Alfonso's Rectifying the Curved: ​A Fourteenth-Century Hebrew Geometrical-Philosophical Treatise

Preface
Acknowledgement
Contents
Chapter 1: Introduction
	1 The Manuscript
	2 The Russian Translation and Previous Research
	3 The Author
	4 The Geometrical-Theological Agenda of the Book
	5 The Intellectual Environment of SMA
		The Jewish Background
		The Arabic Background
		The Christian Background
	6 Mathematical and Philosophical Knowledge in SMA
		Greek Authors
		Greek Authors Not Mentioned By Name
		Arabic Authors
		Arabic Authors Not Mentioned by Name who Could have Inspired Alfonso
		Jewish Authors
		Christian Authors
		Contemporary Scholastic Scientists Who Could Have Inspired Alfonso
		Geometrical Compendia
	7 Genre and Language
	8 Appendix: Translation Comparisons174
		Ibn Rushd
			Tahafut al-Tahafut
			Epistle on the Possibility of Conjunction
			Middle Commentary on De Generatione et Corruptione
		Al-Ghazali
			Maqāṣid al-falāsifah
		Euclid
			Elements Postulate 5
		Ptolemy
			Ptolemy’s Theorem, Almagest I.10
	Edition and Translation Conventions
		The Edition
		The Translation
		Notations
Chapter 2: English Translation and Commentary
	0 [Outline]
		Terms
		Commentary: The Title of the Book
	I Part I, Being an Introduction to the Book
		[I.0. Poetical Preface]
			Commentary: Poetical Language in SMA
		[I.1. Presentation of the Problem: Can Equality Between Rectilinear and Circular Areas Be Established]
			[I.1.1. Presenting the Two Positions]
				Terms
					32
					All these words are used by Alfonso to refer to proofs in geometry. When the focus is on the logical structure of the proof, Alfonso uses the term .50 Mofet is the standard Hebrew term for (ἀπόδειξις, demonstratio). It is used
					This word means falsity in general and sometimes sophistry (Arabic al safsaṭa). It was used in this sense in texts from thirteenth century Spain,53 and also here. The Provençal translators Ya‘aqov ben Makhir and Qalonimos ben Qalonimos in
				Commentary
			[I.1.2. Drawing on Archimedes and Euclid]
				Commentary
			[I.1.3. Drawing on Bryson]
				Terms
					Euclid defined the terms “inscribed” and “circumscribing” (eggraphesthai eis and perigraphesthai peri) in book IV of the Elements. Euclid uses consistently the same two verbs in the construction propositions of Book IV: eggrapsai (“to
					Alfonso uses the terms when he discusses the geometrical aspects of a proof; he uses the term (, syllogism) when he considers the logical structure of a proof. He refers specifically to:
				Historical Commentary
				Mathematical Commentary
			[I.1.4. Drawing Further on Archimedes]
				Terms
					Translated here as product. See Section III.0, commentary on terms.
				Commentary
		[I.2. Introducing Alfonso’s Concept of Mental Superposition]
			[I.2.1. Two Examples of Non-isometric Superposition]
				Terms
					Alfonso uses Bar-Ḥiyya’s term for obtuse angle. Among the Provençal translators Moshe Ibn Tibbon uses, whereas Ya‘aqov ben Makhir uses the term .
				Commentary
			[I.2.2. Modifying Euclid’s Common Notion of Superposition]
				Commentary
			[I.2.3. Mental Superposition in Hippocrates]
				Commentary
			[I.2.4. Mental Superposition in Archimedes]
				Commentary
			[I.2.5. Methodological Remark]
				Terms
					Verbs and adjectives derived from this root occur 24 times in Parts I–II of SMA, but not in Part III. In Parts I–II has two meanings that are obviously related. Usually means “to limit” or “to bound.” Here and in I.5.4
					The Hebrew word is biblical (Isa. 44:13). The root is Talmudic.138 Similar instruments are mentioned in some Arabic texts.139 The two words were not in common use in medieval mathematical texts. The suggested translation of
					The Greek kataskeuē, i.e., the construction part of the Euclidean proof, was translated into Arabic as and to Hebrew by Moshe Ibn Tibbon as .140 The word occurs four times in SMA,141 and in all four cases the meaning accords wit
				Commentary
		[I.3. Turning to Plato]
			[I.3.1. Alfonso’s “Intellectual and Moral” Principle]
				[Translation: Gad Freudenthal149]
				Commentary
			[I.3.2. Three Theses Ascribed to Plato: First Presentation]
				[Three Theses Ascribed to Plato]
				[Five Questions Ascribed to Aristotle]
				Terms
					The two terms Hiuli and geramim (in the plural) appear seven times on SMA, always when referring to Plato.152
					This term was coined by Abraham Bar-Hiyya, as the equivalent of the Arabic . It was used for both pyramid and cone.165 Alfonso uses or for pyramid and or for cone.
