The Compendious Book on Calculation by Completion and Balancing/Notes

by Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī

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NOTES.

 

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Page 1, line 2-5.

The neglected state of the manuscript, in which most diacritical points are wanting, makes me very doubtful whether I have correctly understood the author’s meaning in several passages of his preface.

In the introductory lines, I have considered the words التي باداء ما افترض منها علي من يعبده من خلقه as an amplification of what might briefly have been expressed by التي بادائها “through the performance of which.” I conceive the author to mean, that God has prescribed to man certain duties, ان الله قد افترض علي الناس شيئا من المحامد, and that by performing these (&c. باداء ما افترض) we express our thankfulness (نقع اسم الشكر) &c.

Since my translation was made, I have had the advantage of consulting Mr. Shakespear about this passage. He prefers to read تستوجب ,تقع and تومن instead of نستوجب ,نقع and نومن, and proposes to translate as follows: “Praise to God for his favours in that which is proper for him from among his laudable deeds, which in the performance of what he has rendered indispensible from (or by reason of) them on (the part of) whoever of his creatures worships him, gives the name of thanksgiving, and secures the increase, and preserves from deterioration.”

The construction here assumed is evidently easier than that adopted by myself, in as far as the relative pronoun التي representing محامدة, is made the subject of the three subsequent verbs تقع, &c., whilst my translation presumes a transition from the third person (as in ما هو اهله, and in من يعبده) to the first (as in نقع, &c.).

A marginal note in the manuscript explains the words لعل تقديره ونومن صاحبه من الغير by ونومن من الغير “The meaning may be: we preserve from change him who enjoys it,” (viz. the divine bounty, taking صاحبه for صاحب نعم الله. The change here spoken of is the forfeiture of the divine mercy by bad actions; for “God does not change the mercy which he bestows on men, as long as they do not change that which is within themselves.” بأن الله لم يك مغيرًا نعمةً أنعمها على قوم حتى يغيروا ما بأنفسهم (Coran, Sur. VIII. v. 55, ed. Hinck.).

Page 1, line 7.

على حين فترة من الرسل] (See Coran, Sur. v. v. 22, ed. Hinck.).

Page 1, line 14, 15.

I am particularly doubtful whether I have correctly read and translated the words of the text from واحتسابا to وذكره Instead of احتسابا للاجر I should have preferred احسانا للاخر “benefitting others,” if the verb could be construed with the preposition ل.

Page 2, line 1.

To the words رجل سبق a marginal note is given in the manuscript, which is too much mutilated to be here transcribed, but which mentions the names of several authors who first wrote on certain branches of science, and concludes with asserting, that the author of the present treatise was the first that ever composed a book on Algebra.

Page 2, line 4.

An interlinear note in the manuscript explains فلم شعثه by جمع مفترقه.

Page 2, line 10.

Mohammed gives no definition of the science which he intends to treat of, nor does he explain the words جبر jebr, and مقابلة mokābalah, by which he designates certain operations peculiar to the solution of equations, and which, combined, he repeatedly employs as an expression for this entire branch of mathematics. As the former of these words has, under various shapes, been introduced into the several languages of Europe, and is now universally used as the designation of an important division of mathematical science, I shall here subjoin a few remarks on its original sense, and on its use in Arabic mathematical works.

The verb جبر jabar of which the جبْرٌ jebr is derived, properly signifies to restore something broken, especially to cure a fractured bone. It is thus used in the following passage from Motanabbi (p. 143, 144, ed. Calcutt.)

ومن اعوذ به ممـا احـــادره		يا من الوذ بـه فيما اوملــه
جودا وان عطاياه جواهره		ومن توهمت أن البحر راحته
يد البلا وذوي في السجن ناضره		ارحم شباب فتي أودت بجدته
ولا يهيضون عظما انت جابره		لا يجبر الناس عظما انت كاسره

“O thou on whom I rely in whatever I hope, with whom I seek refuge from all that I dread; whose bounteous hand seems to me like the sea, as thy gifts are like its pearls: pity the youthfulness of one, whose prime has been wasted by the hand of adversity, and whose bloom has been stifled in the prison. Men will not heal a bone which thou hast broken, nor will they break one which thou hast healed.”

Hence the Spanish and Portuguese expression algebrista for a person who heals fractures, or sets right a dislocated limb.

In mathematical language, the verb means, to make perfect, or to complete any quantity that is incomplete or liable to a diminution; i.e. when applied to equations, to transpose negative quantities to the opposite side by changing their signs. The negative quantity thus removed is construed with the particle ب: thus, if $x^{2}-6=23$ shall be changed into $x^{2}=29$, the direction is اجبر المال بالستة وزدها على الثلثة والعشرين i.e. literally “Restore the square from (the deficiency occasioned to it by) the six, and add these to the twenty-three.”

The verb جبر is not likewise used, when in an equation an integer is substituted for a fractional power of the unknown quantity: the proper expression for this is either the second or fourth conjugation of, or the second كمل, of تمّ.

The word مقابلة mokābalah is a noun of action of the verb قبل to be in front of a thing, which in the third conjugation is used in a reciprocal sense of two objects being opposite one another or standing face to face; and in the transitive sense of putting two things face to face, of confronting or comparing two things with one another.

In mathematical language it is employed to express the comparison between positive and negative terms in a compound quantity, and the reduction subsequent to such comparison. Thus $100+10x-10x+x^{2}$ is reduced to $100+x^{2}$ بعد ان قابلنا به “after we have made a comparison.”

When applied to equations, it signifies, to take away such quantities as are the same and equal on both sides. Thus the direction for reducing $x^{2}+x=x^{2}+4$ to $x=4$ will be expressed by قابل.

In either application the verb requires the preposition ب before a pronoun implying the entire equation or compound quantity, within which the comparison and subsequent reduction is to take place.

