# Subject index

#### 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)
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
Heron’s formula for a triangle   IV.4
medial   X.21
arithmetic, fundamental theorem of   VII.31
arithmetic mean or average   V.25
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

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

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

#### C

cancellation
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,
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.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
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
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

#### 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 subtraction   VII.7
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
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

#### G

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.

#### H

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
horn angle. See angle, horn.
hyperbola   XI.Def.18
hyperbolic geometry   I.29

#### I

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

#### JKL

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
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

#### 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
medial
line   X.21
rectangle   X.21
Mersenne, Marin (1588–1648).
Mersenne primes   IX.36
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
1 as a number   V.Def.5
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
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
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
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
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
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

Q.E.D. and Q.E.F.   I.1
quadratic equation, solution by application of areas   II.5,   II.6,   VI.28,   VI.29
of circles   II.14,   XII.2,
of lunes   VI.31
of rectilinear figures   II.14
Varignon parallelogram of a   XI.9

#### R

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
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.

#### 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
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
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
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
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
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

#### UVWXYZ

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

#### Numbers and Symbols

1 as a number   V.Def.5
36°-72°-72° triangle   IV.10
>=<   V.Def.5