Bidat ki Haqeeqat

Image result for bidat meaning and type

The issue of Bid`ah or innovations in religion is a hotly debated one in current times.  It has been argued, sometimes with much vigour and polemic that practises in the religion of Islam that were not current at the time of the Prophet sallallahu alayhi wa sallam or his companions and should be rejected and could even lead to kufr or disbelief.  It has been the opinion of the vast majority of the Ulama throughout the ages that there is Bid’ah is of two types, that which is permissible, and that which is not.  It is the purpose of this article to reiterate the correct position, that innovations or newly introduced practises in the Din of Islam can not only be permissible, but also rewarded, hopefully providing clarification to the many people who have been confused about the issue.

The Definition of the word Bid`ah

The word Bid`ah in Arabic is derived from the root word Bada`ah, literally meaning to create a new thing without precedence.  It is synonymous with the word Khalk that means to create something out of something else.  The attributive name Al Badi is also derived from the same root to denote Allah as the Creator of things that had no previous existence.  In the Qur’an Allah is Badi ussamawaati wal ard i.e. the Creator of the Heavens and Earth (out of nothing).  Therefore, in its literal sense, the word Bid`ah has no negative connotations, it plainly refers to anything that comes into existence that is novel or not previously known.

In the technical sense, in the way it is used in the Shariah it means an addition to the Din of Islam that was not known or practised at the time of the Prophet sallallahu alayhi wa sallam or his companions.

The concept of Bid`ah in the Qur’an.

The Holy Qur’an, the primary source of Knowledge in Islam, has a most important proof of the permissibility of beneficial introductions into the Din. In surah Al-Hadid, Allah says:

As for monasticism, they invented it themselves, for we had not enjoined it on them, seeking thereby to please Allah; but they did not observe it faithfully.  We rewarded only those who were truly faithful, but many of them were transgressors.

The word ‘invented’ used in the above passage is a translation of the Arabic word Ibtada’uha which literally means ‘they made a Bid’ah.’  The verse tells us that monasticism (Rahbaniyat) was instituted by the followers of the Prophet Isa alayhi salaam after him as a new act, as a Bid’ah, for the purpose of seeking the pleasure of Allah. Allah does not condemn this act but rather tells us that after its adoption it was not followed properly.  It is clear that this verse contains an implied permission granted to them for this new act.  If one reads the words carefully, it is apparent that if Allah were condemning the new act, then there would be no need to remark that they did not observe it faithfully.  Having introduced this new act of monasticism, they should have fulfilled its conditions and requirements to achieve the purpose for which they had adopted in the first place.  Instead Allah condemns those who, having adopted monasticism, did not perform it in the proper way: but many of them were transgressors.  In fact not only was the new act permitted, but it was also rewarded, as the verse tells us: We rewarded only those who were truly faithful.  In the context of the preceding part, this would refer to those who were true believers and fulfilled the conditions of the new act and thus achieving the target of seeking thereby to please Allah.

There is an important point to consider here.  The practice of monasticism has been abrogated and cancelled in Islam, but the principle contained in this verse of the acceptability of a new act performed with the correct intention and fulfilling certain conditions is not abrogated, but remains.  The new practice introduced for the pleasure of Allah, in the principles of Islamic jurisprudence becomes a Bid’ah of guidance; that which violates the laws of Shari’ah becomes a Bid’ah of misguidance (see later).

The concept of Bid’ah in the Hadith.

It is related from the route of Jarir Ibn Abdullah that the Prophet sallallahu alayhi wa sallam said:
Whosoever introduced a beneficiary action in Islam will be rewarded for his practice as well as for the practice of the people who follow him, without lessening their reward.  Whosoever introduced a bad practice in Islam will take the sin for it as well as the sin of the people who follow him, without lessening their sin. (Muslim).