The verb قابل is not likewise used, when the reduction of an equation is to be performed by means of a division: the proper term for this operation being ردّ.

The mathematical application of the substantives جبر and مقابلة will appear from the following extracts.

1. A marginal note on one of the first leaves of the Oxford manuscript lays down the following distinction:

من اما الجبر وهو اتمام كل شيء ناقص بما يتم من غير جنسه والمقابلة من المفاعلة وهو المواجهة ولهذا يقال للمصلي القبلة اذا واجهها فلما صار لهذا الحساب جزيل عمله جبر الناقص [بما] نقص منه وزيادة مثل ما جبر به الناقص علي الجنس المقابل لتقابل الزيادة مثلما جبر به الناقص وكثر الاستعمال في ذلك فسمي جبرا ومقابلة لانه يجبر كل شيء بما نقص منه و تقابل الاجناس بعضها الي بعض ...... وقد صارت المقابلة ايضا تعرف [عند] اهل الحساب حذف المقادير المتشابهة

“Jebr is the restoration of anything defective by means of what is complete of another kind. Mokābalah, a noun of action of the third conjugation, is the facing a thing: whence it is applied to one praying, who turns his face towards the kiblah. In this branch of calculation, the method commonly employed is the restoring of something defective in its deficiency, and the adding of an amount equal to this restoration to the other side, so as to make the completion (on the one side) and this addition (on the other side) to face (or to balance) one another. As this method is frequently resorted to, it has been named jebr and mokābalah (or Restoring and Balancing), since here every thing is made complete if it is deficient, and the opposite sides are made to balance one another. . . . . . . Mathematicians also take the word mokābalah in the sense of the removal of equal quantities (from both sides of an equation).”

2. Haji Khalfa, in his bibliographical work (MS. of the British Museum, fol. 167, recto ^[1].). gives the following explanation:

ومعني الجبر زيادة قدر ما نقص في الجملة المعادلة بالاستثناء في الجملة الاخري ليتعادلا ومعني المقابلة اسقاط الزايد من احدي الجملتين للتعادل

“Jebr is the adding to one side what is negative on the other side of an equation, owing to a subtraction, so as to equalize them. Mokābalah is the removal of what is positive from either sum, so as to make them equal.”

A little farther on Haji Khalfa gives further illustration of this by an example:

كما في قولنا عشرة الا شيئا يعدل اربعة اشياء فالجبر رفع الاستثناء بان يزاد مثل المستثني علي المستثني منه فيجعل العشرة كاملة كانه يجبر نقصانه ويزاد مثل المستثني علي عديله كزيادة الشيء في المثال بعد جبر العشرة علي أربعة اشياء حتي تصير خمسة فالمقابلة أن تنقص الاجناس من الطرفين بعدة واحدة قيل هي تقابل بعض الاشيآء ببعض علي المساوات كما في مثال المذكور اذا قوبلت العشرة بالخمسة علي المساوات وسمي العلم بهذين العلمين ..... علم الجبر والمقابلة لكثرة وقوعها فيه

For instance if we say: ‘Ten less one thing equal to four things;’ then jebr is the removal of the subtraction, which is performed by adding to the minuend an amount equal to the subtrahend: hereby the ten are made complete, that which was defective in them being restored. An amount equal to the subtrahend is then added to the other side of the equation as in the above instance, after the ten have been made complete, one thing must be added to the four things, which thus become five things. Mokābalah consists in withdrawing the same amount from quantities of the same kind on both sides of the equation; or as others say, it is the balancing of certain things against others, so as to equalize them. Thus, in the above example, the ten are balanced against the five with a view to equalize them. This science has therefore been called by the name of these two rules, namely, the rule of jebr or restoration, and of mokābalah or reduction, on account of the frequent use that is made of them.”

3. The following is an extract from a treatise by Abu Abdallah al-Hosain ben Ahmed,^[2] entitled, المقدمة الكافية في اصول الجبر والمقابلة or “A complete introduction to the elements of algebra.”

باب تفسير الجبر والمقابلة. اعلم ان الحساب انما سموا هذا النوع جبرا لانهم وضعوه علي معادلة . . . . . . . فلما كانوا وضعوه علي المعادلة اداهم العمل في اكثر مسائلة الي معادلة الناقص بغير الناقص فلم يكن بد من جبر ذلك الناقص بما ينقص وزيادة مثل ذلك علي ما عدله فلما كثر ذلك فيه سموه جبرا فهذا معني الجبر وعلة تسميتهم به هذا النوع. فاما المقابلة فهو حذف المقادير المتشابهة من الجهتين.

“On the original meaning of the words jebr and mokābalah. This species of calculation is called jebr (or completion) because the question is first brought to an equation . . . . . . And as, after the equation has been formed, the practice leads in most instances to equalize something defective with what is not defective, that defective quantity must be completed where it is defective; and an addition of the same amount must be made to what is equalized to it. As this operation is frequently employed (in this kind of calculation), it has been called jebr: such is the original meaning of this word, and such the reason why it has been applied to this kind of calculation. Mokābalah is the removal of equal magnitudes on both sides (of the equation).”

4. In the Kholāset al Hisāb, a compendium of arithmetic and geometry by Bahia-eddin Mohammed ben al Hosain (died a.h. 1031, i. e. 1575 a.d.) the Arabic text of which, together with a Persian commentary by Roshan Ali, was printed at Calcutta^[3] (1812. 8vo.) the following explanation is given (pp. 334. 335.),

“The side (of the equation) on which something is to be subtracted, is made complete, and as much is added to the other side: this is jebr; again those cognate quantities which are equal on both sides are removed, and this is mokābalah”. The examples which soon follow, and the solution of which Baha-eddin shows at full length, afford ample illustration of these definitions. In page 338, $1500-{\tfrac {1}{4}}x=x$ is reduced to $1500=1{\tfrac {1}{4}}x$; this he says is affected by jebr. In page 341, $7x={\tfrac {1}{2}}x^{2}+{\tfrac {1}{2}}x$ is reduced to $13x=x^{2}$, and this he states to be the result of both jebr and mokābalah.

