ISSN: 1735-0573

Tarikh-e Elm

Iranian Journal for the History of Science

Vol. 11(2), 2014-2015 (in Sequence: No. 15)

In English:

Kam�l al-D�n al-F!rs��s Additions to Book XIII of al-�"s��s Ta�r�r (1-19)Mohamed Mahdi Abdeljaouad

In Persian:

Ibn al-Haytham, Kam!l al-D�n al-F!rs� and Wave Motion of Light (159-190) Soheyla Pazari

The Place of Oud and its Various Types

in the Islamic Eras of Persian Music (191-206) Narges Zaker Jafari

A Short Review of the Contents and Sources

of Manuscript No. 3447 in National Library of Tabriz (207-242)Younes Karamati

Diversity and Variety of Hay a Books in Islamic Civilization (243-290) Amir Mohammad Gamini

Iterative Ratios in the Fractal Geometry of Urch�n Domes (291-310)Mohammad Mashayekhi, Farhad Tehrani

Institute for the History of Science

University of Tehran

Tarikh-e Elm Iranian Journal for the History of Science Vol. 11(2), 2014-2015

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Tehran

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Tehran

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Table of Contents

In English:

Kam l al-D n al-F rs !s Additions to Book XIII 1

of al- s !s Ta r r (1-19)

Mohamed Mahdi Abdeljaouad

English Abstracts of Persian Articles 21

In Persian:

Ibn al-Haytham, Kam l al-D n al-F rs 159

and Wave Motion of Light Soheyla Pazari

The Place of Oud and its Various Types 191

in the Islamic Eras of Persian Music Narges Zaker Jafari

A Short Review of the Contents and Sourses 207

of Manuscript No. 3447 in National Library of Tabriz Younes Karamati

Diversity and Variety of Hay a Books 243

in Islamic Civilization Amir Mohammad Gamini

Iterative Ratios in the Fractal Geometry 291

of Urch n Domes Mohammad Mashayekhi, Farhad Tehrani

Persian Abstract of English Articles 311

Tarikh-e Elm, Vol. 11(2) (2014-2015), pp. 1-19

Kam l al-D n al-F rs !s additions to Book XIII of al- s !s Ta r r

Mohamed Mahdi Abdeljaouad

Association Tunisienne des Sciences math & eacute; matiques mahdi.abdeljaouad@gmail.com

(received: July 2015, accepted: September 2015)

AbstractThis article is devoted to Kam l al-D n al-F rs !s (d. 1319)

additions to some remarks contained in Book XIII of al- s s!

Ta r r u l al-handasa, concerning the construction of a semi-

regular polyhedron inscribed into a sphere using the movement

as a way for the construction. This treatise is one treatise among

ten found in a codex preserved at the Bibliothèque nationale de

Tunis.

Keywords: F rs , s ,!polyhedron,!sphere,!geometry.

2/ Kam l al-D n al-Fars !s additions"

Introduction

The treatise we are examining here, manuscript Tunis 16167/6

(73a/74a), is a commentary by Kam l al-D n al-F rs (d. 1319) on some

remarks occurring in Na r al-D n al- s !s Ta r r u l al-handasa that

concern the construction of semi-regular polyhedra inscribed into a

sphere.

The Tunis codex starts with an indication that F rs intends to

comment on the last paragraphs of the thirteenth maq la of al- s !s

Ta r r u l uql dis [Thirteenth Book of al- s !s Exposition of Euclid!s

Elements]1 starting with the sentence � it is necessary that no more than

two angles � � and finishing with the end of the book. Here is exactly

what al- s writes:

Even if it is not required that the faces of a solid belong to

a single species, it is necessary that no more than two

angles <ending at each of the vertices> be of the same kind,

so that the solid does not lose its similarities <i.e. its

symmetries> and hence cannot be inscribed within a sphere.

