Subject Index

acute angle. See angle, acute.

algorithm, Euclidean. See Euclidean algorithm.

alternate angles I.27

alternate proportions and ratios definition V.Def.12 for magnitudes V.16 for numbers VII.13

amicable numbers VII.Def.22

angle (plane). See also solid angle. acute angle I.Def.12 alternate angles I.27 bisection I.9 construction I.23 definition I.Def.8, I.Def.9 exterior angle I.16, I.32 horn angle I.Def.8, III.16, V.Def.4 angles as magnitudes I.Def.9 proportional to arc VI.33 in a segment III.Def.8 obtuse angle I.Def.11 of a segment III.Def.7 on a circumference III.Def.9, III.26, III.27 rectilinear angle I.Def.9 right angle I.Def.10 right angles are equal Post.4 angles about a transversal I.27, I.28, I.29 trisection Post.2 two right angles are straight I.13, I.14 vertical angles I.15

antecedents in proportions V.Def.11

antenaresis. See Euclidean algorithm.

application of areas in an angle I.42, I.44, I.45 exceeding by a parallelogram VI.29 exceeding by a square II.6 falling short by a parallelogram VI.27, VI.28 falling short by a square II.5

approximation of circles by polygons XII.2

Apollonius of Perge (ca. 250–175 B.C.E.) terms for conic sections XI.Def.18

arc proportional to angle VI.33

Archimedes of Syracuse (ca. 287–212 B.C.E.) angle trisection Post.2 neusis Post.2 property of magnitudes X.1

area Heron’s formula for a triangle IV.4 medial X.21

arithmetic, fundamental theorem of VII.31

arithmetic mean or average V.25

associativity of addition for magnitudes C.N.

associativity of multiplication for magnitudes V.3

average, arithmetic and geometric V.25

authenticity of the Elements I.Def.1 of I.40 of V.19 of X.10

axiom axiom of comparability V.Def.4 for magnitudes C.N.

axis of a cone XI.Def.19 of a cylinder XI.Def.22 of a sphere XI.Def.15

base of a cone XI.Def.19 of a cylinder XI.Def.23 of a triangle I.4

bisect an angle I.9 a circumference (arc) III.30 a line I.10

boundary I.Def.13

Brouwer, L.E.J. (1881–1966) nonconstructive fixed point theorem I.6

Byrne, Oliver (1810–1890) edition of the Elements References on the web

cancellation for addition C.N. in proportions V.9

carpenter’s square II.Def.2

center of a circle characterization III.9 construction III.1 definition I.Def.16 intersecting circles have distinct centers III.5 tangent circles have distinct centers III.6

Chrysippus (280 207) 1 as a number VII.Def.1-2

circumference a circumference (arc) III.30

circle area of XII.2 central angle double angle at circumference III.20 chord inside circle III.2 center of. See center, of a circle. construct circle from segment III.25 construction Post.3 definition I.Def.15 diameter of. See diameter. equal angles in segments III.21 equal chords at equal distances III.14 equal circles III.Def.1 intersection of circles III.10 product of secants III.37 product of secants equals tangent² III.36 products of chord sections III.35 proportional to diameter² XII.2 radius of. See radius, of a circle. right angle in semicircle III.31 sector of. See sector, of a circle. segment of. See segment, of a circle. tangent to. See tangent.

circumcenter of a triangle IV.5

circumcircle of a triangle IV.5

circumference proportional to angle VI.33

circumscribed figures circle circumscribed about a pentagon IV.14 circle circumscribed about a rectilinear figure IV.Def.6 circle circumscribed about a square IV.9 circle circumscribed about a triangle IV.5 pentagon circumscribed about a circle IV.12 rectilinear figure circumscribed about a circle IV.Def.4 rectilinear figure circumscribed about a rectilinear figure IV.Def.2 square circumscribed about a circle IV.7 triangle circumscribed about a circle IV.3

commensurable definition X.Def.I.1 and numerical ratios V.Def.5 in square X.Def.I.2 magnitudes and numerical ratios X.5, X.6, X.7, X.8

common notions C.N.

commutativity for addition of magnitudes C.N. of multiplication VII.15, VII.18

compass construction Post.3

componendo V.Def.14

composite numbers definition VII.Def.13 divisible by a prime VII.31

cone axis XI.Def.18 base XI.Def.19 cone one third of cylinder XII.10 definition XI.Def.20 proportional to base XII.11 proportional to height XII.14 reciprocally proportional XII.15 right-angled, acute-angled, obtuse angled XI.Def.18 similar cones XI.Def.24, XII.12

congruent figures I.4 solids XI.Def.10

congruence propositions for triangles. See triangle.

