1.1. Construction of the Table of Sines

The first known table of chords, which is lost, was that of Hipparchus;Footnote ¹⁹ its degree of accuracy is unknown. The second known table was made by Menelaus; this table is also lost. But, the important extant table of chords was that of Ptolemy.Footnote ²⁰ In this table, the values are given in three sexagesimal places for every half degree, but there is an error in the third sexagesimal place.

The earliest Indian “sine table”, with radius equal to 3438, is found in the work of Āryabhaṭa (476–550) on astronomy. The inventors of it are unknown, but it is of Indian origin.Footnote ²¹ They tabulated the values of half the chord, and gave it the name ardhajya, and shortened to jya. It became jayb Footnote ²² in Arabic. The Europeans translated it into sinus.Footnote ²³ But Indians first knew the addition formulas for sines in the year 1150 by Bhaskara II.Footnote ²⁴ After the sine function, the versed sineFootnote ²⁵ was the next function that interested the astronomer, and not the cosine. The versed sine is still used by surveyors. Āryabhaṭa computed a table for it. During the time of Abū al-Wafāʾ, all six trigonometric functions were introduced.

Basic Values: Since the radius is always divided into 60 equal parts, the values of the sine function (defined as line BC in Fig. 2) lie between 0 and 60 (i.e. between 0 and 1 in a unit circle). Abū al-Wafāʾ calls the value 60 the greatest sine. For simplification purposes, throughout the paper a circle of radius 1 will be used.Footnote ²⁶ In his Almagest, he has recourse to sin 90° = 1 in many places, but he never mentions this obvious result (he does not even mention it in his tables) sin 0° = 0.

Fig. 2:

The sine and the cosine.

In addition to this, he also gives this basic result: sin 30° = 0; 30.Footnote ²⁷

Before deducing this value, Abū al-Wafāʾ, in the second section of chapter five of Book I, states that:Footnote ²⁸

The next basic value which is necessary for Abū al-Wafāʾ to construct his table of sines is sin 36° (or sin 18°).

One of these values is sufficient. Abū al-Wafāʾ computes both values in different ways with the technique of the inscribed regular polygon.Footnote ³⁰ To determine the side of a regular pentagon (or regular decagon) inscribed in a circle, he again makes use of Euclid's Elements and applies the idea of the “golden section” (a line cut in extreme and mean ratio). To determine crd 36°, he states the rule in general as follows (taken from proposition 9 of book XIII of the Elements):

Abū al-Wafāʾ does not present a proof for it but just a justification, as follows:

If the side of the hexagon (let it be r) and the side of the decagon (let it be x), inscribed in the same circle, are connected in a straight line, as in Fig. 3, then they form a line cut in extreme and mean ratio at the connection point, and then the following holds:

The positive solution of this quadratic equation gives the side of the decagon as stated above (here r is equal to half d); Abū al-Wafāʾ does not show how he solves the equation, but just mentions the rule. Abū al-Wafāʾ's original text is:Footnote ³¹

Fig. 3:

The side of the decagon.

To determine the side of the pentagon, he presents three methods, two of them depend on the side of the decagon, and the third does not. One of the first two methods is identical to proposition 10 of Book XIII of the Elements (Abū al-Wafāʾ therefore did not give a proof for it):

The second method of Abū al-Wafāʾ for the side of the pentagon is:

The original formulation is:

His complete proof is:

In Fig. 4, let OB be the side of the decagon, i.e. OB = crd(decagon), and AC is the side of the square, i.e. AC = crd(square), then AB is the side of the pentagon,Footnote ³³i.e. AB = crd(pentagon). Since AC ² = OA ² + CO ², and CO ² = CB · OB (because CB is cut into extreme and mean ratio at O), and since CB·OB = OB ² + CO · OB, then

Fig. 4:

The side of the pentagon.

But AB ² = AO ² + OB ². Therefore AC ² = AB ² + CO · OB.

His third method is formulated as follows (without giving a proofFootnote ³⁴):

In modern notation:

In the numerical examples, his answers are:

Footnote ³⁵

Footnote ³⁶

By this, and are both determined, and his results are correct to 11 decimal places. These basic values mentioned here will be taken into account in the approximation of sine 1.

Formulas: In section 3 of chapter 5 of Book I, Abū al-Wafāʾ uses the definition of the cosine (the complement of the sine to ninety degrees) and the Pythagorean Theorem (the right angle triangle OBC in Fig. 2) to deduce the following fundamental trigonometric relation:

and to obtain new values by using the already computed basic values, e.g.:

Again his result is correct to 11 decimal places.Footnote ³⁷ In this regard, Abū al-Wafāʾ's original text is:

In section 5 of chapter 5 of Book I, Abū al-Wafāʾ derives the half and double angle formulas for sines. These formulas are necessary and will both be used in his approximation of sin 1°. His proof on the double formula was as follows:

In Fig. 5, G and H are midpoints of arcs AB and AD respectively,Footnote ³⁸ and GI and HJ are perpendicular. Since triangles ABE and ABD are similar, this means:Footnote ³⁹

where . This means:

and in general the double formula as stated by him is (he uses this in his numerical examples):

Fig. 5:

The double and half formulas for sine.