This hadith which is of sound classification is very clear and unambiguous and is a foundation for proving the validity of good innovations in Islam.  The criterion used as to whether or not a new action is accepted is that it should be hasanah,  or beneficial.  If the action is beneficial then there is an immense reward for it.. New introductions that are bad are punished severely. Scholars of Islam, as will be seen later have derived the conditions for a new act to be considered beneficial or bad.

Although the context of this hadith relates to a specific incident during the time of the Prophet sallallahu alayhi wa sallam when some companions came forward to offer charity to some poverty-stricken new arrivals at Madinah, the meaning is general.  It is not permissible to claim that this Hadith applies only to charity as a general term was used: Whosoever introduced a beneficiary action in Islam.  The Prophet sallallahu alayhi wa sallam did not restrict the reward to ‘He who spends in charity.’  It is the rule among the scholars of Islam that if an ayah of hadith was revealed for a specific incident or reason yet a general term were used in it then its application would be general and not restricted to that incident.

Some people translate the word sunnatan as the specific Sunnah of the Prophet sallallahu alayhi wa sallam himself, instead the general word ‘action’ or ‘practice.’ In other words, whoever revived a Sunnah of the Prophet sallallahu alayhi wa sallam will be rewarded etc.  However this is a gross mistranslation of the Hadith.  It is impossible to differentiate such a thing as a good Sunnah, as all the practices of the Holy Prophet sallallahu alayhi wa sallam were good, and the concept of a ‘bad Sunnah’ for obvious reasons cannot be entertained at all. Therefore it is impossible for this Hadith to apply to the Sunnah of the Prophet sallallahu alayhi wa sallam.

The authentic Hadith of the Prophet sallallahu alayhi wa sallam translated as:
Abstain from innovations, for every kind of innovation is a Bid’ah, and every Bid’ah is misguidance and all misguidance leads to hellfire is often used in an attempt to prove that all new things introduced in Islam are forbidden.  This Hadith would apparently contradict with that given above, however one must study the whole Hadith of which this is only a portion, and thus read it in context to the rest. One must also interpret this Hadith according to the other evidence from either the Qur’an or Hadith instead of giving a meaning from our own (mis)understanding..  The whole Hadith is:
I command you to have Taqwa, and to be obedient to those appointed leader over you, even if it be an Abysinnian slave.  O my companions, those who live after me will, very soon, see a lot of differences among you.  Stick to my path and the path of the Rightly Guided Khalifas.  Abstain from innovations, for every kind of innovation is a Bid’ah, and every Bid’ah is misguidance and all misguidance leads to hellfire.

This Hadith is a warning about events to come very soon after the Prophet’s sallallahu alayhi wa sallam passing on; events characterised by differences among the companions.  The Prophet’s advice was to stick to his path and that of the Rightly Guided Khalifas, indicating that there will be differences of opinion against Hazrat Abu Bakr, Hazrat Umar, Hazrat Usman, and Hazrat Ali, and that when these arise, the people should follow them and also the Sunnah of the Prophet sallallahu alayhi wa sallam.  In fact, the time immediately after the death of the Prophet sallallahu alayhi wa sallam was a time of great disruption and tribulation for the Muslims.  There came several people claiming to be prophets after the Holy Prophet sallallahu alayhi wa sallam who fought against Hazrat Abu Bakr Siddique.  There were groups of Muslims who denied the paying of Zakat, and there were people who abandoned Islam and challenged the authority of the Prophet sallallahu alayhi wa sallam, becoming apostates.  Hazrat Abu Bakr said he would fight those people who claimed to be prophets, who did not pay Zakat or became apostates.  After him came people who denied the Kkalifate of Hazrat Usman, and that of Hazrat Ali.  The Khwarij sect came about which fought against Hazrat Ali.  In all, it was an extremely volatile time.  It is clear that the ‘innovations’ mentioned in this Hadith refer to major disruptions that occurred, including people declaring prophethood after the Prophet sallallahu alayhi wa sallam, people denying the paying of Zakat, and the distorted beliefs of the Khwarij.  These were the kinds of ‘innovation’ referred to by the Prophet sallallahu alayhi wa sallam that were misguidance and therefore leading to Hellfire.