connected figure I.Def.14

consequents in proportions V.Def.11

constructions, 2- and 3-dimensional XI.20

continued proportion V.Def.8, VIII.1 in lowest terms VIII.1, VIII.2, VIII.3, VIII.4 sum of a IX.35

contradiction, proof by I.5

contrapositive proposition I.27

converse of a proposition I.5, I.27

conversion of a proportion or ratio definition V.Def.16 proposition for magnitudes V.19

convertendo V.Def.16

convex figure I.Def.14

cosines, law of II.12, II.13

cross multiplication of proportions for lines VI.16 for numbers VII.19

cube construction XIII.15 definition XI.Def.25 relation to dodecahedron XIII.17 relation to tetrahedron XIII.15

cubic numbers VII.Def.19, IX.3, IX.4, IX.5, IX.6

cut into extreme and mean ratio. See extreme and mean ratio.

cylinder axis of XI.Def.22 bases of XI.Def.23 cone one third of cylinder XII.10 definition XI.Def.21 proportional to base XII.11 proportional to height XII.13, XII.14 reciprocally proportional XII.15 similar cylinders XI.Def.24, XII.12

decagon, regular (10-gon) side of hexagon to side of decagon XIII.9 sides of pentagon, hexagon, & decagon XIII.10

Descartes (1591–1661) geometric algebra VI.12

diameter of a circle bisecting chord III.3 definition I.Def.17 diameter is greatest chord III.15

distance, line to point III.Def.4

distributivity of division over addition VII.5 of division over subtraction VII.7 of multiplication over addition for lines II.1, II.2 for magnitudes V.1, V.2 for numbers VII.6, VII.8 of multiplication over subtraction for magnitudes V.5, V.6

divine proportion, Pacioli’s name for the extreme and mean ratio XIII.16

divisor of a number VII.Def.3

dodecahedron construction XIII.17 definition XI.Def.28 relation to cube XIII.17

dual of a polyhedron XIII.14

duplicate ratio V.Def.9

elegance in mathematics I.30

ellipse XI.Def.18

elliptic geometry I.16

equal circles III.Def.11 equal and similar solids XI.Def.10

equilateral triangle (60°-60°-60° triangle) construction I.1 definition I.Def.20 side of XIII.12

equivalence relation V.Def.3 equality as an equivalence relation C.N. proportion as an equivalence relation V.Def.5

Euclid (fl. ca. 300 B.C.E.).

Euclidean algorithm VII.2, VII.3, X.3 characterization of incommensurability of magnitudes X.2 test for relatively prime numbers VII.1

Eudoxus (ca. 408–355 B.C.E) definition or proportion V.Def.6 principle of exhaustion XII.2 property of magnitudes X.1

even even number VII.Def.6, IX.21, IX.24, IX.27, IX.28, IX.30 even-times even number VII.Def.8, IX.32, IX.34 even-times odd number VII.Def.9, IX.33, IX.34

ex aequali ratios and proportions definition V.Def.17 for magnitudes V.22 for numbers VII.14

excircle of a triangle IV.4

exhaustion, principle of XII.2

exterior angle greater than opposite interior angle of triangle I.16 sum of opposite interior angles of triangle I.32

extreme and mean ratio algebra on segments XIII.1, XIII.2, XIII.3, XIII.4, XIII.5 construction II.11, VI.30 divine proportion, Pacioli’s name for this ratio XIII.16 definition VI.Def.3 is irrational called apotome XIII.6 in a 36°-72°-72° triangle IV.10 in a pentagram IV.11, XIII.8 side of hexagon to side of decagon XIII.9

face of a solid XI.Def.2

figure I.Def.14 connected I.Def.14 convex I.Def.14 rectilinear I.Def.19 simply connected I.Def.14

fit a straight line into a circle, construction IV.1 into a circle, definition IV.Def.7 into a diagram Post.2

Fermat, Pierre de (1601–1665). Fermat primes IV.16 Mersenne primes and perfect numbers IX.36

fourth proportionals V.18

friendly numbers VII.Def.22

fundamental theorem of arithmetic VII.31

Gauss, Carl Friedrich (1777–1855). regular polygons IV.16

GCD. See greatest common divisor.

geometric mean or average V.25, VI.13

geometric progression or sequence. See continued proportion.

geometry elliptic I.16 hyperbolic I.29 nonEuclidean Post.5

gnomon II.Def.2

golden ratio. See extreme and mean ratio.

greatest common divisor Euclidean algorithm for VII.3, VII.2 for several numbers VII.4

greatest common measure of several commensurable magnitudes X.4 of two commensurable magnitudes X.3

group C.N.

