- 01Purpose of the edition and modern geometry
- 02Assistance from Professor Townsend
- 03Questions for examination and exercises
- 04Revisions in the second edition
- 05Careful proofreading and intended use
Elements
Axiomatic reasoning and foundational geometry.
30Chapters
109Lessons
278Topics
Certificate of completion included
This course was produced by Cavua from a public-domain text.
Skills this course leaves you with.
- Apply Euclid's axioms and postulates to build geometric arguments
- Prove basic theorems about angles, triangles, and parallel lines
- Solve classical construction and theorem problems from Book I and Book II
- Use properties of rectangles and parallelograms in geometric proofs
- Analyze circle definitions and derive results from propositions on circles
Curriculum
The full table of contents.
Every chapter, every lesson, every topic — laid out before you enroll. Click any chapter to expand it.
- Chapters
- 30
- Lessons
- 109
- Topics
- 278
- 01Contents
- 01Geometry as figured space and plane geometry
- 02Point, line, plane, and plane figures
- 03Angles, concurrent lines, and triangles
- 01Polygon definitions and classifications
- 02Circle, radius, and diameter
- 03Euclid's postulates
- 04Basic axioms of equality
- 01Axioms and superposition
- 02Theorems, problems, and converse theorems
- 01Prop. I.—Problem
- 01Prop. II.—Problem
- 01Triangle congruence by superposition
- 02Isosceles triangle base angles
- 01Corollary: equilateral triangles are equiangular
- 02Axis of symmetry
- 03Exercises
- 01Prop. VI — Equal angles in a triangle imply equal opposite sides
- 02Prop. VII — Unequal conterminous sides on the same base
- 01Triangle congruence by side-side-base
- 02Bisecting a rectilineal angle
- 01Perpendicular from a point to a line
- 02Adjacent angles on a straight line
- 03Continuation of a straight line
- 04Opposite angles formed by intersecting lines
- 05Exterior angle and triangle angle inequalities
- 01Theorem: greater angle opposite greater side
- 02Triangle side-sum theorem
- 01Triangle inequality for two lines from a point
- 02Construction of a triangle from three given sides
- 01Triangle inequality and circle intersection conditions
- 02Constructing an angle equal to a given angle at a point on a line
- 03Greater side opposite greater angle in triangles
- 04Greater base implies greater contained angle
- 01Congruence of triangles with two equal angles and a side
- 01Definitions of parallel lines and related quadrilaterals
- 02Alternate, corresponding, and interior angles
- 03Criteria for parallel lines from angle equality
- 04Angle properties of parallel lines
- 05Exercises on parallel lines
- 06Transitive parallelism and drawing a parallel through a point
- 01Construction of a parallel line by equal angles
- 02External angle theorem and angle-sum corollaries
- 01Exercises on perpendiculars, angle bisectors, and triangle constructions
- 02Equal and parallel joining lines
- 03Parallelogram properties and converses
- 01Corollaries on parallelograms and lozenges
- 02Exercises on parallelograms and quadrilaterals
- 03Parallelograms on the same base and between the same parallels
- 01Equal parallelograms on equal bases
- 02Triangles on the same base between the same parallels
- 03Two triangles on equal bases between the same parallels
- 01Intercepts and midpoint triangle area results
- 02Equal triangles between the same parallels
- 03Altitude of a triangle
- 04Exercises on triangle and quadrilateral properties
- 05Parallelogram and triangle area relation
- 01Equal areas with equal altitudes
- 02Constructing a parallelogram equal to a triangle
- 03Parallelogram complements on a diagonal
- 01Collinearity of the middle points of three diagonals
- 02Apply a parallelogram equal to a given triangle
- 01PROP. XLV.—PROBLEM
- 01Constructing a square on a given line
- 02Pythagorean theorem for a right triangle
- 01Exercises on right-triangle and square relations
- 02Converse of the Pythagorean theorem
- 03Questions for Examination on Book I
- 01Book I exercises on triangles, parallels, and loci
- 01Preliminary explanations of square measure
- 01Rectangle area as product of sides
- 02Definitions of segments, rectangles, and gnomons
- 03Prop. I: rectangle of a whole line and parts
- 01Distributive law for rectangles
- 02Triangle area as half a rectangle
- 03Square on a divided line
- 01Square on a divided line
- 01Corollaries on squares and line segments
- 02Proposition V: rectangle of unequal parts and square on the middle segment
- 01PROP. VI.—THEOREM
- 01Proposition VII: sum of squares and rectangle
- 01Square on the sum and difference of two lines
- 02Prop. VIII: line divided into two parts
- 03Direct sequence and exercises
- 04Prop. IX: bisected line divided unequally
- 01External division of a bisected line
- 01Exercises on sums of squares and loci
- 02Division of a line in extreme and mean ratio
- 01Incommensurability of extreme and mean ratio segments
- 02Exercises on extreme and mean ratio
- 03Obtuse triangle theorem on the square of the opposite side
- 01Theorem on an acute triangle and the perpendicular intercept
- 01Constructing a square equal to a given rectilineal figure
- 01Definitions of circle elements and related figures
