The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

Thābit ibn Qurra Thābit ibn Qurra (full name: , ar, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, la, Thebit/Thebith/Tebit); 826 or 836 – February 19, 901, was a mathematician, physician, astronomer, and translator who ...

, though the

authorship An author is the writer of a book, article, play, mostly written work. A broader definition of the word "author" states: "''An author is "the person who originated or gave existence to anything" and whose authorship determines responsibility f ...

of the book is questionable. It consists of fifteen propositions (

lemmas Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...

) on

circles A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

History

Translations

The ''Book of Lemmas'' was first introduced in

Arabic Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...

by Thābit ibn Qurra; he attributed the work to Archimedes. In 1661, the Arabic manuscript was translated into

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

Abraham Ecchellensis Ibrahim al-Haqilani (February 18, 1605July 15, 1664; Latinized as Abraham Ecchellensis) was a Maronite Catholic philosopher and linguist involved in the translation of the Bible into Arabic. He translated several Arabic works into Latin, the most ...

and edited by Giovanni A. Borelli. The Latin version was published under the name ''Liber Assumptorum''.

T. L. Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath transla ...

translated Heiburg's Latin work into

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

in his ''The Works of Archimedes''. A more recently discovered manuscript copy of Thābit ibn Qurra's Arabic translation was translated into English by Emre Coşkun in 2018.

Authorship

The original authorship of the ''Book of Lemmas'' has been in question because in proposition four, the book refers to Archimedes in

third person Third person, or third-person, may refer to: * Third person (grammar), a point of view (in English, ''he'', ''she'', ''it'', and ''they'') ** Illeism, the act of referring to oneself in the third person * Third-person narrative, a perspective in p ...

; however, it has been suggested that it may have been added by the translator. Another possibility is that the ''Book of Lemmas'' may be a collection of propositions by Archimedes later collected by a Greek writer.

New geometrical figures

The Book of Lemmas introduces several new

geometrical figure A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

Arbelos

Archimedes first introduced the arbelos (shoemaker's knife) in proposition four of his book: The figure is used in propositions four through eight. In propositions five, Archimedes introduces the

Archimedes's twin circles In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diame ...

, and in proposition eight, he makes use what would be the

Pappus chain In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction The arbelos is defined by two circles, ''C''U and ''C''V, which are tangent at the point A a ...

, formally introduced by

Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

Salinon

Archimedes first introduced the salinon (

salt cellar A salt cellar (also called a salt, salt-box and a salt pig) is an article of tableware for holding and dispensing salt. In British English, the term is normally used for what in North American English are called salt shakers. Salt cellars can be ...

) in proposition fourteen of his book: Archimedes proved that the salinon and the circle are equal in area.

Propositions

#If two circles touch at A, and if CD, EF be parallel diameters in them, ADF is a straight line. #Let AB be the diameter of a semicircle, and let the tangents to it at B and at any other point D on it meet in T. If now DE be drawn perpendicular to AB, and if AT, DE meet in F, then DF = FE. #Let P be any point on a segment of a circle whose base is AB, and let PN be perpendicular to AB. Take D on AB so that AN = ND. If now PQ be an arc equal to the arc PA, and BQ be joined, then BQ, BD shall be equal. #If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called αρβηλος"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P. #Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal. #Let AB, the diameter of a semicircle, be divided at C so that AC = 3/2 × CB r in any ratio Describe semicircles within the first semicircle and on AC, CB as diameters, and suppose a circle drawn touching the all three semicircles. If GH be the diameter of this circle, to find relation between GH and AB. #If circles are circumscribed about and inscribed in a square, the circumscribed circle is double of the inscribed square. #If AB be any chord of a circle whose centre is O, and if AB be produced to C so that BC is equal to the radius; if further CO meets the circle in D and be produced to meet the circle the second time in E, the arc AE will be equal to three times the arc BD. #If in a circle two chords AB, CD which do not pass through the centre intersect at right angles, then (arc AD) + (arc CB) = (arc AC) + (arc DB). #Suppose that TA, TB are two tangents to a circle, while TC cuts it. Let BD be the chord through B parallel to TC, and let AD meet TC in E. Then, if EH be drawn perpendicular to BD, it will bisect it in H. #If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre, then AO² + BO² + CO² + DO² = (diameter)². #If AB be the diameter of a semicircle, and TP, TQ the tangents to it from any point T, and if AQ, BP be joined meeting in R, then TR is perpendicular to AB. #If a diameter AB of a circle meet any chord CD, not a diameter, in E, and if AM, BN be drawn perpendicular to CD, then CN = DM. #Let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively. On AD, BE as diameters describe semicircles on the side towards C, and on DE as diameter a semicircle on the opposite side. Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively. Then shall the area of the figure bounded by the circumferences of all the semicircles be equal to the area of the circle on CF as diameter. #Let AB be the diameter of a circle., AC a side of an inscribed regular pentagon, D the middle point of the arc AC. Join CD and produce it to meet BA produced in E; join AC, DB meeting in F, and Draw FM perpendicular to AB. Then EM = (radius of circle).

References

{{Archimedes Ancient Greek mathematical works Works by Archimedes Arbelos Euclidean geometry Lemmas