What should be the “poles” for irregular shaped bodies?

What should be the “poles” for irregular shaped bodies?

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Continuation of: What is the definition of a "pole" of a celestial body?

From uhoh's answer, we can conclude that a distinct bodies should have a center of mass. If the body is spherical, then the COM will be near the middle of the body. By definition, the body's rotational axis should pass though COM and the extreme ends of this rotational axis should be considered "poles".

But what about bodies whose COM lies outside the body itself? For instance, COM for crazy C-shaped objects lies outside the body, the axis of rotation does not intersect the body's surface and so technically, there are no "poles" for such bodies. So, how should we describe the "extreme ends" of the body? How are COM measured for such bodies? Is there any list for such bodies where COM lies outside the bodies?

Note: There are a few bodies where there are no stable poles but they have their reasons. Saturn's moon Hyperion and the asteroid 4179 Toutatis lack a stable north pole. They rotate chaotically because of their irregular shape and gravitational influences from nearby planets and moons, and as a result the instantaneous pole wanders over their surface, and may momentarily vanish altogether.

Al-Khāzinī, Abu’l-Fath ‘Abd Al-Raḥmān [Sometimes Abū Manṣūr ’ Abd Al-Raḥmān or ’Abd Al-Rahmān Manṣūr]

A slavae-boy of Byzantine origin (a castrato, according to the edition of al-Bayhaqī by Shafī, who reads majbũb for mahbũb), al-Khāzinī was owned by Abu‘l-Husayn (Abu’-Hasan, according to Shafi’) (Aliī ibn Muhammad al-Khāzin al-Marwazī, whose name indicatges that he was treasurer of the court at chancellor there (or, according to meyerhof’s translation of al-Bayhaqī, a religious judge, qādi being read for mādi 0, Because of the owner’s rank the form “s al-Khāzini,”, which denotes a relationship to the Khāzin, should probably be preferred to “alKhāzin,” a form which, however, is encountered very often. His master gave the young man the best possible education in mathematical and philosophical (‘aqliyya) discplines. Al-Khāziniī “became perfect” in the geometrical sciences and pursued a career as a mathematical practitioner under the patronage of the Seljuk court. His work seems to have been done at Merv. 1 That city was then a capital of Khurāsān and from 1097 to 1157 was a seat of the Seljuk ruler Sanjar ibn Malikshāh, who held power first as emir of Khurāsān, then as sultan of the Seljuk empire. it became a brilliant center of literary and scientific activity and by the end of this period was renowned for its Libraries. Al-Khāzinī’s book of astronomical tables was composed for Sanjar, and his balance was constructed for Sanjar’s treasury.

Noted for his asceticism, al-Khāzinī dressed as a Sũfī mystic and ate “the food of pious men”—meat but three times a week and otherwise two cakes of bread a day. Rewards he refused: he handed back 1,000 dinars sent him by the wife of the emir Lājī ākhur beg al-Kabīr the same amount, presented to him by Sanjar through the emir Shāfi ibn Abd al-Rashīd (a pupil of al-Ghazālī, d. 1146/1147), presumably on the occasion of his completing the astronomical tables, was also returned. he had, he said, ten dinars already and lived on three a year, for in his household there was only a cat. Al-Khāzinā had students, but only one name has survived, an otherwise unknown al-Hasan al-sumarquandī.

Scarcely anything else is known of al-Khāzinī’s life (although his own works have not been fully searched). The basic biographical account is that by al-Bayhaqī (d. 1169), who seems to have been personally acquainted with al-Khāzinī. (meyerhof’s translation of the notice must be preferred to that by Wiedemann, who wrote before the publication of Shafī’s critical edition.) Al-Shahrazũrī adds nothing significant and subtracts a good deal Hā jjī Khalīfa has only a few lines with nothing new. Tāshköprüzāde merely mentions an “al-Khāzinī” in connection with astronomical instruments. he does not appear among the 266 “Abd al-Rahmāns” in al-Safadī. 2

At various times al-Khāzinī has been mistakenly identified with Alhazen (i.e., Ibn al-Hytham), Abũ Ja far al-Khāzin (especially in connection with the treatise on astronomical instruments see below), and Abu’l-Fath al-Khāzimī [or al-Hāzimī](a twelfthcentury astronomer of Baghded). 3 There is no evidence that al-Khāzini ever worked in Baghdad assertions that he did must be based on the false assumption that the Seljuk court would be there.

One doubtful passage (Qutb al-Dīn al-Shīrāzī [d. 1311], Nihāyat al-idrāk . . .) indicates that he made astronomical observations at Isfahān “at Isfahān,” however, seems to be an addition of unknown origin or authority. 4 chronology makes it extremely unlikely that al-Khāzinī was a member of the staff of the observatory which was established by the Seljuk Sultan Malikshāh in Isfahān and whichlasted but a short while after the founder’s death in 1092 Umar al-Khayyāīl al-Asfizārī(Omer Khayyam d. 1131[?]) and al-Muzaffar ibn Ismā īl al-Asfizārī(mentioned below in connection with al-Khāzinī’s balace), both a generation older than al-Khāzinī, had in fact been there. 5 Indeed, no evidence shows al-khāzinī to have been associated with any observatory, that is, as a member of a group of researchers attached to an actual astronomical institution. 6 In calculating his zil (book of astronomical tables) al-Khāzinī was said to have worked with Husām al-Dīn Sālār (otherwise dated only as writing between the times of al-Biũnī [d. 1051 or after] and NaŞīr al-Dīn al-Ţũsī [d. 1274) but the source is the sixteenth-century Persian historian Hasan-i ũmlũ, who also associates al-Khāzinī with the poet Anwarī. But Anwarī, astronomically learned though he was, and patronized by Sanjar, almost certainly lived at least a generation later.

Al-Khāzinī, al-Khāzimī, and Anwarī are also among those variously reported to have been involved in the unfortunate astrological prediction of devastating windstorms in 1186 (the entire year was so calm in Khurāsān that the grain crop could not be properly winnowed) but al-Khāzinī’s involvement, again on chronological grounds, is hardly likely. 7

Al-Khāzinī’s Scientific Accomplishments. The known works of al-Khāzinī, seemingly all extant, are the following: al-Zij al-Sanjarī (“The Astronomical tables for Sanjar”), also in a summary (wajīz) by the author Risāla fiaposl-ālāt (“Treatise on [Astronomical] Instruments”), which actually may not be the work mentioned by the biobibliographers (see below al-Bayhaqī does not refer to it) and Kitāb Mīzān al-hikma (“Book of the Balance of Wisdom”), a wide-ranging work that deals primarily with the science of weights and the art of constructing balaces. To the manuscripts listed by Brockelmann should be added 1) Sipahsālār Mosque [madrasa] Library (Teheran) 681-682 (cataloged as “Zīj-i Sanjari” but containing a collection of al-khaāzinī’s works including Risāla fi’l-āt but not the complete zīj) 8 and 2) the manuscript used for the Cairo edition of Kitāb mīzān al-hikma (see below). The contents of the works are discussed later.

It is hard to assess the importance of al-Khāzinī. His hydrostatic balace can leave no doubt that as a maker of scientific instruments he is among the greatest of any time. As a student of statics and hydrostatics, even in their most practical saspects, he is heavily dependent upon earlier workers and borrows especially from al-Asfizārī but his competence is not to be denied, and Kitāb mizamacrn al-hikma is of outstanding imporance to the historian of mechanics, whatever its claims to originality or comprehensiveness may prove to be. In astronomy, as in mechanics, al-Khāzinī. His takes its place in the Eastern Islamic astronomical tradition after those of al-Bīũnī and ldquoUmar al-Khayyāmī and is cucceeded by those produced by the labors of the Marāgha Observatory (Nasīr al-Dīn al-Tũsī and Qutb al-Dīn al-Shīrāzī) and the Samarkand observatory (al-Kāshī [d. ca 1430] and Ulugh Beg [the sultan d. 1449]). Al-Khāzinī is one of twenty-odd Islamic astronomers known to have performed original observations. 9 Kennedy rates his zij very highly and, in suggesting eclipse and visibility theory as subjects that would particularly reward monographic treatment, names topics — particularly visibility theory — for which al-Khāznī’s tables are an especially rich source. 10

In mechanics no works are known that follow in the tradition of Kitān al-hikma treatments of balances or the science of weights become mere manuals for craftsmen who make simple scales or steelyards, or for merchants or inspectors who use them or check them. That branch of learning ceases to be a part of the scientific tradition.