The Persians have borrowed the words jebr and mokābalah, together with the greater part of their mathematical terminology, from the Arabs. The following extract from a short treatise on Algebra in Persian verse, by Mohammed Nadjm-eddin Khan, appended to the Calcutta edition of the Kholāset al Hisāb, will serve as an illustration of this remark.

طرفي که دروست حرف الا
تكمیل کن و مثل آن را
بر طرف دگر فزون كن اي حبر
در مصطلح است نام این جبر
هنگام معادله تو بشناس
افتد اگر این که بعض اجناس
با وصف تجانس از سویت
در هر طرف اند بي مزيت
باید که زهر دو سو براني
نامش تو مقابله بخواني

“Complete the side in which the expression illā (less, minus) occurs, and add as much to the other side, O learned man: this is in correct language called jebr. In making the equation mark this: it may happen that some terms are cognate and equal on each side, without distinction; these you must on both sides remove, and this you call mokābalah.”

With the knowledge of Algebra, its Arabic name was introduced into Europe. Leonardo Bonacci of Pisa, when beginning to treat of it in the third part of his treatise of arithmetic, says: Incipit pars tertia de solutione quarundam quæstionum secundum modum Algebræ et Al-mucabulæ, scilicet oppositionis et restaurationis. That the sense of the Arabic terms is here given in the inverted order, has been remarked by Cossali. The definitions of jebr and mokābalah given by another early Italian writer, Lucas Paciolus, or Lucas de Burgo, are thus reported by Cossali: Il commune oggetto dell’ operar loro è recare la equazione alla sua maggior unità. Gli uffizj loro per questo commune intento sono contrarj: quello dell’ Algebra è di restorare li extremi dei diminuti; e quello di Almucabala di levare da li extremi i superflui. Intende Fra Luca per extremi i membri dell’ equazione.

Since the commencement of the sixteenth century, the word mokābalah does no longer appear in the title of Algebraic works. Hieronymus Cardan’s Latin treatise, first published in 1545, is inscribed: Artis magnæ sive de regulis algebraicis liber unus. A work by John Scheubelius, printed at Paris in 1552, is entitled: Algebræ compendiosa facilisque descriptio, qua depromuntur magna Arithmetices miracula. (See Hutton’s Tracts, &c. 11. pp. 241-243.) Pelletier’s Algebra appeared at Paris in 1558, under the title: De occulta parte numerorum quam Algebram vocant, libri duo. (Hutton, l. c. p. 245. Montucla, hist. des math. I. p. 613.) A Portuguese treatise, by Pedro Nuñez or Nonius, printed at Amberez in 1567, is entitled: Libro de Algebra y Arithmetica y Geomètria. (Montucla, I. c. p. 615.)

In Feizi’s Persian translation of the Lilavati (written in 1587, printed for the first time at Calcutta in 1827, 8vo.) I do not recollect ever to have met with the word جبر; but مقابله is several times used in the same sense as in the above Persian extract.

Page 3, line 3, seqq.

In the formation of the numerals, the thousand is not, like the ten and the hundred, multiplied by the units only, but likewise by any number of a higher order, such as tens and hundreds: there being no special words in Arabic (as is the case in Sanscrit) for ten-thousand, hundred-thousand, &c.

From this passage, and another on page 10, it would appear that our author uses the word عقد, plur. عقود, knot or tie, as a general expression for all numerals of a higher order than that of the units. Baron S. de Sacy, in his Arabic Grammar, (vol. 1. § 741) when explaining the terms of Arabic grammar relative to numerals, translates عقود by næuds, and remarks: Ce sont les noms des dixaines, depuis vingt jusqu’à quatre-vingt-dix.

Page 3, line 9-11.

The forms of algebraic expression employed by Leonardo are thus reported by Cossaui (Origine, &c. dell’ Algebra, I. p. 1.): Tre considerazioni distingue Leonardo nel numero: una assoluta, o semplice, ed è quella del numero in se stesso; le altre due relative, e sono quelle di radice e di quadrato. Nominando il quadrato soggiugne QUI VIDELICET CENSUS DICITUR, ed il nome di censo è quello di cui in seguito si serve. That Leonardo seems to have chosen the expression census on account of its acceptation, which is correspondent to that of the Arabic مال, has already been remarked by Mr. Colebrooke (Algebra, &c., Dissertation, p. liv.)

Paciolo, who wrote in Italian, used the words numero, cosa, and censo; and this notation was retained by Tartaglia. From the term cosa for the unknown number, exactly corresponding in its acceptation to the Arabic شيء thing, are derived the expressions Ars cossica and the German die Coss, both ancient names of the science of Algebra. Cardan’s Latin terminology is numerus, quadratum, and res, for the latter also positio or quantitas ignota.

Page 3, line 17.

I have added from conjecture the words وجذور تعدل عددا which are not in the manuscript. There occur several instances of such omissions in the work.

The order in which our author treats of the simple equations is, 1st. $x^{2}=px$; 2d. $x^{2}=n$; 3d. $px=n$. Lednardo had them in the same order. (See Cossali, 1. c. p. 2.) In the Kholāset al Hisūb the arrangement is, 1st. $n=px$; 2d. $px=x^{2}$; 3d. $n=x^{2}$.

Page 5, line 9.

In the Lilavati, the rule for the solution of the case $cx^{2}+bx=a$ is expressed in the following stanza.