His original text is:

To proof the half formula, Abū al-Wafāʾ does the following:

In Fig. 5, triangles ABE and ABD are similar; then

this means

i.e.

and in general the formula as stated by him is (he uses this in his numerical examples):

Notice that, and by definition: versin x = 1 − cos x. So, the formula actually is:

which Abū al-Wafāʾ does not mention in his Almagest.Footnote ⁴¹ Under the half formula, Abū al-Wafāʾ has already talked about the calculation of the value

by using the already determined values and by applying the half formula several times.Footnote ⁴² He did not mention why, but it seems he was preparing the reader for the approximation of sin 1° as he used it there. He also uses the same half formula as double formula, i.e.

this is equivalent to the formula (which he does not mention):

One might wonder why Abū al-Wafāʾ avoided dealing with this formula, and preferred the half formula with versed sine. This might be because the first could give negative values for some values of x.

As for the sum and difference formulas, Abū al-Wafāʾ presents two formulas with complete proofs, but because of the absence of the symbolism he cannot see that they are the same formula, and thus regards them as two different methods and gives a numerical example for each of them.Footnote ⁴³ The two formulas are:

His original text for the second one is:

And his proof is as follows:

In Fig. 6, where E is the center of the circle, BN is perpendicular to AE, and BD is perpendicular to CE. Since I and H are midpoints of BN and BD respectively, then in the first figure, or in the second figure.Footnote ⁴⁴ This means thatFootnote ⁴⁵.

Fig. 6:

The sum and difference formulas for sine.

The triangles HGB and BEI are similar. This is because angles BGH and BIE are right angles, and angles H and E are equal, because they subtend BI. Therefore

But BH, BE and EI are known ( is given, , and BE is the radius). So, GH is known.

Moreover, triangles BGI and EBH are similar. This is because angles H and G are right angles, and angles E and I are equal, because they subtend BH. Therefore

But BI, BE and HE are known ( is given, and BE is the radius). So, IG is known. Therefore IH is known.Footnote ⁴⁶

As an application to this formula, Abū al-Wafāʾ computed the following values:

where all places are correct.Footnote ⁴⁷ Under this point, it is worth mentioning that Abū al-Wafāʾ also derives sum and difference formula for chords, and that his formulas differ from those of Ptolemy. Ptolemy's sum formula for chords is similar to the modern sum formula for cosine,Footnote ⁴⁸ while Abū al-Wafāʾ's is similar to the sum formula for sine.

Approximation of sin 1°: To do this, Abū al-Wafāʾ makes use of the following lemma:

If the arcs AB, AC, AD,…< 90° are given such that: AC − AB = AD − AC = L, then .

His proof was as follows:

In Fig. 7, let where line AF is a diameter.

Fig. 7:

A lemma for the approximation of the sine of one degree.

Since angle CDH is bigger than BCI, make angle CDK equal to angle BCI. Since angles BCI and CDK are equal, and angles I and K are right angles, and BC equals CD, the triangles BCI and CDK are congruent. Therefore DK = CI.

But DJ < DM and DM < DK = CI ⇒ DJ ≪ CI.Footnote ⁴⁹

Therefore DH − CG < CG − BE.

With the help of this lemma, the basic values, and the formulas, Abū al-Wafāʾ is able to approximate the value of one degree to a good degree of accuracy.Footnote ⁵⁰ His method is as follows:

In Fig. 8,Footnote ⁵¹ suppose thatFootnote ⁵², and ; divide arc BC into six equal arcs. This means that and .

Fig. 8:

The approximation of the sine of one degree.

The following values

can be determined as follows:Footnote ⁵³

by applying the half formula on the side of the decagon. The valueFootnote ⁵⁴

was obtained before.Footnote ⁵⁵ As for the value

he suggests two methods:

They can be expressed as or as , and by half formula

Footnote ⁵⁶

Now since and , then

and

therefore

Similarly

This meansFootnote ⁵⁷

By taking the mean value he obtainsFootnote ⁵⁸

Now sin 1° can be computed from by applying the double formula and he obtainsFootnote ⁵⁹

In the proof, Abū al-Wafāʾ mentions that one cannot determine directly from arcs AB, AG, and AC (as done for example in the case of ). And therefore, the value should be approximated:

Abū al-Wafāʾ does not give a justification for his comment, and what he means is that the half cannot be written as an integer combination of the given three arcs , i.e. interpreted to modern notations.Footnote ⁶⁰

The proof of this is trivial. What is noticeable is that it seems Abū al-Wafāʾ does not theoretically study the above problem deeply; otherwise he would only mention two of the three values, as the third can be expressed as an integer combination of the other two.

This original method of approximating sine of one degree by Abū al-Wafāʾ is considered to be one of his great contributions to trigonometry.Footnote ⁶¹ The best-known results achieved before and after him were:Footnote ⁶²

1. 1) The approximation made by Ptolemy in his Almagest, correct to two places. His method could be summarized as:

   

2. 2) The approximation made by the Egyptian Ibn Yūnus (d. 1005) in his Great Hakemite Table in sexagesimal with base 60 was:

   

   His method is inspired in a way from that of Ptolemy. The method included:

   

3. 3) Al-Kāshī (d. 1436) in his Treatise on Chord and Sine approximated sin 1°, correct to 17 decimal places by means of an iterative method. The method involves the approximate solution of a cubic equation associated with the problem of trisection of an angle. The equation is of the form ax − b = x ³, where x is sin 1°. This method was considered as one of the best achievements in medieval algebra. But he was not the first to approximate the solution of the mentioned equation; al-Bīrūnī did that before him.