Although al-Khāzinī’s publications were wellknown in the Islamic world, and particularly in the Iranian part of it, they do not seem to have been used elsewhere save in Byzantium. The Sanjari zij (ξήξℓ ∑αvT ξaρήs) was utilized, at least for is tables of stars, by George chrysococces (sl. Trebizond, ca. 135- ca. 1346), an astribiner abd geographer, and through him by Theodore Meliteniotes, an astronomer in Constantinople (fl. ca,. 1360- ca. 1388). 11

Works: the Aastronomical Tables. The Sanjarī zij, whose full title is al-zij al-mu’tabar al-Sanjarī al-Sultānī (“The Compared [or “Tested”] Astronomical Tables Relating to Saulan Sanjar”) is also called by shorter forms of the same title (al-Zij al-sultānī refers to other works, however) (“Collection of Chronologies for Sanjar,” if Sanjar can be called al-Sinjarī, after his native town) — the last title resulting from the large amjount of calendrical material and the tables of holidays and fasts and rulers and prophets 12 The known manuscripts are Vatican Library cod. Ar. 761 and British Museum cod. Or. 6669 the work runs to 192 folios in the Vatican manuscript, which is sometimes considered an autograph. Hamdallāh al-Qazwīnī, in Nuzhat al-qulũb, presebts a table to use in conjunction with the Indian dial for determining the qibla (direction if Mecca) for most places in Iran. He indicates that it was produced by al-Khāzinī on the order of Sultan Sanjar. One would expect to find such a table in the zij but it is missing — as are geographical coordinates of cities — from both the Vatican and British Museum manuscripts (the latter being nearly complete, despite LeStrange’remark — the table of contents at the beginning of the codex, however, mits many sections). 13

In 1130/1131 (A.H. 525) al-Khāzinī wote an abridgement of his tables called waijiz al-Zij al-mu’tabar al-sulţānī 14 that year presumably marks a terminus ante quem for the tables themselves. The British Museum and Vatican manuscripts of the zij have no date in the obvious places. The year A.H. 530 is assigned by Suter and taken over by Sayili without basis. 15 Keenedy, Destombes, and Nallino have produced no precise dating. Nallino, using the Vatican manuscript (folios 191v-192r), describes the star tabgles as having longistudes and latitudes of forty-three fixed stars for A.H. 509 (1115/1116) 16 Kennedy, using the same manuscript, describes the same table as providing latitudes and longitudes, temperaments, and magnitudes of forty-six stars for A.H. 500 (1106/1107), presumably on the basis of the parameters. 17 Destombes says —as Nallino does, but using the British Museum manuscript — that the star table is for forty-three stars for A.D. 1115. 18 Thence Destombes presumes that the Zij was written in 1115 and “corrects” the date of ca. 1120 attached to the tables by Kennedy without discussion. 19 The tables are, however, dedicated to Sanjar, who was sultan of the empire only from 1118 but he had been emir of Khurāsān since 1097, and the use of the title “sultān” in the zij is in any case problematical. Sayili does cite coronation. 20 Nallino, however, had long sice pointed out a reference to the calip Mustarshid bi’llāh, who occupied the office in 1118-1135. 21 One is left, then, with the interval 118-1131 for the completion of the zij, all subsidiary evidence pointing to the beginning of this period

That al-Khāzinī made a certain number of actual observations is not questioned probably they were done at Merv independently of any observatory. Quŗb al-Dīn al-Shīrāzī discusees the measurements of the obliquity of the ecliptic by al-Khāzinī and indicates that they were very careful—praise which suggests high technical competence and good intruments. 22 In the Wajiz (fol.1v) al-Khāzinī states that he compared observed and found disagreement for all of them. 23 In fact the word mu’tabar in the title suggests just such a comparison, indeed a testing or “experimental verification.” But al-Bayhaqī in his biographical notice says that the mean motions (awsāŗ) and equations (tadīlāt) determined by al-Khāzinī need further study—except in the case of Mercury, especially in its retrograde motion, for which the positions had been observed and tried.

The Indian theory of cycles (i.e., those which culminate in the “world day,” the period which the cosmos takes to return to any given state) as reported in the Sindhind and in Abu Mashar’s al-Hazāzinī (“the Thousands”) greatly interested al-Khāzinī despite al-Bīũnī’s unequivocal strictures against that sort of astronomy. 24 It is possible to deduce those cycles from the motions one observes, al-Khāzinī claims, but difficult because of the amount of calculation. 25 The Sanjarī zīj has a fair amount of such material, but al-Khāzinī keeps all his computations strictly within the Islamic ptolemaic tradition (as far as can be said).

Among his predecessors in astronomy, apart from al-Bīānī it is Thābit ibn Qurra and al-Battānī whose jij’s seem to have concerned him most. 26 He reproduces Thābit’s work on lunar visibility before presenting his own exceptionally detailed treatment, and frequently he reports the methods or conclusions of Thābit or al-Battā in other connections. For his value of the obliquity of the ecliptic al-Khāzinī, like al-Battānī, chooses 23deg35’—but only after discussing the discrepancies among the results obtained by others, mentioning difficulties due to refraction, and then rejecting both decreasing and alternately increasing and decreasing values of the obliquity. 27 Unlike any other Islamic astronomer except Habash alHāsib al-Marwazī, al-Khāzinī uses the canonical religious date for the Hijrī epoch. 28

Al-Khāzinī’ zij general is very rich. The chronologies and the section on visibility have already been mentioned. 29 The latter, besides tabulating the arcs of visibility for the five planets (perhaps calculated in an original fashion) as well as those for the moon, also presents differences according to clime and incorporates historical material. Tables of trigonometric functions, of astronomical parameters generally, and especially of planetary mean motions (including those of sun, moon, and lunar nodes) are thorough and highly precise—the planetary mean motions, for example, are given in degrees or revolutions per day to eight or more significant sexagesimal (fourteen or more decimal) figures and the tables relating to eclipse theory are also greatly elaborated. The absence of material on terrestrial geography has been noted, and the star tables have been described in connection with the dating of the Zij. There are, finally, a number of tables of astrological quantities. positions are recorded here for “al-Kayd,” perhaps a comet. 30

Treatise on Instruments. The Treatise on Instruments (Risāla fi’l-ālāt), found by Sayili in codices 682 and 681 of the library of the Sipahsālār Mosque in Teheran, is a short work, occupying seventeen folios in the manuscript. 31 It is probably the same as al-ālāt al-’ajiba (al-rasadiyya) (“The Remarkable [Observational] Instruments”), which was noted by Ibn al-Akfānī, Tāshköprüde, and Hājjī Khalīfa. 32 Sayili ascribes the work to “Abd al-Rahman al-akfāzinī. So does Brockelmann, following Wiedemann, “Beiträge . . . IX” Wiedemann repeats this ascription in “Beitrāge . . . LVII, Rdquo but in his articles for the Encyclopaedia of Islam on “al-Khāzini . . .” he allots the work without comment (although, indeed, with citation of the passages in the “Beiträge. . .”) to al-Khāzin, an astronomer, mathematician, and instrument maker of the mid-tenth certury 33 Ibn al-Akfānī, Tāshköprüzādeh, and Hājjī Khalifa (at both places) ascribe a treatise of that title to “al-khāzinī” without further identification but this carries no weight, for Hājjī Khalīfa refers to “Abu Ja’far al-Akfānī, four times, to “Abu’l-Fath ’Abd al-Khāzinī” once, and otherwise to “al-Khāzinī”—so that Flügel’ note, following upon Ibn Khaldü n’ mention of “Abu Ja ’ far, but so also are both Kitāb mizān al-hikma and zij al-safā ih (the former now known to be by Abd al-Rahm’n, the latter, by Abu Ja’far). 34 Since the treatise on instruments is a minor one al-Bayhaqī’s failure to note it as a work of Abd al-Rahmān al-Khāzinī means nothing similarly, the absence of the title from the frequent references of al-Birünī to works by Abu Jafar al-Khāzin can produce no certainty in the other direction The incidental mentions by the biobibliographers seem to suggest the later man, but the only concrete evidence for assigning the work to Abd al-Rahmān al-Khānī is that of the Teheran manuscript, mentioned above, which was copied in A.H. 683 (1284/1285).

The Risāla has seven parts, each devoted to a different instrument: a triquetrum, a dioptra, a “triangular instrument,” a quadrant, devices involving reflection, an astrolabe, and simple helps for the naked eye. The quadrant is in face called a suds, or “sextant,” and performs the functions of the sextant, although its arc is 90deg. Apart from describing the devices and their use, the treatise also demonstrates their geometrical basis.