Then, the number of solid angles <i.e. vertices> has to be

even, exactly four, since two <angles> cannot constitute <a

solid angle>, while six or more <vertices> would exceed

four right angles. And the species of one of the faces has to

be a triangle for the same reason. When <the faces of the

polyhedron> are composed of triangles and squares, the

figure has fourteen faces, eight triangles and six squares. It

is as if it were composed of a cube and an octahedron and

its side will be equal to the side of the hexagon occurring in

the great circle of the sphere. When <the faces of the

polyhedron> are triangles and pentagons, the figure has

thirty-two faces, twenty triangles and twelve pentagons. It

is as if it were composed of these two figures and its side

will be equal to side of the decagon occurring in the great

circle of the sphere. It results from this that the number of

solids inscribed into a sphere is seven. We have ended the

1. In this paper we refer to the second lithograph edition of al- !s Exposition of the

Elements. Tehran 1880.

Tarikh-e Elm, Vol. 11(2) (2014-2015) /3

thirteenth maq la which is the end of the book (Ta r r,

197). 1

In these paragraphs, al- s shows that there are only seven

polyhedrons which can be inscribed in a sphere, adding to the five

regular Platonic solids two Archimedean ones: the cubo-octahedron

(which has fourteen faces, eight equilateral triangles and six squares)

and the icosi-dodecahedron (which has thirty-two faces, twenty

triangles and twelve squares)2.

F rs !s intention here is to construct a representative example of a

family of semi-regular polyhedra (the prisms) inscribed in a given

sphere.

F rs s!place!in!the!Arabic!Euclidean!tradition

A brilliant student of Qu b al-D n al-Shir z (d. 710/1310), Kam l al-

D n al-F rs (d. 718/1318) was one the great Persian specialist of optics,

the author of Tanq al-man dhir, a critical commentary of Ibn al-

Haytham!s Kit b al-man dhir. He is also known for his work on

amicable numbers and his commentary on Ibn al-Khaww m!s (d.

724/1324) Al-Faw �id al-bah �iyya f l-qaw �id al- is biyya, an

important text book in arithmetic, algebra and practical geometry. But,

as far as we know, none of the standard bio-bliographical sources has

credited him with any substantial work in Euclidean geometry; only

some of his short commentaries and glosses in this field are extant.

These are:

.

.

.

.

.

.

.

. )(.

2. De Young [2008, 181-191] shows that the anonymous author of Book XVI to Euclid!s

Elements has provided a construction of these two semi-regular polyhedrons and has proven

them to be inscribed in a sphere. (Propositions: 7 to 11).

4/ Kam l al-D n al-Fars !s additions"

- Ris la �ala Ta r r al-Abhar f l-mas�ala al-mashh ra min kit b

uql dis (Treatise on Abhar !s exposition on the well-known

problem of Euclid!s book).1

- Ris la f l-z wiyya (Treatise on angle).2

- An-nadhar f qawl al- s f khir al-maq la al-th lithah "ashar

(Discussion of what al- s said at the end of Book XIII). This

is the title we propose for the treatise we are presenting in this

paper.

F rs s!propositions

Constructing a prism inscribed in a given sphere

F rs s!treatise!is!composed!of!two!independent propositions. He places

himself in the continuity of al- s s! argumentation,! introducing! his!

additions! by! a! simple! sentence:! "this! <i.e.! subject>! needs! to! be!

discussed! here�3 and he does not seem to be compelled to add any

theoretical arguments for justifying the new construction.

The first proposition is not formally enunciated; however it might be

reconstructed in four parts: (1) Construction of a triangular prism

inscribed in a sphere. (2) Construction of an equilateral triangular prism

inscribed in a sphere. (3) Construction of any equilateral polygonal

prism inscribed in a sphere. (4) Equilateral prisms inscribed in a sphere

are limitless.

The first construction starts directly by its first step:

1. Draw a sphere and one of its diameters.

2. Cut the sphere with a plane perpendicular to the diameter so that

the section is a small circle.

3. Draw an equilateral triangle inscribed into the circle.

1. We have prepared an edition and a translation into English of this anonymous treatise which

has been proposed for publication.

2. For two recent editions of this!treatise,!see!Maw lid ![2014].

.

Tarikh-e Elm, Vol. 11(2) (2014-2015) /5

4. On the sphere and on the other side of its diameter, draw a circle

equal to the first circle.

5. Draw perpendiculars to the first circle from each vertex of the

triangle. They end at the vertices on the circumference of the

second circle and make a triangle equal to the first.