Heath, Thomas Little (1861–1940) edition of the Elements About the Text, References on the web

height of a figure VI.Def.4

Heiberg, Johan Ludvig (1854–1928) edition of the Elements About the Text, References on the web, I.Def.1

Heron of Alexandria (ca. 1st century C.E.) definition of equal and similar solids XI.Def.10 Heron’s formula for area of a triangle IV.4 minimum distance problem I.20

hexagon, regular inscribed in a circle IV.15 side of hexagon to side of decagon XIII.9 sides of pentagon, hexagon, & decagon XIII.10

hexahedron, regular. See cube.

Hilbert, David (1862–1943) Foundations of Geometry I.4

Hippocrates of Chios (fl. ca. 430 B.C.E.). his Elements I.3 quadrature of lunes VI.31

horn angle. See angle, horn.

hyperbola XI.Def.18

hyperbolic geometry I.29

icosahedron construction XIII.16 definition XI.Def.27

incenter of a triangle IV.4

incircle of a triangle IV.4

inclination line to a line. See angle. line to a plane XI.Def.5 plane to a plane XI.Def.6 similar XI.Def.7

incommensurable. See commensurable.

infinitude of prime numbers IX.20

inscribed figures 15-gon inscribed in a circle IV.16 circle in a pentagon IV.13 circle in a rectilinear figure IV.Def.5 circle inscribed in a square IV.8 circle inscribed in a triangle IV.4 hexagon inscribed in a circle IV.15 pentagon inscribed in a circle IV.11 rectilinear figure in a circle IV.Def.3 rectilinear figure in a rectilinear figure IV.Def.1 square inscribed in a circle IV.6 triangle inscribed in a circle IV.2

inverse proportions and ratios definition V.Def.13 proposition V.7

inverse proposition I.27

irrational. See rational.

irrationality of surds VIII.8

isosceles triangle definition I.Def.20 has equal base angles I.5, I.5 larger vertex angle & larger base I.24, I.24

jointly ratios and magnitudes taken jointly V.Def.14, V.17, V.18

law of cosines II.12, II.13

law of sines I.19

law of trichotomy. See trichotomy.

LCM. See least common multiple.

least common multiple VII.33, VII.34, VII.35 of several numbers VII.36

Lindemann, Ferdinand (1852–1939) transcendence of π II.14

line. See also straight line. definition I.Def.2 ends of a line I.Def.3 medial X.21

lowest terms VII.20 are are relatively prime VII.21, VII.22, VIII.1 reduce to VII.33

lunes, quadrature VI.31

magnitude V.Def.1 commensurable. See commensurable. infinite and infinitesimal magnitudes V.Def.4 multiple of a magnitude V.Def.2 part of a magnitude V.Def.1 proportional magnitudes V.Def.5 ratio of magnitudes V.Def.3, V.Def.4 magnitudes in the same ratio V.Def.5

marginal references I.1

mean and extreme ratio. See extreme and mean ratio.

mean, arithmetic and geometric V.25

medial line X.21 rectangle X.21

Mersenne, Marin (1588–1648). Mersenne primes IX.36

monad, definition VII.Def.1

modern analysis, method of VI.1

multilateral figure I.Def.19. See polygon.

multiple a magnitude V.Def.2 of a number VII.Def.5

multiplication of numbers VII.Def.15

neusis Post.2

nonEuclidean geometry Post.5

number amicable numbers VII.Def.22 composite number VII.Def.13 cubic number VII.Def.19 definition VII.Def.2 divisible by a prime VII.32 divisor of a number VII.Def.3 even. See even, number. even-times even. See even, number. even-times odd. See even, number. friendly numbers VII.Def.22 multiple of a number VII.Def.5 odd. See odd, number. odd-times odd. See odd, number. part of a number VII.Def.3 parts of a number VII.Def.4 perfect number VII.Def.22 plane number VII.Def.16 prime number VII.Def.11 relatively composite numbers VII.Def.14 relatively prime numbers VII.Def.12 sides of a plane number VII.Def.16 sides of a solid number VII.Def.17 similar plane and solid numbers VII.Def.21 solid number VII.Def.17 square number VII.Def.18 triangular number VII.Def.16 1 as a number V.Def.5

number theory foundations of VII.1 Peano's axioms VII.Def.1

oblong I.Def.22

obtuse angle. See angle, obtuse.