गुणधमूलोनयुतस्य राशे

दृष्टस्य युक्तस्य गुणार्धकृत्या १

मूलं गुणार्धेन युतं विहीनं

वर्गीकृतं प्रष्ठुरभीष्ट राशिः ११

i.e. rendered literally into Latin:

Per multiplicatam radicem diminutæ [vel] auctæ quantitatis

Manifestæ, additæ ad dimidiati multiplicatoris quadratum

Radix, dimidiato multiplicatore addito [vel] subtracto,

In quadratum ducta-est interrogantis desiderata quantitas.

The same is afterwards explained in prose:

यो राशिः स्वमूलेन केनचित् गुणितेन उनो युतो वा दृष्टस्तस्य मूलस्य गुणार्धकृत्या युक्तस्य दृष्टस्य यत् पदं तद्गुणार्धेन युतं यदि मूलोनो दृष्टो राशिर्भवति यदि गुणमूलयुतो दृष्टस्तर्हि विहीनं कार्य तस्य वर्गो राशिः स्यात् ११

i.e. “A quantity, increased or diminished by its square-root multiplied by some number, is given. Then add the square of half the multiplier of the root to the given number: and extract the square-root of the sum. Add half the multiplier, if the difference were given; or subtract it, if the sum were so. The square of the result will be the quantity sought.” (Mr. Colebrooke’s translation.)

Feizi’s Persian translation of this passage runs thus:

هرگاه شخصي عددي را مضمر کرد و جذر او را يا كسري از جذر او را در عددي ضرب کرد و نام مضروب فيه بیان کرد و حاصل ضرب را با عدد مضمر جمع کرد یا از وي نقصان کرد آنچه بعد از جمع یا نقصان حاصل شده است آنرا نیز ظاهر کرد طریق دانستن آن عدد چنان است که مضروب فيه مذكور را تنصیف کرده مجذور او بگیرند و با حاصل جمع يا باقي نقصان که ظاهر کرده بود جمع کرده جذرش بگیرند بعد از آن نصف مضروب فيه مذكور را با جذر مذکور جمع کنند اگر سائل نقصان کرده باشد و نقصان کنند اگر او جمع کرده است بعد از آن مجمع يا باقي را مجذور بگیرند بعينه همان عدد مضمر خواهد بود

With the above Sanskrit stanza from the Lilavati some readers will perhaps be interested to compare the following Latin verses, which Montucla (1. p. 590) quotes from Lucas Paciolus:

Si res et census numero coæquantur, a rebus
Dimidio sumpto, censum producere debes,
Addereque numero, cujus a radice totiens
Tolle semis rerum, census latusque redibit.

Page 6, line 16.

فنصّف الاجذار تكون خمسة] Such instances of the common instead of the apocopate future, after the imperative, are too frequent in this work, than that they could be ascribed to a mere mistake of the copyist: I have accordingly given them as I found them in the manuscript.

Page 7, line 1.

وكذلك فافعل] The same structure occurs page 21, line 15.

Page 8, line 11.

فهذه الستة الضروب] Hadji Khalfa, in his article on Algebra, quotes the following observation from Ibn Khaldun

قال ابن خلدون وقد بلغنا أن بعض ائمة التعاليم من اهل المشرق انتهي المعادلات الي اكثر من هذه الستة وبلغها الي فوق العشرين واستخرج لها كلها اعمالا وثيقة ببراهين هندسية

“Ibn Khaldun remarks: A report has reached us, that some great scholars of the east have increased the number of cases beyond six, and have brought them to upwards of twenty, producing their accurate solutions together with geometrical demonstrations.”

Page 8, line 17.

See Leonardo’s geometrical illustration of the three cases involving an affected square, as reported by Cossali (I. p. 2.), and hence by Hutton (Tracts, &c., II. p. 198.) Cardan, in the introduction of his Ars magna, distinctly refers to the demonstrations of the three cases given by our author, and distinguishes them from others which are his own. At etiam demonstrationes, præter tres Mahometis et duas Lodovici (Lewis Ferrari, Cardan’s pupil), omnes nostræ sunt.—In another passage (page 20) he blames our author for having given the demonstration of only one solution of the case $cx^{2}+a=bx$. Nec admireris, says he, hanc secundam demonstrationem aliter quam a Mahumete explicatam, nam ille immutata figura magis ex re ostendit, sed tamem obscurius, nec nisi unam partem eamque pluribus.

Page 17, line 11-13.

The words from والا سدسا في to وسدس السدس are write ten twice over in the manuscript.

Page 19, line 12.

جذر مال معلوم او اصم] “The root of a rational or irrational number.” In the Kholāset al Hisāb, p. 128. 137. 369, the expression منطق (lit. audible) is used instead of معلوم, which stands in a more distinct opposition to اصم (lit. inaudible, surd). Baha-eddin applies the same expressions also to fractions, calling منطق those for which there are peculiar expressions in Arabic, e.g. ثلث one-third, and اصم those which must be expressed periphrastically by means of the word جزء a part, e.g. من خمسة وعشرين ثلثة اجزاء three twenty-fifths. See Kholāset al Hisāb, p. 150.

Page 19, line 15.

The manuscript has مثلي ذلك المال. The context requires the insertion of جذر after مثلي, which I have added from conjecture.

Page 20, line 15-17.

ما يصيب الواحد] “What is proportionate to the unit,” i.e. the quotient. This expression will be explained by Baha-Eddin’s definition of division (Kholāset al Hisāb, P. 105).

القسمة طلب عدد نسبته الي الواحد كنسبة المقسوم الي المقسوم عليه

“Division is the finding a number which bears the same proportion to the unit, as the dividend bears to the divisor.”

Page 21, line 17.

جذري] The MS. has جذر.

Page 24, line 6.

تمكننا لها صورة لا تحسن] An attempt at constructing a figure to illustrate the case of $[100+x^{2}-20x]+[50+10x-2x^{2}]$ has been made on the margin of the manuscript.