6. Join the vertices of the two triangles, each one to the one below

it.

We get a right prism inscribed in a sphere with triangular bases and

three rectangular faces. F rs does not conclude that he got what he

wanted; on the contrary he adds here: !the height of the prism is greater

than the side of the triangular base".

For the second part, he offers a more precise construction:

Imagine now the movement toward the center <of the sphere> of the

two cutting surfaces unchanged in their relation to the diameter, moving

with similar movements, making the sides of the triangles greater and

the sides of the rectangles smaller, until they all become equal.

F rs asks the reader to imagine a simultaneous movement of the two

bases toward the center of the sphere. It makes the sides of the triangular

bases greater and the sides of the rectangular faces smaller. The

movement is stopped when the edges all become equal.

Now the conclusion is explicitly stated: !The produced prism has

therefore equilateral faces; it is circumscribed by a sphere and each of

its angles is made of three plane angles: two angles of a square and one

of an <equilateral> triangle".

The third part is a first porism: !The same method can be used for

constructing pentagonal prism with five square lateral faces".

The fourth part is another porism:" Solids with all edges equal and

inscribed in a circle are limitless".

It is quite remarkable that no diagrams are used in this proof and the

procedure is not instantiated in any particular case.

The following diagram shows the final figure in modern perspective.

6/ Kam l al-D n al-Fars !s additions"

Figure 1. Imagining a movement of a plane

In order to get squares as lateral faces, F rs uses the movement as a

way for the solution. For purist Euclidian geometers, motion was not

allowed in demonstrations; however F rs is not the first Arabic

mathematician to do it: before him, Th bit ibn Qurra introduced motion

in proofs when he tried to demonstrate the parallel postulate. The same

approach is found in al-Sijz !s Kit b f tahsil al-subul l istikhr j al-

ashk l al-handasiyya.1 When they use motion in a proof, both authors

introduce the verb "imagine#. For example, Th bit writes: "When we

imagine a solid moving in a unique direction according to uniform and

rectilinear movement, any point of the solid describes a straight line# 2

and similarly, al-Sizj writes: "Imagine a straight line moving �#3.

Later when trying to prove the parallel postulate, Ibn al-Haytham

introduces also the movement of the finite straight line perpendicular to

a fixed line, the extremity of which describing a straight line parallel to

the fixed one. F rs uses also motion in imagination and it was perhaps

due to his familiarity with Ibn al-Haytham!s works that he thought it

suitable to follow his ideas and found an original way to construct a

new class of polygonal prisms inscribed within a sphere unrelated to the

classical techniques used in the Archimedean tradition.4

1. These works are edited, commented on and translated into French in Rashed [2009, 688-

931].

.

.

. !

4. %Umar al-Khayyam rejected completely the introduction of motion in geometry (See his

"Commentary on the Difficulties of Certain Postulates of Euclid!s Book�, in [Rashed R. and

Vahabzadeh B. 2000, pp. 219-220]. Vitrac [2007] gives a complete analysis of the use of

movement in the Euclidean geometrical tradition and an important bibliography on the subject.

Tarikh-e Elm, Vol. 11(2) (2014-2015) /7

F rs s!second!proposition

In this section of the treatise, F rs is more conventional: he enunciates

the problem: To inscribe in a sphere a polygonal prism similar to a given

polygonal prism. The construction is done in two phases. In the first

phase, given the polygonal prism whose side is equal to the given length

AB, he finds the sphere in which this polygonal prism can be inscribed.

In the second phase, the polygonal prism similar to the above polygonal

prism is inscribed in the given sphere.

The first phase can be divided into 9 steps:

Let AB be the circle in

which a polygon to the

side AB is inscribed, let

Z be its center.

Take AD perpendicular

to the plane of the circle

and AD = AB. Let E be

the midpoint of AD.

Let EH be parallel to AZ

contained in the plane

DA, AZ, and EH = AZ.

Then EH is collinear to

the diameter of the

sphere and EH is

contained in the plane,

D belongs to the surface

of the sphere.

HZ is perpendicular on

the surface of the circle

as is AE.

Figure2

Then H is the center of the sphere.

Then DH is the radius of the sphere and the diagonal of rectangular

triangle DEH.

