determine the side of a regular

nonagon, but this method remains unknown to us. Al-Kāshī proposed an original iterative

method of approximate solution, which can be summed up as follows: Assume that the

equation possesses a very small positive root x; for the first approximation, take ; for the

second approximation, ; for the third, , and generally x0 = 0.

It may be proved that this process is convergent in the neighborhood of values of . Al-

Kāshī used a somewhat different procedure: he obtained x1 by dividing q by p as the first

sexagesimal place of the desired root, then calculated not the approximations x2, x3, . . .

themselves but the corresponding corrections, that is, the successive sexagesimal places of x.

The starting point of al-Kāshī’s computation was the value of sin 3°, which can be calculated

by elementary operations from the chord of 72° (the side of a regular inscribed pentagon) and

the chord of 60°. The sin 1° for a radius of 60 is obtained as a root of the equation

The sexagesimal value of sin 1° for a radius of 60 is 1 2I 49II 43III 11IV 14V 44VI 16VII

26VIII 17IX; and the corresponding decimal fraction for a radius of 1 is

0.017452406437283571. All figures in both cases are correct.

Al-Kāshī’s method of numerical solution of the trisection equation, whose variants

were also presented by Ulugh Bēg, Qādi Zāde, and his grandson Maḥmūd ibn Muḥammad

Mīrīm Chelebī (who worked in Turkey), requires a relatively small number of operations and

shows the exactness of the approximation at each stage of the computation. Doubtless it was

one of the best achievements in medieval algebra. H. Hankel has written that this method

“concedes nothing in subtlety or elegance to any of the methods of approximation discovered

in the West after Viéte.” But all these discoveries of al-Kāshīs’s were long unknown in

Europe and were studied only in the nineteenth and twentieth centuries by such historians of

science as Sédillot, Hankel, Luckey, Kary-Niyazov, and Kennedy.

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3. Multiple Algorithms and Multiple Solution

Five different multiplication algorithms are given in Treatise I, Chapter III. Here we describe

two of them briefly without all the details given by al-Kashi. The first one using a lattice

contains as many rows as the number of multiplicand digits, and as many columns as the

number of multiplier digits; then we divide each cell diagonally into two triangles. We place

the digits of the multiplicand above the columns and the digits of the multiplier beside the

rows. Then we perform the multiplication and we put the ones of each product in the lower

triangle and the tens in the upper triangle. After that we add diagonally. For example, the

multiplication of 7806 by 175 is represented by

Figure 1: 𝟕𝟖𝟎𝟔×𝟏𝟕𝟓 multiplication process represented by lattice from al-Nabulsi (1977, p.57).

Another way: we start by multiplying the first digit from the multiplier by every digit from

the multiplicand, and we place the ones of the second product under the tens of the first

product and the ones of the third under the tens of the second, and so on. Then we multiply

the second digit from the multiplier by every digit from the multiplicand, and we place the

ones of the first product above the tens of the first product of the first digit, then we put the

ones of the second product above the tens of the second product of the first digit, and so on.

10

We do that for all digits.

The multiplication

of 358 by 624 is

represented by the

following:

Figure 2: 𝟑𝟓𝟖×𝟔𝟐𝟒 multiplication process representationfrom al-Nabulsi (1977, p. 59).

Chapter IV in Treatise V is devoted to examples. Section I contains 25 general word

problems, Section II discusses seven word problems in inheritance solved algebraically with

extensive fraction arithmetic, and Section III is devoted to solving word problems

geometrically.

The sixth problem in Section I is:

A piece of jewellery is made from gold and pearl. Its weight is three methqals3

The Key to Arithmetic introduces four different methods to solve this problem. In modern

symbols these become and its price is twenty-four dinars. The price of one methqal of gold is

five dinars, and of pearl is fifteen dinars. We want to know the weight of each kind.

By algebra: let 𝑥 be the gold weight. Then the pearl weight becomes 3−𝑥.

The pearl price =15(3−𝑥)=45−15𝑥The jewellery price =45−15𝑥+5𝑥=45−10𝑥=24

We get 𝑥=2.1 methqals, the gold weight. Then the pearl weight is 0.9 methqal.

In this first approach, an algebraic equation is formulated with the gold weight as unknown,

and then it is solved by a standard use of algebra.

Chapter III in Treatise V consists of fifty rules for ratios, geometry, algebra,

number theory, and other topics. One of al-Kashi’s methods to solve word problems

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is using these rules; he refers to this as ‘maftoohat’ ( .(مفتوحات

By maftoohat: the gold weight is (translating the original Arabic words to mathematical

symbols)

In this second approach al-Kashi considers the jewellery as made entirely from pearl; then the

price is increased by 21 dinars, and then he divides this increase by the difference between

the two prices, to get the gold weight.