Page 30, line 10.

فخذ ما شئت] A marginal note in the manuscript defines this in the following manner.

يعني اقسم العشرة كيف شئت اربعة حنطة وستة شعيرا او ستة حنطة واربعة شعيرا او ثلثة حنطة وسبعة شعيرا او عكس ذلك او كيف ما شئت فانه يصح العمل فيه حاشية من شرح المزيحفي

“He means to say: divide the ten in any manner you like, taking four of wheat and six of barley, or four of barley and six of wheat, or three of wheat and seven of barley, or vice versa, or in any other way: for the solution will hold good in all these cases. (Note from Al Mozaihafi’s Commentary).”

Page 42, line 8.

The manuscript has a marginal note to this passage, from which it appears that the inconvenience attending the solution of this problem has already been felt by Arabic readers of the work.

Page 45, line 16.

This instance from Mohammed’s work is quoted by Cardan (Ars Mugna, p. 22, edit. Basil.) As the passage is of some interest in ascertaining the identity of the present work with that considered as Mohammed’s production by the early propagators of Algebra in Europe, I will here insert part of it. Nunc autem, says Cardan, subjungemus aliquas quæstiones, duas ex Mahumete, reliquas nostras. Then follows Quæstio I. Est numerus a cujus quadrato si abjeceris ${\tfrac {1}{3}}$ et ${\tfrac {1}{4}}$ ipsius quadrati, atque insuper 4, residuum autem in se duxeris, fiet productum æquale quadrato illius numeri et etiam 12. Pones itaque quadratum numeri incogniti quem quæris esse 1 rem, abjice ${\tfrac {1}{3}}$ et ${\tfrac {1}{4}}$ ejus, cs insuper 4, fiet ${\tfrac {5}{12}}$ rei m: 4, duc in se, fit ${\tfrac {25}{144}}$ quadrati p: 16 m: ${\tfrac {1}{33}}$ rebus, et hoc est æquali uni rei et 12; abjice similia, fiet 1 res æqualis ${\tfrac {25}{144}}$ quadrati p: 4 m: $3{\tfrac {1}{3}}$ rebus, &c.

The problem of the Quæstio II. is in the following terms, Fuerunt duo duces quorum unusquisque divisit militibus suis aureos 48. Porro unus ex his habuit milites duos plus altero, el illi qui milites habuit duos minus contigit ut aureos quatuor plus singulis militibus daret; quæritur quot unicuique milites fuerint. In the present copy of Mohammed’s algebra, no such instance occurs. Yet Cardan distinctly intimates that he derived it from our author, by introducing the problem which immediately follows it, with the words: Nunc autem proponamus quæstiones nostras.

Page 46, line 18.

The manuscript has the following marginal note to this passage:

هذه المسئلة تعمل بالكعب وطريقه ان تاخذ مالا وتلقي ثلثه يبقي ثلثا مال تضرب ذلك في ثلثة اجذار فيكون كعبين يعدلان مالا فزده مرتين علي قدر المال يكون جذرين يعدلان درهما والجذر نصف المال والمال اذا القيت ثلثه بقي سدس اذا ضربت ذلك في ثلثة اجذاره وهي درهم ونصف بلغ ذلك ربع درهم مثل المال كما ذكر ربع

“This instance may also be solved by means of a cube. The computation then is, that you take the square, and remove one-third from it; there remain two-thirds of a square. Multiply this by three roots; you find two cubes equal to one square. Extracting twice the square-root of this, it will be two roots equal to a dirhem. Accordingly one root is one-half, and the square one-fourth.^[4] If you remove one-third of this, there remains one-sixth, and if you multiply this by three roots, that is by one dirhem and a half, it amounts to one-fourth of a dirhem, which is the square as he had stated.”

Page 50, line 52.

I am uncertain whether my translation of the definition which Mohammed gives of mensuration be correct. Though the diacritical points are partly wanting in the manuscript, there can, I believe, be no doubt as to the reading of the passage.

Page 51, line 12.

I have simply translated the words اهل الهندسة by “geometricians,” though from the manner in which Mohammed here uses that expression it would appear that be took it in a more specific sense.

Firuzabadi (Kamus, p. 814, ed. Calcutt.) says that the word handasah (الهَنْدَسَةُ) is originally Persian, and that it signifies “the determining by measurement where canals for water shall be dug.”

The Persians themselves assign yet another meaning to the word هندسه hindisah, as they pronounce it: they use it in the sense of decimal notation of numerals.^[5] It is a fact well known, and admitted by the Arabs themselves, that the decimal notation is a discovery for which they are indebted to the Hindus.^[6] At what time the communication took place, has, I believe, never yet been ascertained. But it seems natural to suppose that it was at the same period, when, after the accession of the Abbaside dynasty to the caliphat, a most lively interest for mathematical and astronomical science first arose among the Arabs. Not only the most important foreign works on these sciences were then translated into Arabic, but learned foreigners even lived at the court of Bagdad, and held conspicuous situations in those scientific establishments which the noble ardour of the caliphs had called forth. History has transmitted to us the names of several distinguished scholars, neither Arabs by birth nor Mohammedans by their profession, who were thus attached to the court of Almansur and Almamun; and we know from good authority, that Hindu mathematicians and astronomers were among their number.

If we presume that the Arabic word handaseh might, as the Persian hindisah, be taken in the sense of decimal notation, the passage now before us will appear in an entirely new light. The اهل الهندسة, to whom our author ascribes two particular formulas for finding the circumference of a circle from its diameter, will then appear to be the Hindu Mathematicians who had brought the decimal notation with them;—and the اهل النجوم منهم, to whom the second and most accurate of these methods is attributed, will be the Astronomers among these Hindu Mathematicians.