Extend EH by ET with T on the sphere

Then HT = HD.

The second phase of the construction takes place in a great circle of

the given sphere:

8/ Kam l al-D n al-Fars !s additions"

In the given sphere, draw

great arc KSL with KL its

diameter and M its center.

Let KM be cut at N so that

the ratio of KN to NM is

equal the ratio of TE to EH.

From N, draw the

perpendicular NS to KL.

Then the arcs DT and SK are

similar.

So the ratio of SN to NM is

equal to the ratio of DE to

EH.

Figure 3

Let us show the similarity of the arcs DT and SK in modern

notations.

KN : NM TE : EH (definition of the position of N on line KL)

KM : NM TH : EH (Porism of Euclid�s Prop. V-19)

MS : NM HD : EH (since KM = MS and TH = HD as radius in

their respective circles)

So the right triangles MNS and HED are similar and the interior

angle KMS is equal the interior angle THD. This is equivalent to say

that le great arc SK is similar to the great arc DT which also implies

that SN : NM DE : EH.

We now continue the construction:

Construct trough S a plane perpendicular to SN.

It produces on the sphere a circle with a radius equal to NM.

Extend SN toward P such that P be on the circle.

Therefore, we have SP : NM DA : AZ , since SP = 2SN and DA =

2DE and EH = AZ (by specification).

Let SY be the edge of the constructed polygon inscribed in the circle

produced by the plane passing through S on the given sphere. Then,

since the radius of this circle is equal NM, so the ratio of the radius of

the circle to the edge of the polygon, noted KM : SY, is the same as AZ

: AB.

Tarikh-e Elm, Vol. 11(2) (2014-2015) /9

Then F rs concludes that, !ex aequali" (bi�l mus w t) 1, the ratio of

SP to SY is the same as the ratio of DA to AB. Indeed, he has proved

that:

SP : KM DA : AZ and KM : SY AZ : AB, thus SP : SY DA :

AB.

And since par specification, DA = AB, we have SY = SP. The edge

produced is equal SP.

Once the base inscribed in a circle passing through the point P is

similarly produced, we get the prism2 as required.

It is clear that the first phase of the construction is intended at

analyzing the geometrical characteristics of a sphere circumscribed to a

given polygonal prism. Ultimately, the aim is to fix the ratio of the edge

of the polygon to the radius of the circle. The second phase aims at

constructing the specified prism and proving that all its edges are equal.

In this text, F rs uses with great mastery the Euclidian techniques,

imagining figures in the space and manipulating proportions and he

does not think necessary the insertion of explicit references to Euclid!s

or to al- !s Recension of the Elements in his elaborate proofs.

Discussions on semiregular polyhedra inscribed in a sphere

In the Arabic Euclidean tradition, F rs s!text!is!not!the!only!attempt!to!

construct semi-regular polyhedra inscribed in a sphere. There is at least

another extant construction included in the so-called Book XVI, an

anonymous! addition! to! Euclid s! Elements presented by De Young3.

This treatise contains

"nineteen propositions describing techniques for

constructing polyhedra within other polyhedra or within

spheres.!(#)!The!contents!are,!however,!in!the!tradition!of!

Archimedes! rather! that! Euclid.! (#)! The! addendum! ends!

with what may be the earliest discussion of the construction

of a representative example from each of the classes of

semiregular polyhedra known today as prisms and

antiprisms.$![De!Young!2008,!133]

1. The!expression!"ex!aequali$! is!based!on!Euclid s!V-22 and indicates an inference of the

following kind:

If a : b d : e and b : c e : f then a : c d : f.

2. The!word!used!by!the!author!is!"us uw nah$!which!literally!means!cylinder.

3. De Young [2008, 133-209] presents this Book XVI appended to a manuscript containing

al- s s!Tahr r. It is a unique copy dated 1593/4 and its author is unknown.