Kitāb Mīzān al-Hikma. The most interesting and important of al-Khāzinī’s writings, both in itself and as a source of information on earlier work— if only because it is a much rarer sort of book than a zij— is Kitāb mīzān al-hikma, the Book of the Balance of Wisdom. A long treatise (the Hyderabad ed. has 165 large octavo pages of Arabic text, exclusive of figures and tables), it studies the hydrostatic balance, its construction and uses, and the theories of statics and hydrostatics that lie behind it, as well as other topics both related and unrelated. Written in A.H. 515 (1121/1122) for Sultan Sanjar’s treasury, 35 Kitāb mīzān al-hikma has survived in four manuscripts, of which three are independent. The treatise has been published, partially edited and largely translated.

Study of the Kitāb mīzān al-Hikma may begin from either the edition of selected parts, accompanied by sometimes inaccurate English translations, that was produced in 1859 by Khanikoff and the editors of the Journal of the American Oriental Society— or from the uncritical but serviceable text of the Hyderabad edition, which was made on the basis of the two related Indian manuscripts and a photocopy of the one used by Khanikoff. 36 (It is Khanikoff’s manuscript that seems to be the oldest.) Variants can be sought from the Cairo edition, which is a rather unprofessional transcription of an additional manuscript, from East Jerusalem 37 the text, of which up to half is missing, seems closer to Khanikoff’s copy of the work than to the Indian ones. 38

Those parts of his not full complete manuscript left untranslated by Khanikoff were almost entirely rendered into German by Wiedemann, 39 who, however, provided no Arabic text and occasionally abridged and paraphrased without sufficient indication. Of the long studies, that by Ibel is helpful Bauerreiss’ thesis demands caution save when he is describing the apparatus. The commentary in Khanikoff’s article is by now badly dated.

An elaborate literary conceit (three pages in the Hyderabad edition) on the name mīzān al-hikma— thus far translated as “the balance of wisdom”—opens the book and the phrase does indeed repay consideration. The hydrostatic balance built by al-Asfizārī (who was a generation older than al-Khāzinī) had been called mīzān al-hikma 40 an improvement upon earlier instruments of the type first constructed by Archimedes, it was likewise intended to detect alloys passing for gold, and other frauds. Created for Sultan Sanjar, the scales was destroyed, out of fear by his treasurer (not the one who was al-Khāzinī’s master) and al-Asfizāzrī “died of grief.” 41 Al-Khāzinī subsequently built a similar balance, further refined, for Sanjar’s treasury this he called al-mīzān al-jāmi’ (the “comprehensive” or “combined balance”) and mīzān al-hikma, in honor, presumably, of al-Asfizārī. 42 The primary meaning, then, of mīzān al-hikma is “balance of true judgment,” of accurate discrimination between pure and adulterated metals, between real gems and fakes. The name in fact consciously echoes the Koranic balance with the long beam that is to be erected on the Day of Judgment. 43

The first words of Kitā mīzān al-hikma are praises to God the Wise (all-Hakīm), the Just (al-Adl)—or, in variants, the Judgment (al-Hukm), the (al-Haqq), the Justice (al-’Adl lexically distinct from the form above). 44 Words derived from the root H-K-M are then cleverly woven into the text, together with form the root-D-L (which denotes justice in the sense of equitability and evenhandedness, and one of whose derivatives, itadala, means “to balance” and is specifically applied to weights on a scale). “Justice,” says al-Khāzinī, “is the support of all virtues and the foundation of all excellencies. For perfect virtue is wisdom and has two parts, knowledge (ilm) and action (amal), and two halves, religion and the world, perfect knowledge and proficient (muhkam) activity (fil) and justice is the combination of [those] two and the union of the two perfections of it [wisdom], by which is conferred the limit of every greatness and by means of which is attained precedence in every excellence.” God in his Mercy, continues al-Khāzinī, has set up among men three arbiters [huķķām] of justice: the glorious Koran, to which the Traditions of the Prophet are the sequel the rightly guided and well-versed scholars (ulamā’), among whom is the just governor, alluded to in the words of the Blessed, “the sulţān, the shadow of the Most-High God upon Earth, the refuge of the injured, and the judge (hākim)” 45 and the balance, which is the tongue of justice, the just judgment whose decision satisfies all, the order and justice in human conduct and transactions—the balance which God Himself has associated with his very Koran (as al-Khāzinī shows with a surprising number of strongly worded and explicit textual proofs from the Koran). 46

But these religious (and political) themes must not obscure the fact that for contemporary students of the sciences hikma meant not only wisdom but particularly philosophy (that is to say, Islamic Peripateticism), with its two divisions, theoretical and practical, answering to the two virtues and which may reasonably if not perfectly be associated with the divisions of knowledge and action, religion and the world, stated by al-Khāzinī. Certainly he proceeds to describe what a later age called a “philosophical balance”—the other possible translation of mīzān al-hikma. Al-Khāzinī writes:

This just balance is founded upon geometrical demonstrations and deduced from physical causes, in two aspects: 1) as regards centers of gravity, the most elevated and noble division of the mathematical sciences, which is knowledge that the weights of heavy things differ according to the distances they are placed [from a fulcrum]—the foundation of the steelyard and 2)[as regards] knowledge that the weights of heavy things differ according to the rarity or density of the fluids in which the thing weighed is immersed—the foundation of the mīzān al-hikma. 47

When al-Khāzinī lists the advantages of his balance, “which is something worked out by the human intellect and perfected by trying out and testing” and which “performs th functions of skilled craftsmen,” he names benefits variously theoretical and practical—precision, ability to distinguish pure metal from alloy and to determine the content of binary alloys, use fulness in calculations relating to a treasury, gains due to ease and versatility in use (for instance, the possibility of recourse to any reference liquid—from the broad scope of its applications comes its other name, “the comprehensive balance”), and the seventh and last advantage, “the gain above all others, “that in enables judging true gems from false, 48 His is a philosophical balance desirable both for the superior theory in its construction and the range and excellence of uses to which it can be put and of its practical virtues the greatest is the ability to judge genuine from fraudulent. Among al-Khāzinī’s great Islamic precursors in this art al-Rāzī(Lat., Rhazes the famous physician) had physical balance,” that is, as pertaining to physician prinicples), whereas ’Umar al-Khayyāmī had designated his highly developed steelyard al-qustās al-mustaqīm (“the upright [or “honest”] balance”) al-Asfizārī and al-Khāzinī had found a name which included both aspects: mīzān al-hikma— the balance of wisdom—meaning the “balance of right judgment” and “the philosophical balance.”.

Al-Khāzinī is perfectly explicit in stating what sort of book he is composing. As a preliminary he divides the fundamental principles of any art into three classes those which are acquired in early childhood an youth, after on sensation or several sensations, spontaneously, and which are called first things and common knowledge those which are obtained by trying out and by assiduous investigation (In the area of the art itself). So it is, then with the art of the balance, which has principles both geometrical and physical (considering as it does the categories both of quantity and of quality) but th author will not mention the obvious principles belonging to common knowledge and will refer only in passing, as necessary, to principles taken over from other disciplines or obtained by investigation. 49

Even though he presents proposition and general theorems of statics and hydrostatics in books I and II, al_Khāzinī supplies no proofs and frequently no explanations he employs demonstration in his treatise only when it is required in connection with designing or using the balance of wisdom or another instrument. It is not deductive work of mathematical science but rather a technical presentation of the art of the philosophical (or scientific) balance.

Country to what is assumed about most medieval authors, al-Khāzimīwas well aware of the historical progress made in his art—the introduction to kitābmizān al-hikma contains two sections 50 which report the the invention of the hydrostatic balance by Archimedes (following Menelaus’ account) and the modifications and perfections introduced by later workers up to al-Khāzinī himself. He states, indeed, that “the knowledge of the relations [in specific weight] of one metal to another depends upon that perfecting of the balance through delicate and detailed devising by all who have studied it, or developed it by fixing the marks for specific gravities of metals relative to a particular sort of water,” 51 Hence he had seen fit “to assemble on this subject whatever we have gained from the works of the ancients and of later philosophers who have followed them, in addition to what [our own] thought with God’s aid and giving of success had granted.” 52 In fact much of Kitābmixā al-hikma is composed of extracts most of what is original relates to the “balance of wisdom” itself or to its applications.

Contents of Kitā Mizān al-Hikma. . The Book of the Balance of Wisdom comprises eight books (maqālāt) divided into fifty chapters (abwāb) larger, intermediate, and smaller divisions of the text are initial summary and table of contents by al-Khāzinī, as given by the manuscripts, differ from each other nd from the headings in the actual text.