Table of sines: Abū al-Wafāʾ was not satisfied with the results achieved by his predecessors, and he aimed to correct the trigonometric tables. In the preface of his Almagest, he states clearly that:

His table of sines consists of 5 columns. The first column is for the arcs at intervals of a quarter of a degree up to 90 degrees, starting from a quarter of a degree. The second column is for the corresponding values of sines. After computing sin 1°, and with the help of the formulas, Abū al-Wafāʾ can easily fill the table by determining the values for n = 1, 2, 3, … , 360, where the answers are correct at least to 8 decimal places.Footnote ⁶³

The third column is for the increments per minute for each interval of the second column.Footnote ⁶⁴ The fourth column is for the corresponding values of the versed sines. The fifth column is for the increments per minute for each interval of the fourth column. Unfortunately, all his trigonometric tables are lost. Regarding the values that are not in the table, Abū al-Wafāʾ, following his predecessors, uses linear interpolation to approximate their sine values. The average increments in columns three and five, which are called sixtieths, are used in the linear approximation. Since the obliquity of the ecliptic (23; 35) and the latitude of Baghdad (33; 25), which he determined by observation, as will be shown later, are not in the table of sines, he approximates their trigonometric values using linear approximation.Footnote ⁶⁵ Abū al-Wafāʾ likewise make use of linear interpolation to approximate the inverse sine.Footnote ⁶⁶

1.2. Construction of the Table of Tangents and Cotangents

Definitions and Relationships: Abū al-Wafāʾ introduces Fig. 9 to present the relationship among the six trigonometric functions and their definitions.Footnote ⁶⁷ In Fig. 9, arc AB is given.Footnote ⁶⁸ The tangent of arc AB is defined as the line AF, the cotangent is line CE (complement of the tangent to ninety degrees), the secant is defined as line OF, and the cosecant is defined as line OE, the sine is defined as line BD, and the cosine is defined as line OD (the complement of the sine to ninety degrees). What could attract attention, from an historical point of view, is that Abū al-Wafāʾ, when he introduces these functions and their relationships, ignores the versed sine; he also ignores it in his spherical theorems, and in his applications in astronomy,Footnote ⁶⁹ and instead of it he uses the cosine on many occasions.

Fig. 9:

The definitions of the six trigonometric functions.

Using Fig. 9, Abū al-Wafāʾ can easily derive the following relations:Footnote ⁷⁰

With these definitions and relations, Abū al-Wafāʾ is considered to be the first to introduce the trigonometric functions in these clear and definite forms:

With the help of these relations he constructs the table of tangents and cotangents and works out the spherical theorems (particularly the tangent rule).

Numerical Examples: In some of the examples, when he assumes the radius to be 1, his results are identical to our modern results, as these values which were appearing for the first time:

But when he tabulates the tangents and cotangents he uses a circle of radius 60 to make the calculations easy, as the 60 was common that time. But he immediately comments:

Table of Tangents and Cotangents: This table is more elegant than that of sines. In this table, he tabulated the tangents and the cotangents in reverse order. One might wonder why he did not tabulate the sines and cosines in the same way.

The table consists of four columns. The first one is for arcs of interval one-fourth of degree, i.e. arcs of the form for n = 1, 2, …, 360, and this first column is for tangents. The second column is again for the arcs but in the reverse order, this second column is for cotangents. The third column is the corresponding tangents, or the corresponding cotangents. The fourth column is for the average increment per minute for each interval in the third column (to be used for linear approximations). Regarding the table of tangents and cotangents, Abū al-Wafāʾ mentions that the first column ends with 90, and that the second column starts with 89; 45. The table is unfortunately lost, and it would have been interesting to know whether Abū al-Wafāʾ left the places of tan 90° and cot 0° empty or if he wrote “infinity”:

1.3. Spherical Theorems

After discussing plane trigonometry and constructing the trigonometric tables with high accuracy in Book I, Abū al-Wafāʾ discusses the spherical theorems in chapter 1 of Book II. This second book of his Almagest consists of ten chapters. In the first chapter, which is the most important from a historical point of view, Abū al-Wafāʾ introduces four spherical theorems (mentioned in the introduction), which he applies tens of times throughout his Almagest in spherical astronomy. During the Arabic-Islamic civilization, the sine rule was known as al-shakl al-mughnī (The theorem which dispenses one from the need [of Menelaus's theorem]). It was KūshyārFootnote ⁷⁴ who first gave the theorem this name. The expression al-shakl al-mughnī was also dedicated to the law of sines, i.e. even though they looked different in form, they were both known under the same name.Footnote ⁷⁵ This could cause confusion or misunderstanding in writing the history of these two spherical rules. Al-Khujandī called the sine rule Qānūn al-Hayʾa (The Law of Astronomy), but as far as we know, this name was neither common nor popular. In his Almagest, Abū al-Wafāʾ has no special name for these theorems, he just calls them theorems, shakl, and assigns to them numbers, and in his applications, he refers to the sine rule as theorem 1 and to the law of sines as theorem 7, according to their appearance in the first chapter of Book II. So, the mentioned names were probably given after Abū al-Wafāʾ completed his Almagest.