10/ Kam l al-D n al-Fars !s additions"

Among these propositions, the eighteenth describes the construction

in a sphere of �a polyhedron having equilateral faces, two of which are

specified figures occurring in a single circle and the remainder are

squares# [De Young 2008, 200]. Therefore, the anonymous author of

Book XVI discusses exactly the same problem as F rs , but they differ

in their approaches as it appears from De Young"s report:

�In this proposition, we construct a decagonal prism. The figure

consists of two planes parallel to a great circle such that two planes cut

the sphere forming equal circles. In these two circles we construct our

desired equilateral figures # in this case decagons. We arrange these

figures so that the vertices of one lie directly over the vertices of the

other plane and connect the two vertices by lines between the two

planes. We show that these connecting lines are perpendicular to the

planes of the circles and that they are equal to one another.! [De Young

2008, 201]

Let us summarize the steps of this construction:

Consider first a decagon inscribed in a circle.

Construct a rectangle WEZH with WE equal the diameter and EZ

equal the edge of the decagon.

In a great circle of the given sphere, inscribe a rectangle ABGD

similar to WEZH, with AD corresponding to WE.

Draw the two circles obtained by the intersection of the given sphere

with the planes perpendicular in A and D to AD.

Draw in each of the two circles decagons and make their beginning

points A and B.

Then we get a decagonal prism inscribed in the given sphere.

The following diagrams show the different steps in modern

perspective:

Tarikh-e Elm, Vol. 11(2) (2014-2015) /11

Figure 4

The anonymous author assumes implicitly two lemmas: the first is a

direct consequence of Euclid�s proposition VI-4 concerning similar

regular polygons inscribed in different circles; it says that for all of

them, the ratio of the edge of the polygon to its diameter is the same.

The second lemma is Proposition 17 of Book XVI that shows that it is

always possible to draw inside a circle a quadrilateral similar to a given

right-angled parallelogram.

Thus, the rectangle WEZH chosen at the beginning of the proof has

the good dimensions: EZ is equal to the edge of a decagon inscribed in

a circle that has a diameter equal to WE and the prism made at the end

of the proof has also the good dimensions: AB is equal the edge of the

octagon inscribed in the circles making the bases of the cylinder. 1

Final remarks

Three types of constructions of right prisms inscribed in a given sphere,

with different proofs, have been presented here; F rs �s first one is

highly interesting since it uses motion but it proves the existence of

1. De Young [2008, 199-202 and 165-166] presents an Arabic edition of the propositions 17

and 18 of Book XVI and their translation into English with commentaries and notes.

12/ Kam l al-D n al-Fars !s additions"

these objects. Like the anonymous proof, his second proof is more

traditional, taking its inspiration and techniques in the Euclidean

tradition, but we have shown that the two are different in their

approaches.

In his paper, De Young presents the history of regular and

semiregular polyhedra and asserts that Propositions 18 and 19 �may be

the earliest discussion of the construction of a representative example

from each of the classes of semiregular polyhedra known today as

prisms and antiprisms# [De Young 2008, 133]. We do not know if F rs

had read the anonymous Book XVI and attempted to add his own

constructions or if he did not read it. Anyhow, this treatise confirms that

he was an exceptional mathematician.

References Brentjes S. (2008). Elements: Reception of Euclid's Elements in the Islamic

World. Encyclopaedia of the History of Science, Technology, and Medicine in

Non-Western Cultures. New York: Springer-Verlag Berlin Heidelberg.

De Young G. (2008). Book XVI. A Medieval Arabic Addendum to Euclid"s

Elements. SCIAMVS, 9. pp. 133-209.

Mawaldi M. (2014). Ris la f l-z wiyya li Kam l al-D n al-F rs (Treatise on

the angle by Kam l al-D n al-F rs ), London: Al-Furq n.

Na r al-D n al- s . 1298 H. [1880]. Ta r r u l uql dis. [Exposition of

Euclid�s Elements], Second lithograph edition. Tehran.

Radawy Modaras. (1975) Al-F rs . Bulletin of the Imperial Iranian Perennis,

Teheran: Academy of philosophy.

Rashed R. (2002). Les mathématiques infinitésimales du IXe au XIe siècle, Vol.

4: Ibn al-Haytham. London: Al-Furq n.

Rashed R. and Vahabzadeh B. (2000). Omar Khayyam the Mathematician, New

York: Bibliotheca Persica Press.