Further evidence for this comes from another sound Hadith related by Ibn Abbas.  The word ‘innovation’ used in the Hadith quoted above is a translation of the word Muhdasa, which is derived from the word Ihdas, meaning disruption.  The following Hadith gives us the Prophet’s interpretation of this word:
O people, you will be gathered on the Day of Judgement in the same way you were born (naked).  The first person to be given the dress of the hereafter will be Hazrat Ibrahim.  Some people from my ummah will be brought in front of me, and taken toward hell.  I will recognise them and I will say, “These are my companions.”  An angel will say, “Don’t you know what kinds of disruption (Ihdasa) they committed after you?  Although they embraced Islam in your life, soon after your demise they became apostates and turned towards kufr.

This Hadith of the Prophet sallallahu alayhi wa sallam therefore defines what sort of innovation in the Din of Islam is a misguidance, that is something major in the fundamentals or belief system of Islam, typical of those innovations that occurred not long after his time.  This argument enables us to understand the following Hadith:
He who innovates something in this matter of ours that is not of it will have it rejected. (Agreed)

The same word Ahdasa is used here which is translated as ‘innovates.’  Using the hadith about the companions sent to Hellfire who committed grave disruptions to interpret the word Ahdasa, the Hadith is also referring to major additions or alterations to the Din of Islam, that are not of it.  Another variation of this Hadith related by Muslim is as follows:

He who does an act which our matter is not (in agreement) with, will have it rejected.

The same word Ahdasa is used here which is translated as ‘innovates.’  Using the hadith about the companions sent to Hellfire who committed grave disruptions to interpret the word Ahdasa, the Hadith is also referring to major additions or alterations to the Din of Islam, that are not of it.  Another variation of this Hadith related by Muslim is as follows:
He who does an act which our matter is not (in agreement) with, will have it rejected.

This Hadith gives us a criterion by which every new act must be judged, namely that it should not go against the Shariah and be compatible with the Qur’an and Sunnah.  Therefore every new act is not condemned but rather should be evaluated on its merits to see whether it is in agreement with the Qur’an and Sunnah.

A final point regarding the interpretation of Hadith needs to be mentioned..  If interpretation is attempted without proper knowledge, one may find apparent contradictions between various Hadith.  If one interpreted the last few Hadith as meaning every new act in Islam is a misguidance, this would be in contradiction to the first hadith mentioned about the rewards of introducing beneficial practises into Islam and the punishments for introducing bad practises. All the Hadith mentioned above are of sound classification; in reality, there are no contradictions if the Hadiths are interpreted properly.  This is what the great Scholars of Islam have done.  By interpreting correctly and with proper knowledge, they have conformed and bridged the meanings between the Hadith.  This concept is very well known in the science of Hadith exegesis, for example, takhsis al-amm is a frequent procedure of usul al-fiqh by which an apparently unqualified statement is qualified to avoid the contradiction of another necessary principle.

The concept of Bid’ah according to Scholars of Islam.