This conjecture is singularly supported by the curious fact, that the two methods here ascribed by Mohammed to the اهل الهندسة actually do occur in ancient Sanskrit mathematical works. The first formula, $p={\sqrt {10d^{2}}}$, occurs in the Vijaganita (Colebrooke’s translation, p. 308, 309.); the second, $p={\tfrac {d\times 62832}{20000}}$, is reducible to ${\tfrac {d\times 3927}{1250}}$, the proportion given in the following stanza of Bhaskara’s Lilavati:

व्यासे भनंदाग्निहते विभक्ते

खबाणसूर्यैः परिधिस्तु सूक्ष्मः १

द्वाविंशतिघ्ने विहृतेऽथ शैलैः

स्थूलेभोऽथवा स्याद्व्यवहारयोग्यः ११

“When the diameter of a circle is multiplied by three thousand nine hundred and sventy-seven, and divided by twelve hundred and fifty, the quotient is the near circumference: or multiplied by twenty-two and divided by seven, it is the gross circumference adapted to practice.”^[7] (Colebrooke’s translation, page 87. See Ferzi’s Persian translation, p. 126, 127.)

The coincidence of d×62832/20000 with d×3927/1250 is so striking, and the formula is at the same time so accurate, that it seems extremely improbable that the Arabs should by mere accident have discovered the same proportion as the Hindus: particularly if we bear in mind, that the Arabs themselves do not seem to have troubled themselves much about finding an exact method.^[8]

Page 57, line 5-8.

The words between brackets are not in the manuscript: I have supplied the apparent hiatus from conjecture.

Page 61, line 4.

A triangle of the same proportion is used to illustrate this case in the Lilavati (Feizi’s Persian transl. p. 121. Colebrooke’s transl. of the Lilavati, p. 71. and of the Vijagenita, p. 203.)

Page 65, line 12-14.

The words between brackets are in the manuscript written on the margin. I think that the context warrants me sufficiently for haying received them into the text.

Page 86, line 5.

The words between brackets are not in the text, I give them merely from my own conjecture.

Page 71, line 8, 9.

The author says, that the capital must be divided into 219320 parts: this I considered faulty, and altered it in my translation into 964080, to make it agree with the computation furnished in the note. But having recently had an opportunity of re-examining the Oxford manuscript, I perceive from the copious marginal notes appended to this passage, that even among the Arabian readers considerable variety of opinion must have existed as to the common denominator, by means of which the several shares of the capital in this case may be expressed.

One says:

انظر لمال يكون لسدسه ربع والربعه ثلث وما بقي يتقسم علي ماية وخمسة وتسعين ولا يوجد ذلك في اقل اربعة وعشرين فاضرب اربعة وعشرين في ماية وخمسة وتسعين يصح من ذلك أربعة الاف وستماية وثمانون ومنه يصح

“Find a number, one-sixth of which may be divided into fourths, and one-fourth of which may be divided into thirds; and what thus comes forth let be divisible by hundred and ninety-five. This you cannot accomplish with any number less than twenty-four. Multiply twenty-four by one hundred and ninety-five: you obtain four thousand six hundred and eighty, and this will answer the purpose.”

Another:

وفي وجه آخر انك تجعل ماية وستة وخمسين ^[9] سدس المال وتضربها في ٦ فيكون ٩٣٦ واذا استخرجت نصيب الابن وهو الثلث والربع وجدته ٥۴٦ ولا خمس ليا فاضربها في ٥ يكون ۴٦۸۰ للام من ذلك ۴٣٥ وللزوج ۷۸۰ وللابن ۲۸۸ ولصاحب الخمسين ۱۴۹۳ والصاحب الربع ٦٩٥

“According to another method, you say take one hundred and fifty-six for the one-sixth of the capital. Multiply this by six; you find nine hundred and thirty-six. Taking from this the share of the son, which is one-third and one-fourth, you find it five hundred and forty-six. This is not divisible by five: therefore multiply the whole number of parts by five: it will then be four thousand six hundred and eighty. Of this the mother receives four hundred and twenty-five, the husband seven hundred and eighty, the son two hundred and eighty-eight (twelve hundred and eighty-eight?), the legatee, who is to receive the two-fifths, fourteen hundred and ninety-two, and the legatee to whom the one-fourth is bequeathed, six hundred and ninety-five.”

Another:

وفي [وجه] آخر يصح من تسعة ألاف وثلثماية وستين ووجه العمل في ذلك أن [تقسم] الفريضة في اثني عشر للام سهمان وللزوج ثلثة وللابن سبعة فتضربها في ٢٠ لذكر الخمسي والربع فيكون مايتين واربعين فتاخذها سدسها اربعين للام والثلث جائز عليها وليس للاربعين ثلث فتضرب اصل المسئلة في ثلثة لذلك فيكون سبعماية وعشرين فتاخذ سدسها للام ماية وعشرين فيخرج من ذلك الثلث لاصحاب الوصايا وهو اربعون مقسوم علي ثلثة عشر لا يصح فاضرب المسئلة في ذلك ١٣ يكون ۹۳٦۰ [۹۰٦۳ .MS] لما ذكرنا للام من ذلك ثماني ماية وخمسون وللابن الفان وخمسماية وستة وسبعون وللزوج الف خمسماية وستون ولصاحب الخمسين الفان وتسعماية واربعة وثمانون ولصاحب الربع الف وثلثماية وتسعون والله اعلم

“According to another method, the number of parts is nine thousand three hundred and sixty. The computation then is, that you divide the property left into twelve shares; of these the mother receives two, the husband three, and the son seven. This (number of parts) you multiply by twenty, since two-fifths and one-fourth are required by the statement. Thus you find two hundred and forty. Take the sixth of this, namely forty, for the mother. One-third out of this she must give up. Now, forty is not divisible by three. You accordingly multiply the whole number of parts by three, which makes them seven hundred and twenty. The one-sixth of this for the mother is one hundred and twenty. One-third of this, namely forty, goes to the legatees, and should be divided by thirteen; but as this is impossible, you multiply the whole number of parts by thirteen, which makes them nine thousand three hundred and sixty, as we said above. Of this the mother receives eight hundred and fifty, the son two thousand five hundred and seventy-six, the husband one thousand five hundred and sixty, the legatee to whom the two-fifths are bequeathed, two thousand nine hundred and eighty-four, and the legatee who is to receive one-fourth, one thousand three hundred and ninety.”