- 01Concentric circles, concyclic points, cyclic quadrilaterals, and directed angles
- 02Finding the centre of a circle
- 03A chord lies inside the circle and outside its extension
- 04A diameter perpendicular to a chord bisects it
- 05Two chords cannot bisect each other
- 01Chords that bisect each other and non-concentric circles
- 02Lines from an interior point to a circle
- 03Lines from an exterior point to a circle
- 01Equal lines from a point to a circle
- 02Centers of circles and contact theorems
- 03Equal chords in equal circles
- 01Greatest chord and chord lengths by distance from the centre
- 01Perpendicularity of the tangent to the radius
- 02Construction of a tangent from an external point
- 01Perpendicular from the centre to a tangent
- 02Converse theorem for tangents through the point of contact
- 01Angle at the centre and angle in the circumference
- 02Angles in the same segment
- 01Opposite angles of a cyclic quadrilateral
- 01Similar segments on the same chord
- 02Similar segments on equal chords
- 03Constructing a circle from an arc
- 04Memory scheme for Propositions XXVI–XXIX
- 01Equal angles and equal arcs in equal circles
- 01Equal arcs subtend equal chords
- 02Bisecting a given arc
- 03Angles in a semicircle and segments
- 01Tangent and chord angles in alternate segments
- 01Angle between a tangent and a chord
- 02Constructing a circular segment with a given angle
- 03Cutting off a segment from a given circle
- 01Intersecting chords and segment rectangles
- 02Concyclic extremities from equal segment products
- 03Equiangular triangles and noncorresponding sides
- 04Exercises on chord products and secants
- 01Secant-segment rectangle constant for a fixed point
- 02Converse tangent criterion
- 01Tangent from a point to a circle
- 02Exercises and examination questions on Book III
- 01Definitions of inscription, circumscription, and regular figures
- 02Proposition I: chord equal to a given line in a circle
- 03Proposition II: inscribing a triangle equiangular to a given triangle
- 01Constructing a triangle equiangular to a given triangle
- 02Inscribing a circle in a given triangle
- 01Circumcircle of a triangle
- 01Inscribing a square in a given circle
- 02Describing a square about a given circle
- 03Inscribing a circle in a given square
- 01Circle described about a given square
- 02Isosceles triangle with base angles double the vertical angle
- 03Inscribing a regular pentagon in a given circle
- 01Regular pentagon circumscribed about a circle
- 02Inscribing a circle in a regular pentagon
- 01Circumscribed circle about a regular pentagon
- 02Inscribing a regular hexagon in a given circle
- 01Inscribing a regular pentagon and equilateral triangle
- 02Regular polygon of fifteen sides
- 03Questions for Examination on Book IV
- 01Euclid’s definitions of ratio and proportion
- 02Continued proportion and compound ratio
- 01Compound, duplicate, and triplicate ratios
- 02Homologous terms and derived proportion propositions
- 01Invertendo: equal ratios and reciprocals
- 02Multiple and submultiple ratios
- 03Equal magnitudes and ratios to a third magnitude
- 04Comparing ratios of unequal magnitudes
- 05Equal ratios imply equal magnitudes
- 06Greater ratio implies greater magnitude
- 01Ratios equal to the same ratio
- 02Sum of equal ratios
- 03Comparison of equal ratios
- 04Antecedent and consequent comparison
- 01Equimultiples and proportional transformations
- 01Proposition XIX: remainder ratios
- 02Proposition XX: excess above the second
- 03Proposition XXI: comparison of two triples
- 01Proposition XXI: transverse order ratios and comparison of extremes
- 02Proposition XXII: ex aequali and continued proportion
- 03Proposition XXIII: ex aequo perturbato
- 01Ratios compounded of equal ratios
- 02Addition of magnitudes in proportion
- 03Sum of extremes and means in proportionals
- 04Questions and exercises on Book V
- 01Definitions of similar figures and proportion
- 02Triangles and parallelograms with the same altitude
- 01Proportional segments in a triangle
- 02Angle bisector theorem
- 03Exercises on proportional division and angle bisectors
- 01Harmonic division properties
- 02Proportional sides of equiangular triangles
- 03Parallel lines meeting at infinity
- 04Exercises
- 01Triangles with proportional sides are equiangular
- 02One equal angle and proportional sides about it
- 03One equal angle, proportional sides, and same-species remaining angles
- 01Right triangle altitude and similar triangles
- 02Cutting off submultiples and dividing lines similarly
- 03Third and fourth proportionals
- 04Mean proportional constructions and applications
- 05Reciprocal proportionality of parallelograms and triangles
- 06Rectangles and squares in proportion
- 07Exercises on chord and rectangle relations
- 01Construction of similar polygons
- 02Directly and inversely similar figures
- 03Corollaries on similar polygons
- 04Similar triangles and duplicate ratio of areas
- 01Similar polygons divided into similar triangles
- 01Homologous points in similar figures
- 01Centre of similitude and homothetic figures
- 01Similar rectilineal figures similar to the same figure
- 02Proportional figures on proportional lines
- 01PROP. XXIII.—T H E O R E M .