Menelaus's theorem (the rule of six quantities), which was known as al-shakl al-qaṭṭāʿ (the quadrilateral), was the only spherical theorem used in spherical astronomy till the time of Abū al-Wafāʾ.Footnote ⁷⁶ Since the theorem contained six quantities and a compound ratio, expressed with words, and no symbols, the application of this theorem was in many cases difficult and cumbersome. When the theorem became known to Arabs, they made many studies on it, sometimes as commentaries on the Almagest of Ptolemy and the Spherica of Menelaus, and sometimes as independent treatises. The purpose of these studies was to facilitate the use of this theorem. In his book, The Keys of Astronomy, the young al-Bīrūnī mentioned many of those Arab scholars, and the most famous among them were Thābit ibn Qurra (d. 901), Abū al-ʿAbbās al-Nayrīzī, Abū Jaʿfar al-Khāzin, and Abū Naṣr ibn ʿIrāq. Thābit wrote a work on the compound ratio, and another one on the quadrilateral.

Before discussing the disputes about the history of the sine rule and the law of sines, we should note that Abū al-Wafāʾ in his Almagest ignored Menelaus's theorem completely, as if it did not exist.Footnote ⁷⁷ He used it neither in his applications nor in his proofs for the new theorems. Later, after Abū al-Wafāʾ's death, some Muslim scientists, like Ibn al-Haytham, in solving the problem of the Qibla, applied Menelaus's theorem.

The answer to the question regarding who was the first to discover the rule of sine is limited to three scientists, namely: Abū Naṣr, al-Khujandī, and Abū al-Wafāʾ. Since all these were distinguished scientists, and there is a lack of sources, and they all strongly claimed the priority, it is enormously difficult to give a final judgment on who first discovered the sine rule. Therefore a list of the main elements that might be crucial in making a judgement, would be more practical and more helpful. They are as follows:Footnote ⁷⁸

• • Many scientists worked on Menelaus's theorem to make it easy to use, such as:

  • Thābit; books on the compound ratio and quadrilateral;Footnote ⁷⁹

  • Abū al-ʿAbbās al-Nayrīzī; commentary on Ptolemy's Almagest;

  • Abū Jaʿfar al-Khāzin; commentary on Ptolemy's Almagest;

  • Abū Saʿīd al-Sijzī;

  • Abū Naṣr ibn ʿIrāq; the book Tahdhīb.

• • In their commentaries, al-Nayrīzī and al-Khāzin demonstrated particular cases of Menelaus's theorem, each of which led to the sine rule (just by using a very simple trickFootnote ⁸⁰). Unfortunately, they didn't transpose these particular cases to a general rule; i.e. they did not aim to make these cases to replace the quadrilateral.

• • The sine rule employs only arcs, and the technique of proving it was common to all scientists (straight lines inside the sphere and the similarity between plane triangles), and a particular case of it is also a particular case of Menelaus's theorem.

Accordingly, the three scientists in this dispute could in reality have discovered the sine rule independently; especially if they were aware of the achievements of al-Nayrīzī and al-Khāzin. For example, in his proof, al-Khujandī used the above mentioned simple trick. And this independence could explain the strong claim by each of the three scientists.

As for the law of sines, the case is totally different:

• • The dispute over priority was only between Abū al-Wafāʾ and Abū Naṣr.

• • New concepts and terms were employed, namely: a spherical angle and the sine of a spherical angle. This theorem dealt with sides and angles of a spherical triangle, whereas the sine rule dealt only with spherical arcs. So, in this case, the crucial issue is the initial thinking of formulating a theorem for a general spherical triangle employing its angles.Footnote ⁸¹

• • A new technique of proving was introduced, i.e. by working entirely on the surface of the sphere. Before using this technique in his proof of the law of sines, Abū al-Wafāʾ also used it twice: in proving the cosine rule and in deducing another form of the sine rule itself.

• • Abū al-Wafāʾ was the first to apply it; he applied it 13 times in his Almagest.

• • Abū Naṣr was strongly supported by his pupil, al-Bīrūnī, to be the first to discover the theorem.

Accordingly, it would be difficult to believe that they both formulated the theorem and proved it independently at the same time. Also worth mentioning is the fact that the law of sines can be derived from the sine rule by using simple tricks, and in this sense, the one who discovered the sine rule might think that he also had the right to claim discovery of the law of sines. The formulation is the key issue here.

The Sine Rule: Since it is his main tool in spherical astronomy,Footnote ⁸² Abū al-Wafāʾ presents different geometrical figures, different forms, and two different proofs for the sine rule.Footnote ⁸³ One of his formulations is as follows:

His first proof is as follows (notice that Abū al-Wafāʾ does not draw the lines, in dot form, but just lines as if the figure were two dimensionalFootnote ⁸⁴):

In Fig. 10, all arcs are of great circles, and angles E and C are right angles, and point G is the center of the sphere. Line BH is perpendicular to GC, and line DI is perpendicular to GE. They are also perpendicular to plane ACG. And lines BJ and DK are perpendicular to line AG.

Fig. 10:

The spherical sine rule.

Since BH is parallel to DI and BJ is parallel to DK, the angles JBH and KDI are equal. But angles JHB and KID are right angles. Therefore the triangles JBH and DIK are similar. This means

But .