octahedron, regular construction XIII.14 definition XI.Def.26

odd odd number VII.Def.7, IX.22, IX.23, IX.25, IX.26, IX.27, IX.29, IX.30, IX.31 odd-times odd number VII.Def.10

Luca Pacioli (1445–1517) construction of the icosahedron XIII.16

Pappus of Alexandria (fl. ca. 320 C.E.) proof of I.5

parabola XI.Def.18

parallel lines I.Def.23, I.31 planes XI.Def.8 postulate Post.5 transitivity of parallelism I.30, XI.9

parallelogram area of I.35, I.36 basic properties I.34 definition I.34 about the diameter I.43 equiangular parallelograms proportional to sides VI.23 proportional to base VI.1 reciprocally proportional parallelograms VI.14 similar parallelograms about the diameter VI.24, VI.26

parallelepiped (parallelepipedal solid) bisected by diagonal XI.28 construct similar one XI.27 definition XI.24 equal XI.29, XI.30, XI.31 proportional to base XI.25, XI.32 proportional to sides XI.33, XI.36, XI.37 reciprocally proportional parallelepipeds XI.34

part of a magnitude definition V.Def.1 problem of parts V.5

part of a number definition VII.Def.3

parts of a number definition VII.Def.4

Peano, Giuseppe (1858–1932). Peano's axioms for number theory VII.Def.1

pentagon, regular circumscribed about a circle IV.12 criterion of regularity XIII.7 diagonals cut in extreme and mean ratio XIII.8 inscribed in a circle IV.11 Richmond’s construction IV.11 sides of pentagon, hexagon, & decagon XIII.10 side of pentagon is irrational called minor XIII.11

perfect number definition VII.Def.22 construction IX.36

perpendicular, line to a line construction given a point I.11, I.12 definition I.Def.10

perpendicular, line to a plane definition XI.Def.3 propositions XI.4, XI.6, XI.8, XI.11, XI.12, XI.13

perturbed proportion definition V.Def.18 proposition V.22

plane definition I.Def.7 determined by intersecting lines XI.2 determined by triangle XI.2 inclination to a line XI.Def.5 inclination to a plane XI.Def.6 intersection of two planes XI.3 parallel planes XI.Def.8, XI.14, XI.15, XI.16, XI.17 perpendicular to a line XI.Def.3, XI.14 perpendicular to a plane XI.Def.4, XI.18, XI.19

plane angle. See angle.

plane number definition VII.Def.16 similar plane numbers VII.Def.21, VIII.26, IX.1, IX.2 proportional to sides VIII.5

Playfair axiom of parallels I.30

point definition I.Def.1

polygons approximating circles XII.2 areas of similar polygons VI.20, XII.1 constructible regular polygons IV.16

polyhedra, regular See tetrahedron, cube, octahedron, icosahedron, and dodecahedron. classification XIII.18 duals of XIII.14

Pons Asinorum I.5

postulates Post.1-5

powers of 2 IX.32

prime numbers definition VII.Def.11 dividing products VII.30 Fermat primes IV.16 infinitude of IX.20 Mersenne primes IX.36 powers of IX.13 products of IX.14 relatively prime VII.Def.12

principle of exhaustion XII.2

prism. See also parallelepiped. definition XI.Def.13 equal prisms XI.39 triangular prism partitioned into three equal pyramids XII.5

Proclus (410–485 C.E.) Commentary on Book I I.3

proof by contradiction I.5 nonconstructive I.5

progression, geometric. See continued proportion.