Another scholium briefly says:

ماية الف واحد وعشرين الف وستماية وثمانين في لفظ شرح المزيحفي فاذا اردت اختصارها فارجعها الي نصف جزء من ثلثة عشر

“With one hundred and twenty-one thousand six hundred and eighty, according to Mozahafi's commentary. If you want to express it briefly, you may reduce it by taking moieties of thirteenths.”

Page 85, line 8.

The manuscript has the following marginal note to this passage:

وأن شئت في عمل هذه المسئلة فاجعل الوصية الاولي نصيبا لانه أوصي له بنصيب ولم يستشن عليه شيئا واجعل الوصيتين الآخرين شيئا وزد علي انصباء الورثة يكون الجميع سبعة انصباء وشيئا واعمل على ما تقدم تخرج النصيب ۴۹ والشيء ٥۳

“If you prefer, you may also, in solving this problem, make the first legacy a share, since the testator has bequeathed a whole share without any deduction; and call the two other legacies thing. Add this to the shares of the heirs: the total amount will be seven shares and thing. Then proceed as above: you will find the share to be forty-nine, and the thing fifty-three,”

Page 93, line 1.

The following is a marginal note of the manuscript:

معني السؤال في هذه التكملة أن قوله بتكملة خمس المال بنصيب بنت أي اوصت له بخمس المال الا نصيب بنت وذلك قوله بتكملة ربع المال بنصيب الام أي أوصت له بربع المال الا نصيب الام

“The purpose of the question about such a completement is this. If the author says: as much as must be added to the share of a daughter to make it equal to one-fifth of the capital, he means to say, that the testatrix bequeathed to the legatee one-fifth of the capital, less the share of the daughter; again, if he says: as much as must be added to the share of the mother to make it one-fourth of the capital, he intends, that the testatrix bequeathed to the legatee one-fourth of the capital, less the share of the mother.”

Page 95, line 14, 19.

The manuscript has here the following note.

قال الفقيه احمد بن عباس (^[10]) اقول ان التكملة هذه المسئلة ١٣ سهما والاستثناء من التكملة هو ربع ما يبقي من المال بعد رفع التكملة من المال والذي يبقي من المال بعد رفع التكملة منه ٥٦ وربعها ۱۴ اذا نزعت منها نصيب بنت وهو ٥ بقي منها ۹ وهي الاستثناء من التكملة إذا رفعتها من التكملة وهي ١٣ بقي منها أربعة وهو الوصية كما ذكر والله أعلم

“The Fakih Ahmed Ben Abbas (^[10]) says: I hold, that the completement in this instance is thirteen parts, and the deduction from the completement is one-fourth of what remains of the capital after the completement has been taken from the capital. This remainder of the capital, after subtracting the completement, is fifty-six, and its fourth is fourteen. If you subtract from this the portion of a daughter, which is five, by to all there remains nine of it, and this is the deduction from the completement. Subtracting it from the completement, which is thirteen, there remains four, and this is the legacy, as the author has said,”

Page 98, line 8.

The word مثلها which I have omitted in my translation of this and of two following passages, is in the manuscript explained by the following scholium:

مثلها متساوية لها في الحسن والسن والنسب والمال والبلد والعصر . . . . . والبكارة

“Adequate, i. e. corresponding to her beauty, her age, her family, her fortune, her country, the state of the times, . . . . and her virginity,” (Part of the gloss is to me illegible.) The dowry varies according to any difference in all the circumstances referred to by the scholium. See Hamilton’s Hedaya, vol. 1. page 148.

Page 113, line 7.

The manuscript has the following marginal note

العقر في الامة بمنزلة مهر المثل في الحرة وهو ما تتزوج عليه مثلها في الاوصاف المعتبرة في المماثلة

“The Okr of a slave girl corresponds to the adequate dowry of a free-born woman; it is a sum of money on payment of which one of distinguished qualities corresponding to her would be married.” See Hamilton’s Hedaya, vol. 11. page 71.

I am very doubtful whether I have well understood the words in which our author quotes Abu Hanifah’s opinion.

Abu Hanifah al No’man ben Thabet is well known as an old Mohammedan lawyer of high authority. He was born at Kufa, A.II. 80 (A.D. 690), and died A.H. 150 (A.D. 767). Ebn Khallikan has given a full account of his life, and relates some interesting anecdotes of him which bear testimony to the integrity and independence of his character.

Page 113, line 16.

The marginal notes on this chapter of the manuscript give an account of what the computation of the cases here related would be according to the precepts of different Arabian lawyers, e. g. Shafei, Abu Yussuf, &c. The following extract of a note on the second case will be sufficient as a specimen:

الجواب الذي ذكره الخوارزمي في هذه المسئلة انما هو علي مذهب ابي يوسف وزفر (^[11]) واحد الوجوه لاصحاب الشافعي فاما ابو حنيفة فانه يجعل ما لزم الواهب بسبب وطئه وصية ايضا فتكون الوصية علي قوله شيئا وثلثا وهو احد الوجوه علي مذهب الشافعي وعند محمد الجيس (^[11]) تجعل وطء الواهب لما وهب منه والا يلزمه شيء بسبب ذلك وهو احد الوجوه علي هذهب الشافعي فعلي هذا الوجه تصح الهبة في ثلثها وتبطل في ثلثيها ولا دور لان التركة علي حالها وعلي قول ابن حنيفة تعمل لما فعلت علي مذهب أبي يوسف وزفر (^[11]) فاذا صار بايدي الورثة ثلثماية الا شيئا وثلث شيء يعدل شيئين وثلثي شيء لان الذي لزمه بالعمر وصية ايضا فاذا جبرت وقابلت عدل الشيء خمسة وسبعين درهما وهو ربع الجارية فنصح الهبة في ربعها وتبطل في ثلثة ارباعها