The vast majority of the classical Scholars of Islam make a distinction between innovations that are acceptable, that may be called innovations of guidance, and those that are not, that may be called innovations of misguidance. Imam ash-Shafiyy wrote, “There are two kinds of introduced matters.  One is that which contradicts a text of the Qur’an, or the Sunnah, or a report from the early Muslims, or the consensus of the Muslims: this is an innovation of misguidance (bid’at dalala).  The second kind is that which is in itself good and entails no contradiction of any of these authorities: this is a ‘non-reprehensible innovation’ (bid’a ghayr madhmuma).” (Ibn Asakir, Tabyin Khadib al-Muftari (Damascus, 1347), 97, tr. Abdul Hakim Murad. Similar definitions have been expounded by other great classical scholars, such as Imam al-Bayhaqiyy, Imam an-Nawwawiyy, and Izzudin Ibn Abdus-Salaam and Hafiz Ibn Hajar al-Asqalaniyy, among others.  Izzudin Ibn Abdus-Salaam (one of the greatest mujtahids) categorised innovations into five types: the obligatory (wajib), the recommended (mandub), the permissible (mubah), the offensive (makruh), and the forbidden (haram). Quoted in Muhammad al-Jurdani, al-Jawahir al-lu’lu’iyyah fi sharh al-Arba’in al- Nawawiya (Damascus, 1328), 220-1.  Among the obligatory innovations Ibn Abdus-Salaam cites the following examples: recording the Qur’an and the laws of Islam in writing at a time when it was feared they would be lost, studying Arabic Grammar in order to resolve controversies over the Qur’an, and developing philosophical theology (kalam) to refute the claims of the Mu’tazilites.  Under recommended innovation come activities such as building madrassas, writing books on beneficial Islamic subjects, and in-depth studies of Arabic linguistics.  Permissible innovations include worldly activities such as sifting flour, and constructing houses in various styles not known in Madinah.  Reprehensible innovations include overdecorating mosques or the Qur’an.  The category of forbidden innovations includes unlawful taxes, giving judgeships to those unqualified to hold them, and sectarian beliefs and practices that explicitly contravene the known principles of the Qur’an and Sunna.

Innovations of Guidance and Innovations of Misguidance.

With the concept of Bid’ah being clarified somewhat, the reader may want to know what practices fall with the domains of innovations of guidance, which are permissible and rewarded, and innovations of misguidance, which are forbidden and punishable. For innovations of guidance, it would be fair to say that every single Muslim practices these innovations, knowingly or otherwise, and the list is long.  A few examples have been mentioned above.

For examples of innovations of misguidance it would be useful to look at the aforementioned Hadith about Bid’ah referring to the time soon after the death of the Prophet sallallahu alayhi wa sallam when there came false prophets, apostates and people who did not pay Zakat.  Therefore, if one were to declare or follow another prophet after the Holy Prophet sallallahu alayhi wa sallam this would be an innovation of misguidance.  Following on from this, any change in the major beliefs and tenets of Islam would be in the same category.  This could include for example, denying the attributes of Allah, denying the existence of angels etc.  Any change in the basic practises of Islam would also be an innovation of misguidance, such as reducing or increasing the number of salaats in a day or changing the number of rakaats, fasting on forbidden days.  Decreeing those things that are Halaal as Haraam or vice versa would also be an innovation of misguidance as would be adding verses to the Qur’an or falsifying Hadith.  As can be seen these are major sins and lead to Shirk and even Kufr.  These things are not necessarily far-fetched as they seem as the history of Islam bears witness to a number of stray sects of Islam that adopted certain of these practices and beliefs.


There is an oft repeated concept held by some Muslims today, that any practice in religion that was not done by the Prophet sallallahu alayhi wa sallam or his companions should be rejected as it is a misguidance and therefore punishable in Hellfire.  However one must go beyond slogans and oversimplifications and reach a correct opinion by examining the facts based upon the Qur’an and Sunnah.  As we have seen, new practices are not rejected, but are accepted and even rewarded.  However, the practice concerned should be compatible with the dictates of the Shari’ah, otherwise it will be rejected.  The opinion of those who condemn any new act without qualification comes from a misunderstanding of the sources of the Qur’an and Hadith, for example by quoting passages out of context or without the true meaning.  It is apparent that the classical scholars, who probably had a greater knowledge of Qur’anic or Hadith exegesis than any living person today decreed that newly introduced practices are allowed as long as they do not contradict the Qur’an or Sunnah.  This stands in marked contrast to the opinion of many so-called learned people today.  They should be careful of condemning an act as Haraam or prohibited if it is not specifically prohibited by the Qur’an or Sunnah, as judging a permissible act as Haraam may lead to Shirk.  In fact, the introduction of new things into the deen ensures that Islam can apply itself to any given time and situation, and some new things have even been essential for its preservation and propagation.