Because cross-reference or heading are often missing from the translation of Kitaāb mizā al-hikma making them hard to use, page references are supplied here. All the translation follow the Khanikoff manuscript. Wied. = Wiedemann B = Wiedemann, “Beiträge . . .”[numbers in parentheses refer to reprint] Khan. = Khanikoff edition. I: 1.1 = book I, chapter 1, section 1. All numbers are inclusive.)

Al-Khāzinī’s long introduction and his own summary and table of contents preceded the body of th treatise.

Introduction: Khan., 3-16 extracts in Clagett, The Science of Mechanics in the Middle Ages, 56-58. Summary of contents: Khan., 16-18. Table of contents: Kahn., 18-24 Ibel, 80-83 compare table of contents drawn up according to heading in the text, at end of Hyderabad edition.

Book I sets forth geometrical and physical principles underlying the hydrostatic balance: theorems on centers of gravity from works by Ibn al-Haytham and Abu Sahl al-Qũhī (ch. 1) theorems from Arabic translations of works entitled “On the Heavyand the Light”—by Archimedes (a fragment of “On Floating Bodies”) (ch. 2), by Euclid (ch. 3), and by Menelause (ch. 4) repetition or summary of important theorems (ch. 5) and propositions on sinking and floating (ch. 6), following Archimedes (?). 53 Thus far no proofs or discussions.

Chapter 7 is a detailed description of the construction and use of Pappus’ araeometer, 54 an instrument for determining specific gravities of liquids here geometrical demonstrations are indicated, in accordance with al-Khāzinī’ aims.

I:1, Khan., 25-33, retranslated in Ibel, 85-88, reprinted in Clagett, 58-61. I:2, B VII compare Clagett, 52-55. I:3, Ibel, 37-39 see also Ernest A. Moody and Marshall Clagett, The Medieval Science of Weights (Madison, Wis., 1960), pp. 23-31. I:4, not translated, but see Ibel, 77-78, 181-185. I:5, Kahn., 34-38 largely reprinted in Clagett, 61-63 compare, for I:3 3. Wied., “Inhalt”. I:6, B XVI , 133-135 (I, 492-494). I:7, Khan., 40-52 (and the notes) compare Bauerreis, 95-108.

Book II begins with a discussion of the balancing of weights and its various causes, taken from a work of Thaābit ibn Qurra. 55 The rest of this book derives from al-Asfizārī and treats, without demonstration, the following topics: constrained motion of the centers of gravity of bodies the equilibrium of a balance beam, geometrical or physical, with application of the results to a spear held in the hand the construction of a steelyard, the graduation of its beam, and the methods of weighing with it and the conversion of steelyards form one system of weights to another.

II, entire, B XVI , 136-158 (I, 495-517).

Book III has three parts. Part I (chapter 1-3) comes from al-Bīũnī’s [Maqāla] fīl-nisab [allātī] bayna’l-filizzāt wa’-jawāhir fi’l-hajm (“On the Relations [in Weight] Among Metals and Precious Stones With Respect to [a Given] Volume”): specific gravities—or water-equivalents (weights of water equal in volume to reference weights of water equal in volume ro reference weights of the given materials)— of metals, precious stones, and other substances of interest. Al-Bīrũnī here describes his “cone-shaped instrument” (al-āla al-makrmţīya)—a pyconmetic metal vessal shaped like an Erlenmeyer flask, with a handle and a spout that is a narrow tube projecting out and down from the neck—and explains its use in measuring the weight of a volume of water equal to the simple, which has been introduced into the flask and displaces water through the spout into one of the pans of a balance. The neck of the flask is about the diameter of a man’s little finger the spout is perforated all along its sides to minimize the effects of surface tension.

This part of book III and the remaining two parts, also by al-Bīrũnī, have elaborate tables of values and detailed indications of procedure.

Part 2 records how he weight was obtained of a cubic cubit of water by making an exact hollow cube of brass, determining its internal volume through a precise measurement of its dimensions, weighing the water necessary to fill it, and multiplying the result by the appropriate ratio. The weights of a cubic cubit of several metals are then found, using their water equivalents. This part ends with the calculation of the weight of gold required to fill the volume of the earth (chapter 4, section 3). Part 3 (which is chapter 5) continues in similar vein with problems about dirham doubled successively on each square of a chessboard, starting with a single dirham on the first square—their total number, the number of chests to hold them, the length of time to spend them.

III :1.1-.3 and III :1.5-.6, missing in Khanikoff’s manuscript, now in the Hyderabad edition no translation, but compare Khan., 53-56 Wied., “Bêrûnîshe Gefäss” and Wied.,” Mīzān”, p. 534, III :4.1-.2 see B XXXIV . III :4.3- III :5, B XIV see also Julius Ruska, “Ķazwīnīstudien,” 254-257.

Book IV is historical. First come descriptions of the hydrostatic balances of Archimedes and of Menelaus, an explanation of the latter’s methods of analyzing alloys, and a summary of the values he found for specific gravities (thus far according to Menelaus). Then follow presentations of the “physical balance” of Muhammad ibn Zakariyyā al-Rāzī and of the water balance of Umar al-Khayyāmī, with detailed diagrams, based on works by those authors.

IV :1, Archimedes, according to Menelaus, Ibel, 185-186. IV :2- IV :3, Menelaus B XV , 107-112 (I , 466-471). IV :4, al-Rāzī, Ibel, 153-156. IV :3 and IV :4 are reversed in the Hyderabad edition. in the Khanikoff manuscript IV:4.2-3 was displaced to the end of the book, section 3 being abridged the full text of section 3 is in the Hyderabad edition, pages 85-86, without abridgement of disordering. III :5, B XV, 113-117 I, 472-476) III:5.1 also in Ibel, 1589-159.

In Books V and VI , Kitāb mizān al-hikama becomes a manual of the “balance of wisdom.” Starting with the instruction al-Asfizārī had left, the discussion becomes al-Khāzinī own after the first chapter of book V (or perhaps after the first section of that chapter). Diagrams, illustrations, and tables are outstandingly rich. Book V explains the fabrication of the parts of the balance, their arrangement and assembly, and the adjustment and checking of the balance, pointing out defects that may be found or mistakes that may be made. Book VI, the longest of the work (although only a fifth of the whole), sets forth the operation of the balance: selecting the counterpoises, leveling the beam, and weighing, then graduating the balance in order to use it for measuring specific gravities. After that has been done, a number of special procedures can be exploited: testing the genuineness of metals and precious stones by use of the two movable scale pans (a method restricted to the “balance of wisdom”) and discovering the ratio of the constituents of a two-element alloy or other mixture (perhaps to determine a correct monetary value) assaying and appraising by another technique, that of tajrīd (“isolation”), which involves a single movable bowl and algebraic calculation and finding specific gravities of substances by computation from their weights in air and in water. Other, related procedures are also mentioned, and some special theorems are introduced. The end of book VI (chapter 10) is an appendix on the prices of gems in times past, taken from al-Bīũnī’s Kitāb a-jamāhir fi marifat al-jawāhir (“Book of Gatherings on Knowledge of Precious Stones”).

Many of the methods of using the balance are provided with geometrical proofs. The suspension of the the instrument, its indicator tongue, and the placing of the instrument, its indicator tongue, and the placing of the marks on the beam are treated with special care.

V: Ibel, 112-136. V:1.4, also Khan., 88-94, VI:1-VI:4, Ibel, 136-151 (also: VI:2.5, Khan., 98-99, and VI:4.1-.2, Khan., 100-104.) VI:5-V:9, B XV, 117-132 (I, 476-491) (beginning with the method of tajrīd). VI:10, Wied., “. . . Wert von Edelsteinen. . .”

Books VII and VII treat special modifications of the “balance of wisdom” and other specialized balances. There are abundant diagrams and tables. Differently graduated and without the extra scale pans that make it a hydrostatic balance, al-Khāzinī’s instrument can be applied to exchanging among different coinages. To his discussion of the adjustment and use of the exchanging balance (VII:2-VII:4) the author prefixes an unattributed analysis of proportion (VII:1.1-.5) and other mathematical material (VII:1.6-.7) The text on proportions is nearly identical to that by al-Bīũnī in Kitāb al-tafhīm. 56 relating to the mint and of exchange.