Here is the proof of the sine rule as it is given by Abū al-Wafāʾ:Footnote ⁸⁵

This theorem allows Abū al-Wafā to deduce another form of the rule. In Fig. 11, all arcs are of great circles, angles E and C are again right angles. Make arcs AH and AI quadrants and apply the already proved sine rule to obtain

Fig. 11:

Another form of the spherical sine rule.

But . Therefore .

The Tangent Rule: This rule was credited to Abū al-Wafāʾ by his contemporaries and later Muslim scientists. Combining the rule together with the definitions and the relationships among the trigonometric functions mentioned before, the most significant in this rule is that Abū al-Wafāʾ, by applying it many times in spherical astronomy, made the functions tangent and cotangent as famous as sine and cosine.Footnote ⁸⁷ Al-Bīrūnī noticed its importance and adopted it immediately, describing it as a great contribution to astronomy (and it is), and explained the rule in detail in his works distinguishing between the tangent as shadow and the tangent as function of an arc. At first, the tangent rule was controversial; e.g. al-Khujandī and Kūshyār did not believe that it was true, but the young al-Bīrūnī convinced them later. In his The Quadrilateral, al-Ṭūsī, again discussed the cause that made the theorem controversial in detail.Footnote ⁸⁸ Abū al-Wafāʾ's formulation is as follows:

His proof is as follows:

In Fig. 12, point I is the centre of the sphere, and the spherical angles D and B are right angles, and all arcs are of great circles, and BH and DJ are perpendicular to plane ABD. HK and JM are perpendicular to AI. It is clear that

Fig. 12:

The spherical tangent rule.

Since HK and HB are parallel to JM and JD, angles KHB and MJD are equal. And angles HBK and JDM are right angles. Therefore triangles KHB and MJD are similar. This means:

Here is the proof of the tangent rule as it is given by Abū al-Wafāʾ:

Abū al-Wafāʾ uses this rule tens of times in his applications, and by this he makes the tangent and cotangent as famous as the sine and cosine (see the last discussion under the law of sines).

The Cosine Rule: This was less applied than the first two. But it bears two important indications with respect to history. It first indicates that Abū al-Wafāʾ had thought about making rules for spherical triangles,Footnote ⁹¹ and also that Abū al-Wafāʾ deduced it directly by applying the sine rule. Later, al-Bīrūnī classified the spherical triangles by their angles in a good systematic way, giving a formula for each, and enabling one to determine the unknown sides and angles (of the spherical triangle) from the known sides and angles. This was also done by al-Ṭūsī in his The Quadrilateral by dividing a general spherical triangle into two right angled spherical triangles; this enabled him to determine all the unknown quantities of the triangle from any three known quantities (by applying the tangent rule or law of sines).Footnote ⁹² In his applications, Abū al-Wafāʾ generally mentions (twice on two different occasions) that because two sides of a spherical triangle are known and one angle is a right angle, the third side is also known. Abū al-Wafāʾ's formulation of the theorem is as follows:

His proof for the cosine rule is as follows:

In Fig. 13, all arcs are of great circles and angle C of the spherical triangle ABC is a right angle, and arcs AD and AE are quadrants, this means angles D and E are right angles.Footnote ⁹³ By sine rule:

Fig. 13:

The cosine rule for a right-angled spherical triangle.

But . This means:

The Law of Sines: Abū al-Wafāʾ used this theorem only 13 times out of about 250 applications in spherical astronomy, one of these 13 times was in solving the problem of the Qibla. He actually looked at it as a reserve theorem, and applied it only in case he wanted to work out another method or find an alternative; he preferred to use sine and tangent rules. According to the appearance of the applications in his Almagest, the first application ever of this theorem was in determination of the ortive amplitude. This theorem appears at the end of the first chapter of Book II but Abū al-Wafāʾ introduces the definition of a spherical angle at the beginning of the chapter:

Later, at the end of the chapter, and after demonstrating the above spherical theorems, Abū al-Wafāʾ formulates the spherical law of sines as follows:

For the proof of the theorem, Abū al-Wafāʾ presents three geometrical figures for three different cases of a spherical triangle; Fig. 14 is one of them.Footnote ⁹⁴ His proof is as follows:

Fig. 14:

The law of sines for spherical triangles.

In Fig. 14, ABC is a spherical triangle of great arcs, point B is a pole for circle DE, and point C is a pole for circle EG. This means that angles G and D are right angles, and point E is a pole for circle GD, and then angle J is right angle. Sine rule gives:

ButFootnote ⁹⁵ and . Therefore

Straight after he derived the spherical theorems, Abū al-Wafāʾ goes back to the relationship between the tangent and cotangent and proves that they are reciprocal, i.e. he discusses and proves the cases

where a is a constant, and he assumes that the radius of the circle is 1. These relations allows Abū al-Wafāʾ to derive “different” formulas for the same astronomical problem. These techniques were mentioned because they will be used in the conclusion.

