Subject index for the Elements

Headings

A B C D E F G H I J K L M N O P Q R S T U V W Y Z Numbers & symbols

A

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.
    obtuse 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 of a triangle
    Heron's formula   IV.4
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
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

B

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
    nonconstructive fixed point theorem   I.5

C

cancellation
    for addition C.N.
    in proportions   V.9
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.
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.1
    and numerical ratios   V.Def.5
    in square   X.Def.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.
consequents in proportions   V.Def.11
constructions, 2- and 3-dimensional   XI.20
continued proportion   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
conversion of a proportion or ratio
    definition  V.Def.16
    proposition for magnitudes V.19
convertendo   V.Def.16
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

D

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
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

E

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
    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

F

face of a solid   XI.Def.2
figure   I.Def.14
    rectilinear   I.Def.19
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

G

Gauss, Carl Friedrich (1777-1855).
    regular polygons   IV.16
GCD. See greatest common divisor.
geometric mean or average   V.25
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 algorthim 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.

H

height of a figure VI.Def.4
Heron of Alexandria (ca. 1st century C.E.)
    defintion 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

I

icosahedron
    construction   XIII.16
    definition   XI.Def.27
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

J

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

K L

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 pi   II.14
line
    See also straight line.
    definition   I.Def.2
    ends of a line   I.Def.3
lowest terms   VII.20
    are are relatively prime   VII.21,   VII.22,   VIII.1
    reduce to   VII.33
lunes, quadrature   VI.31

M

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
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

N

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

O

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

P

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
    constructable 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.
    defintion   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
    defintion   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

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

R

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.3
    number   V.Def.3
    numbers and commensurable magnitudes   X.5,   X.6,   X.7,   X.8
    squares and areas   X.Def.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
rectilinear figure
    definition   I.Def.19
reflexive relation. See equivalence relation.
regular polygons, constructable   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.

S

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
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
    defintion   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.

T

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
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
    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
    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

U

unit, definition   VII.Def.1

V

Varignon (1654-1722)
    Varignon parallelogram of a quadrilateral   XI.9
vertical angles   I.15

W

word order   I.18

X Y Z

Zeno of Sidon (1st century B.C.E)
    criticism of proposition I.1
Zhou bi suan jing
    Pythagorean theorem   I.47

Symbols

36°-72°-72° triangle   IV.10
>=<   V.Def.5

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