He then (VII:6 to the end of book VIII) considers the other, special-purpose balances: (VII:6) a scales for weighing dirhams and dinars without counterpoises (VII:7) balances for use in leveling, measuring differences in level, and smoothing vertical surfaces 57 (VII:8) the “righteous steelyard” of Umar al-Khayyāmī, which can weigh from a grain to a thousand dirhams or a thousand dinars by using an indicator tongue and three counterpoises associated with three different graduations of the beam and the clock-balance, to which al-Khāzomī devotes a rather long notice (thirteen pages plus figures, all of book VIII in the Hyderavbad edition) that pays special attention to the water or sand reservoir and to the measurement of sort intervals (for instance, for astronomical purpose), down to seconds of time.

VII and VII: the Hyderabad text and al-Khāzinī’s own list of chapters assign eight chapters to book VII that enumeration is followed here. Al-Khāzinī’s brief summary of the work and Khanikoff’s altered table of contents give five chapters to book VII, moving the next three to the beginning of book VIII.

VII:1-VII:6 (except VII:1.1-.5), B LXVIII, 6-15 (II, 220-229) VII:1.1-.5, ibid., 3-6 (II, 317-220). VII:7, Ibel, 159-160. VII:8, Ibel, 107-110. VIII:1-VIII:4 and first paragraph of VIII:5 (according to al-Khāzinī’s list of chapters VIII, part 1, chapters 1-4 and first paragraph of part 2, chapter 1, in Hyderabad edition and in Khanifoff manuscript), B XXXVII. The Khanikoff MS ends there the Hyderabad edition, pp. 164-165, gives the other two paragraphs of VIII, part 2, chapter 1, and the short chapter 2, which seems to end the book properly VIII, part 2, chapters 1 and 2, seems to contain what is suggested by the title of VIII:5 in al-Khāzinī’s list of chapters.

Kitāb mizān al-hikma is also well-stocked with miscellaneous incidental statements of interest—on the rising and sinking of mountains, for example, and the natural production of gold out of lead.

Al-Khāzinī and the Science of Weights in Islam . Much of what is most interesting in the work comes from other authors, Greek or Islamic. For his theorems in geometry and physics al-Khāzinī draws upon Euclid, Archimedes, Menelaus, and, without citation and perhaps indirectly, Pseudo Aristole’s Mechanica problemata, chapters 1 and 2, 58 among the Greeks, and from Thābit ibn Qurra, Abu Sahl al-Qũhī, Ibn al-Haytham, al-Bīũnī, and al-Asfizārī. Exactly what al-Khāzinī does to the extracts he incorporates is a matter for detailed study, but there seems to be nothing in the way of basic physical theory that is his own.

He is especially indebted for significant material to al-Bīũnī. The very careful explanations (in book III) of refined instruments and methods for determining specific gravities come from al-Bīũnī’s “On the Relations Among Metals . . .” and, when discussing the determination of specific gravities by “isolation” (in VI:5), al-Khāzinī uses data from a lost work by al-Bīũnī. 59 The treatment of (mathematical) proportion (beginning of book VII) almost certainly comes, as was noted, from the Kitāb al-tafhim and the numerical problems involving large numbers (end of book III) are taken from al-Bīũnī’s writings. How much (if any) of al-Khāzinī’s material on the exchanging balance or the chronometric balance derives from al-Bīũnī’s lost treatises on those subjects cannot, of course, be known. 60 The historically valuable notice on the prices of gems at various times and places (VI: 10) is also from al-Bīũnī.

When al-Khāzinī traces his scientific lineage in the art of th hydrostatic balance, he first names Archimedes (giving Menelaus’ account of the assay of the crown presented to Hiero II, tyrant of Syracuse) and Menelaus, who is said to have been attempting to solve the problem of a three-component alloy. As his first precursors in the Muslim world he lists Sanad ibn Alī, Yũhannaā, who were contemporaries in the mid-ninth century. (Those three and only they are mentioned as his Islamic predecessors by al-Bīũnī, in “On the Relations Among Metals . . .,” in the study of specific gravities. Only Sanad ibn Ali is otherwise known, although there was a Yũhannā ibn Yũsuf al-Qass, a scientist who died in 980/981.) Then he mentions al-Rāzī (who included in one of his works—not extant save insofar as it is presented in al-Khāzinī—a chapter on his water balance this is cited by al-Bīũnī in the study just mentioned) and, surprisingly, Ibn al-‘Amīd (d. 969/970) and Ibn Sīnā(Avicenna d. 1037), neither of whom is known to have worked in this art next, al-Bīũnī, then Abu Hafs [sin ususally Abu l-Fath] Umar al-khayyāmī and last, al-Asfizārī, who had died before al-Khāzinī composed Kitāb mīzān al-hikma and before “reducing all his views on the subject to writing.”

It was al-Rāzī who added the indicator tongue to the hydrostatic balance 61 (see Figure 1), and a third scale pan was attached not later than the time of al-Bīũnī. Al-Asfizārī put on the two movable scale pans and indicted the possibility of cutting specificgravity marks into the beam. Al-Khāzinī made further refinements, mainly, it seems, in marking the beam for specific gravities of various substances for more than one reference liquid. 62

The grounds for excluding, as practitioners in this art, Euclid, Pappus, the Banũ Mũsā, al-Kindī Thābit ibn Qurra, Abu Sahl ai-Qũhī, and Ibn al-Haytham maybe that al-Khāzinī regards them as not actually having worked with a hydrostatic balance. (The only balances considered in books IV and V are the ones of Archimedes, Menelaus, al-Rāzī, Umar al-Khayyāmī, and al-Asfizārī besides his own.) That can scarcely be said, however, about abũ Mansũr al-Nayrīzī, whose work on the determination of specific gravities was used by al-Khāzinī. 63

No real successors to al-Khāzinī in the art of the balance seem to have arisen in the Islamic world. The Book of the Balance of Wisdom is used as a source, however, in several encyclopedias and mineralogical compilations. Fakhr al-Dīn al-Rāzī (d. 1209) has long extracts in one of his persian encyclopedias of the sciences, Jāmi al-ulũm. 64 In the lapidary (Azhār al-afkār fi jawāhir al-açhjār) by Ahmad ibn Yũsuf al-Tīfāshiī (d, 1253) and in the mineralogical section of the Cosmography (Ajā ib al-makhlũqāt) by Zakariyyā ibn Muhammad al-Qazwīnī (d. 1283) are passages parallel to ones found in both al-Bīũnī and al-Khāzinī. 65 Many later mineralogical works are heavily indebted to material by al-Bīũnī how many of the authors were familiar with al-Khāzinī’s supplementary endeavors on the specific gravities of gems is unclear. The later medieval Islamic literature concerning specific gravities, in either the mathematical or the lapidary tradition, is treated by Bauerreiss, J. J. Clément-Mullet, and Wiedemann. 66

Al-Khāzinī’s Archimedean World Picture. Kitāb mīzān al-hikma has no integrated exposition of the theories of mechanics, but the theorems and excerpts on physical fundamentals that compose books I and II have a very definite cast. Most important for determining al-Khāzinī’s theoretical framework is chapter 5 of book I, for he himself seems to have selected the theorems that are repeated there for emphasis. Certain conceptual foundations, however, must be found in chapter 1 of book I. Heaviness (al-thiqal) one is told, is the force (al-quwwa), an inherent force by which any heavy body is moved towards the center of the world, and in no other direction, without cease, until (and only until—compare section 6) it reaches the center (I: 1.1). The force of a heavy body varies according to its density (al-kathāfa) or rarity (al-sakhāfa)(I: 1.2). Also missing from the recapitulation on chapter 5 are the theorems in I: 1.4-.9 on centers of gravity and the law of the lever, notably the axiomatization in section 5 of the balancing of two heavy bodies (relative to a given point or plane) according to the pattern of book I of Euclid’s Elements. The whole of I:1 is discussed in Clagett’s commentary 67 although Clagett does not reedit the text or revise Khanikoff’s deficient translation (sections 7 and 8 are particularly troubling), his analysis has not greatly suffered.

Two subjects of great interest emerge from the presentation of book I, chapter 5. The first is al-Khāzinī’s idea of “gravity.” His conception of heaviness, which is obviously Aristotelian, is here fitted to a picture of the subcelestial world that is purely “hydrostatic” and, in this sense, Archimedean. A heavy body becomes heavier in a rarer medium, lighter in a denser medium two bodies of different substances but of the same weight in some given medium differ in weight elsewhere, the body of smaller volume being the heavier in a denser, and the lighter in a rarer, medium (I: 5.1). In section 2, theorems 6 and 7 make explicit hat heavy bodies are essentially heavier than they are found to be in air and are heavier in a rarer air, lighter in a denser one. 68

A general relationship is then stated (I:5.3): the weight of any heavy body varies according to its distance from the center of the world, and the relation of weight (al-thiqal meaning precisely the weight as measured in the medium at that distance) to weight is as the relation of distance from the center to distance from the center. A body thus has its maximum and essential weight where there is no interfering medium, and has zero weight at the center. (No concept of “essential lightness” is found nor is there any question of “mass,” although “density” [al-Kathāfa] is in one way closely related.)