New Results In The Research On Some Mathematical Works Of Nasir Al-Din Al-Tusi

This work was supported by the Science Development Foundation, The President of the Republic of Azerbaijan-Grant, EIF-2011-1(3)-82/19/1.

The bibliographic work of G.P. Matvievskaya and B.A. Rosenfeld[1] contains the titles and storage locations of 29 manuscripts of Nasir al-Din Abu Jafar Muhammad ibn Hasan Abu Bakr al-Tusi (1201-1274: see more on his life and work in Nasir al-Din al-Tusi): 4 on mathematics, 22 on astronomy, as well as 4 in physics, 5 in logic and philosophy, and one manuscript in economics, music, mineralogy, and poetry. We will focus hereafter on the four mathematical works of Al-Tusi.


1. Al-Tusi’s Tahrir of Euclid‘s Elements

Fig 1. Nasir al-Din al-Tusi is pictured at his writing desk at the Maragha observatory, which opened in 1259. (Source)

One of the most famous works of Al-Tusi is his “Exposition of Euclid’s Elements” (Tahrir al-Usul al-Handasiya li-Uqlidis). There are two versions of this treatise. The first version was published in 1594 in Arabic in Rome, and then in 1657 in Latin in London. The second version was published in Arabic in Tehran in 1881.

In the fifties of the last century, Azerbaijani researchers H. Zarinazade and G. Mammadbeyli translated the second version of the Exposition into Azerbaijani language, but for various reasons they did not finish the work of editing and publishing. Finally, in 2001, this work has been completed by A. Guliyev, E. Babaev and A. Babaev.[2]

The style of Al-Tusi’s Tahrir of Euclid’s Elements is characterized by citing the text of Euclid and producing comments on them, starting with the words: “I’m talking about.” In this process, Al-Tusi gives different proofs of the theorems than those produced by in the the Euclidean text.

The study of Al-Tusi’s comments, in terms of geometric concepts, terminology and demonstrations of theorems leads us to the conclusion that, at the time of writing the Tahrir, the nature and conception of of geometry has changed.

If the geometry of Euclid, as defined by the words of S.A.Yanovsky, was “the geometry of compass and ruler, but idealized ruler and compass,”[3]and according to the translator and commentator of the Elements D. Mordukhai-Boltovsky had a “constructive” nature, the “geometry of Al-Tusi” is the geometry of ideal entities. If the “existence” of Euclid is the ability to “build” even with “idealized compasses and rulers,” the “existence” of Al-Tusi has logically inferred an ideal character.

This is indicated by the following postulates, which Al-Tusi formulates before the Euclidean ones:

1. “Point, line, plane and surface by the most important way exist, and a circle exists.”

2. “On any line and on a surface we can take a point.”

3. “On any surface we can assume a line.”

4. “No matter how the point is, we can assume the line passing through this point.”

5. “Any point, line segment and a planar surface on the applicability of its similarity.”

Fig 2. Al-Tusi’s record of Euclid’s proof of the Pythagorean theorem. (Source)

In the XIth book of the Elements on solid geometry, Euclid gives no stereometric axioms. It was thought that the first stereometric axioms were formulated only in the 17th century. Nasir al-Tusi formulated three stereometric axioms:

1. “Through a straight line can be drawn a plane.”

2. “Through a straight line and a point lying outside of it can be drawn (only one) plane.”

3. “Two straight lines do not include space.”

As it is known, up to the 15th century, the number “one” (the unit) was not considered as a number, because it was presented as a quantitative expression of the monad, and the number was defined aa a set of units (in the Greek tradition). Numerical characteristic of the units are not detected. In his commentary on the VIIth book of the Elements, Al-Tusi wrote:

“I say, the number is called something that takes place in a row of account. By this definition, the “unit” should be a number.”[4]

 So, the property of the numbers (starting with one) is to be in the row of the account (i.e., to be a characteristic of factoring in one-to-one correspondence). 