Vitrac B. (2007). Quelques remarques sur l�usage du mouvement en géométrie

dans la tradition euclidienne : de Platon et Aristote à "Umar al-Khayyam.

https://hal.archives-ouvertes.fr/hal-00174944.

Tarikh-e Elm, Vol. 11(2) (2014-2015) /13

The manuscript Tunis Mss-16167/6

The short work under consideration here belongs to the codex Tunis

Mss-16167 (also known as Ahmadiyya 8452) and is the sixth unit (73a-

74a) among ten all devoted to commentaries on Euclid�s Elements.1

Rashed [2002, 736] presents a short description of the codex but ignores

the existence of this particular F rs �s treatise. We discovered another

copy of this treatise: the fourteenth unit of the codex Leiden Or.14,

copied around 1667. After examining the two manuscripts2, it appears

that the Leiden manuscript is a copy of the Tunis one3, but it contains

significant scribal errors that make it unsuitable for the edition of the

text. Modaras Radwy [1975] indicates that there is another copy of this

treatise in Mashhad (Iran), however we did not get a copy of it.

The Tunis codex is composed of 90 folios, 13x21,5 cm, 23 lines each

and with nasta�liq script, and has been written by a unique copyist:

Darw sh Ahmad al-Kar mi who ended copying it in 869/1464. F rs �s

treatise contains only two diagrams placed in a unique rectangular

!window", but the first diagram is difficult to read for it encroaches on

the text and letters of the diagram are mixed with those of the text. The

Leiden copy was no help since the !windows" stayed empty of

diagrams; however, we used it in several instances in order to remove a

doubt concerning ambiguous words or, if needed, to correct and adjust

the meaning of a sentence.

1. This volume also contains the well known Ibn al-Haytham�s (d. 1038) Shar� mus dar t

uql dis l-Ibn al-Haytham [Commentary on the Premises of Euclid�s Elements] (ff. 1b-59b), Al-

#Abb s ibn Sa� d al-Jawhar �s (d. 835) Ziy d t al-!Abb s ibn Sa� d f l-maq la al-kh misa min

uql dis [Additions to the Fifth Book of Euclid�s Elements] (ff. 60b-61a) and Th bit b. Qurra�s

(d. 901) Fi l-!illati l-lati lah rattaba uql dis ashk l kit bihi dh lika l-tart bi [Treatise on the

Cause of why Euclid disposed Propositions of his book in such order] (ff. 86b-90b). Most of

the treatises of this collection of manuscripts have been analyzed; some have even been edited

and translated into French, English or Persian.

2. I obtained a copy of Leiden 14/14 thanks to Professor Pierre Ageron (University of Caen).

3. Rashed [2002, 737] shows that four other units (18-19-20-21) of the codex Leiden 14 have

also been copied directly from the Tunisian codex.

14/ Kam l al-D n al-Fars !s additions"

Edition and translation of MSS_16167-6

He said: In the name of Allah, the most merciful, the most gracious.

The greatest, the supreme, the guide of the greatest sages, the chief of a

cohort of scientists, the completeness in the state and the religion, al-

Hasan al-F rs , may God receives him in his garden and gives him its

fresh water, said: what The sage knowledgably man of science, N ir

Al-Dawla wa l-D n, said at the end of Book XIII: !It cannot exceed two

right angles", to the end of the phrase, needs to be discussed.

We draw the sphere and one of its diameters. We imagine a plane

surface perpendicular to this diameter and cutting the sphere; it

produces a very small circle, as for example a circle with a diameter of

10 while the diameter of the sphere is 120 degrees. <We draw> in this

circle an equilateral triangle and in the other side of the diameter an

identical circle. Then, from the vertices of the triangle we draw three

perpendiculars to its surface. They end on the circumference of the

second circle. Joining the three ends we get another triangle equal to the

first. From the three perpendiculars and the sides of the two triangles,

three equal rectangles are produced. The five surfaces produced are the

faces of a prism which is such that its height is greater than the sides of

its base.

Imagine now the movement toward the center <of the sphere> of the

two cutting surfaces unchanged in their relation to the diameter, moving

with similar movements, making the sides of the triangles greater and

the sides of the rectangles smaller, until they all become equals.