				Commentary
				Three Theses Ascribed to Plato (Marked [i] to [iii] in the Text)
					Thesis [i]: On hiuli
					Thesis [ii]: On Dimensional Reduction
					Thesis [iii]: On Weight
				Five Questions Ascribed to Aristotle (Marked [a] to [e] in the Text)
			[I.3.3. Correction of the First Presentation]
				Commentary
			[I.3.4. Three Theses Ascribed to Plato: Second Presentation]
				[The Three Theses Ascribed to Plato (Order of Presentation Changed)]
				Terms
					Alfonso uses the combination seven times in SMA. Both verbs and are often followed by the specification “by one another” and refer to the act of measuring. It is not clear at this stage why Alfonso uses insistently the two
				Commentary
					Thesis [i]: On hiuli
					Thesis [iii]: On Weight
					Thesis [ii]: On Dimensional Reduction
			[I.3.5. The Confusion About Indivisible Parts]
				Historical Commentary
				Mathematical Commentary
		[I.4. Turning to Aristotle]
			[I.4.1. An Intermediate Existence Between Potentiality and Actuality]
				Terms
					The term, translated here as “distance,” was used in SMA to denote [i] distance in philosophical discussions of divisibility222; [ii] distance between two lines,223 a concept that was not defined by Euclid, but introduced by some o
				Commentary
			[I.4.2. Defining Mental Superposition]
				Terms
					The word occurs 42 times in SMA and the verb () 13 more times. Most frequently, Ṣiyyur applies to operations: to motion (27 times), to superposition (11 times), to division (twice). A few times it applies to magnitudes: to lines
					These terms are used several times in I.4.2 and I.5.3. We translate as “equally divided,” as “similarly divided.”239 The meaning of these terms is explained in the commentary below.
					was the standard term for ratio in the mathematical vocabulary based on Bar-Hiyya and was widely used by Sephardic authors; was the common term used by the Provençal translators. is not a part of Alfonso’s vocabulary and occu
				Commentary
			[I.4.3. Identifying Plato’s Elementary geramim with Aristotle’s Minimal Parts]
				Commentary
			[I.4.4. Excursus on Void]248
				Commentary
		[I.5. Adding Two New Superposition Postulates to Geometry]
			[I.5.1. First Addition: A Postulate of Imagining Motion]
				Commentary
			[I.5.2. The Postulate of Imagining Motion Should Replace the Parallels Postulate]
				Commentary
			[I.5.3. Second Addition: The Postulate of Measurement]
				Terms
					Here and in III.33 the term galgal refers to a spherical shell (of the sun, moon, Venus, and Mercury) and is translated “orb.” In I.3.4 galgal refers to the outermost sphere and is translated there as “sphere.”
					The verb is the Hebrew parallel of the Arabic .282 It occurs only twice in SMA (in the outline and here) in the plural form Shevarim, and is translated here as “sections.”
				Commentary
			[I.5.4. Concluding Remark on the Aim of the Book]
				Terms
					See II.5.5 below.
					See I.2.5 above.
				Commentary
	II Part II, On Accidents Associated with Motion, Which Prevent its Imagining from Being a First Principle of Geometry, and how Several Early Sages were Confused by it
		[II.0. Foreword]
			Terms
				The subject of Part II is “accidents concomitant with motion,” which is an unusual expression. The word (accident, Arabic) is used in Aristotelian as well as in Kalām philosophy mainly to denote an attribute of substance,295 not
		[II.1. Motion in Geometry: Essential Errors]
			[II.1.1. First Error: Rotating a Finite Line in an Infinite Plane]
				Terms
					The reason for the repetition “an extremity and end” is not clear. The same expression is used by Iṣḥaq Israeli in his rendering of Euclid’s Elements I definition 3, that the point is the extremity of a line.296
					The literal translation is “to be on a straightness of a point,” we translate “to be on the straight extension of.” A line segment AB is said to be “on the straight extension” of a point P when P falls on the straight extension of AB.