2. Trigonometry in the Shakl al-Qatta’

Fig 3. Front cover of Contemplation and Action: The Spiritual Autobiography of a Muslim Scholar: Nasir al-Din Tusi. Translated by Seyyed Jalal Hosseini Badakhchani. London: I. B. Tauris, 1999.

The work of Nasir al-Din al-Tusi Treatise on the Complete Quadrilateral(Shakl al-Qatta’)[5] is in essence the first mathematical work on trigonometry as a distinct science. Until Al-Tusi, trigonometry was considered as part of astronomy. In this treatise, Al-Tusi introduced the concept of the polar triangle and gives the calculations for it. In addition, he developed the theory of ratios of Eudoxus for incommensurable quantities and introduced the numerical characteristic of one ratio (the “measure” of ratio).

This treatise was known to European scholars, particularly to Regiomontanus (15th century), and in 1952 it was translated into Russian by G.D. Mammadbeyli and B.A.Rozenfeld.

V.N. Molodshy,[6]referring to the Shakl al-Qatta’, wrote: “In the 13th century, Azerbaijani astronomer and mathematician Nasir al-Din al-Tusi defined the concept of a positive real number, just like Newton (i.e., Al-Tusi defined that 400 years before Newton).” We must recall that the number in Newton’s definition is the relation of one quantity to another of the same kind, taken as a unit.

The new in Al-Tusi’s discovery in this work is that we could trace how he developed the notion of “measure” of ratio, what led him to the extension of the notion of rational number. 

3. The parallels problem

The development of the theme of parallel lines was ongoing for two thousand years. The theory of parallels was substantially advanced in the works of the 9th-14th-centuries scholars of the Islamic world. Al-Tusi’s treatise on parallels is called Al-Risala al-shafiya ‘an al-shak fi al-khutut mutawaziyya (Treatise healing the doubt about the parallel lines). This is a very known work of Al-Tusi. The new in the investigation of this treatise is the possibility to study the Tusi view on axioms and postulates, because this logical problem was reduced to the question of “Y Postulate”. The translation into Russian of the treatise, carried by B. A. Rosenfeld and A. P. Yushkevich, was published in The Historico-Mathematical Investigation.”[7]

In this treatise, in addition to his theory of parallel, Al-Tusi produced also some of the results of ‘Umar al-Khayyam, Al-Hassan ibn al-Haytham and others. 

4. The treatise of arithmetic Jami’ al-hisab

Fig 4. Tashkent manuscript of Nasir al-Din al-Tusi’s treatise The Collection of Arithmetic (Jami’ al-hisab bi-‘l-Takht wa-‘l-turab), folio 120.

The treatise of Nasir al-Din al-Tusi The Collection of Arithmetic (Jami’ al-hisab bi-‘l-Takht wa-‘l-turab) was written in 1265. A fragment of this treatise (the 11th section of the first part) was translated into Russian in the 1960’s by S. A. Akhmedov and B. A. Rosenfeld.[8] The treatise was translated in full into Azerbaijani by scholars of the National Academy of Sciences of Azerbaijan A. Amirahmedov, E. Mamedov and A.Babayev (the author of this paper), from Tashkent and St. Petersburg manuscripts, and was published in 2008.[9]

Then the treatise was translated into Russian and completed with Azerbaijani translation, but verified against  the Arabic original by the scientific workers of the Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan A. A. Babaev, E. M. Mamedov, and V. F. Medzhlumbekova.[10]

Let us explain, first of all, the name of the treatise. In the East, a board covered with dust was used, from antiquity to the Middle Ages, as a means of calculation. On this board, with the pointy sticks (as a handle) arithmetic operations are recorded. Intermediate results were erased and replaced by others, so as to define the algorithmic character of the arithmetic operations. Thus a set of algorithms turned the board into a kind of computing device. Thus the Collection text was an excellent textbook lacking of methodological omissions.