As it appears, in his proofs Abū al-Wafāʾ was independent and did not make any direct or indirect use of old spherical theorems. In addition to this, despite the fact that he derived the above spherical theorems, and also those related to plane trigonometry, and made a great contribution to both spherical and plane trigonometry, and despite the fact that the “trigonometry” part of his Almagest is separated from the “astronomy” part, Abū al-Wafāʾ was not aiming to make the “trigonometry” an independent science. With respect to him, like his predecessorsFootnote ⁹⁶ and contemporaries,Footnote ⁹⁷ “trigonometry” was a part of spherical astronomy. But Abū al-Wafāʾ surely paved the way for his successors to achieve total independence from astronomy. The main steps were made by Muʿādh al-Jayyānī (d. 1097), a Spanish-Arab mathematician, in his treatise Kitāb Majhūlat qisiy al-kura (The Book on Unknown Arcs of a Sphere),Footnote ⁹⁸ Naṣīr al-Dīn al-Ṭūsī (d. 1274) in his treatise on The Quadrilateral,Footnote ⁹⁹ and Regiomontanus (d. 1476) in his De Triangulis.

1.4. Calculation of the latitude of a location

Observation Instruments: The first practical problems that faced Abū al-Wafāʾ was the determination of the maximum inclination (the obliquity of the ecliptic) and the latitude of Baghdad;Footnote ¹⁰⁰ he found them by determining the summer and winter solstices by observation. In his Almagest, Abū al-Wafāʾ describes in words three instruments for observation without giving geometrical figures. He does not tell us which one he used or which one he recommended. Below there are geometrical figures of these instruments and descriptions on how they work. Abū al-Wafāʾ spent a good part of his life observing stars and planets, and he was chosen to be a member of Baghdad's observatory. This observatory was set up on the request of Sharaf al-DawlaFootnote ¹⁰¹ and opened c. 988. Sharaf al-Dawla died in 989 and the observatory was closed. The observatory included some instruments such as a quadrant more than 6 meters long (one of his instruments below is a quadrant). Abū al-Wafāʾ is considered among historians to be the first to build a wall quadrant for observation. The original description of the instruments by Abū al-Wafāʾ is:

Complete Circle with two Holes: In Fig. 15, the circle ABCD is drawn on a plate of solid matter with center O in the plane of the meridian. The circumference is divided into 360 degrees, and each degree into 60 minutes, and so on as much as possible. The opposite points F and E (the holes) are connected by the arm FE. This arm, which is fixed at O, can be rotated, either by an alidade or by putting the holes E and F on a circle, which can be rotated, concentric with circle ABCD. The diameter BD is parallel to the horizon, and point A represents the zenith of the location.

Fig. 15:

The complete circle; an observation instrument.

By rotating the arm EF so that the ray of the sun enters both E and F, one can measure the altitude of the sun at noon by measuring the degrees between E and B. He also generalizes this method to all other stars and planets. In this case, if there is no ray, the star or the planet should be seen through both holes.

Quadrant with two Holes: In Fig. 16, the square ABCD is of solid matter, where AD and BC are parallel to the horizon, and point A (hole) is the center of the quadrant BED, and E is a hole. The quadrant is divided into 90 degrees, and each degree is divided into 60 minutes, and so on as much as possible. By rotating the alidade (along arm AE fixed at A) until the ray of the sun enters both A and E, the latitude of the sun at noon is determined by measuring arc ED. From a technological point of view, this instrument is better than the previous one. In this instrument, one uses a smaller area but with a larger arc, and thus achieves more accurate results. In his zīj, al-Battānī described in detail two instruments, one of which is similar to this one, which he called al-labina (a squared plate made of copper, stone, or wood), but it is not as definite and clear as that of Abū al-Wafāʾ. Al-Battānī's second instrument was al-ʿaḍāda al-ṭawīla (the long alidade and uses three rulers).

Fig. 16:

The quadrant; an observation instrument.

The Gnomon: This instrument is an old one, but is worth mentioning as Abū al-Wafāʾ determines the altitude of the sun at noon with the help of this method of calculation. Abū al-Wafāʾ employs his tangent (cotangent) in an attractive way:

In Fig. 17, ED is the gnomon which is set horizontally and DF is its shadow, and AC is the gnomon which is set vertically and CB is its shadow. Abū al-Wafāʾ says only one sentence about this method: “If we determined the inverse tangent (cotangent) of the shadow, the result is the wanted altitude at noon”. That means:

Fig. 17:

The gnomon; an observation instrument.

Observing the Solstices at Baghdad: Abū al-Wafāʾ did not talk much about this point, but just mentioned, during several years of observation at Baghdad, that he determined the maximum altitude of the sun at noon at the summer solstice and it was 80 degrees and 10 minutes (i.e. 80; 10), and that he determined its minimum altitude at noon at winter solstice and it was 33 degrees. The difference, which is 47; 10, is the great arc between the solstices, and its half, which is 23; 35, is the maximum inclination of the sun from the equator (the obliquity of the ecliptic). According to al-Battānī, in his zīj, the difference between the maximum and minimum altitudes used by Hipparchus and Ptolemy is 47; 42, and his own difference, observed from the city al-Raqqa (a city in northern Syria), is 47; 10 (the same result as that of Abū al-Wafāʾ achieved). But al-Battānī's maximum altitude is 59; 36, and the minimum is 12; 26, observed by means of his mentioned instrument, the long alidade. Abū al-Wafāʾ's original text about this point and his next point is:

Calculation of the Obliquity of the Ecliptic: In Fig. 18,Footnote ¹⁰³ points W and S on the meridian circle are respectively the winter solstice (the minimum latitude of the sun at noon), call it m (arc WH), and the summer solstice (the maximum altitude of the sun at noon), call it M (arc SH). Point Z is the zenith of the location, and N is the North Pole; the figure is seen from east. After finding m and M by observation, to calculate the obliquity of the ecliptic, call it ε, Abū al-Wafāʾ suggests the following (and this is only for the northern locations):

or

Fig. 18:

The obliquity of the ecliptic.