proportion alternate proportions V.Def.12, V.16, VII.13 antecedents in proportions V.Def.11 consequents in proportions V.Def.11 continued. See continued proportion. conversion of a proportion V.Def.16, VII.19 cross multiplication VII.19 definition V.Def.6 proportions as equivalence relations V.Def.5 proportions ex aequali V.Def.17, V.22, VII.14 inverse proportions V.Def.13, V.7 magnitudes V.Def.6 numbers VII.Def.20 proportions taken jointly V.Def.14, V.17, V.18 perturbed proportion V.Def.18, V.22 proportions taken separately V.Def.15, V.17, V.18 operations on proportions V.Def.3 proportion in three terms V.Def.8 reciprocal. See reciprocal, proportion. transitivity V.11

proportional construct third proportional VI.11 construct fourth proportional VI.12 construct mean proportional VI.13 fourth proportionals V.18 fourth proportional of numbers IX.19 magnitudes V.Def.6 mean proportionals between cubic numbers VIII.12 mean proportional between similar plane numbers VIII.18, VIII.20 mean proportionals between similar solid numbers VIII.19, VIII.21 mean proportional between square numbers VIII.11 numbers VII.Def.20 third proportional of numbers IX.18

proposition contrapositive I.27 converse of I.5 inverse of I.27

pyramid. See also tetrahedron, regular. definition XI.Def.12 pyramids proportional to their sides XII.8 pyramids proportional to their bases XII.5, XII.6 pyramid third of prism with same base XII.5 reciprocally proportional pyramids XII.9

Pythagorean theorem I.47 converse I.48 generalized to similar figures VI.31

Pythagorean triples X.29.Lemma1

Q.E.D. and Q.E.F. I.1

quadratic equation, solution by application of areas II.5, II.6, VI.28, VI.29

quadrilateral figure I.Def.19

quadrature of circles II.14, XII.2 of lunes VI.31 of rectilinear figures II.14

quadrilateral Varignon parallelogram of a XI.9

radius of a circle definition I.Def.15 perpendicular to tangent III.18, III.19

ratio alternate ratio V.Def.12, V.16, VII.13 compounded ratio V.Def.3, VIII.5 conversion of a ratio V.Def.16, VII.19 definition V.Def.3 duplicate ratio V.Def.9 extreme and mean. See extreme and mean ratio. ratios ex aequali V.Def.17, V.22, VII.14 greater ratio V.Def.7 inverse ratio V.Def.13 ratios taken jointly V.Def.14, V.17, V.18 in lowest terms VII.20 ratios of magnitudes V.Def.4 magnitudes in the same ratio V.Def.5 mixed ratio V.Def.3 nature of ratios V.Def.3 numerical ratio VII.Def.20, V.Def.5 operations on ratios V.Def.3 ratios taken separately V.Def.15, V.17, V.18 ratios of more than two terms V.Def.3 ratios of various kinds V.Def.3 triplicate ratio V.Def.9

rational line X.Def.I.3 number V.Def.3 numbers and commensurable magnitudes X.5, X.6, X.7, X.8 squares and areas X.Def.I.4

reciprocally proportional figures definition VI.Def.2 parallelograms VI.14 pyramids XII.9 triangles VI.15

rectangle (rectangular parallelogram) contained by sides II.Def.1 medial X.21

rectilinear figure definition I.Def.19

reflexive relation. See equivalence relation.

regular polygons, constructible IV.16

relation equivalence relation V.Def.3 reflexive relation V.Def.3 symmetric relation V.Def.3 transitive relation V.Def.3

relatively composite numbers VII.Def.14

relatively prime numbers definition VII.Def.12 are in lowest terms VII.21, VII.22 numbers dividing them are VII.23 primes are VII.29 products of VII.24, VII.25, VII.26, VII.27 sums of VII.28

revolution, solid of XI.Def.14

rhombus & rhomboid I.Def.22

right triangles. See triangles, right.

scalene triangle definition I.Def.20

section into extreme and mean ratio. See extreme and mean ratio.

sector of a circle definition III.Def.10

segment of a circle definition III.Def.6 angle in III.Def.8, III.31 angle of III.Def.7 construct circle from segment III.25 equal angles in segments III.21 equal segments III.24 similar segments III.Def.11

separately ratios taken separately V.Def.15, V.17, V.18, VII.11

separando V.Def.15

sequence, geometric. See continued proportion.

series (sum), geometric IX.35

sides of a plane number VII.Def.16 of a solid number VII.Def.17

semicircle definition I.Def.18

semigroup C.N.

similar areas of similar polygons   VI.20 figures on proportional lines   VI.22 equal and similar solids XI.Def.10 plane and solid numbers VII.Def.21 rectilinear figures construction VI.18 similar cylinders and cones XI.Def.24 definition VI.Def.1 construction of given area VI.25 segments of circles III.Def.11 solids XI.Def.9 transitivity of similarity   VI.21 triangles. See triangles, similar.