				Commentary
			[II.1.2. Explaining the Error]
				Terms
					Alfonso uses these three terms to denote impossibility. Shmuel Ibn Tibbon in his glossary distinguishes between two of the terms: is a stronger term than, and is used to denote logical impossibility. Perhaps Alfonso was aware of th
				Commentary
			[II.1.3. Second Error: Moving a Ray Along Itself]
				Commentary
			[II.1.4. Explaining the Error]
				Commentary
			[II.1.5. Third Error: Moving an Infinite Line Parallel to Itself (the Attempt of Moshe of Seville and Others to Prove the Parallels Postulate)]
				Historical Commentary
				Mathematical Commentary
			[II.1.6. Moshe of Seville’s Justification of the Use of Motion in Geometry]
				Commentary
			[II.1.7. Explaining the Error]
				Terms
					The adjective, translated here as “determined,” appears three times in SMA, all three in this paragraph in the combinations . The expression was used also by Falaquera.334
				Commentary
		[II.2. Motion in Geometry: Accidental Errors]
			[II.2.1. Ibn al-Haytham’s and Others’ Attempt to Prove the Parallels Postulate]
				Historical Commentary
				Mathematical Commentary
			[II.2.2. Explaining the Error]
				Commentary
			[II.2.3. Al-Nayrizi’s Attempt to Prove the Axiom of Parallels]
				[Al-Nayrizi’s First Lemma]
				[Al-Nayrizi’s Second Lemma]
				Terms
					The word haqdama (Arabic) is the standard term for a premise (or “postulate” or “principle”).361 Here it is used to translate al-Nayrizi’s,362 namely proposition. In Part II Section 3 below Alfonso uses the Hebrew for t
					The pair of words (how) and (its demonstration) are used in Part II and Part III of SMA as titles for the exposition and the demonstration parts of the Euclidean proposition. The common Arabic titles are masʼalah and burhān-hu
				Historical Commentary
				Mathematical Commentary
					The First Lemma
					The Second Lemma
			[II.2.4. Explaining the Error]
				Commentary
		[II.3. Alfonso’s “Proof” of the Parallels Postulate by Means of Absolute Geometry with No Added Premises]
			[II.3.0. Foreword]
				Terms
				Commentary
			[II.3.1. The First Figure]
				Commentary
			[II.3.2. The Second Figure]
				Commentary
				[The Examination of the Three Cases]
				Commentary
				Commentary
				Terms
					The word appears in SMA four times, in three of which (twice here and in III.15) it seems to be used to refer to a diagram.383
				Commentary
				Commentary
				The Error in Alfonso’s Proof
				[Completing the Argument Using Simple Superposition]
				Commentary
			[II.3.3. The Third Figure]
				Commentary
			[II.3.4. The Fourth Figure [Thabit’s Theorem]]
				Mathematical Commentary
			[II.3.5. The Fifth Figure]
				Mathematical Commentary
			[II.3.6. The Sixth Figure]
				Mathematical Commentary
			[II.3.7. Concluding Remark]
				Terms
					Alfonso’s refers to Euclid’s Elements as, namely The Principles of Geometry. This title is not common. Moshe Ibn Tibbon and Ya‘aqov ben Makhir refer to it as or .
				Commentary
		[II.4. Motion in Geometry: Essential Problems (Continued)]
			[II.4.1. Dismissing the Possibility of Actually Infinite Magnitudes]
				Commentary
			[II.4.2. Theological Remark]
				Commentary
		[II.5. Motion in Geometry: Accidental Problems (Continued from II.2)]
			[II.5.1. Measuring the Circumference of a Circle by Rolling]
				Commentary
			[II.5.2. The Continuity of the Rolling Motion]
				Historical Commentary
				Terms
					Alfonso uses this term for picking at random a point on a line or on a circumference of a circle. We translate it “pick” although literally the meaning is “throw.”
				Mathematical Commentary
			[II.5.3. It Is Impossible to Measure the Circumference of the Circle by Rolling (the Wheel Paradox)]
				Commentary
			[II.5.4. Rolling Defined a One-to-One Correspondence Between a Circle and a Line Segment, but Such a Correspondence Does Not Imply Equality]
				Historical Commentary
				Mathematical Commentary
			[II.5.5. It is Impossible to Measure the Circumference of the Circle by Thread]
				Terms
					Alfonso uses these two terms to denote rational and irrational ratios, respectively.440 The Greek rhētos that is used in the definitions of Elements X is translated into Arabic muntaqah by Iṣḥaq-Thābit, and into Hebrew in Moshe I
					This term is used in II.2.4 above in the usual sense of definition. Here it is used in the sense of root. This is one of the senses of coined by Bar-Hiyya.443
				Commentary
			[II.5.6. Concluding Remark]
				Terms
					Alfonso identifies here as theoretical geometry and as of practical geometry. In I.5.3, however, he associates with, a concept which plays a major role in his theoretical geometry.450 Bar Ḥiyya uses for the scie
				Commentary
	III Part III
	The Third Part, on the Properties of Rectilinear Magnitudes and Areas which are Useful in this Science
		[III.0. Preface]
			Commentary
			Terms
		The Properties
			III.1 Properties 1, 10–13: Expansions
				[1. The First Property]
					Terms
						See commentary on terms after I.4.1.
						Typical of the style of Part III is the comparison of ratios or of expansions using the comparative prefix (meaning “as”) rather than using the explicit expression “equal to.” This style is typically Euclidean. occurs in Part III 5
					Mathematical Commentary
				III.[10] Property 10485
					Mathematical Commentary
				III.[11] Property 11
					Mathematical Commentary
				III.[12] Property 12 [Ptolemy’s Theorem]
					Mathematical Commentary
					Historical Commentary
				III.[13] Property 13
					Mathematical Commentary
			Properties 14–22: Right-Angled Triangles and Circumscribed Polygons
				III.[14] Property 14
					Mathematical Commentary
				III.[15] Property 15
					Terms
						Alfonso’s word for tangent is ; Bar-Hiyya uses,500 and the Provençal translators .501 Although Properties 14 and 15 refer to the same diagram, the term appears only in Property 15. In Property 14 Alfonso used the verb
						(v.) A verb formed of the word, namely to be the side opposite the right angle (i.e., to be the hypotenuse). The word appears twice, here and in III.17.502
					Mathematical Commentary
				III.[16] Property 16
					Mathematical Commentary
				III.[17] Property 17
					Mathematical Commentary
				III.[18] Property 18
					Mathematical Commentary
					Historical Commentary
					Note
				III.[19] Property 19
					Mathematical Commentary
				III.[20] Property 20
					Mathematical Commentary
				III.[21] Property 21
					Mathematical Commentary
				III.[22] Property 22
					Mathematical Commentary
			Property 23: Quadrature of Lunes
				III.[23] Property 23
					Historical Commentary
					Mathematical Commentary
						The First Construction
						The Second Construction
			Properties 24–27: Theory of Proportion
				III.[24] Property 24
					Terms
						The word in the sense of ratio is not a part of the mathematical vocabulary of SMA. Alfonso uses (176 times) the older term that was coined by Bar-Hiyya and commonly used in the Iberian Peninsula. The term, used here in a s
						The word is used five times in Parts I–II meaning “composed.” In Part III is used as a mathematical term for compound ratio. Following the Arabic the word was used by Abraham Ibn Ezra to denote both the result of additi
					Mathematical Commentary
				III.[25] Property 25
					Mathematical Commentary
				III.[26] Property 26
					Terms
						Summation. Here it is the summation of the antecedents and of consequents, when a sequence of ratios is given. According to Elements V.12, if all the ratios = p, then the ratio of the sum of antecedents to the sum of consequents is also =
					Mathematical Commentary
				III.[27] Property 27
					Mathematical Commentary
		Appendix A. Properties 26–7 and Euclid’s Definition of Proportion
		Appendix B. Weak and Strict Inequalities
			Property 28: Spherical Trigonometry
				III.[28] Property 28
				Terms
					See Section III.0 above.
					Mathematical Commentary
					Historical Commentary
			Properties 29–32: The Conchoid and Its Applications574
				III.[29] Property 29
					Terms
						The term “conchoid” follows the Greek and Latin word for mussel shell.589 The use of the term qaw paruṣ for conchoid may be unique in Hebrew. The word paruṣ means cracked or broken into, and in his extant Hebrew book Theshuva la-Meḥaref
					Mathematical Commentary
				III.[30] Property 30
					Mathematical Commentary
				III.[31] The 31st Proposition
					Mathematical Commentary
					Historical Commentary
				III.[32] The 32nd Proposition
					Terms
						This expression means duplicate ratio. This concept is defined in Elements V.9. Moshe Ibn Tibbon uses the expression .
					Mathematical Commentary
					Historical Commentary
			Property 33: The Ṭūsī Couple
				III.[33] Property 33
					Historical Commentary
					Mathematical Commentary
						Argument I. The Planet Moves on the Diameter of the Outer Circle
						Argument II. The Planet Does Not Rest at the Endpoints of Its Oscillation
		Appendix. Tūsī’s Argument (From the Tadhkira)641
	Part IV to Explain How a Body is Divided into Surfaces, and the Surface into Lines, as Plato Taught, and the Way of Evaluating and Measuring Them by One Another, and in it 16 Principles
		Commentary
		[4.1] The First Principle
			Terms
			Commentary
Chapter 3: Hebrew Text
Glossaries
	A.1.	Hebrew-English Glossary
	A.2.	 List of Verbs Frequently Used in Proofs and Constructions (in the First Person Plural, as they Appear in the Text)
	A.3.	 Arabic-English Glossary (Spelling as it Appears in the Text)
References
Primary Sources
Alfonso de Valladolid
Al-Ghazali
Al-Nayrizi
Aristotle
Archimedes
Euclid 
Elements
Fibonacci
Ibn Rushd (Averroes)
Oresme Nicole
Proclus
Ptolemy
Secondary Sources
Index