The treatise consists of three parts. The first part is devoted to whole numbers and operations on them; the second part studies fractions and calculations over them; the third part is devoted to the operations on the fractions in the sexagesimal system of calculation, which was used by astronomers.

In the study of the treatise we found marvelous facts, previously unknown and changing the dating of some mathematical statements. Thus, in the 8th section, Part 1, Al-Tusi puts the table to denote the degrees (Table 1). In this table the letter designations of degrees are given. We compared this table with the table of Chapter 11 of John Wallis’ The Historical and Practical Treatise on Algebra (Table 2).[11]

This table shows the designations used by mathematicians in the 16th and 17th centuries, to denote the root, square, cube, from the 4th to the 16th degrees. In Table 1, the second column is the designation of Wyeth, the third column is the designation of Outred, the forth column is the designation of Garriot and the fifth column is the designation of Descartes. We compared the designations of Outred (1574-1660) and those of Tusi (Table 1).

In Table 1 r, l, b are the last letters of the Arabic words judhur (root), mal (square), ka’b (cube) respectively. Outred’s letters q, c are the first letters of words Quadratum, Cubus (square and cube). The designation A is not correlated with any word.

Table 1. Translation of Tusi’s table

Table 2: The table from Wallis work

Taking into consideration the fact that the Arabic words are written from right to left, the principle of designation of Al-Tusi and Outred are the same. In the Tusi’s table there are the designations of “negative” power – the sheaves of degrees.

Note that it is not known any use of such symbols, not only by the predecessors of Al-Tusi, but also by Arab mathematicians of the later period. For example, there is nothing like it in the famous work The Key to Arithmetic” of Al-Kashi, who lived two centuries later.

Table 3: The table of the comparison (Tusi-Outred)

Fig 5. TFolio From The Akhlaq-i Nasiri of Nasir al-Din al-Tusi: School courtyard with boys reading and writing. Lahore, Mughal period, circa 1595 CE. (Source)

Another impressive fact is Al-Tusi’s remark about the “criterion” in the first part of the 12th chapter of his treatise. Here we present a validation of the results of arithmetic operations with the help of this “criterion.” The method of “criterions” involves comparing the remainders, obtained by dividing by 9, 7, 11, etc. of the result of operation on digits of terms of arithmetic action and of number of the resulting of arithmetic operation. These remainders are called “criterions”. In the case of the division by 9 (“criterion” on 9) this operation on digits is the summation of the digits of the number. This method was known to the Greeks and to the Indians.

It was thought that the coincidence of those “criterions” of the initial numbers and of the result is the necessary and sufficient condition for conviction in the correctness of the calculation. This is indicated in the works of such great medieval mathematicians as Abu ‘l-Hasan ibn Ahmad al-Nasawi (10th century)[12] and Ahmad ibn al-Banna (13th-14th centuries).[13] However, this condition is necessary, but not sufficient. In the history of mathematics the first indication of insufficiency of that condition dates back to the 15th century.

In the twelfth chapter of the Collection, Nasir al-Din al-Tusi wrote:

“Calculators have a way to check, known as a” criterion.” If the calculation was carried out correctly, the “criterions” also coincide; if the “criterions” are not equal, then the computation was also carried correctly. We can not say that if the “criterions” are equal, then the calculation was carried correctly, (or) if the calculation was carried out correctly, the “criterions” do not coincide.”

Thus, the assertion that the condition of coincidence of criterions is insufficient appeared in the 11th century.[14]

In the 9th section of the first part, Al-Tusi gives the algorithm to extract the square root, in the 10th section – algorithm for cube root, and in the 11th – algorithm for root of any degree.

Fig 6. Page from Sharh Usul Ashkal Kitab Uqlidis fi ‘Ilm al-Handasa dated Sha’ban 1074/March 1664.

It can be argued that the algorithms for extracting the root of fourth and higher degrees have been found in this work of Al-Tusi. However, ‘Umar al-Khayyam[15] indicates that in his own book Problems of Arithmetic he gave the method of extracting the root of the fourth and higher.