“The solution of this question given by the Khowarezmian is according to the school of Abu Yussuf Wazfar, and one of the methods of Shafei’s followers. Abu Hanifan calls the sum which the donor has to pay on account of having cohabited with the slave-girl likewise a legacy; thus, according to him, the legacy is one and one-third of thing: this is another method of Shafei’s school. According to Mohammed ben al Jaisi, the donor has nothing to pay on account of having cohabited with the slave girl:^[12] and this is again a method adopted by the school of Shafei. After this method, one-third of the donation is really paid, whilst two-thirds become extinct: and there is no return, as the heritage has remained unchanged. According to Abu Hanifaii, you proceed in the same manner as after the precepts of Abu Yussuf Wazfar. Thus the heirs obtain three hundred less one and one-third of thing, which is equal to two things and two-thirds: for what he (the donor) has to pay on behalf of the dowry, is likewise a legacy. Completing and reducing this, one thing is equal to seventy-five dirhems: this is one-fourth for the slave-girl; one-fourth of the donation is actually paid, and three-fourths become extinct.”

---

1. This manuscript is apparently only an abridgement of Haji Khalfa’s work.
2. I have not been able to find any information about this writer. The copy of the work to which I refer is comprized in the same volume with Mohammed ben Musa’s work in the Bodleian library. It bears no date.
3. A full account of this work by Mr. Strachey will be found in the twelfth volume of the Asiatic Researches, and in Hutton’s Tracts on mathematical and philosophical subjects, vol. II. pp. 179-193. See also Hutton’s Mathematical Dictionary, art. Algebra.
4. ${\begin{aligned}(x^{2}-{\tfrac {1}{3}}x^{2})\times 3x&=x^{2}\\2x^{3}&=x^{2}\\2x&=1\\x&={\tfrac {1}{2}}.\end{aligned}}$.
5.  

   هندسه بكسر اول وثالث وفتح سين بي نقطه بمعني
   اندازه و شکل باشد و ارقامی را نیز گویند که در زیر حروف کلمات نویسند همچو ابجد هوز حطي
   

   ٠ ١ ٢ ٣ ۴ ٥ ٦ ٧ ٨ ٩ ١٠

   “Hindisah is used in the sense of measurement and size; the same word is also applied to the signs which are written instead of the words (for numbers) as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,” Burhani Kati.

6. It is almost unnecessary to adduce further evidence in support of this remark. Baha-eddin, after a few preliminary remarks on numbers, says

   وقد وضع لها حكماء الهند الارقام التسعة المشهورة

   “Learned Hindus have invented the well known nine figures for them.” (Kholáset al-Hisáb, p. 16.)
   In a treatise on arithmetic, entitled متن النزهة في علم الحساب which forms part of Sir W. Ouseley’s most valuable collection of Oriental manuscripts, the nine figures are simply called الاشكال الهندية.
   See, on the subject generally, Professor von Bohlen’s work, Das alte Indien, (Königsberg, 1830. 1831. 8.) vol. II. p. 221, and Alexander von Humboldt’s most interesting dissertation: Ueber die bei verschiedenen Völkern üblichen Systeme von Zahlzeichen, &c. (Berlin, 1829. 4) page 24.

7. The Sanskrit original of this passage affords an instance of the figurative method of the Hindus of expressing numbers by the names of objects of which a certain number is known. the expressions for the units and the lower ranks of numbers always preceding those for the higher ones. भ (lunar mansion) stands for 27; नन्द (treasure of Kuvera) for 9; and अग्नि (sacred fire) for 3: therefore भनन्दानि = 3927, Again, ख (cypher) is 0; वाण (arrow of Kamadeva) stands for 5; सूर्य (the sun in the several months of the year), for 12: therefore खवाण सूर्य = 1250. For further examples, see As. Res. vol. x1. p. 281, ed. Cale. and the title-pages or conclusions of several of the Sanskrit works printed at Calcutta;—e. g. the Sutras of Panini and the Siddhantakaumudi.
8. This would appear from the very manner in which our author introduces the several methods; but still more from the following marginal note of the manuscript to the present passage:
   وهو تقريب

   لا تحقيق ولا يقف احد علي حقيقة ذلك ولا يعلم دورها الا الله لان الخط ليس بمستقيم فيوقف علي حقيقته وانما قيل ذلك تقريب كما قيل في جذر الاصم انه تقريب لا تحقيق لان جذره لا يعلمه الا الله واحسن ما في هذه الاقوال ان تضرب القطر في ثلثة وسبع لانه اخف وأسرع والله اعلم

   “This is an approximation, not the exact truth itself: nobody can ascertain the exact truth of this, and find the real circumference, except the Omniscient: for the line is not straight so that its exact length might be found, This is called an approximation, in the same manner as it is said of the square-roots of irrational numbers that they are an approximation, and not the exact truth: for God alone knows whet the exact root is. The best method here given is, that you multiply the diameter by three and one-seventh; for it is the easiest and quickest. God knows best!”

9. The numbers in this and in part of the following scholium are in the MS. expressed by figures, which are never used in the text of the work.
10. The name is written very indistinctly in the manuscript.
11. These names are very indistinctly written in the manuscript.
12. I doubt whether this is the meaning of the original, the words from محمد till يلزمه being very indistinctly written in the MS.