The produced prism has therefore as bases equilateral triangles, it is

inscribed into a sphere and each of its angles are made of three plane

angles two of them are angles of a square and one of them is the angle

of a triangle. The same can be said concerning a cylinder with bases

which are pentagonal or any other regular polygon with equal angles

and sides. Since every angle of a polygon is smaller than two right

angles, it can produce with two right angles belonging to a rectangle

always a solid angle. Therefore, the kinds of solids with all edges equal

and inscribed in a sphere can be indefinitely great. That is what we

wanted.

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16/ Kam l al-D n al-Fars !s additions"

Then when we want to construct one of the above mentioned solids in

a sphere, we draw a circle and construct in it a figure similar to the base

of the solid. Let AB be the circle, Z its center and AB a side of the

polygon inscribed in it. Join AZ and take AD <perpendicular to the

circle> and equal to AB. Let it be bisected at E. From E draw the parallel

EH to AZ contained in the plane DA, AR. Let EH = AZ.

It is clear that if the circle AB where one of the two bases of a

cylinder inscribed in a sphere, EH would be collinear to the diameter of

the sphere, because the plane parallel to the base and bisecting the

cylinder would necessarily pass through the center of the sphere, the

two circles of the bases being equal. Therefore EH is contained in this

plane and D belongs to the surface of the sphere. Join HZ, which is

perpendicular on the surface of the circle as is AE.

Then H is the center of the sphere. Join DH; it is the radius of the

sphere and (�) on DE and EH. We extend EH by HT equal to HD and

we draw the arc DT of a great circle of the sphere with H as a center

and HD as a radius.

Then we first draw on the given sphere the great circle KSL with KL

as its diameter and M as its center. Let KM be cut at N so that the ratio

of KN to NM be the same as the ratio of TE to EH. And from N <we

draw> the perpendicular NS and we join SM and SK.

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18/ Kam l al-D n al-Fars !s additions"

Since the ratio of KN to KM is the same as the ratio of TE to TH, then

the arcs DT and SK are similar and the ratio of SN to NM is the same

as the ratio of DE to EH. We construct through S a plane perpendicular

to SN; it produces a circle with a radius equal NM. We extend SN

toward P; then the ratio of SP to its radius is the same as the ratio of DA

to AZ and the ratio of its radius to the edge of the figure, constructed in

the sphere, that has a base similar to the base of the cylinder is the same

as the ratio of ZA to AB. Then ex aequalia, the ratio of the BP to the

edge of the figure constructed in the sphere is as the ratio of DA to AB.

Then the constructed edge is equal to SP. The same can be said for the

base inscribed in a circle passing through the point P, and once the

figure drawn, we get the cylinder as posed. That is what we wanted.

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Tarikh-e Elm, Vol. 11(2) (2015), pp. 21-24

Abstracts of Persian Articles

Ibn al-Haytham, Kam l al-D n al-F rs and Wave Motion of

Light Soheyla Pazari

Ph.D. in History of Science

Institute for the Humanities and Cultural Studies

Apart expressing the cause of the reflection of light from the surface of

polished objects, Ibn al-Haytham in his al-Man ir claims that the

acceptance or rejection of light by these objects is only due to their

polished surfaces and have nothing to do with their solidity, because the

reflection of light from the surface of soft polished objects such as water

is also possible. Kam l al-D n al-F rs in the Tanq al-Man zir

criticizes this theory. He denies Ibn al-Haytham!s idea about

simultaneous reflection and refraction from soft polished objects. In

other words F rs thinks that Ibn al-Haytham!s explanation for

reflection from solid surfaces cannot be used in the case of soft surfaces.

He also believes that the propagation of light is similar to the

propagation of sound and not to the motion of material objects. In 16th

and 17th centuries Europe there is a similar discussion between those

who believes in the particle theory and those who believing in the wave

theory. This article proposes to study this challenge.

Keywords: Ibn al-Haytham, Kamal al-D n al-F rs , Light, the motion

of objects, the transition of sound.

The Place of Oud and its Various Types in the Islamic Eras of

Persian Music History Narges Zaker Jafari

University of Guilan

One of the chapters in the history of Iranian music is the place and

significance of the musical instrument called Oud. This instrument,

initially named "Barbat#, emerged during the Sassanid dynasty.