And according to the data which he already determined, he obtains:

Calculation of the Latitude of Baghdad: After determining ε, he easily found the latitude of the given location, call it ϕ. He uses the following methods for northern locations (Fig. 18):

or

or

And according to his data, the latitude of Baghdad is:

Footnote ¹⁰⁴

Abū al-Wafāʾ's original text about the calculation of the latitude is:

This number 33; 25 (the latitude of Baghdad) will be used in the example of the Qibla. Since it is neither in the table of sines nor in the table of tangents, Abū al-Wafāʾ approximates its trigonometric values and his results are as follows:

Footnote ¹⁰⁶

Footnote ¹⁰⁷

Footnote ¹⁰⁸

Footnote ¹⁰⁹

Determination of the Direction of True North: Despite the fact that the line of meridian (or the four directions at a given location) is basic for the problem of the Qibla (as a practical problem), in his Almagest, Abū al-Wafāʾ does not discuss it. On the contrary, al-Battānī, in his works, which Abū al-Wafāʾ studied in his youth, describes some practical methods for determining the line of meridian. One of these methods described by al-Battānī is as follows:

In Fig. 19, let circle ABD be horizontal, and place at its centre O a straight thin stick with sharp peak vertically.Footnote ¹¹⁰ When the shadow of the stick is decreasing (during the morning hours), mark a point (point A) on the circumference of the circle as soon as the shadow tries to enter the circle, and when the shadow is increasing (during the afternoon hours), mark another point (point B) on the circumference of the circle as soon as the shadow tries to go out the circle. Let point C be the midpoint of arc AB,Footnote ¹¹¹ then the diameter DC is the meridian line, i.e. the directions of true south and true north. To locate the directions of east and west, draw a diameter perpendicular to DC.

Fig. 19:

The direction of the true North.

1.5. Determination of the Distance between two Locations

In his Almagest, Abū al-Wafāʾ discusses this subject in chapter 15 of Book IV, as it appears in the table of contents at the beginning of Book IV. But, this chapter is lost. In fact, the copyist forgot it, and after he finished copying chapter 14 he started immediately with chapter 16, this is according to the manuscript of the Almagest (MS Paris, BN, no Ar. 2494). But a good part of this lost chapter is found in another Arabic manuscript, also in Paris BN, no Ar. 5968, named Dustūr al-Munajjimīn. Footnote ¹¹² The author of Dustūr al-Munajjimīn was only interested in the methods and the numerical examples Abū al-Wafāʾ produced, and dropped the proofs. It is not difficult to reconstruct Abū al-Wafāʾ's proofs. Following Abū al-Wafāʾ's way in proving the astronomical methods as appeared in his applications in Books II, III, and IV, one can easily discover how and which of the spherical rules Abū al-Wafāʾ used in his demonstrations to prove his two methods for distance between locations.

The Fig. 20Footnote ¹¹³ will clarify the terms defined by Abū al-Wafāʾ. Points B and M are two northern locations, and circle CAE is the horizon of B, and circle NBC is the meridian of B, and circle NMH is the meridian of M, and circle GED is the equator with pole N, and in this case E is a pole for the meridian of B. All circles are great circles. The figure is seen from the west and it coincides with the case Baghdad-Mecca.Footnote ¹¹⁴

Fig. 20:

The distance between two locations.

So, arc BM is the distance between B and M, which is the wanted; arc GH is the difference between their longitudes; the circle FME is the circle of the equation of longitude; arc FM is called the equation of longitude, call it λ _e; arc FG is called the equation of latitude,Footnote ¹¹⁵ call it ϕ _e; arc BG is the latitude of B, and arc MH is the latitude of M.

To determine arc BM, where the latitudes and longitudes of B and M are known,Footnote ¹¹⁶ Abū al-Wafāʾ works out two methods. The first method depends on the equation of latitude, arc FG, and the equation of longitude, arc FM. So, determining arcs FG and FM is a first step. Throughout his Almagest, Abū al-Wafāʾ follows this way to build his work:

1. 1) Proving the astronomical problem with the help of a geometrical figure applying the proven spherical theorems. This part of the work is directed to geometers. He calls this part “Method”.

2. 2) Stating the method generally. This part is directed to the calculators. He calls this part “Computation”.

3. 3) Giving a numerical example on the problem (usually Baghdad was the location). This part is directed to beginners. He calls this part “Example”.

4. 4) Finding, if possible, another method or more for the same astronomical problem i.e. repeating again the points 1 to 3.