In the 3rd section, there is a table compiled for the binomial coefficients and algorithm to determine them. Although it is believed that the author of the table of the binomial coefficients (triangle of Pascal) was a famous French mathematician and physicist Blaise Pascal (1623-1662). In support of this, we give a quote from the Encyclopædia of Mathematics published in 1998 in Moscow: “to calculate the binomial coefficients, Pascal developed a method (“Pascal’s triangle”).[16] However, such table is found in the work of al-Samaw’al (12th century).[17]

Another interesting fact contained in the 3rd section-Part II is the method of finding a common denominator as the least common multiple. All previous mathematicians and even Ahmad ibn al-Banna,[18] sixty years younger than Al-Tusi, in order to find a common denominator, they simply multiply the denominators of the terms. Finding a common denominator as the least common multiple dates from the second half of the 16th century (Tartaglia and Clavius).[19]

The scientific work of Al-Tusi is an invaluable source for the study of mathematical thought in the Eastern Middle Ages and for rethinking of many mathematical ideas. The works of Al-Tusi as well as those of his great predecessors – Ibn Sina, al-Khwarizmi, ‘Umar al-Khayyam, etc. – refute the perception among some researchers that the mathematics of the Eastern Middle Ages was purely practical, and that there was a regression in mathematical theoretical thought in comparison with the Ancient period.

5. Notes and references   

[1] G. P. Matviyevskaya, B. A. Rosenfeld, Mathematicians and Astronomers of Medieval Islam and their Works(VIII-XVII centuries), vol. 2, Moscow 1983.

[2] Nasir al-Din al-Tusi, Tahriri oglidis. Baku 2001.

[3] S. A. Yanovski, “About the Axioms”, Historico-Mathematical Investigations. Moscow, No XIII, 1959.

[4] Al-Tusi, Tahriri oglidis, Baku 2001.

[5] Al-Tusi, The Treatise on the complete quadrilateral (Shyaklul Gita). Baku 1952. See also the reprint of this text in A Collection of mathematical and astronomical Treatises as revised by Nasiraddin al-Tusi, Frankfurt: Institute for the History of Arabic-Islamic Science, 1998, pp. 363-434.

[6] V. N. Molodshy, Foundations of the number’s doctrine in the XVIII and early XIX century. Moscow 1963.

[7] Nasir al-Din al-Tusi, “Treatise healing doubts about parallel lines” Historico-mathematical investigations, No. XIII, Moscow 1960.

[8] Al-Tusi, “Collection of arithmetic with the help of the board and the dust, 1st part,” Historico-mathematical investigations, Moscow 1999.

[9] Al-Tusi, Collection of arithmetic with the help of the board and the dust. Baku 2008 (Azeri), 2011( Russian).

[10] Ibidem.

[11] T. A. Tokareva, “On the John Wallis historical and practical treatise on algebra”, Historical-mathematical investigations, No. XXVII, Moskow 1983.

[12]Ali ibn Ahmad al-Nasawi, “Enough about Indian arithmetic,” Historical-mathematical investigations, No. XV, Moscow 1963.

[13] Ahmad ibn al-Banna, “Summary of arithmetic,” Historical-mathematical investigations, Series II, Vol. 9 (44), Moscow, 2005.

[14] Ahmad ibn al-Banna, “Summary of arithmetic,” op. cit.

[15] Omar Khayyam, “On the evidence of problems of algebra and almukabaly,” Historical-mathematical investigations, No. 6, Moscow 1953.

[16] Great Encyclopedic Dictionary of Mathematics, Moscow 1998, p. 175.

[17] B. A. Rosenfeld, “Algebraic treatise of al-Samaw’al,” Historical-mathematical investigations, No XX, 1975.

[18] Ahmad ibn al-Banna. “Summary of arithmetic,” op. cit.

[19] N.A. Aleksandrov, Mathematical terms, Moscow 1978