Another way: the pearl weight is

Then the gold weight is 2.1 methqals.

Similarly, in this third approach, al-Kashi

supposes that the jewellery is made entirely

from gold, and he proceeds as in the second

way.

By geometry: We represent the problem as in the following picture.

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Figure 6: Geometric illustration from al-Nabulsi (1977, p. 500).

This last approach presents a way to think geometrically about this problem, by considering

weight and price magnitudes represented by sides of a rectangle, and the product of these

magnitudes is the rectangle area.

4. Law of Cosines and Jamshid al-Kashi

The law of cosines may be used for calculating the length of one side of a triangle

when the angle of the opposite this side, and the length of the other two sides, are known. The

law may be expressed as c 2 = a2 + b 2 - 2abcos(C), where a, b, and c are triangle side lengths

and C is an angle between sides a and b.

Because of its generality, the application of the law ranges from land surveying to

calculating the flight paths of aircraft. Notice how the law of cosines becomes the

Pythagorean Theorem

c 2 = a2 + b 2 for right triangles, when C

becomes 90° and the cosine becomes

zero. Also note that if all three side lengths

of a triangle are known, we can use the

law of cosines to compute the angles of a

triangle.

Euclid's Elements (300 B.C.) contains the seeds of concepts that lead to the law of

cosines. In the fifteenth century, the Persian astronomer and mathematician al-Kashi provided

accurate trigonometric tables and expressed the theorem in a form suitable for modem usage.

French mathematician Franois Viete discovered the law independently of al-Kashi. In French,

the law of cosines is named Theoreme d'Al-Kashi, after al-Kashi's unification of existing

works on the subject.

Figure: Triangle.

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The angles α, β, and γ are respectively opposite the sides a, b, and c.

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) is a

statement about a general triagnle that relates the lengths of its sides to the cosine if one of its

angles. Using notation as in Fig., the law of cosines states that

where γ denotes the angle contained between sides of lengths a and b and opposite the side of

length c.

The law of cosines generalizes the Pythagorean theorem which holds only for right triangles:

if the angle γ is a right angle (of measure 900or π/2 radians), then cos(γ) = 0, and thus the law

of cosines reduces to

The law of cosines is useful for computing the third side of a triangle when two sides and

their enclosed angle are known, and in computing the angles of a triangle if all three sides are

known.

By changing which sides of the triangle play the roles of a, b, and c in the original formula,

one discovers that the following two formulas also state the law of cosines:

Application of Law of Cosines: An ExampleTwo cars leave a city at the same time and travel along straight highways that differ in

direction by 800. One car averages 60 miles per hour and the other averages 50 miles per

hour. How far apart will the cars be after 90 minutes?

Solution:

Determine how far each car has traveled during the 90 minutes

d1 = distance of car 1

d2 = distance of car 2

r1 = rate of car 1 = 60 mph

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r2 = rate of car 2 = 50 mph

t = time traveled = 90 minutes = 1.5 hours

d1 = r1 ×t , d2 = r2 × t

d1 = (60 mph) × (1.5 hrs), d2 = (50 mph) × (1.5 hrs)

d1 = 90 mi, d2 = 75 mi

Draw diagram of situation

Calculation: c2= a2+b2-2abcosc

c2= (75)2+ (90)2-2×75×90×cos800

c2=11380.6224

c=106.68≈ 107 mi

5. Fixed Point Iteration Method

He wrote The Reckoners’ Key which summarizes arithmetic and contains work on algebra

and geometry. In another work, al’Kashi applied the method now known as fixed-point

iteration to solve a cubic equation having as a root. Generally, for an equation of the

form x = f(x).

We define the iteration

where x0 is some initial “guess”. If the iterations converge, then it must be a solution of the

equation. Such a method is called fixed

point iteration. Another more famous

fixed point iteration coming much

later is Newton’s Method-

.

15

He also worked on solutions of systems of equations and developed methods for finding the

nth root of a number – Horner’s method today.

Fixed point iteration method Example: Finding roots of

Solution:

Here

f =

We Know according to fixed point method

Now, we assume that the first point is 10.

Therefore,

When, n=0:

n=1:

n=2:

n=3:

n=4:

16

After that when we take n=5 the value of will also be 5.0000. That is we got a converge

point.

Hence the one of the solution of is 5.0000.

In similar way we can find another solution of the equation.

6. Calculation of PI

Al-Kāshī wrote his most important works in Samarkand. In July 1424 he completed

Risāla al-muḥitiyya (“The Treatise on the Circumference”), masterpiece of computational

technique resulting in the determination of 2π to sixteen decimal places.