The First Method for the Distance: To determine the distance BM, δ, Abū al-Wafāʾ applies the cosine rule on the right angle spherical triangle FBM (Fig. 20) and obtains:

Notice that arc BF is the difference between arcs BG and FG, i.e. . Suppose that M(ϕ ₁, λ ₁) and B(ϕ ₂, λ ₂), where ϕ ₁ < ϕ ₂, the rule in general as stated by Abū al-Wafāʾ isFootnote ¹¹⁷:

To determine λ _e and ϕ _e, he suggests the following: Since angles G and F are right angles, the sine rule gives

But arc NH is a quadrant and arc NM is the complement of arc MH, this means:

And in general as stated by Abū al-Wafāʾ is:

As for ϕ_e, since angles G and H are right angles, the sine rule gives

But arc EF is a quadrant and arc EM is the complement of arc MF, this means

And in general as stated by Abū al-Wafāʾ:

Footnote ¹¹⁸

The Second Method for the Distance: The method does not employ λ _e and ϕ _e, and it is generally described by Abū al-Wafāʾ as follows:Footnote ¹¹⁹

where Δϕ is the difference in latitude, and Δ λ is the difference in longitude. The original text is as follows:

Example: Baghdad – Mecca: Abū al-Wafāʾ's data for Baghdad are:Footnote ¹²¹ϕ ₂ = 33; 25, λ ₂ = 70; and for Mecca: ϕ ₁ = 22, λ₁ = 67.

His calculation for determining λ _e is:

Footnote ¹²²

Footnote ¹²³

His calculation for determining ϕ _e is:

Footnote ¹²⁴

His calculation for determining the distance between Baghdad and Mecca is:

Footnote ¹²⁵

Footnote ¹²⁶

1.6. Determination of the Qibla

Abū al-Wafāʾ discusses this subject in chapter 16 of Book IV of his Almagest. He starts the chapter by introducing the terms supplied with a geometrical figure. The important terms are the distance between Mecca and the given location and the direction of the Qibla, which is usually called the azimuth. Fig. 21, which he produced, is actually part of his complete figure (Fig. 20). In Fig. 21, arc CAE is the horizon of the given location B with meridian circle DFC, M is Mecca and circle FME is the circle of the equation of longitude. Then arc BM is the distance between the location B and Mecca, and arc AC is the direction of Mecca.Footnote ¹²⁷ Abū al-Wafāʾ demonstrates three methods to find arc AC. Two methods depend on the distance between the locations, but the third does not.

Fig. 21:

The direction of the Qibla.

Besides the pure mathematical solutions of the problem of the Qibla, scientists during the Arabic-Islamic civilization discussed other “non-mathematical” methods. One of these famous methods, discussed for example by al-Ṭūsī, was as follows: since the latitude of Mecca is less than the maximum inclination (the obliquity of the ecliptic), the sun passes above Mecca twice in a year.Footnote ¹²⁸ Such a thing happens when the declination of sun is equal to the latitude of Mecca. So, when the sun at noontime is above Mecca (at its zenith), the direction of the sun is the direction of Mecca. But this method works only for the hemisphere of Mecca, because on the opposite hemisphere the sun will not be visible. In applying this method one should take into consideration the time difference between Mecca and the given location.

The First Method for the Qibla: Since the angles E and J are right angles, the sine rule gives:Footnote ¹²⁹

The general form as stated by Abū al-Wafāʾ is:

where q is the wanted Qibla. Footnote ¹³⁰ Abū al-Wafāʾ's own formulation is:

The Second Method for the Qibla: In the spherical triangle BMN, the law of sines gives:Footnote ¹³¹

where arc MN is the complement of MH (the latitude of Mecca).

The general form as stated by Abū al-Wafāʾ is:Footnote ¹³²

Abū al-Wafāʾ's own formulation is:

The Third Method for the Qibla: This third method, which will be discussed in the conclusion, is different than the first two methods. It does not use the distance between the locations, but the equation of longitude, λ _e, and the equation of latitude, ϕ _e, and latitude of the location, ϕ ₂. Here, Abū al-Wafāʾ demonstrates methods for computation of λ _e and ϕ _e different than the previous ones; this time he only applies the tangent rule.

In Fig. 20, angles H and G are right angles, and tangent rule gives:Footnote ¹³³

Abū al-Wafāʾ's own formulation is:

In Fig. 20, angles F and G are right angles, and tangent rule gives:

Abū al-Wafāʾ's own formulation is:

In Fig. 20, angles C and F are right angles, and tangent rule gives:Footnote ¹³⁴

Footnote ¹³⁵

Abū al-Wafāʾ's own formulation of this method was:

Abū al-Wafāʾ calls this method:

The following diagram, Fig. 22, sketches the main steps of Abū al-Wafāʾ's work.

Fig. 22:

Abū al-Wafāʾ's methods for the Qibla.

Summary: Given a northern location B(ϕ ₂, λ ₂), and Mecca M(ϕ ₁, λ ₁), where e.g. ϕ ₁< ϕ ₂.Footnote ¹³⁶ The distance between M and B is denoted by δ, and Δλ = |λ ₂ − λ ₁|, and q is the Qibla. Abū al-Wafāʾs work can be summarized as follows:

Example (The Qibla at Baghdad): Given Baghdad with latitude ϕ ₂ = 33; 25 and longitude λ₂ = 70, and Mecca with latitude ϕ ₁ = 22 and longitude λ₁ = 67. Abū al-Wafāʾ produces three numerical examples, an example for each method. The example on the third is as follows:

Starting with the equation of latitude, ϕ _e, his calculation is:

Footnote ¹³⁷

As for the equation of longitude, λ_e, his calculation is:

Footnote ¹³⁸

And for the Qibla his calculation is:

Footnote ¹³⁹

Footnote ¹⁴⁰

According to his data for longitudes and latitudes of Baghdad and Mecca and his methods, and according to our trigonometric tables, the more accurate answer for the Qibla is: