Glossary of Greek, Latin & Technical Terms

Greek, Latin, Arabic, and technical mathematical terms tagged across Heath’s Elements, with translations and folio occurrences.

a fortiori (a-fortiori)

Language: Latin (Mathematical) | Translation: a fortiori

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Latin (Mathematical)).

Occurrences: Vol. II, p. 17, Vol. II, p. 30, Vol. II, p. 32, Vol. II, p. 123, Vol. II, p. 245, Vol. III, p. 350, Vol. III, p. 376, Vol. III, p. 433, Vol. III, p. 434, Vol. III, p. 511

acute angle (acute-angle)

Language: Euclidean Definition | Translation: acute angle

Book I: 12. An acute angle is an angle less than a right angle.

Occurrences: Vol. I, p. 153

ὀξυγώνιος (acute-angled)

Language: Greek | Translation: acute angled

Heath Greek Index: ὀξυγώνιος — acute-angled 187

Occurrences: Vol. III, p. 270

acute-angled (acute-angled-cone)

Language: Euclidean Definition | Translation: acute angled cone

Book XI: And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

Occurrences: Vol. III, p. 261

acute-angled triangle (acute-angled-triangle)

Language: Euclidean Definition | Translation: acute angled triangle

Book I: 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Occurrences: Vol. I, p. 154

ad infinitum (ad-infinitum)

Language: Latin (Mathematical) | Translation: ad infinitum

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Latin (Mathematical)).

Occurrences: Vol. III, p. 254, Vol. III, p. 255

ἀδιαίρετον (adiaireton)

Language: Greek | Translation: adiaireton

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 279

Alternando (ἐναλλάξ) (alternando)

Language: Euclidean Proportion Operation (Book V, Def. 12) | Translation: Alternate ratio (taking antecedent to antecedent as consequent to consequent)

Book V, Def. 12: If A : B = C : D, then alternando A : C = B : D (proved in V.16).

Occurrences: Vol. II, p. 114, Vol. II, p. 134, Vol. II, p. 231, Vol. II, p. 353, Vol. III, p. 502

ἀμβλυγώνιον (amblygonion)

Language: Greek | Translation: amblygonion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 187

ἀναλογία (analogia)

Language: Greek | Translation: analogia

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 119, Vol. II, p. 137, Vol. II, p. 112, Vol. II, p. 292, Vol. II, p. 293, Vol. II, p. 432, Vol. II, p. 434, Vol. II, p. 435

Angle (γωνία) (angle)

Language: Euclidean Definition (Book I, Def. 8–12) | Translation: γωνία

The inclination to one another of two lines in a plane which meet one another and do not lie in a straight line (classified by Aristotle, Syrianus, and Proclus under quality, quantity, or relation).

Occurrences: Vol. I, p. 160, Vol. I, p. 176, Vol. I, p. 181, Vol. I, p. 187, Vol. I, p. 188, Vol. I, p. 248, Vol. I, p. 249, Vol. I, p. 252, Vol. I, p. 278, Vol. I, p. 280, Vol. I, p. 283, Vol. I, p. 308, Vol. I, p. 350, Vol. I, p. 370, Vol. I, p. 38, Vol. III, p. 263, Vol. III, p. 264, Vol. III, p. 267

angle in a segment (angle-in-segment)

Language: Euclidean Definition | Translation: angle in segment

Book III: 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.

Occurrences: Vol. II, p. 1

angle of contact (angle-of-contact)

Language: Mathematical Term | Translation: angle of contact

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. II, p. 41, Vol. II, p. 42

angle of a segment (angle-of-segment)

Language: Euclidean Definition | Translation: angle of segment

Book III: 7. An angle of a segment is that contained by a straight line and a circumference of a circle.

Occurrences: Vol. II, p. 1, Vol. II, p. 4

προσαρμόζουσα (annex)

Language: Greek | Translation: annex

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 159

ἀνταναίρεσις (antanairesis)

Language: Greek | Translation: antanairesis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 120, Vol. II, p. 121, Vol. II, p. 431

ἀνθυφαίρεσις (anthyphairesis)

Language: Greek | Translation: anthyphairesis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 121

ἀντιστροφή (antistrophe)

Language: Greek | Translation: antistrophe

Heath Greek Index: ἀντιστροφή — conversion 256–7: leading variety, ἡ προηγουμένη or ἡ κυρίως, ibid.

Occurrences: Vol. I, p. 257

ἀπαγωγή (apagoge)

Language: Greek | Translation: apagoge

Heath Greek Index: ἀπαγωγή — reduction 135: εἰς τὸ ἀδύνατον 136

Occurrences: Vol. I, p. 430, Vol. I, p. 135

Apodeixis (ἀπόδειξις) (apodeixis)

Language: Greek (Formal Division of a Proposition) | Translation: Proof / Demonstration

The fifth formal division of a Euclidean proposition, drawing the deductive chain of assertions from the hypotheses, constructions, axioms, and prior propositions.

Occurrences: Vol. I, p. 429, Vol. I, p. 129, Vol. I, p. 136

Apotome (ἀποτομή) (apotome)

Language: Euclidean Definition (Book X. 73) | Translation: ἀποτομή

Book X.73: The irrational remainder when a rational straight line commensurable in square only with the whole is subtracted from a rational straight line; classified into six species in Book X Defs. III.

Occurrences: Vol. III, p. 7, Vol. III, p. 158, Vol. III, p. 159

ἄρρητος (arrhetos)

Language: Greek | Translation: arrhetos

Heath Greek Index: ἄρρητος — irrational: of λόγος 137: of diameter (diagonal) 399

Occurrences: Vol. I, p. 426, Vol. I, p. 413, Vol. I, p. 137

ἀσύμπτωτοι (asymptotos)

Language: Greek | Translation: asymptotos

Heath Greek Index: ἀσύμπτωτος — not-meeting, non-secant, asymptotic 40, 161, 203

Occurrences: Vol. I, p. 161, Vol. I, p. 203, Vol. I, p. 427, Vol. I, p. 40

ἄτομοι γραμμαί (atomoi-grammai)

Language: Greek | Translation: atomoi grammai

Heath Greek Index: ἄτομοι γραμμαί — “indivisible lines” 268

Occurrences: Vol. I, p. 268, Vol. I, p. 426

axis of the cone (axis-of-cone)

Language: Euclidean Definition | Translation: axis of cone

Book XI: 19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

Occurrences: Vol. III, p. 261

axis of the cylinder (axis-of-cylinder)

Language: Euclidean Definition | Translation: axis of cylinder

Book XI: 22. The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.

Occurrences: Vol. III, p. 262

axis of the sphere (axis-of-sphere)

Language: Euclidean Definition | Translation: axis of sphere

Book XI: 15. The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.

Occurrences: Vol. III, p. 261

base (base-of-cone)

Language: Euclidean Definition | Translation: base of cone

Book XI: 20. And the base is the circle described by the straight line which is carried round.

Occurrences: Vol. III, p. 262

base of the segment (base-of-segment)

Language: Euclidean Definition | Translation: base of segment

Book III: 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.

Occurrences: Vol. II, p. 1

bases (bases-of-cylinder)

Language: Euclidean Definition | Translation: bases of cylinder

Book XI: 23. And the bases are the circles described by the two sides opposite to one another which are carried round.

Occurrences: Vol. III, p. 262

βάσις (basis)

Language: Greek | Translation: basis

Heath Greek Index: βάσις — base 248–9

Occurrences: Vol. I, p. 252, Vol. II, p. 4, Vol. II, p. 192

Binomial straight line (ἐκ δύο ὀνομάτων) (binomial)

Language: Euclidean Definition (Book X. 36) | Translation: ἐκ δύο ὀνομάτων

Book X.36: The irrational sum of two rational straight lines commensurable in square only; classified into six species in Book X Defs. II.

Occurrences: Vol. III, p. 7, Vol. III, p. 83, Vol. III, p. 84

boundary

Language: Euclidean Definition | Translation: boundary

Book I: 13. A boundary is that which is an extremity of anything.

Occurrences: Vol. I, p. 153

πλάτος (breadth)

Language: Greek | Translation: breadth

Heath Greek Index: πλάτος — breadth 158–9

Occurrences: Vol. I, p. 159, Vol. I, p. 169, Vol. III, p. 262

Centre of the circle (κέντρον τοῦ κύκλου) (centre)

Language: Euclidean Definition (Book I, Def. 16) | Translation: Κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται

Book I, Def. 16: ‘And the point is called the centre of the circle.’

Occurrences: Vol. I, p. 184, Vol. I, p. 185, Vol. I, p. 186, Vol. I, p. 199, Vol. I, p. 154, Vol. I, p. 423, Vol. I, p. 0003, Vol. II, p. 2, Vol. II, p. 3

centre of the sphere (centre-of-sphere)

Language: Euclidean Definition | Translation: centre of sphere

Book XI: 16. The centre of the sphere is the same as that of the semicircle.

Occurrences: Vol. III, p. 261

Circle (κύκλος) (circle)

Language: Euclidean Definition (Book I, Def. 15) | Translation: Κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς περιεχόμενον…

Book I, Def. 15: ‘A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.’

Occurrences: Vol. I, p. 183, Vol. I, p. 184, Vol. I, p. 185, Vol. I, p. 186, Vol. I, p. 153, Vol. I, p. 242, Vol. I, p. 50, Vol. II, p. 2, Vol. II, p. 4, Vol. II, p. 5, Vol. II, p. 40, Vol. II, p. 79

inscribed in a figure (circle-inscribed-in-figure)

Language: Euclidean Definition | Translation: circle inscribed in figure

Book IV: 5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.

Occurrences: Vol. II, p. 78

touch one another (circles-touching)

Language: Euclidean Definition | Translation: circles touching

Book III: 3. Circles are said to touch one another which, meeting one another, do not cut one another.

Occurrences: Vol. II, p. 1

circumscribed about a circle (circumscribed-about-circle)

Language: Euclidean Definition | Translation: circumscribed about circle

Book IV: 4. A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.

Occurrences: Vol. II, p. 78

circumscribed about a figure (circumscribed-about-figure)

Language: Euclidean Definition | Translation: circumscribed about figure

Book IV: 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.

Occurrences: Vol. II, p. 78

Commensurable (σύμμετρος) (commensurable)

Language: Euclidean Definition (Book X, Def. 1) | Translation: Σύμμετρα μεγέθη λέγεται τὰ τῷ αὐτῷ μέτρῳ μετρούμενα

Book X, Def. 1: ‘Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.’

Occurrences: Vol. III, p. 10, Vol. III, p. 11

commensurable in length (commensurable-in-length)

Language: Mathematical Term | Translation: commensurable in length

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. III, p. 11, Vol. III, p. 34, Vol. III, p. 48

commensurable in square (commensurable-in-square)

Language: Euclidean Definition | Translation: commensurable in square

Book X: 2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.

Occurrences: Vol. III, p. 10

commensurable in square only (commensurable-in-square-only)

Language: Mathematical Term | Translation: commensurable in square only

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. III, p. 11, Vol. III, p. 48, Vol. III, p. 71

Componendo (σύνθεσις λόγου / συνθέντι) (componendo)

Language: Euclidean Proportion Operation (Book V, Def. 14) | Translation: Composition of a ratio (antecedent + consequent to consequent)

Book V, Def. 14: If A : B = C : D, then componendo (A + B) : B = (C + D) : D (proved in V.18).

Occurrences: Vol. II, p. 88, Vol. II, p. 115, Vol. II, p. 135, Vol. II, p. 168, Vol. II, p. 169, Vol. II, p. 174, Vol. II, p. 175, Vol. II, p. 184, Vol. II, p. 213, Vol. II, p. 251, Vol. II, p. 432, Vol. III, p. 519, Vol. III, p. 83, Vol. III, p. 85, Vol. III, p. 116, Vol. III, p. 149, Vol. III, p. 232, Vol. III, p. 392, Vol. III, p. 393, Vol. III, p. 449, Vol. III, p. 463, Vol. III, p. 464, Vol. III, p. 465

composite number (composite-number)

Language: Euclidean Definition | Translation: composite number

Book VII: 13. A composite number is that which is measured by some number.

Occurrences: Vol. II, p. 278, Vol. II, p. 286

Numbers composite to one another (composite-to-one-another)

Language: Euclidean Definition | Translation: composite to one another

Book VII: 14. Numbers composite to one another are those which are measured by some number as a common measure.

Occurrences: Vol. II, p. 278

compound ratio (compound-ratio)

Language: Mathematical Term | Translation: compound ratio

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. II, p. 132, Vol. II, p. 134

cone

Language: Euclidean Definition | Translation: cone

Book XI: 18. When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone.

Occurrences: Vol. III, p. 261, Vol. III, p. 270

contained (contained-rectangle)

Language: Euclidean Definition | Translation: contained rectangle

Book II: 1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.

Occurrences: Vol. I, p. 370

Convertendo (ἀναστροφὴ λόγου / ἀναστρέψαντι) (convertendo)

Language: Euclidean Proportion Operation (Book V, Def. 16) | Translation: Conversion of a ratio (antecedent to antecedent − consequent)

Book V, Def. 16: If A : B = C : D, then convertendo A : (A − B) = C : (C − D) (V.19 Porism).

Occurrences: Vol. II, p. 115, Vol. II, p. 135, Vol. II, p. 174, Vol. II, p. 175, Vol. II, p. 432, Vol. III, p. 38, Vol. III, p. 67, Vol. III, p. 68, Vol. III, p. 103, Vol. III, p. 105, Vol. III, p. 108, Vol. III, p. 110, Vol. III, p. 111, Vol. III, p. 114, Vol. III, p. 163, Vol. III, p. 164, Vol. III, p. 178, Vol. III, p. 179, Vol. III, p. 180, Vol. III, p. 183, Vol. III, p. 184, Vol. III, p. 185, Vol. III, p. 187, Vol. III, p. 189, Vol. III, p. 190, …and 4 more

corresponding magnitudes (corresponding-magnitudes)

Language: Euclidean Definition | Translation: corresponding magnitudes

Book V: 11. The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

Occurrences: Vol. II, p. 114, Vol. II, p. 134, Vol. II, p. 238

cube

Language: Euclidean Definition | Translation: cube

Book XI: 25. A cube is a solid figure contained by six equal squares.

Occurrences: Vol. III, p. 262

cube (cube-number)

Language: Euclidean Definition | Translation: cube number

Book VII: 19. And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

Occurrences: Vol. II, p. 278

cylinder

Language: Euclidean Definition | Translation: cylinder

Book XI: 21. When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder.

Occurrences: Vol. III, p. 262

βάθος (depth)

Language: Greek | Translation: depth

Heath Greek Index: βάθος — depth 158–9

Occurrences: Vol. I, p. 159, Vol. III, p. 262

Diameter of the circle (διάμετρος τοῦ κύκλου) (diameter)

Language: Euclidean Definition (Book I, Def. 17) | Translation: Διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη…

Book I, Def. 17: ‘A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.’

Occurrences: Vol. I, p. 185, Vol. I, p. 186, Vol. I, p. 154, Vol. I, p. 370, Vol. I, p. 399, Vol. I, p. 424, Vol. I, p. 413, Vol. II, p. 256, Vol. III, p. 12

diameter of the sphere (diameter-of-sphere)

Language: Euclidean Definition | Translation: diameter of sphere

Book XI: 17. A diameter of the sphere is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.

Occurrences: Vol. III, p. 261

διάστημα (diastema)

Language: Greek | Translation: diastema

Heath Greek Index: διάστημα — distance 166, 167, 207: (of radius of circle) 199: (of an angle) = divergence 176–7

Occurrences: Vol. I, p. 161, Vol. I, p. 166, Vol. I, p. 177, Vol. I, p. 207, Vol. I, p. 424, Vol. II, p. 295

διαστάσεις (dimensions)

Language: Greek | Translation: dimensions

Heath Greek Index: διαστάσεις — almost = dimensions 157, 158

Occurrences: Vol. I, p. 158, Vol. I, p. 424, Vol. III, p. 262

Diorismos (διορισμός) (diorismos)

Language: Greek (Formal Division of a Proposition) | Translation: Definition / Specification (or Condition of Possibility)

Either (1) the third formal division of a proposition restating the goal in terms of the particular figure, or (2) the determination of the limits/conditions under which a problem admits of a solution.

Occurrences: Vol. I, p. 234, Vol. I, p. 243, Vol. I, p. 293, Vol. I, p. 337, Vol. I, p. 344, Vol. I, p. 421, Vol. I, p. 424, Vol. I, p. 40, Vol. I, p. 128, Vol. I, p. 129, Vol. I, p. 130, Vol. I, p. 131, Vol. I, p. 142, Vol. I, p. 51, Vol. II, p. 186, Vol. II, p. 259, Vol. II, p. 262, Vol. II, p. 267, Vol. II, p. 435

dodecahedron

Language: Euclidean Definition | Translation: dodecahedron

Book XI: 28. A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.

Occurrences: Vol. III, p. 262

δοκίς (dokis)

Language: Greek | Translation: dokis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 291

duplicate ratio (duplicate-ratio)

Language: Euclidean Definition | Translation: duplicate ratio

Book V: 9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

Occurrences: Vol. II, p. 114

δυνάμεις (dynameis)

Language: Greek | Translation: dynameis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 1, Vol. III, p. 3

δυναμένη (dynamene)

Language: Greek | Translation: dynamene

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 13, Vol. III, p. 119

δύναμις (dynamis)

Language: Greek | Translation: dynamis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 282, Vol. II, p. 283, Vol. II, p. 290, Vol. II, p. 294, Vol. III, p. 11

δύνασθαι (dynasthai)

Language: Greek | Translation: dynasthai

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 13

ἡ εἰς τὸ ἀδύνατον ἀπαγωγή (eis-to-adynaton)

Language: Greek | Translation: Reduction to the impossible (reductio ad absurdum)

Indirect demonstration proving a proposition by showing that its contradiction leads to an impossibility.

Occurrences: Vol. I, p. 136

Ekthesis (ἔκθεσις) (ekthesis)

Language: Greek (Formal Division of a Proposition) | Translation: Setting-out

The second formal division of a Euclidean proposition, instantiating the given data in a concrete geometric figure with lettered elements.

Occurrences: Vol. I, p. 337, Vol. I, p. 430, Vol. I, p. 129, Vol. I, p. 130, Vol. I, p. 50, Vol. I, p. 51, Vol. II, p. 399

ἔλλειψις (elleipsis)

Language: Greek | Translation: elleipsis

Heath Greek Index: ἔλλειψις — falling-short (with reference to application of areas) 36, 343–5, 383–4

Occurrences: Vol. I, p. 343, Vol. I, p. 36

ἔνστασις (enstasis)

Language: Greek | Translation: enstasis

Heath Greek Index: ἔνστασις — objection 135

Occurrences: Vol. I, p. 255, Vol. I, p. 428, Vol. I, p. 23, Vol. I, p. 135

ἐφάπτεσθαι (ephaptesthai)

Language: Greek | Translation: ephaptesthai

Heath Greek Index: ἐφάπτεσθαι — to touch 57

Occurrences: Vol. II, p. 2, Vol. II, p. 3, Vol. II, p. 79

ἐφαρμόζειν (epharmozein)

Language: Greek | Translation: epharmozein

Heath Greek Index: ἐφαρμόζειν — to coincide, ἐφαρμόζεσθαι, to be applied to 168, 224–5, 249

Occurrences: Vol. I, p. 224, Vol. I, p. 225, Vol. I, p. 249, Vol. I, p. 84

ἐπίπεδον (epipedon)

Language: Greek | Translation: epipedon

Heath Greek Index: ἐπίπεδον — plane in Euclid, used for surface also in Plato and Aristotle 169

Occurrences: Vol. I, p. 169, Vol. I, p. 170, Vol. I, p. 183, Vol. I, p. 184, Vol. I, p. 431

Equal and similar solid figures (equal-and-similar-solid-figures)

Language: Euclidean Definition | Translation: equal and similar solid figures

Book XI: 10. Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude.

Occurrences: Vol. III, p. 261

Equal circles (equal-circles)

Language: Euclidean Definition | Translation: equal circles

Book III: 1. Equal circles are those the diameters of which are equal, or the radii of which are equal.

Occurrences: Vol. II, p. 1

ἴσον ἰσάκις (equal-multiplied-by-equal)

Language: Greek | Translation: equal multiplied by equal

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 3

equally distant from the centre (equally-distant-from-centre)

Language: Euclidean Definition | Translation: equally distant from centre

Book III: 4. In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.

Occurrences: Vol. II, p. 1

equilateral triangle (equilateral-triangle)

Language: Euclidean Definition | Translation: equilateral triangle

Book I: 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Occurrences: Vol. I, p. 154

even number (even-number)

Language: Euclidean Definition | Translation: even number

Book VII: 6. An even number is that which is divisible into two equal parts.

Occurrences: Vol. II, p. 277, Vol. II, p. 281

even-times even number (even-times-even-number)

Language: Euclidean Definition | Translation: even times even number

Book VII: 8. An even-times even number is that which is measured by an even number according to an even number.

Occurrences: Vol. II, p. 277, Vol. II, p. 281, Vol. II, p. 282, Vol. II, p. 283, Vol. II, p. 285

even-times odd number (even-times-odd-number)

Language: Euclidean Definition | Translation: even times odd number

Book VII: 9. An even-times odd number is that which is measured by an even number according to an odd number.

Occurrences: Vol. II, p. 277, Vol. II, p. 282, Vol. II, p. 283, Vol. II, p. 284, Vol. II, p. 285

Ex aequali (δι’ ἴσου λόγος) (ex-aequali)

Language: Euclidean Proportion Operation (Book V, Def. 17) | Translation: Ratio ex aequali (taking extreme to extreme across ordered series of ratios)

Book V, Def. 17: If A : B = D : E and B : C = E : F, then ex aequali A : C = D : F (proved in V.22; perturbed proportion in V.23).

Occurrences: Vol. II, p. 396, Vol. II, p. 409, Vol. II, p. 421, Vol. II, p. 423, Vol. II, p. 424, Vol. II, p. 425, Vol. II, p. 115, Vol. II, p. 136, Vol. II, p. 137, Vol. II, p. 140, Vol. II, p. 141, Vol. II, p. 175, Vol. II, p. 176, Vol. II, p. 178, Vol. II, p. 179, Vol. II, p. 180, Vol. II, p. 182, Vol. II, p. 184, Vol. II, p. 194, Vol. II, p. 202, Vol. II, p. 231, Vol. II, p. 236, Vol. II, p. 237, Vol. II, p. 241, Vol. II, p. 243, …and 36 more

διαστατόν (extended)

Language: Greek | Translation: extended

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 263

Extreme and mean ratio (ἄκρος καὶ μέσος λόγος) (extreme-and-mean-ratio)

Language: Euclidean Definition (Book VI, Def. 3) | Translation: Golden Section / Divine Proportion

Book VI, Def. 3: ‘A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.’ (Constructed in II.11 and VI.30; applied to the pentagon, icosahedron, and dodecahedron in Book XIII).

Occurrences: Vol. II, p. 188

extremities (extremity)

Language: Euclidean Definition | Translation: extremity

Book I: 3. The extremities of a line are points.

Occurrences: Vol. I, p. 153

fifth apotome (fifth-apotome)

Language: Euclidean Definition | Translation: fifth apotome

Book X: 5. if the annex be so commensurable, a fifth ;

Occurrences: Vol. III, p. 112, Vol. III, p. 177, Vol. III, p. 187

fifth binomial (fifth-binomial)

Language: Euclidean Definition | Translation: fifth binomial

Book X: 5. if the lesser, a fifth binomial;

Occurrences: Vol. III, p. 6, Vol. III, p. 102, Vol. III, p. 112, Vol. III, p. 143, Vol. III, p. 155

figure

Language: Euclidean Definition | Translation: figure

Book I: 14. A figure is that which is contained by any boundary or boundaries.

Occurrences: Vol. I, p. 153

first apotome (first-apotome)

Language: Euclidean Definition | Translation: first apotome

Book X: 1. Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a first apotome.

Occurrences: Vol. III, p. 9, Vol. III, p. 104, Vol. III, p. 120, Vol. III, p. 177, Vol. III, p. 179, Vol. III, p. 193, Vol. III, p. 214, Vol. III, p. 215, Vol. III, p. 236, Vol. III, p. 451

first apotome of a medial (first-apotome-of-a-medial-straight-line)

Language: Euclidean Definition | Translation: first apotome of a medial straight line

Book X: And let it be called a first apotome of a medial straight line.

Occurrences: Vol. III, p. 7, Vol. III, p. 85, Vol. III, p. 123, Vol. III, p. 160, Vol. III, p. 169, Vol. III, p. 198, Vol. III, p. 238

first bimedial (first-bimedial)

Language: Euclidean Definition | Translation: first bimedial

Book X: If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line.

Occurrences: Vol. III, p. 7, Vol. III, p. 84, Vol. III, p. 85, Vol. III, p. 136, Vol. III, p. 155, Vol. III, p. 160

first binomial (first-binomial)

Language: Mathematical Term | Translation: first binomial

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. III, p. 5, Vol. III, p. 103, Vol. III, p. 104, Vol. III, p. 119, Vol. III, p. 120, Vol. III, p. 134, Vol. III, p. 135, Vol. III, p. 155

first binomial straight line (first-binomial-straight-line)

Language: Euclidean Definition | Translation: first binomial straight line

Book X: 1. Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a first binomial straight line;

Occurrences: Vol. III, p. 101

fitted into a circle (fitted-into-circle)

Language: Euclidean Definition | Translation: fitted into circle

Book IV: 7. A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.

Occurrences: Vol. II, p. 78

fourth apotome (fourth-apotome)

Language: Euclidean Definition | Translation: fourth apotome

Book X: 4. Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome ;

Occurrences: Vol. III, p. 110, Vol. III, p. 177, Vol. III, p. 185, Vol. III, p. 186, Vol. III, p. 236, Vol. III, p. 465

fourth binomial (fourth-binomial)

Language: Euclidean Definition | Translation: fourth binomial

Book X: 4. Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a fourth binomial;

Occurrences: Vol. III, p. 6, Vol. III, p. 102, Vol. III, p. 110, Vol. III, p. 141, Vol. III, p. 155

γαμήλιον διάγραμμα (gamelion-diagramma)

Language: Greek | Translation: gamelion diagramma

Heath Greek Index: γαμήλιον διάγραμμα — “Figure of Marriage” (Plutarch), term for Pythagorean triangle (3, 4, 5), 417

Occurrences: Vol. I, p. 417

Gnomon (γνώμων) (gnomon)

Language: Euclidean Definition (Book II, Def. 2) | Translation: γνώμων

Book II, Def. 2: ‘In any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.’

Occurrences: Vol. I, p. 370, Vol. I, p. 371, Vol. I, p. 64, Vol. II, p. 40

golden section (golden-section)

Language: Mathematical Term | Translation: golden section

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. II, p. 99

at a greater distance (greater-distance-from-centre)

Language: Euclidean Definition | Translation: greater distance from centre

Book III: 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls.

Occurrences: Vol. II, p. 1

have a greater ratio (greater-ratio)

Language: Euclidean Definition | Translation: greater ratio

Book V: 7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

Occurrences: Vol. II, p. 114

have a ratio (have-a-ratio)

Language: Euclidean Definition | Translation: have a ratio

Book V: 4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

Occurrences: Vol. II, p. 114, Vol. II, p. 129, Vol. II, p. 133

ἡγούμενα (hegoumenon)

Language: Greek | Translation: hegoumenon

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 134

The height (height)

Language: Euclidean Definition | Translation: height

Book VI: 4. The height of any figure is the perpendicular drawn from the vertex to the base.

Occurrences: Vol. II, p. 188, Vol. II, p. 189

ἑπόμενα (hepomenon)

Language: Greek | Translation: hepomenon

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 134

ἑτερόμηκες (heteromekes)

Language: Greek | Translation: heteromekes

Heath Greek Index: ἑτερόμηκες — oblong 151, 188

Occurrences: Vol. I, p. 188, Vol. I, p. 151

ἑτερομήκης (heteromekes-arithmos)

Language: Greek | Translation: heteromekes arithmos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 288, Vol. II, p. 289, Vol. II, p. 290, Vol. II, p. 293, Vol. II, p. 434

ἵππου πέδη (hippou-pede)

Language: Greek | Translation: hippou pede

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 162, Vol. I, p. 426

ὁμοιομερεῖς (homoiomeres)

Language: Greek | Translation: homoiomeres

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 162, Vol. I, p. 40

ὅπερ ἔδει δεῖξαι (Q.E.D.) (hoper-edei-deixai)

Language: Greek | Translation: Being what it was required to prove (Quod erat demonstrandum)

Euclid’s canonical closing formula for every theorem.

Occurrences: Vol. I, p. 252, Vol. I, p. 55, Vol. I, p. 57, Vol. II, p. 64, Vol. II, p. 211

ὅπερ ἔδει ποιῆσαι (Q.E.F.) (hoper-edei-poiesai)

Language: Greek | Translation: Being what it was required to do (Quod erat faciendum)

Euclid’s canonical closing formula for every construction problem.

Occurrences: Vol. I, p. 252, Vol. I, p. 55, Vol. II, p. 8, Vol. II, p. 268

κερατοειδής (hornlike-angle)

Language: Greek | Translation: hornlike angle

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 4, Vol. II, p. 39, Vol. II, p. 40, Vol. II, p. 433

ὅρος (ὅροι) (horos)

Language: Greek | Translation: Definition / Boundary

A definition delimiting the meaning of a geometric or arithmetic term.

Occurrences: Vol. I, p. 182, Vol. I, p. 183, Vol. I, p. 423, Vol. I, p. 424, Vol. I, p. 129, Vol. I, p. 143

ὑπερβολή (hyperbole)

Language: Greek | Translation: hyperbole

Heath Greek Index: ὑπερβολή — exceeding, with reference to method of application of areas 36, 343–5, 386–7

Occurrences: Vol. I, p. 343, Vol. I, p. 36

icosahedron

Language: Euclidean Definition | Translation: icosahedron

Book XI: 27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

Occurrences: Vol. III, p. 262

μήκει (in-length)

Language: Greek | Translation: in length

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 11, Vol. III, p. 48

δυνάμει (in-square)

Language: Greek | Translation: in square

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 11

be in the same ratio (in-the-same-ratio)

Language: Euclidean Definition | Translation: in the same ratio

Book V: 5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Occurrences: Vol. II, p. 114

κλίσις (inclination)

Language: Greek | Translation: inclination

Heath Greek Index: κλίσις — inclination, 176

Occurrences: Vol. III, p. 263, Vol. III, p. 264, Vol. III, p. 267

inclination of a straight line to a plane (inclination-of-line-to-plane)

Language: Euclidean Definition | Translation: inclination of line to plane

Book XI: 5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

Occurrences: Vol. III, p. 260

inclination of a plane to a plane (inclination-of-plane-to-plane)

Language: Euclidean Definition | Translation: inclination of plane to plane

Book XI: 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.

Occurrences: Vol. III, p. 260

Incommensurable (ἀσύμμετρος) (incommensurable)

Language: Euclidean Definition (Book X, Def. 1) | Translation: ἀσύμμετρα μεγέθη

Book X, Def. 1: Magnitudes which cannot have any common measure.

Occurrences: Vol. III, p. 10

incommensurable in length (incommensurable-in-length)

Language: Mathematical Term | Translation: incommensurable in length

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. III, p. 11, Vol. III, p. 197

incommensurable in square (incommensurable-in-square)

Language: Euclidean Definition | Translation: incommensurable in square

Book X: 2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.

Occurrences: Vol. III, p. 10

inscribed in a circle (inscribed-in-circle)

Language: Euclidean Definition | Translation: inscribed in circle

Book IV: 3. A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.

Occurrences: Vol. II, p. 78

inscribed in a rectilineal figure (inscribed-in-rectilineal-figure)

Language: Euclidean Definition | Translation: inscribed in rectilineal figure

Book IV: 1. A rectilineal figure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.

Occurrences: Vol. II, p. 78

Invertendo (ἀνάπαλιν) (invertendo)

Language: Euclidean Proportion Operation (Book V, Def. 13) | Translation: Inverse ratio (taking consequent to antecedent)

Book V, Def. 13: If A : B = C : D, then invertendo B : A = D : C (V.7 Porism).

Occurrences: Vol. II, p. 114, Vol. II, p. 134, Vol. II, p. 433

Irrational straight line (ἄλογος) (irrational)

Language: Euclidean Definition (Book X, Def. 3) | Translation: ἄλογος εὐθεῖα

Book X, Def. 3: A straight line incommensurable both in length and in square with the assigned rational straight line.

Occurrences: Vol. I, p. 413, Vol. III, p. 10, Vol. III, p. 12

ἄρρητος διάμετρος τῆς πεμπάδος (irrational-diameter-of-five)

Language: Greek | Translation: irrational diameter of five

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 12

ἰσάκις (isakis)

Language: Greek | Translation: isakis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 120, Vol. II, p. 129, Vol. II, p. 143, Vol. II, p. 288, Vol. II, p. 290, Vol. II, p. 291, Vol. II, p. 292, Vol. II, p. 432

ἰσόπλευρον (isopleuron)

Language: Greek | Translation: isopleuron

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 187, Vol. I, p. 188

isosceles triangle (isosceles-triangle)

Language: Euclidean Definition | Translation: isosceles triangle

Book I: 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Occurrences: Vol. I, p. 154

Kataskeue (κατασκευή) (kataskeue)

Language: Greek (Formal Division of a Proposition) | Translation: Construction / Machinery

The fourth formal division of a Euclidean proposition, adding auxiliary lines and circles justified by the Postulates or prior problems.

Occurrences: Vol. I, p. 337, Vol. I, p. 424, Vol. I, p. 129, Vol. I, p. 51

κάθετος (kathetos)

Language: Greek | Translation: kathetos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 181, Vol. I, p. 271, Vol. I, p. 425, Vol. I, p. 428

κερατοειδής (keratoeides)

Language: Greek | Translation: keratoeides

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 177, Vol. I, p. 178, Vol. I, p. 421, Vol. I, p. 426

κλίσις (klisis)

Language: Greek | Translation: klisis

Heath Greek Index: κλίσις — inclination, 176

Occurrences: Vol. I, p. 176

κοιναὶ δόξαι (koinai-doxai)

Language: Greek | Translation: koinai doxai

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 221, Vol. I, p. 120

κοιναὶ ἔννοιαι (koinai-ennoiai)

Language: Greek | Translation: Common Notions (Axioms)

Euclid’s title for universal, self-evident axioms shared across sciences (called τὰ κοινά or ἀξιώματα by Aristotle).

Occurrences: Vol. I, p. 121, Vol. I, p. 62

κορυφή (koryphe)

Language: Greek | Translation: koryphe

Heath Greek Index: κορυφή — vertex: κατὰ κορυφήν, vertical (angles) 278

Occurrences: Vol. I, p. 278

μῆκος (length)

Language: Greek | Translation: length

Heath Greek Index: μῆκος — length, 158–9

Occurrences: Vol. I, p. 158, Vol. I, p. 159, Vol. I, p. 169, Vol. III, p. 3, Vol. III, p. 262

Line (γραμμή) (line)

Language: Euclidean Definition (Book I, Def. 2) | Translation: Γραμμὴ δὲ μῆκος ἀπλατές

Book I, Def. 2: ‘A line is breadthless length.’ Book I, Def. 3: ‘The extremities of a line are points.’

Occurrences: Vol. I, p. 156, Vol. I, p. 159, Vol. I, p. 165, Vol. I, p. 168, Vol. I, p. 171, Vol. I, p. 176, Vol. I, p. 183, Vol. I, p. 199, Vol. I, p. 153, Vol. I, p. 271, Vol. I, p. 325, Vol. I, p. 329, Vol. I, p. 413, Vol. I, p. 150, Vol. I, p. 60, Vol. I, p. 62

μείζων (major)

Language: Euclidean Definition | Translation: major

Book X: If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole straight line is irrational : and let it be called major.

Occurrences: Vol. III, p. 7, Vol. III, p. 87, Vol. III, p. 88, Vol. III, p. 270

μαθήματα (mathemata)

Language: Greek | Translation: mathemata

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 262

Medial straight line (μέση) (medial)

Language: Euclidean Definition (Book X. 21) | Translation: μέση εὐθεῖα

Book X.21: The side of an irrational rectangle contained by two rational straight lines commensurable in square only (equivalent to ⁴√k · ρ).

Occurrences: Vol. III, p. 50

μέγεθος (megethos)

Language: Greek | Translation: megethos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 117, Vol. II, p. 129, Vol. II, p. 135, Vol. II, p. 165, Vol. II, p. 168, Vol. II, p. 280, Vol. II, p. 281, Vol. II, p. 312

μηνοειδές (menoeides)

Language: Greek | Translation: menoeides

Heath Greek Index: μηνοειδής — lune-like (of angle) 26, 201: τὸ μηνοειδὲς (σχῆμα), lune 187

Occurrences: Vol. I, p. 187, Vol. I, p. 421

μεσότητες (mesotes)

Language: Greek | Translation: mesotes

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 292

ἐφόδιον (method)

Language: Greek | Translation: method

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 366

Method of exhaustion (method-of-exhaustion)

Language: Mathematical Method (Book XII) | Translation: Exhaustion by inscribed/circumscribed polygons or polyhedra (Eudoxus & Archimedes)

Rigorous Greek limiting method grounded in Book X.1 (and Book V Def. 4) avoiding actual infinitesimals via double reductio ad absurdum.

Occurrences: Vol. III, p. 365, Vol. III, p. 366

ἐλάσσων (minor)

Language: Euclidean Definition | Translation: minor

Book X: And let it be called minor. Q. E. D.

Occurrences: Vol. III, p. 7, Vol. III, p. 163

μονὰς προσλαβοῦσα θέσιν (monas-proslabousa-thesin)

Language: Greek | Translation: monas proslabousa thesin

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 155

μονόστροφος ἕλιξ (monostrophos-helix)

Language: Greek | Translation: monostrophos helix

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 165

multilateral

Language: Euclidean Definition | Translation: multilateral

Book I: 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Occurrences: Vol. I, p. 154

multiple

Language: Euclidean Definition | Translation: multiple

Book V: 2. The greater is a multiple of the less when it is measured by the less.

Occurrences: Vol. II, p. 113, Vol. II, p. 120, Vol. II, p. 129, Vol. II, p. 143, Vol. II, p. 277, Vol. II, p. 292, Vol. II, p. 432

multiply

Language: Euclidean Definition | Translation: multiply

Book VII: 15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

Occurrences: Vol. II, p. 278, Vol. II, p. 287

mutatis mutandis (mutatis-mutandis)

Language: Latin (Mathematical) | Translation: mutatis mutandis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Latin (Mathematical)).

Occurrences: Vol. II, p. 28, Vol. II, p. 31, Vol. II, p. 169, Vol. II, p. 290

νεύσεις (neuseis)

Language: Greek | Translation: neuseis

Heath Greek Index: νεύσεις — inclinations, a class of problems 150–1: νεύειν, to verge 118, 150

Occurrences: Vol. I, p. 422, Vol. I, p. 150, Vol. I, p. 151

Number (ἀριθμός) (number)

Language: Euclidean Definition (Book VII, Def. 2) | Translation: Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος

Book VII, Def. 2: ‘A number is a multitude composed of units.’

Occurrences: Vol. II, p. 115, Vol. II, p. 277, Vol. II, p. 279, Vol. II, p. 280, Vol. II, p. 281, Vol. II, p. 282, Vol. II, p. 284, Vol. II, p. 285, Vol. II, p. 286, Vol. II, p. 287, Vol. II, p. 288, Vol. II, p. 290, Vol. II, p. 291, Vol. II, p. 293, Vol. II, p. 295, Vol. II, p. 312

oblong

Language: Euclidean Definition | Translation: oblong

Book I: 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Occurrences: Vol. I, p. 154

obtuse angle (obtuse-angle)

Language: Euclidean Definition | Translation: obtuse angle

Book I: 11. An obtuse angle is an angle greater than a right angle.

Occurrences: Vol. I, p. 153

ἀμβλυγώνιος (obtuse-angled)

Language: Greek | Translation: obtuse angled

Heath Greek Index: ἀμβλυγώνιος — obtuse-angled 187

Occurrences: Vol. III, p. 270

obtuse-angled (obtuse-angled-cone)

Language: Euclidean Definition | Translation: obtuse angled cone

Book XI: And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

Occurrences: Vol. III, p. 261

obtuse-angled triangle (obtuse-angled-triangle)

Language: Euclidean Definition | Translation: obtuse angled triangle

Book I: 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Occurrences: Vol. I, p. 154

octahedron

Language: Euclidean Definition | Translation: octahedron

Book XI: 26. An octahedron is a solid figure contained by eight equal and equilateral triangles.

Occurrences: Vol. III, p. 262

odd number (odd-number)

Language: Euclidean Definition | Translation: odd number

Book VII: 7. An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

Occurrences: Vol. II, p. 277, Vol. II, p. 282

odd-even (odd-times-even-number)

Language: Mathematical Term | Translation: odd times even number

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. II, p. 283, Vol. II, p. 284, Vol. II, p. 285

odd-times odd number (odd-times-odd-number)

Language: Euclidean Definition | Translation: odd times odd number

Book VII: 10. An odd-times odd number is that which is measured by an odd number according to an odd number.

Occurrences: Vol. II, p. 277, Vol. II, p. 284, Vol. II, p. 285

ὀρθογώνιον (orthogonion)

Language: Greek | Translation: orthogonion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 187, Vol. I, p. 188, Vol. I, p. 370, Vol. I, p. 380

ὀξυγώνιον (oxygonion)

Language: Greek | Translation: oxygonion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 187

παραβολή (parabole)

Language: Greek | Translation: parabole

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 36

παραβολὴ τῶν χωρίων (parabole-ton-chorion)

Language: Greek | Translation: Application of areas

The Pythagorean geometric-algebraic method (Book I.44–45, Book II, Book VI.27–29) of applying a parallelogram equal to a given area along a given straight line—either exactly (παραβολή), in defect (ἔλλειψις), or in excess (ὑπερβολή).

Occurrences: Vol. I, p. 343

παράδοξος τόπος (paradoxos-topos)

Language: Greek | Translation: paradoxos topos

Heath Greek Index: παράδοξος τόπος — ὁ, “the Treasury of Paradoxes” 329

Occurrences: Vol. I, p. 329

Parallel planes (parallel-planes)

Language: Euclidean Definition | Translation: parallel planes

Book XI: 8. Parallel planes are those which do not meet.

Occurrences: Vol. III, p. 260

Parallel straight lines (παράλληλοι εὐθεῖαι) (parallel-straight-lines)

Language: Euclidean Definition (Book I, Def. 23) | Translation: Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ’ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις

Book I, Def. 23: ‘Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.’

Occurrences: Vol. I, p. 154

παραλληλεπίπεδος (parallelepiped)

Language: Greek | Translation: parallelepiped

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 326

παραλληλόγραμμον (parallelogrammon)

Language: Greek | Translation: parallelogrammon

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 346, Vol. I, p. 370

παραπληρώματα (parapleroma)

Language: Greek | Translation: parapleroma

Heath Greek Index: παραπλήρωμα — complement, q.v.

Occurrences: Vol. I, p. 341, Vol. I, p. 370, Vol. I, p. 423

part (part-aliquot)

Language: Euclidean Definition | Translation: part aliquot

Book V: 1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

Occurrences: Vol. II, p. 113, Vol. II, p. 115, Vol. II, p. 277, Vol. II, p. 280, Vol. II, p. 292

parts (parts-non-aliquot)

Language: Euclidean Definition | Translation: parts non aliquot

Book VII: 4. but parts when it does not measure it.

Occurrences: Vol. II, p. 277, Vol. II, p. 280, Vol. II, p. 292

πηλικότης (pelikotes)

Language: Greek | Translation: pelikotes

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 116, Vol. II, p. 117, Vol. II, p. 132, Vol. II, p. 190, Vol. II, p. 292, Vol. II, p. 432, Vol. II, p. 436

περαίνουσα ποσότης (perainousa-posotes)

Language: Greek | Translation: perainousa posotes

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 279

πέρας (peras)

Language: Greek | Translation: peras

Heath Greek Index: πέρας — extremity 165, 182: πέρας συγκλεῖον (Posidonius’ definition of figure) 183

Occurrences: Vol. I, p. 156, Vol. I, p. 165, Vol. I, p. 168, Vol. I, p. 169, Vol. I, p. 171, Vol. I, p. 182, Vol. I, p. 259, Vol. I, p. 425

πέρας συγκλεῖον (peras-synkleion)

Language: Greek | Translation: peras synkleion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 183

Perfect number (τέλειος ἀριθμός) (perfect-number)

Language: Euclidean Definition (Book VII, Def. 22) | Translation: Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν

Book VII, Def. 22: ‘A perfect number is that which is equal to its own parts.’ (Constructed from prime sums of powers of two in IX.36).

Occurrences: Vol. II, p. 278

περιφερείας (periphereia)

Language: Greek | Translation: periphereia

Heath Greek Index: περιφέρεια — circumference 184

Occurrences: Vol. I, p. 170, Vol. I, p. 183, Vol. I, p. 184, Vol. I, p. 185, Vol. I, p. 186, Vol. I, p. 423, Vol. I, p. 50, Vol. II, p. 3, Vol. II, p. 4, Vol. II, p. 5, Vol. II, p. 79

perpendicular

Language: Euclidean Definition | Translation: perpendicular

Book I: 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Occurrences: Vol. I, p. 153, Vol. III, p. 263

perturbed proportion (perturbed-proportion)

Language: Euclidean Definition | Translation: perturbed proportion

Book V: 18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

Occurrences: Vol. II, p. 115, Vol. II, p. 136, Vol. II, p. 176, Vol. II, p. 177, Vol. II, p. 434

ἐπίπεδον (plane)

Language: Greek | Translation: plane

Heath Greek Index: ἐπίπεδον — plane in Euclid, used for surface also in Plato and Aristotle 169

Occurrences: Vol. III, p. 263, Vol. III, p. 264, Vol. III, p. 265

Plane angle (ἐπίπεδος γωνία) (plane-angle)

Language: Euclidean Definition (Book I, Def. 8) | Translation: Ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων κλίσις

Book I, Def. 8: ‘A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.’

Occurrences: Vol. I, p. 153

plane is at right angles to a plane (plane-at-right-angles-to-plane)

Language: Euclidean Definition | Translation: plane at right angles to plane

Book XI: 4. A plane is at right angles to a plane when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.

Occurrences: Vol. III, p. 260

plane (plane-number)

Language: Euclidean Definition | Translation: plane number

Book VII: 16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

Occurrences: Vol. II, p. 278, Vol. II, p. 287, Vol. II, p. 288, Vol. II, p. 293

τὸ ὑποκείμενον ἐπίπεδον (plane-of-reference)

Language: Greek | Translation: plane of reference

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. III, p. 272

Plane surface (ἐπίπεδος ἐπιφάνεια) (plane-surface)

Language: Euclidean Definition (Book I, Def. 7) | Translation: Ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ’ ἑαυτῆς εὐθείαις κεῖται

Book I, Def. 7: ‘A plane surface is a surface which lies evenly with the straight lines on itself.’

Occurrences: Vol. I, p. 169, Vol. I, p. 153

πλῆθος ὡρισμένον (plethos-horismenon)

Language: Greek | Translation: plethos horismenon

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 280

πλινθίς (plinthis)

Language: Greek | Translation: plinthis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 291

Point (σημεῖον) (point)

Language: Euclidean Definition (Book I, Def. 1) | Translation: Σημεῖόν ἐστιν, οὗ μέρος οὐθέν

Book I, Def. 1: ‘A point is that which has no part.’ Earlier Pythagoreans defined it as a ‘monad having position’ (μονὰς προσλαβοῦσα θέσιν).

Occurrences: Vol. I, p. 155, Vol. I, p. 156, Vol. I, p. 177, Vol. I, p. 183, Vol. I, p. 153, Vol. I, p. 252, Vol. I, p. 259, Vol. I, p. 429

πολύπλευρα (polypleuron)

Language: Greek | Translation: polypleuron

Heath Greek Index: πολύπλευρον — many-sided figure 187

Occurrences: Vol. I, p. 187

πόρισμα (porisma)

Language: Greek | Translation: porisma

Heath Greek Index: πόρισμα — porism q.v.

Occurrences: Vol. I, p. 13

Prime number (πρῶτος ἀριθμός) (prime-number)

Language: Euclidean Definition (Book VII, Def. 11) | Translation: Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος

Book VII, Def. 11: ‘A prime number is that which is measured by a unit alone.’ (Proved infinite in multitude in IX.20).

Occurrences: Vol. II, p. 278, Vol. II, p. 284, Vol. II, p. 285, Vol. II, p. 292

Numbers prime to one another (prime-to-one-another)

Language: Euclidean Definition | Translation: prime to one another

Book VII: 12. Numbers prime to one another are those which are measured by an unit alone as a common measure.

Occurrences: Vol. II, p. 278

prism

Language: Euclidean Definition | Translation: prism

Book XI: 13. A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.

Occurrences: Vol. III, p. 261

πρόβλημα (problema)

Language: Greek | Translation: problema

Heath Greek Index: πρόβλημα — problem q.v.

Occurrences: Vol. I, p. 429, Vol. I, p. 126, Vol. I, p. 129

προμήκης (promekes-arithmos)

Language: Greek | Translation: promekes arithmos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 289, Vol. II, p. 293, Vol. II, p. 434

proportional

Language: Euclidean Definition | Translation: proportional

Book V: 6. Let magnitudes which have the same ratio be called proportional.

Occurrences: Vol. II, p. 114, Vol. II, p. 129, Vol. II, p. 134, Vol. II, p. 135, Vol. II, p. 165, Vol. II, p. 168, Vol. II, p. 188, Vol. II, p. 278, Vol. II, p. 292, Vol. II, p. 293, Vol. II, p. 346, Vol. II, p. 435

Protasis (πρότασις) (protasis)

Language: Greek (Formal Division of a Proposition) | Translation: Enunciation

The first of the six formal divisions of a Euclidean proposition (Proclus, p. 203), stating in general terms what is given and what is sought.

Occurrences: Vol. I, p. 425, Vol. I, p. 129

πτῶσις (ptosis)

Language: Greek | Translation: ptosis

Heath Greek Index: πτῶσις — case 134

Occurrences: Vol. I, p. 134

pyramid

Language: Euclidean Definition | Translation: pyramid

Book XI: 12. A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point.

Occurrences: Vol. III, p. 261

quadrilateral

Language: Euclidean Definition | Translation: quadrilateral

Book I: 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Occurrences: Vol. I, p. 154

quadrinomial

Language: Mathematical Term | Translation: quadrinomial

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Mathematical Term).

Occurrences: Vol. III, p. 256

Ratio (λόγος) (ratio)

Language: Euclidean Definition (Book V, Def. 3 & 4) | Translation: Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα ποιὰ σχέσις

Book V, Def. 3: ‘A ratio is a sort of relation in respect of size between two magnitudes of the same kind.’ Def. 4 requires that the magnitudes, when multiplied, are capable of exceeding one another.

Occurrences: Vol. II, p. 2, Vol. II, p. 114, Vol. II, p. 117, Vol. II, p. 118, Vol. II, p. 119, Vol. II, p. 129, Vol. II, p. 133, Vol. II, p. 134, Vol. II, p. 135, Vol. II, p. 136, Vol. II, p. 189, Vol. II, p. 248, Vol. II, p. 292, Vol. II, p. 293, Vol. II, p. 353, Vol. II, p. 432, Vol. III, p. 263

Rational straight line (ῥητή) (rational)

Language: Euclidean Definition (Book X, Def. 3) | Translation: ῥητὴ εὐθεῖα

Book X, Def. 3: A straight line commensurable either in length and in square, or in square only, with the assigned straight line.

Occurrences: Vol. I, p. 430, Vol. III, p. 10, Vol. III, p. 12

Reciprocally related figures. (reciprocally-related)

Language: Euclidean Definition | Translation: reciprocally related

Book VI: [2. Reciprocally related figures. See note.]

Occurrences: Vol. II, p. 188, Vol. II, p. 189

rectilineal (rectilineal-angle)

Language: Euclidean Definition | Translation: rectilineal angle

Book I: 9. And when the lines containing the angle are straight, the angle is called rectilineal.

Occurrences: Vol. I, p. 153

Rectilineal figures (rectilineal-figure)

Language: Euclidean Definition | Translation: rectilineal figure

Book I: 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Occurrences: Vol. I, p. 154

reductio ad absurdum (reductio-ad-absurdum)

Language: Latin (Logical) | Translation: Reduction to absurdity / the impossible (ἡ εἰς τὸ ἀδύνατον ἀπαγωγή)

Indirect proof establishing a truth by deducing a contradiction from the denial of the conclusion.

Occurrences: Vol. II, p. 8, Vol. II, p. 9, Vol. II, p. 15, Vol. II, p. 21, Vol. II, p. 36, Vol. II, p. 50, Vol. II, p. 52, Vol. II, p. 99, Vol. II, p. 193, Vol. II, p. 200, Vol. II, p. 299, Vol. II, p. 321, Vol. II, p. 335, Vol. II, p. 341, Vol. II, p. 346, Vol. II, p. 356, Vol. II, p. 370, Vol. II, p. 371, Vol. III, p. 2, Vol. III, p. 22, Vol. III, p. 31, Vol. III, p. 65, Vol. III, p. 87, Vol. III, p. 284, Vol. III, p. 299, …and 4 more

rhomboid

Language: Euclidean Definition | Translation: rhomboid

Book I: 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Occurrences: Vol. I, p. 154

ῥόμβος (rhombos)

Language: Greek | Translation: rhombos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 188, Vol. I, p. 189

rhombus

Language: Euclidean Definition | Translation: rhombus

Book I: 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Occurrences: Vol. I, p. 154

Right angle (ὀρθὴ γωνία) & Perpendicular (κάθετος) (right-angle)

Language: Euclidean Definition (Book I, Def. 10) | Translation: ὀρθὴ γωνία / κάθετος

Book I, Def. 10: ‘When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.’

Occurrences: Vol. I, p. 153

ὀρθογώνιος (right-angled)

Language: Greek | Translation: right angled

Heath Greek Index: ὀρθογώνιος — right-angled: as used of quadrilaterals = rectangular 188–9

Occurrences: Vol. III, p. 270

right-angled (right-angled-cone)

Language: Euclidean Definition | Translation: right angled cone

Book XI: And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

Occurrences: Vol. III, p. 261

right-angled triangle (right-angled-triangle)

Language: Euclidean Definition | Translation: right angled triangle

Book I: 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Occurrences: Vol. I, p. 154

at right angles to a plane (right-angles-to-plane)

Language: Euclidean Definition | Translation: right angles to plane

Book XI: 3. A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane.

Occurrences: Vol. III, p. 260

scalene triangle (scalene-triangle)

Language: Euclidean Definition | Translation: scalene triangle

Book I: 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Occurrences: Vol. I, p. 154

σχῆμα (schema)

Language: Greek | Translation: schema

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 169, Vol. I, p. 183, Vol. I, p. 186

σχέσις (schesis)

Language: Greek | Translation: schesis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 116, Vol. II, p. 117, Vol. II, p. 292

second apotome (second-apotome)

Language: Euclidean Definition | Translation: second apotome

Book X: 2. But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome.

Occurrences: Vol. III, p. 106, Vol. III, p. 177, Vol. III, p. 181, Vol. III, p. 217

second apotome of a medial (second-apotome-of-a-medial-straight-line)

Language: Euclidean Definition | Translation: second apotome of a medial straight line

Book X: And let it be called a second apotome of a medial straight line. Q. E. D.

Occurrences: Vol. III, p. 7, Vol. III, p. 87, Vol. III, p. 125, Vol. III, p. 162, Vol. III, p. 163, Vol. III, p. 172, Vol. III, p. 202, Vol. III, p. 240

second bimedial (second-bimedial)

Language: Euclidean Definition | Translation: second bimedial

Book X: If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straight line.

Occurrences: Vol. III, p. 7, Vol. III, p. 85, Vol. III, p. 87, Vol. III, p. 97, Vol. III, p. 125, Vol. III, p. 257

second binomial (second-binomial)

Language: Euclidean Definition | Translation: second binomial

Book X: 2. but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a second binomial;

Occurrences: Vol. III, p. 6, Vol. III, p. 101, Vol. III, p. 122, Vol. III, p. 136, Vol. III, p. 137, Vol. III, p. 155

sector of a circle (sector-of-circle)

Language: Euclidean Definition | Translation: sector of circle

Book III: 10. A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.

Occurrences: Vol. II, p. 2, Vol. II, p. 5

segment of a circle (segment-of-circle)

Language: Euclidean Definition | Translation: segment of circle

Book III: 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle.

Occurrences: Vol. II, p. 1, Vol. II, p. 4, Vol. II, p. 5, Vol. II, p. 189

semicircle

Language: Euclidean Definition | Translation: semicircle

Book I: 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

Occurrences: Vol. I, p. 154

Separando / Dividendo (διαίρεσις λόγου / διελόντι) (separando)

Language: Euclidean Proportion Operation (Book V, Def. 15) | Translation: Separation of a ratio (antecedent − consequent to consequent)

Book V, Def. 15: If A : B = C : D, then separando (A − B) : B = (C − D) : D (proved in V.17).

Occurrences: Vol. II, p. 405, Vol. II, p. 421, Vol. II, p. 115, Vol. II, p. 135, Vol. II, p. 166, Vol. II, p. 168, Vol. II, p. 169, Vol. II, p. 170, Vol. II, p. 174, Vol. II, p. 175, Vol. II, p. 184, Vol. II, p. 436, Vol. III, p. 38, Vol. III, p. 43, Vol. III, p. 46, Vol. III, p. 244, Vol. III, p. 467

ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα (side-of-a-medial-minus-a-medial-area)

Language: Euclidean Definition | Translation: side of a medial minus a medial area

Book X: If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

Occurrences: Vol. III, p. 7, Vol. III, p. 165, Vol. III, p. 166, Vol. III, p. 167

ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα (side-of-a-medial-minus-a-rational-area)

Language: Euclidean Definition | Translation: side of a medial minus a rational area

Book X: If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational: and let it be called that which produces with a rational area a medial whole.

Occurrences: Vol. III, p. 7, Vol. III, p. 164

ῥητὸν καὶ μέσον δυναμένη (side-of-a-rational-plus-a-medial-area)

Language: Euclidean Definition | Translation: side of a rational plus a medial area

Book X: If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole straight line is irrational; and let it be called the side of a rational plus a medial area.

Occurrences: Vol. III, p. 7, Vol. III, p. 88, Vol. III, p. 89

πλευρά (side-of-number)

Language: Euclidean Definition | Translation: side of number

Book VII: 16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

Occurrences: Vol. II, p. 79, Vol. II, p. 189, Vol. II, p. 278, Vol. II, p. 288

ἡ δύο μέσα δυναμένη (side-of-the-sum-of-two-medial-areas)

Language: Euclidean Definition | Translation: side of the sum of two medial areas

Book X: If two straight lines incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole straight line is irrational; and let it be called the side of the sum of two medial areas.

Occurrences: Vol. III, p. 7, Vol. III, p. 89, Vol. III, p. 90

Similar cones and cylinders (similar-cones-and-cylinders)

Language: Euclidean Definition | Translation: similar cones and cylinders

Book XI: 24. Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.

Occurrences: Vol. III, p. 262

Similar plane and solid numbers (similar-plane-and-solid-numbers)

Language: Euclidean Definition | Translation: similar plane and solid numbers

Book VII: 21. Similar plane and solid numbers are those which have their sides proportional.

Occurrences: Vol. II, p. 278

Similar rectilineal figures (similar-rectilineal-figures)

Language: Euclidean Definition | Translation: similar rectilineal figures

Book VI: 1. Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.

Occurrences: Vol. II, p. 188

Similar segments of circles (similar-segments-of-circles)

Language: Euclidean Definition | Translation: similar segments of circles

Book III: 11. Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.

Occurrences: Vol. II, p. 2

Similar solid figures (similar-solid-figures)

Language: Euclidean Definition | Translation: similar solid figures

Book XI: 9. Similar solid figures are those contained by similar planes equal in multitude.

Occurrences: Vol. III, p. 261

similarly inclined (similarly-inclined-planes)

Language: Euclidean Definition | Translation: similarly inclined planes

Book XI: 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations are equal to one another.

Occurrences: Vol. III, p. 260

sixth apotome (sixth-apotome)

Language: Euclidean Definition | Translation: sixth apotome

Book X: 6. and, if neither, a sixth.

Occurrences: Vol. III, p. 114, Vol. III, p. 177, Vol. III, p. 190, Vol. III, p. 240

sixth binomial (sixth-binomial)

Language: Euclidean Definition | Translation: sixth binomial

Book X: 6. and if neither, a sixth binomial.

Occurrences: Vol. III, p. 102, Vol. III, p. 145, Vol. III, p. 157

solid

Language: Euclidean Definition | Translation: solid

Book XI: 1. A solid is that which has length, breadth, and depth.

Occurrences: Vol. III, p. 260

solid angle (solid-angle)

Language: Euclidean Definition | Translation: solid angle

Book XI: 11. A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.

Occurrences: Vol. III, p. 261

solid (solid-number)

Language: Euclidean Definition | Translation: solid number

Book VII: 17. And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.

Occurrences: Vol. II, p. 278, Vol. II, p. 290

sphere

Language: Euclidean Definition | Translation: sphere

Book XI: 14. When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.

Occurrences: Vol. III, p. 261

square

Language: Euclidean Definition | Translation: square

Book I: 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Occurrences: Vol. I, p. 154

square number (square-number)

Language: Euclidean Definition | Translation: square number

Book VII: 18. A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

Occurrences: Vol. II, p. 278

stand upon (stand-upon-circumference)

Language: Euclidean Definition | Translation: stand upon circumference

Book III: 9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.

Occurrences: Vol. II, p. 1

στιγμή (stigme)

Language: Greek | Translation: stigme

Heath Greek Index: στιγμή — point 156

Occurrences: Vol. I, p. 156, Vol. I, p. 429

στοιχεῖα (stoicheion)

Language: Greek | Translation: stoicheion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 32, Vol. I, p. 114, Vol. I, p. 116, Vol. I, p. 117, Vol. I, p. 50

Straight line (εὐθεῖα γραμμή) (straight-line)

Language: Euclidean Definition (Book I, Def. 4) | Translation: Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται

Book I, Def. 4: ‘A straight line is a line which lies evenly with the points on itself.’

Occurrences: Vol. I, p. 166, Vol. I, p. 172, Vol. I, p. 176, Vol. I, p. 181, Vol. I, p. 183, Vol. I, p. 187, Vol. I, p. 190, Vol. I, p. 153, Vol. I, p. 242, Vol. I, p. 248, Vol. I, p. 259, Vol. I, p. 282, Vol. I, p. 370, Vol. I, p. 427, Vol. I, p. 20, Vol. I, p. 60, Vol. III, p. 11, Vol. III, p. 263, Vol. III, p. 269, Vol. III, p. 270, Vol. III, p. 271

Surface (ἐπιφάνεια) (surface)

Language: Euclidean Definition (Book I, Def. 5) | Translation: Ἐπιφάνεια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχει

Book I, Def. 5: ‘A surface is that which has length and breadth only.’ Book I, Def. 6: ‘The extremities of a surface are lines.’

Occurrences: Vol. I, p. 169, Vol. I, p. 176, Vol. I, p. 153, Vol. I, p. 329, Vol. I, p. 431, Vol. I, p. 62, Vol. III, p. 260, Vol. III, p. 263

Symperasma (συμπέρασμα) (symperasma)

Language: Greek (Formal Division of a Proposition) | Translation: Conclusion

The sixth formal division of a Euclidean proposition, reverting to the general or particular statement and closing with Q.E.D. (ὅπερ ἔδει δεῖξαι) or Q.E.F. (ὅπερ ἔδει ποιῆσαι).

Occurrences: Vol. I, p. 423, Vol. I, p. 129, Vol. I, p. 149, Vol. I, p. 58

σύνθεσις (synthesis)

Language: Greek | Translation: synthesis

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. II, p. 135, Vol. II, p. 168

touch a circle (tangent-to-circle)

Language: Euclidean Definition | Translation: tangent to circle

Book III: 2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Occurrences: Vol. II, p. 1

τετραγωνισμός (tetragonismos)

Language: Greek | Translation: tetragonismos

Heath Greek Index: τετραγωνισμός — squaring, definitions of 149–50, 410

Occurrences: Vol. I, p. 410, Vol. I, p. 430, Vol. I, p. 150

τετράγωνον (tetragonon)

Language: Greek | Translation: tetragonon

Heath Greek Index: τετράγωνον — square: sometimes (but not in Euclid) any four-angled figure 188

Occurrences: Vol. I, p. 187, Vol. I, p. 188, Vol. I, p. 189, Vol. I, p. 350

τετράπλευρα (tetrapleuron)

Language: Greek | Translation: tetrapleuron

Heath Greek Index: τετράπλευρον — quadrilateral 187

Occurrences: Vol. I, p. 187, Vol. I, p. 188

θεωρήματα (theorema)

Language: Greek | Translation: theorema

Heath Greek Index: θεώρημα — theorem q.v.: θεώρημα τῆς νύμφης, “Theorem of the Bride,” = Eucl. I. 47, 417

Occurrences: Vol. I, p. 427, Vol. I, p. 412

θεώρημα τῆς νύμφης (theorema-tes-nymphes)

Language: Greek | Translation: theorema tes nymphes

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 417

third apotome (third-apotome)

Language: Euclidean Definition | Translation: third apotome

Book X: 3. But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.

Occurrences: Vol. III, p. 109, Vol. III, p. 177, Vol. III, p. 183, Vol. III, p. 221, Vol. III, p. 240

third binomial (third-binomial)

Language: Euclidean Definition | Translation: third binomial

Book X: 3. and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a third binomial.

Occurrences: Vol. III, p. 102, Vol. III, p. 108, Vol. III, p. 125, Vol. III, p. 139, Vol. III, p. 157

τομή (tome)

Language: Greek | Translation: tome

Heath Greek Index: τομή — section, = point of section 170, 171, 278

Occurrences: Vol. I, p. 170, Vol. I, p. 171, Vol. I, p. 383, Vol. I, p. 430

τοπικὸν θεώρημα (topikon-theorema)

Language: Greek | Translation: topikon theorema

Heath Greek Index: τοπικὸν θεώρημα — locus-theorem 329

Occurrences: Vol. I, p. 329

τόπος (topos)

Language: Greek | Translation: topos

Heath Greek Index: τόπος — locus 329–31: room or space 23 n.: place (where things may be found), thus τόπος ἀναλυόμενος 8, 10: παράδοξος τόπος 329

Occurrences: Vol. I, p. 156, Vol. I, p. 223, Vol. I, p. 329, Vol. I, p. 330, Vol. I, p. 331, Vol. I, p. 427, Vol. I, p. 432, Vol. I, p. 15, Vol. I, p. 23

τόπος ἀναλυόμενος (topos-analyomenos)

Language: Greek | Translation: topos analyomenos

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 329, Vol. I, p. 421, Vol. I, p. 8, Vol. I, p. 10, Vol. I, p. 133

τραπέζια (trapezion)

Language: Greek | Translation: trapezion

Technical geometric or mathematical term in Heath’s edition of Euclid’s Elements (Greek).

Occurrences: Vol. I, p. 188

trapezia (trapezium)

Language: Euclidean Definition | Translation: trapezium

Book I: 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Occurrences: Vol. I, p. 154

trilateral figures (trilateral)

Language: Euclidean Definition | Translation: trilateral

Book I: 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Occurrences: Vol. I, p. 154

τρίπλευρα (tripleuron)

Language: Greek | Translation: tripleuron

Heath Greek Index: τρίπλευρον — three-sided figure 187

Occurrences: Vol. I, p. 187

triplicate ratio (triplicate-ratio)

Language: Euclidean Definition | Translation: triplicate ratio

Book V: 10. When four magnitudes are < continuously > proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

Occurrences: Vol. II, p. 114

Unit (μονάς) (unit)

Language: Euclidean Definition (Book VII, Def. 1) | Translation: Μονάς ἐστιν, καθ’ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται

Book VII, Def. 1: ‘A unit is that by virtue of which each of the things that exist is called one.’

Occurrences: Vol. II, p. 135, Vol. II, p. 277, Vol. II, p. 279, Vol. II, p. 280, Vol. II, p. 285, Vol. II, p. 287, Vol. II, p. 436