sines, law of I.19

simply connected figure I.Def.14

solid congruent solids XI.Def.10 definition XI.Def.1 equal and similar solids XI.Def.10 face of XI.Def.1 of revolution XI.Def.14 similar solids XI.Def.9

solid angle definition XI.Def.11 propositions XI.20, XI.21, XI.23, XI.26

solid number definition VII.Def.17 proposition IX.7 similar solid numbers VII.Def.21, VIII.27

sphere axis of XI.Def.15 center of XI.Def.16 definition XI.Def.14 diameter of XI.Def.17 proportional to diameter³ XII.18 volume XII.18

square construction I.46 definition I.Def.22 of the hypotenuse I.47

square number VII.Def.18

squaring (finding areas). See quadrature.

straight line bisection I.10 construct third proportional VI.11 construct fourth proportional VI.12 construct mean proportional VI.13 cut off line I.3 cut off a part VI.9 cut proportionally VI.10 definition I.Def.4 distance to a point III.Def.4 draw between two points Post.1 equidistant lines I.Def.23 extend a line Post.2 fit in a circle IV.Def.7, IV.1 inclination to a plane XI.Def.5 parallel lines I.Def.23, I.31 planarity of XI.1, XI.5 perpendicular lines. See perpendicular, line to a line. perpendicular to a plane. See perpendicular, line to a plane. place a line I.2 tangent. See tangent.

substitution of equals C.N.

superposition, method of I.4

surface. See also plane. definition I.Def.5 edges of a surface I.Def.6

surds, irrationality of VIII.8

symmetric relation. See equivalence relation.

tangent circles definition III.Def.3 have distinct centers III.6 meet at common diameter III.11, III.12 meet at one point III.13

tangent line to a circle definition III.Def.2 construction III.17 perpendicular to radius III.18, III.19

tetrahedron, regular called a pyramid XI.Def.25 construction XIII.13 relation to cube XIII.15

Thales of Miletus (ca. 624–547 B.C.E.) right angle in semicircle III.31

Theon of Alexandria (ca. 335–ca. 405) editor of the Elements I.Def.1

topology I.Def.13

touch. See tangent.

transitivity. See also equivalence relation. of equality of ratios V.11 of less than I.7 of parallel lines I.30, XI.9 of similarity   VI.21

transversal, angles about a I.27, I.28, I.29

trapezium I.Def.22

triangle 36°-72°-72° triangle IV.10 acute triangle I.Def.21 angle bisector cuts base proportionally VI.3 area of a triangle I.37, I.38 proportional to base VI.1 similar triangles VI.19 circumcenter of a triangle IV.5 circumcircle of a triangle IV.5 congruence proposition angle-angle-side I.26 angle-side-angle I.26 side-angle-side I.4 side-side-angle I.26 side-side-side I.8 construction given 3 sides I.22 equilateral I.Def.20. See equilateral triangle. excircle of a triangle IV.4 exterior angle sum of opposite interior angles I.32 greater side opposite greater angle I.18, I.19 Heron’s formula for area IV.4 incenter of a triangle IV.4 incircle of a triangle IV.4 inscribed in a circle IV.2 isosceles triangle I.Def.20 obtuse triangle I.Def.21 parallel cuts sides proportionally VI.2 reciprocally proportional triangles VI.15 right triangle I.Def.21 perpendicular creates similar right triangles   VI.8 scalene triangle I.Def.20 similar areas in duplicate ratio   VI.19 equiangular triangles are   VI.4 proportional triangles are   VI.5 side-angle-side proposition   VI.6 side-side-angle proposition   VI.7 triangle inequality I.20

triangular number VII.Def.16

trichotomy, law of for magnitudes C.N., V.Def.5 in practice I.5 for ratios V.Def.7

trilateral figure I.Def.19. See triangle.

triplicate ratio V.Def.9

trisection of an angle Post.2, I.9

unit, definition VII.Def.1

Varignon (1654–1722) Varignon parallelogram of a quadrilateral XI.9

vertical angles I.15

word order I.18

Zeno of Sidon (1st century B.C.E) criticism of proposition I.1

Zhou bi suan jing Pythagorean theorem I.47

1 as a number V.Def.5

36°-72°-72° triangle IV.10

>=< V.Def.5