Clagett blames al-Khāzinī for saying that (quoting Clagett) “gravity [i.e., al-thiqul] varies directly as distance from the center of the world” after he has said that “gravity depends on the density of the medium in accordance with Archimedes’ principle.” 69 The assumption behind this criticism is that al-Khāzinī must have the density of the medium vary directly as the distance from the center yet al-Khāzinī is more likely to hold the opposite opinion—that the density of the medium varies roughly as the distance from the periphery of the world—for the density of the medium is the cause of th reduction in weight of a body weighed in it as that body is brought nearer the center. The weight of the body at a given distance from the center of the world less its weight at another distance must be equal to the weight of an equal volume of the medium at the second distance less the weight of the same volume of the (different) medium at the first distance.

The relationships stated by al-Khāzinī in I:5.3 can hardly have been intended as continuos and exact ones. But if he be granted any reasonable assumptions—for example, finite densities at the surface of the earth, zero density at the periphery of the cosmos, finite weights at the periphery, zero weight at the center—then in the idealized continuous case (earth shading off into water, water into air, etc.) weight will be directly proportional to distance from the center of the world, although in the form W = a + br, where a and b are constants and the difference in weight of a body at two distances from the center will be a constant multiple of the difference in distance: ΔW = bΔr. Thus al-Khāzinī’s statement that weight to weight is as distance to distance, although strictly wrong, is not an unreasonable brief presentation of the results of his physical picture of the sublunar cosmos.

In connection with that view of weight, one discovers an important corollary, seen most directly from theorems 3 and 4 of I:1.9 but implicit throughout chapters 1 and 5. The weight of an object, to summarize the text, has (at least) two possible manifestations: through its inherent force that tends toward the center of the world and acts against the interference of the ambient medium and through the force with which it acts against the interference of another body when they are turning about a fulcrum. The rule for instances of the second sort is that, when any two bodies balance each other with reference to a determined point, the weight of one to the weight of the other is inversely as the ratio between the two segments of a horizontal line cut off by vertical lines that pass through the centers of gravity of the two bodies and through the pivot point. Or, weight is to weight as distance from the center is to distance from the center (to wit, from the fulcrum, which is also the combined center of gravity of the balancing bodies). For both kinds of “heaviness,” then, weight is to weight as distance from the center to distance from the center the symmetry is absolute. There is no doubt that it is intentional, and no doubt that al-Khāzinī intends his statement of proportionality about gravity (or heaviness) with regard to distance from the center of the world to be taken as strictly as possible.

Hydrodynamical Ideas. The second topic to be considered is the movement of heavy bodies through a liquid. In I:1.3 their motion is said to be proportional to the fluidity (al-rutũfa) of the medium further, if two bodies unequal in density (al-kathāfa) or rarity (al-sakhāfa) but having the same shape and the same volume move (that is, fall) in the same medium, the denser is faster. Then comes an addition to the Archimedean analysis: in the case of equal volumes and densities, the body of smaller surface moves faster in a given medium. But the treatment cannot be successfully completed in a symmetric manner: if two bodies of the same density but different volumes move in a given medium, one is now told, the larger moves more quickly (all manuscripts). Glossators, Khanikoff, and Clagett prefer “more slowly” 70 —and indeed a greater volume tends to have a greater surface, and certainly does so if the shape is the same but the next section, 1:1.4, states the expected law, that it is the heavier body that moves faster. So the effects of total weight versus those of specific weight need to have been analyzed further. And it is clear that theory has not fully assimilated the effects of shape nor could it have done so.

Al-Khāzini takes over into 1:5.2 the non-Archimedean notion that a cause of the differing forces of the motios of bodies, in liquids and in air, is their difference in shape. A liquid medium interferes (āwaqa) with the motion of a heavy body through it it also reduces the body’s force and heaviness in proportion (bi qadr) to its volume, that is, in proportion to the weight of an equal volume of the liquid medium (Archimedes’ Principle). Whenever the moving body is increased in size, the interference (al-mu’āwaqa) becomes greater. “Interference” here refers to both the kinetic and the static effects. But the interference as regards weight is known to be due to the density (al-Kathāfa) of the medium, and the interference in motion to its liquidity (al-rutũba), inversely—compare the theorems of 1:1.3, reviewed in the preceding paragraph. So a good start has been made on separating static and kinetic effects and distinguishing viscosity from density, even if it has not been carried through completely and consistently. From this vantage al-Khāzini’s observations farther on, in section 2 of chapter 3, become revealing. He is reporting what happens to the beam of the hydrostatic balance during its actual operation: “Yet,” he says, “when a body lies at rest in the water-bowl [the bowl filled with the reference liquid], the beam of the balance rises according to the measure of the volume of the body, not according to its shape” whereas “the rapidity of the motion of the beam is in proportion to the force of the body [and hence its shape], not to its volume.”

Al-Asfizārī on Mechanics. Al-Asfizārī’s discussions of mechanical topics, included in book II of Kitāb mizān al-hikma, have notable examples and make interesting points, even though they supply no proofs. First to be treated is the problem of severl heavy bodies simultaneously seeking the centre of the world. Al-Asfizārī mentions in passing that that centre must in fact be a natural place and not a geometrical point, and asserts that it is the common centre of gravity of those bodies that must come into coincidence with the middle of the cosmos. As heuristics for this idea, al-Asfizārī considers the cases of one and two spheres free to roll in a concave spherical bowl and of one and two spherical bobs freely suspended (for the case of two, from a single point by cords of equal length). In the most difficult instances that he treats, two employed (this must be the intention of the text, which, however, is inexplicit) in both such situations a particular vertical line (the one passing through the bowl perpendicularly to the plane tangent at its lowest point, in the example with the bowl, through the point of suspension in the other) cuts the line joining the centers of gravity of the two spheres at a point such that the two line segments thus formed are in inverse ratio to the weights of the spheres. 71

The second chapter by al-Asfizārī investigates the conditions for equilibrium of a balance. He distinguishes the point of suspension of the beam and the natural motion of falling. To achieve equilibrium, two independent causes of motion of the balance must be made proportionate—the distance of the weights from the centre of suspension, which al-Asfizārī prefers to treat in terms of the arcs described, and the natural heaviness of the weights, their tendency toward the center of the world. However, the case of a physical balance-beam (one that has weight) suspended from a point away from its center and kept even by unequal weights hung from its ends is then handled according to the method of Euclid in “On the Balance,” a procedure related closely to the one used in Thābit ibn Qurra’s Risāla fi’l-qarastũn. 72 In that proof Peripatetic concepts are ignored, and the analysis diverges widely from Archimedes’ as well.

The results thus derived for physical levers are applied to the problem of the forces acting on the hand of someone holding up a spear, well back along the shaft, in a horizontal position. Al-Asfizārī recognizes two components, the (natural) weight of the spear and the unbalanced weight (of the other sort, the kind that produces forced motion about a center) of its front portion. His explanation (bayān, not burhaān [“demonstrative proof”], as throughout these chapters by al-Asfizārī) is well advanced, although wrong—he does not distinguish moments from froces it was too advanced for the scribes, who have muddled the text. 73

Specific Gravities .Much of the sholarly attention that has been paid to Kitāb mizān al-hikma has been stimulated by its tables of specific gravities. The literature has concerned itself especially with their accuracy and with the relationship between al-Bīũnī investigations and al-Khāzinī’s 74

Precautions are taken by al-Bīũnī and al-Khāzinī to assure the purity of the substances whose water equivalents are being meaured, and the difficulties of entrapped air, especially in cavities in gems, are dealt with. But knowledge of chemical identities and physical states (for instance, of alloys) is quite rudireerence liquid, are reasonably well known. 76 (The change of density with temperature is explicitly recognized, but apparently not a change of volume the discussions consequently deserve study from the standpoint of theories of matter.) 77 The need for standardization is accepted, although nothing effective could be done—the lack of a sufficient institutional basis for science prevented it. The specific gravities that are recorded are rarely correct to within 1 percent in cases where precision is possible whatever their source, these uncertainties are two orders of magnitude larger than in necessitated by the balance itself.