At the beginning of the Risāla al-muḥītīyya al-Kāshī points out that all approximate

values of the ratio of the circumference of a circle to its diameter, that is, of π, calculated by

his predecessors gave a very great (absolute) error in the circumference and even greater

errors in the computation of the areas of large circles, Al-Kāshī tackled the problem of a more

accurate computation of this ratio, which he considered to be irrational, with an accuracy

surpassing the practical needs of astronomy, in terms of the then-usual standard of the size of

the visible universe or of the “sphere of fixed stars.” For that purpose he assumed, as had the

Iranian astronomer Qutb al-Din al-Shīrāzī (thirteenth-fourteenth centuries), that the radius of

this sphere is 70,073.5 times the diameter of the earth.

Concretely, al-KĀshī posed the problem of calculating the said ratio with such

precision that the error in the circumference whose diameter is equal to 600,000 diameters of

the earth will be smaller than the thickness of a horse’s hair. Al-Kāshī used the following Old

Iranian units of measurement:

1 parasang (about 6 kilometers) = 12,000 cubits,

1 cubit = 24 inches (or fingers),

1 inch = 6 widths of a medium-size grain of barley,

I width of a barley grain = 6 thicknesses of a horse’s hair.

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The great-circle circumference of the earth is considered to be about 8,000 parasangs,

so al-Kāshī’s requirement is equivalent to the computation of π with an error no greater than

the width of a horse’s hair. This computation was accomplished by means of elementary

operations, including the extraction of square roots, and the technique of reckoning is

elaborated with the greatest care. Al-Kāshī’s measurement of the circumference is based on a

computation of the perimeters of regular inscribed and circumscribed polygons, as had been

done by Archimedes, but it follows a somewhat different procedure. Al-K¯ashi applied his

fundamental theorem to calculate successively the value of the chord cn of the arc αn0 = 180o −

360o/3(2n) (where n ≥ 0). From his fundamental theorem al-K¯ashi obtained the identity

From here he found the lengths of the sides of inscribed and circumscribed regular

polygons each with 3(2n) sides (n ≥ 1), in a given circle. Then he determined the number of

sides of the inscribed regular polygon in a circle whose radius is six hundred times the radius

of the Earth in a

such a way that the

difference

between the

circumference of the circle and the perimeter of the inscribed regular polygon in this circle

will become less than the width of a horse’s hair.

Al Kashi continued to use his fundamental theorem to calculate the value of π, correct

to 16 decimal places, using inscribed and circumscribed polygons, each with 3(228) =

805,306,368 sides. The chord of 60° is equal to r, and so it is possible by means of this

theorem to calculate successively the chords c1, c2, c3. . . of the arcs 120°, 150°, 165°, in

general the value of the chord cn of the arc will be . The chord cn being known, we may,

according to Pythagorean Theorem, find the side of the regular inscribed 3 ×2n sided polygon,

for this side an is also the chord of the supplement of the arc αn° up to 180°. The side bn of a

similar circumscribed polygon is determined by the proportion bn: an = r: h, where h is the

apothem of the inscribed polygon. In the third section of his treatise al-Kāshī ascertains that

the required accuracy will be attained in the case of the regular polygon with 3×2 28 = 805,

306, 368 sides.

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He resumes the computation of the chords in twenty-eight extensive tables; he verifies

the extraction of the roots by squaring and also by checking by 59 (analogous to the checking

by 9 in decimal numeration); and he establishes the number of sexagesimal places to which

the values used must be taken. We can concisely express the chords cn and the sides an by

formulas where the number of radicals is equal to the index n. In the sixth section, by

multiplying a28 by 3×228, one obtains the perimeter p28 of the inscribed 3×228 sided polygon

and then calculates the perimeter p28 of the corresponding similar circumscribed polygon.

Finally the best approximation for 2π r is accepted as the arithmetic mean whose sexagesimal

value for r = 1 is 6 16I 59II 28III 1IV 34V 51VI 46VII14VIII 50IX, where all places are correct. In the

eighth section al-Kāshī translates this value into the decimal fraction 2π=

6.2831853071795865, correct to sixteen decimal places. This superb result far surpassed all

previous determinations of π.

Name of books by al-Kāshī

I. Original Works. Al-Kāshī’s writings were collected as Majmūʿ (“Collection”; Teheran,

1888), an ed. of the matematicheskie issledoveniya, 7 (1954), 9–439, Russian trans. by B.