Generally known under the name "Oud# in Islamic era, it became

grepopular in Islamic countries. In Iran, this instrument enjoyed great

22/ Abstracts of Persian Articles

importance until the fall of the Safavid dynasty. From the early treatises

on music, dating back to the third or fourth century A.H. to the

documents belonging to the Safavid era, Oud was considered the most

preponderant instrument in comparison with other instruments. In this

paper, we propsose to study the structural features and different types

of this instrument. According to the old literature on music, Oud has

had various sizes and different types of Oud used to be called by

different names. Difference in the number of strings has been another

distinctive feature of various types of Oud mentioned in the old

literature.

Keywords: History of Iranian music science, Persian musical treatises,

Oud.

A Short Review of the Contents and Sources of Manuscript No.

3447 in National Library of Tabriz Younes Karamati

Institute for the History of Science, University of Tehran

Manuscript No. 3447 in National Library of Tabriz is a Persian treatise

which covers arithmetic, geometry, algebra, hay a and the topic of

�Masses and Distances. The redaction of the text has been finished in

670AH/1271AD but we don!t know anything about the author of the

treatise. In the two first chapters of his treatise (on arithmetic) the

unknown author draws upon Abd al-Q hir al-Baghd d !s al-Takmila f

Ilm al- is b and in the third chapter (on geometry) upon Ab al-Waf

B zj n !s al-Man zil al-S b . The 4th chapter, which is on algebra, is an

elementary work and despite the advanced works on algebra by the

Muslim scholars at that time this work does not go beyond the quadratic

equations. In the �Conclusion� of the treatise the author adds some

probleems under the title �Rare problems pertaining to each chapter�.

Althouth this work doesn!t contain anything knew in mathematics, it is

important from a historical point of view, as far as it marks a step in redaction

of scientific works in Persian language.

Keywords: Ab al-Waf B zj n , Abd al-Q hir al-Baghd d , Algebra,

Arithmetic, al-Takmila f Ilm al- is b, al-Man zil al-Sab

Tarikh-e Elm, Vol. 11(2) (2014-2015) /23

Diversity and Variety of Hay a Books in Islamic Civilization Amir Mohammad Gamini

Institute for the History of Science, University of Tehran

Once the Greek astronomy's most worthy masterpiece, Ptolemy�s

Almagest, was translated into Arabic, many scientific works were

produced in Islamic civilization. The genre known as Hay a, which

deals with the cosmological aspect of astronomy, gradually seperated

itself from other astronomical branches. While these branches were in

relation with each other, their goals were different. Computational

astronomy, known as science of !Zijes", was developed to produce

arithmetical tables to forecast the position of the heavenly bodies in any

given time. The science of astronomical instruments and science of

timekeeping were two of the other branches of astronomy. Science of

Hay'a uaully presents a non-technical descrpition of the large scale

structure of the universe, i.e. the Earth and its position, configuration

and size of the celestial spheres and the planets. In this article almost all

of the Hay a books of Islamic civilization are listed and classified.

Based on their manuscripts we mention their scientific specifications

and compare them with each other. The golden age of these works in

12th century is studied in more details.

Key words: Almagest, Solid Orbs, Hay a, s , Sh r z , Ur .

Iterative Ratios in the Fractal Geometry of Urch n Domes Mahammad Mashayekhi

M.A. in Architecture

Farhad Tehrani

University of Shahid Behesti, Tehran

In the classification of traditional domes, they are divided into two main

categories: !roc� (cone-shaped) and n r� (hemisphere-like) domes.

Urch n domes are a sub-category of cone-shaped domes which were

common in the past in the region of Mesopotamia and southwest of Iran.

The fractal geometry is the best way to describe the geometrical

structure of these domes, because the existence of self-similar figuresm

and the relations between the parts of the dome, such as the height and

span, create some fractal relations. In this paper we try to make a

detailed study of urch n domes and to discover their geometric

24/ Abstracts of Persian Articles

structures using some numeric sequences in relation with fractal

geometry.

Keywords: Fractal geometry, Geometric proportions, Infinite series,

Urchin domes.

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