As regard metals, the values for mercury lead, and tin are excellent (0.06 to 0.3 percent of) for gold and iron, reasonably correct (about 1 percent) brass and bronze are surprisingly well done, and the value for copper is right, although there, as for other elemental metals as well as alloys, the description of the physicochemical state of the substance is insufficient. The same is true for glass, and more acutely so for the special earths that are tried. Nor are measurements on precious stones successful, except of emeralds, where the result falls in the middle of the actual range (specific gravity 2.68-2.78). The figure for salt is 1.5 percent high. The specific gravity of blood of a healthy man is about 2 percent low. Water at the boil is exact, as far as can be told but ice is 5 percent high and saturated salt water, 6 percent low. The gravimetric precision attained is more than sufficient, however, to discriminate reliably among hot and cold water, hot and cold human urine, and fresh water, sea water, and saturated salt water.

The Balance of Wisdom. In the Islamic world al-Khāzinī’s treatise was particularly valued for its descriptions of the insturments themselves—the arraeometer of Pappus, al-Bīũnī’s pycnometer flask, the earlier hydrostatic balances of book IV, the specialized balances and steelyard of books VII and VIII, and al-Khāzinā’s own balance. His instument must now be explained. (in the account that follows the capital letters refer of figure 1.) 78

The “balance of wisdom”s is a huydrostatic balance of standard form with five scale pans, a rather complicated polyfilar suspension, and a sensitive indicator tongue. The overall length of the beam is four cubits (about two meters or six feet). the length of the tongue, about fifty centimeters. The extraordinary limit of precision of this balance arises from its long beam, very accurate construction, and nearly frictionless suspension, of which the center of gravity and axis of oscillation turn out to be very close together. The double suspension increases stability but is which is such also as to magnify the motion of the tongue (D), which is much more carefully designed than the illustration suggests. Al-Khāzinī claims, and Wiedemann and Bauerreiss leave unchallenged, a sensitivity of one part in 60,000 (or up to 1 in 100,000, depending on the value of the h abba 79 for a weight of about 4.5 kilograms. This would mean a noticeable deflection for a change in weight of about forth-five to seventy-five milligrams. 80 Khanikoff and Wiedemann point out that specific gravities measured with the “balance of wisdom” are as precise as those obtained up to the eighteenth or beginning of the nineteenth century 81 in fact the “balance of wisdom” had the precision of an analytical balance, although it required quite large samples. It was too far in advance of chemical knowledge to be of service to chemistry, however, and belonged to researchers in a largely separate tradition.

The beam, A, is six centimeters thick, strengthened at the middle by the brace C. crossbar B is set in through C corresponding to it are the two lower crosspieces, F, of the fork. The fork is hung by its upper crosspiece on rings encircling a rod, the rod itself being anchored in any convenient manner. The beam is suspended from many paralle threads running between exactly opposite points on the crosspieces B and F. The knob below the center of the beam fastens and adjusts the tongue, a peg in the base of the tongue extending through crosspiece B and the beam. On both halves the beam is graduated with marks cut into its top into the marks fit the points of the precisely made rings of steel from which the scale pans hang. Up to five pans, all of equal weight, are employed, for measuring specific gravities, analyzing binary alloys, or detecting fraudulent gems. Pan H, whose bottom is drawn out into a point to facilitate its sinking into a liquid, is called the “cone-shaped” or “the judge” (al-h kim), for it is there that a suspected object is placed. Pan J, called “the winged” (almujannah), has deeply indented sides so that it can be brought close to other pans. K is a running weight (rummä na sayyä ar), used, when necessary, to adjust the leading of that end of the beam. In many kinds of measurements, after the balance has been brought into equilibrium the desired magnitudes can be read

directly from the divisions on the beam (hence al-Khāziní’s talk of locating the marks for givben substanes and for given liquid media).

The problem of suspending a balance beam is treated in detail (V:1.4). Al-Khamacrziní’s table showing the stability or instability o0f the instrument for diffe3rent positions of the axis of suspension is reproduced in translation in Khankkoff and in Ibel. 82

In determining the composition of alloys or testing the genuineness of precious stones, the great advantage of the “balance of wisdom”, part from its sensitivity, is the directness of the measurement and the consequent avoidance of calculations with dubious paramenters. Thus the most immediate way of testing a suspected gem or metal object or measuring the constituent of a binary alloy—a method possible only with an instrument as advanced as the “balance of wisdom”—is the procedure set forth by al-Khāzinni in VI:41—2. 83

The substance is first weighed, the sample being placed in the air panat the left end of the beam and the .mithqāl’s (the weights) in the air pan at the right end. The movable pans are both hung from the right half of the beam, at distances corr4esponding to the two assumed components of the sample. (The distance, d’ for any such constituent, measured from the center of suspension, is given by the formula d’, l/=W’,w/W’,l is the distance frojm the center of suspension to the point where the air pan at the left end of the beam is hung, w’ is the weight in water of an arbitrary volume of the given substance, and w’ is the weight in air of the same volume of that substance.)

The sample is then placed in the filled water pan, sufficient care being taken to assure that water (or, more generally, the reference liquid) reaches all its parts. The mithqāl’ fromthe air pan at the right-hand end are transferred to one of the movable air pans, then to the other, if necessary if equilibrium results in either case, the material being tested is that to which the given movable pan corresponds-for instance, the pure gem or its likeness in colored paste. For an alloy or other mixed body, however, the mithqaumll’s must be distributed between the two movable pans to create equilibrium (with sand or sifted seeds substituted for the last mithqamull, if needed to achieve an exact balance). The weight contained in each movable pan when equilibrium has been reached is equal to the weight in the mixed body of the component corresponding to the position of that pan.

Khanikoff provides a formula by which that procedure may be expressed (the notation is altered here for clarity): 84

where W denotes air weight and s specific gravity unprimed quantities refer to the mixed material, primes to the component, seconds to the other. Specific gravities, in fact, are neither needed nor used in the method, and they should be replaced by water weight and air weights according to the identity 1 -I/s = Ww/ W (for unprimed, primed, and seconded quantities). The right-had ratio is, of coursem the one involved in the placing of the movable pans. This equation for the weights of the two components is correct upon the assumption that the volume of the mixed body is equal to the sum of the volumes of its constituents, a relation that holds for all mechanical mixtures and is closely true for many alloys.

Magnetic Flux

The concept of magnetic flux is crucial to understanding Faraday’s law, because it relates flux changes to the induced ​electromotive force​ (EMF, commonly called ​voltage​) in the coil of wire or electric circuit. In simple terms, magnetic flux describes the flow of the magnetic field through a surface (although this “surface” isn’t really a physical object it’s really just an abstraction to help quantify the flux), and you can imagine it more easily if you think about how many magnetic field lines are passing through a surface area ​A​. Formally, it’s defined as:

Where ​B​ is the magnetic field strength (the magnetic flux density per unit area) in teslas (T), ​A​ is the area of the surface, and ​θ​ is the angle between the "normal" to the surface area (i.e., the line perpendicular to the surface) and ​B​, the magnetic field. The equation basically says that a stronger magnetic field and a bigger area lead to more flux, along with a field aligned with the normal to the surface in question.

The ​B​ ​∙ ​A​ in the equation is a scalar product (i.e., a “dot product”) of vectors, which is a special mathematical operation for vectors (i.e., quantities with both a magnitude or “size” ​and​ a direction) however, the version with cos (​θ​) and the magnitudes is the same operation.

This simple version works when the magnetic field is uniform (or can be approximated as such) across ​A​, but there is a more complicated definition for cases when the field isn’t uniform. This involves integral calculus, which is a bit more complicated but something you’ll need to learn if you’re studying electromagnetism anyway:

The SI unit of magnetic flux is the weber (Wb), where 1 Wb = T m 2 .

Famous Asteroids

The best-known asteroids are 1 Ceres, which is 952 kilometres across, 2 Pallas (with a diameter of 544 kilometres), and 4 Vesta (roughly 580 km across). These are rocky minor planets, and astronomers have observed them since the 1800s. Ceres is a differentiated asteroid. That means it has a rocky core and a icy outer crust. It might have an internal ocean. Pallas has a very irregular shape, and may be what’s left of an early protoplanet. Vesta is very bright and is likely the leftover of a rocky protoplanet.