A. Rosenfeld and commentaries by Rosenfeld and A. P. Youschkevitch; and Klyuch

arifmeti. Traktat of okruzhnosti ( “The Key of Arithmetic. A Treatise on

Circumference”), trans. bty B. A. Rosenfeld, ed. by V. S. Segal and A. P. Youschkevitch,

commentaries by Rosenfeld and Youschkevitch, with photorepros. of Arabic MSS.

His individual works are the following:

1. Miftāḥ al-ḥisāb (”The Key of Arithmetic”) or Miftāḥ al-ḥussāb fi ’ilm al-ḥisāb

(“The Key of Reckoners in the Science of Arithmetic”). Arabic MSS in Leningrad, Berlin,

Paris, Leiden, London, Istanbul, Teheran, Meshed, Patna, Peshawar, and Rampur, the most

important being Leningrad, Publ. Bibl. 131; Leiden, Univ. 185; Berlin, Preuss. Bibl. 5992

and 2992a, and Inst. Gesch. Med. Natur. 1.2; Paris, BN 5020; and London, BM 419 and India

Office 756. There is a litho. ed. of another MS (Teheran, 1889). Russian trans. are in

“Matematicheskie traktaty,” pp. 13–326; and Klyuch arifmetiki, pp. 7–262, with

photorepro. of Leiden MS on pp. 428–568, There is an ed. of the Leiden MS with

commentaries (Cairo, 1968). See also P. Luckey, “Die Ausziehung dos n-ten Wurzel...” and

“Die Rechenkunst bei Ğamšid b. Masʿud al-Kāašsī...”

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2. Risāal-muḥīitīyya (“Treatise on the Circumference”; 1424). Arabic MSS are in Istanbul,

Teheran, and Meshed, the most important being Istanbul, Ask. müze. 756. There is an ed. of

another MS in Majmūʾ and one of the Istanbul MS with German trans. in P. Luckey, Der

Lehrbrief über den Kreisumfang von Gamšīd b. Masʿūd al-Kāši . Russian trans. are in

“Matematicheskie trakaty,” pp.327–379; and in Klyuch arifmetiki, pp. 263–308, with

photorepro . of Istanbul MS pp. 338–426.

3. Talkhīis al-Miftāah (“Compendium of the Key”). Arabic MSS in London, Tashkent,

Istanbul, Baghdad, Mosul, Teheran, Tabriz, and Patna, the most important being London,

India Office 75; and Tashkent, Inst. vost. 2245.

4. Risāla dar sharḥ-i ālāt-i rasd (”Treatise on the Explanation of Observational

Instruments”; 1416). Persian MSS in Leiden and Teharan, the more important being Leiden,

Univ. 327/12, which has been pub. as a supp. to V. V. Bartold, Ulugbek i ego uremya; and

E. S. Kennedy,” Al-Kāshi’s Treatise on Astronomical Observation Instruments,” pp. 99, 101,

103. There isd an English trans. in Kennedy, “Al-Kāshī’s Treatise...,” pp. 98–104; and a

Russian trans. in V. A. Shishkin, “Observatoriya Ulugebeka i ee issledovanie,” pp. 91–94.

5. Mukhtasar dar ʿlim-i hayʾ at (“Compendium on the Science of Astronomy”) or Risāla

dar hayʾ at ( “Treatise on Astronomy”; 1410–1411).

6. Zij-i Khaqāni fī takmīl-i Zij-i Īlkhānī (“Khaqāni Zij— perfection of īlkhānī Zij”

1413–1414).

7. Risāla al-watar waʾl-jaib (“Treatise on the Chord and Sine”). There is an ed. of a MS in

Majmūʾ.

8. Ilkaḥāt an-Nuzha (“Supplement to the Excursion” 1427). There is an ed. of a MS in

Majmūʾ.

9. Sullam al-samāʿ fi ḥall ishkāl waqaʿa liʾl-muqaddimī fiʾl-abʿād waāl-ajrām (

“The Stairway of Heven, on Resolution of Difficulties Met by Predecessors in the

Determination of Distances and Sizes”; 1407).

10. Nuzha al-ḥadāiq fi kayfiyya sanʿsa al-āla al-musammā bi tabaq al-manātiq

(“The Garden Excursion,; on the Method of Construction of the Instrument Called Plate of

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Heavens”; 1416). Arabic MSS are in London, Dublin, and Bombay, the moist important

being London, India Office Ross 210.

Limitation of our work:

1. The fundamentals books are not available at English version.2. The research work about al-Kashi are not free accessible.3. Al-Kashi is an astronomer so there are many calculations are complex and not easy to

describe by easy process such in pie calculation he uses the sexagesimal system.

Conclusion

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