What should be the &ldquopoles&rdquo for irregular shaped bodies? - Astronomy

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The Science Behind Nature’s Patterns

The curl of a chameleon's tail, the spiral of a pinecone's scales and the ripples created by wind moving grains of sand all have the power to catch the eye and intrigue the mind. When Charles Darwin first proposed the theory of evolution by natural selection in 1859, it encouraged science enthusiasts to find reasons for the natural patterns seen in beasts of the land, birds of the air and creatures of the sea. The peacock's plumage, the spots of a shark must all serve some adaptive purpose, they eagerly surmised.

Yet one person saw all this as "runaway enthusiasm," writes English scientist and writer Philip Ball in his new book, Patterns in Nature: Why the Natural World Looks the Way it Does.  Scottish zoologist D’Arcy Wentworth Thompson was pushed to publish his own treatise in 1917 explaining that even nature's creativity is constrained by laws generated by physical and chemical forces. Thompson's ideas didn't clash with Darwin's theory, but they did point out that other factors were at play. Whereas natural selection might explain the why of a tiger's stripes—a strategy to blend in with shadows in grasslands and forest— the way that chemicals diffuse through developing tissue can explain how pigment ends up in bands of dark and light, as well as why similar patterns can crop up on a sea anemone.

In Patterns in Nature, Ball brings his own background as a physicist and chemist to bear as well as more than 20 years of experience as an editor for the scientific journal Nature. His first book, published in 1999 (The Self-Made Tapestry), and a trilogy, published in 2009 (Nature’s Patterns: Shapes, Flow, Branches), explore the subject of natural patterns, but neither has visuals as rich as his latest.

Patterns in Nature: Why the Natural World Looks the Way It Does

The vivid photographs in the book are vital, Ball explains, because some of the patterns can only be fully appreciated through repetition. "It's when you see several of them side by side in glorious detail that you start to get a sense of how nature takes a theme and runs with it," he says.

The explanations Ball offers are simple and graceful, as when he explains how a soaked patch of ground can dry into a cracked landscape. "The dry layer at the surface tries to shrink relative to the still-moist layer below, and the ground becomes laced with tension throughout," he writes.

Yet he also offers enough detail to intrigue scientists and artists alike. The stunning photographs were curated by the designers at Marshall Editions, a publisher at the Quarto Group in London, which licensed the book to the University of Chicago Press.

Ball spoke to about his book and inspirations.

What exactly is a pattern?

I left it slightly ambiguous in the book, on purpose, because it feels like we know it when we see it. Traditionally, we think of patterns as something that just repeats again and again throughout space in an identical way, sort of like a wallpaper pattern. But many patterns that we see in nature aren't quite like that. We sense that there is something regular or at least not random about them, but that doesn't mean that all the elements are identical. I think a very familiar example of that would be the zebra's stripes. Everyone can recognize that as a pattern, but no stripe is like any other stripe.

I think we can make a case for saying that anything that isn't purely random has a kind of pattern in it. There must be something in that system that has pulled it away from that pure randomness or at the other extreme, from pure uniformity.

Why did you decide to write a book about natural patterns?

At first, it was a result of having been an editor at Nature. There, I started to see a lot of work come through the journal—and through scientific literature more broadly—about this topic. What struck me was that it's a topic that doesn't have any kind of natural disciplinary boundaries. People that are interested in these types of questions might be biologists, might be mathematicians, they might be physicists or chemists. That appealed to me. I always liked subjects that don't respect those traditional boundaries.

But I think also it was the visuals. The patterns are just so striking, beautiful and remarkable.

Then, underpinning that aspect is the question: How does nature without any kind of blueprint or design put together patterns like this? When we make patterns, it is because we planned it that way, putting the elements into place. In nature, there is no planner, but somehow natural forces conspire to bring about something that looks quite beautiful.

Do you have a favorite example of a pattern found in nature?

Perhaps one of the most familiar but really one of the most remarkable is the pattern of the snowflake. They all have the same theme—this six-fold, hexagonal symmetry and yet there just seems to be infinite variety within these snowflakes. It is such a simple process that goes into their formation. It is water vapor freezing out of humid air. There's nothing more to it than that but somehow it creates this incredibly intricate, detailed, beautiful pattern.

Another system we find cropping up again and again in different places, both in the living and the nonliving world, is a pattern that we call Turing structures. They are named after Alan Turing, the mathematician who laid the foundation for the theory of computation. He was very interested in how patterns form. In particular, he was interested in how that happens in a fertilized egg, which is basically a spherical cell that somehow gets patterned into something as complicated as a human as it grows and divides.

Turing came up with a theory that was basically an explanation for how a whole bunch of chemicals that are just kind of floating around in space can interact as to create differences from one bit of space to the next. In this way, the seeds of a pattern will emerge. He expressed that process in very abstract mathematical terms.

Now, it seems that something like this might be responsible for the patterns that form upon animal skins and some patterns we see in insects as well. But it also appears in some quite different systems, in sand dunes and sand ripples forming after wind has blown sand.

In your book, you mention the fact that science and math hasn't fully explained some of these patterns yet. Can you give an example?

We've only really understood how snowflakes get these branched formations since the 1980s even though people have studied and thought about that question for several hundred years. Yet even now it is a bit of a mystery why every arm of the snowflake can be pretty much identical. It is almost as though one arm can communicate with the others to make sure they grow in a special way. That is still surprising.

New forms of patterns are being discovered almost as fast as we can find explanations. There are strange vegetation patterns in semi-arid regions of the world where there are patches of vegetation separated by patches of bare ground. They too seem to have a Turing-like mechanism behind them but that understanding is very recent too.

What do you hope readers will find in the book?

When I started looking into this subject, I started to see patterns everywhere. I remember when I was halfway through writing my first book in 1999 and I was on a beach in Wales, I suddenly realized that everywhere there were patterns. In the clouds and the sky there were different patterns, there were wave patterns and so on in the sea. In the water running down through the sand, there was a different kind of pattern. Even the cliffs themselves weren't purely random.

So, you start to see patterns all around you. I hope that people will find this happening to them that they'll appreciate how much structure surrounding us is patterned. There's just splendor and joy in that.

About Marissa Fessenden

Marissa Fessenden is a freelance science writer and artist who appreciates small things and wide open spaces.

Centre of gravity

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Centre of gravity, in physics, an imaginary point in a body of matter where, for convenience in certain calculations, the total weight of the body may be thought to be concentrated. The concept is sometimes useful in designing static structures (e.g., buildings and bridges) or in predicting the behaviour of a moving body when it is acted on by gravity.

In a uniform gravitational field the centre of gravity is identical to the centre of mass, a term preferred by physicists. The two do not always coincide, however. For example, the Moon’s centre of mass is very close to its geometric centre (it is not exact because the Moon is not a perfect uniform sphere), but its centre of gravity is slightly displaced toward Earth because of the stronger gravitational force on the Moon’s near side.

The location of a body’s centre of gravity may coincide with the geometric centre of the body, especially in a symmetrically shaped object composed of homogeneous material. An asymmetrical object composed of a variety of materials with different masses, however, is likely to have a centre of gravity located at some distance from its geometric centre. In some cases, such as hollow bodies or irregularly shaped objects, the centre of gravity (or centre of mass) may occur in space at a point external to the physical material—e.g., in the centre of a tennis ball or between the legs of a chair.

Published tables and handbooks list the centres of gravity for most common geometric shapes. For a triangular metal plate such as that depicted in the figure, the calculation would involve a summation of the moments of the weights of all the particles that make up the metal plate about point A. By equating this sum to the plate’s weight W, multiplied by the unknown distance from the centre of gravity G to AC, the position of G relative to AC can be determined. The summation of the moments can be obtained easily and precisely by means of integral calculus.

The centre of gravity of any body can also be determined by a simple physical procedure. For example, for the plate in the figure, the point G can be located by suspending the plate by a cord attached at point A and then by a cord attached at C. When the plate is suspended from A, the line AD is vertical when it is suspended from C, the line CE is vertical. The centre of gravity is at the intersection of AD and CE. When an object is suspended from any single point, its centre of gravity lies directly beneath that point.

The Editors of Encyclopaedia Britannica This article was most recently revised and updated by Erik Gregersen, Senior Editor.

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How to Find Scale Factor

This article was co-authored by Mario Banuelos, Ph.D. Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

This article has been viewed 702,241 times.

The scale factor, or linear scale factor, is the ratio of two corresponding side lengths of similar figures. Similar figures have the same shape but are of different sizes. The scale factor is used to solve geometric problems. You can use the scale factor to find the missing side lengths of a figure. Conversely, you can use the side lengths of two similar figures to calculate the scale factor. These problems involve multiplication or require you to simplify fractions.


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