- 01Similar parallelograms with a common angle
- 02Maximum parallelogram in a triangle
- 03Inscribing a parallelogram equal to a given figure
- 01Construction of an escribed parallelogram equal to a given figure
- 01Division of a line in extreme and mean ratio
- 02Similar figures on the sides of a right triangle
- 03Parallel homologous sides in joined triangles
- 01Angles, arcs, and sectors in equal circles
- 02Questions for Examination on Book VI
- 03Mean position and least-squares properties
- 04Construction and locus problems
- 05Transversal and collinearity theorems
- 06Perpendicular bisectors and triangle rectangles
- 07Cyclic quadrilateral and tangent-ratio loci
- 08Pascal’s theorem
- 01Two sides and given area of a triangle
- 02Cyclic quadrilateral area and related triangle identities
- 03Parallel-line and circle locus problems
- 04Regular polygons, equilateral-triangle constructions, and triangle metric relations
- 01Variable circle and tangent intercepts
- 01Locus of a point from perpendicular feet on a polygon
- 01Definitions of coplanar lines, dihedral angles, and solid angles
- 02Propositions on planes and intersecting lines
- 03Perpendiculars to coplanar lines and the normal
- 04Three concurrent lines with a common perpendicular
- 01Normal lines to a plane
- 02Projection of a line on a plane
- 03Parallel lines and a transversal are coplanar
- 04A parallel line to a normal is also normal
- 01Parallel lines and angle theorems
- 02Normals to planes and uniqueness
- 03Parallel planes from a common normal
- 04Parallel planes cut by a third plane
- 05Proportional segments in parallel planes
- 01Plane perpendicular to a normal line
- 02Common section of two perpendicular planes
- 03Sum of two face angles of a trihedral angle
- 04Sum of the face angles of a solid angle
- 01Definitions of polyhedra and solids of revolution
- 02Volume theorems for right prisms and parallelopipeds
- 01Equal-volume parallelepipeds with aligned or offset edges
- 02Volume of a parallelopiped as base times altitude
- 03Diagonal plane dividing a parallelopiped into equal prisms
- 01Diagonal plane bisects a parallelopiped
- 02Volume of prisms from base and altitude
- 03Sections parallel to a pyramid's base are similar
- 04Equal-altitude pyramids with equal bases have equal volumes
- 01Equality of pyramid volumes
- 02Triangular pyramid as one third of a prism
- 03Every pyramid as one third of a prism
- 01Cylinder volume by base area and height
- 02Guldinus’s theorem for a revolving rectangle
- 03Cone as one-third of a cylinder
- 04Sphere as two-thirds of a circumscribed cylinder
- 01Surface of a sphere equals the circumscribed cylinder
- 01Modern theory of parallel lines
- 02Legendre’s proof of Euclid I. XXXII
- 03Hamilton’s quaternion proof of Euclid I. XXXII
- 04Inscribing a regular seventeen-sided polygon in a circle
- 01Finding two mean proportionals between two given lines
- 01Proof that AB, DE, FA, CD are in continued proportion
- 02Philo’s Line and its minimum property
- 01Trisection of an angle
- 01Quadrature of the circle and approximating π
- 01Critical notices of Casey's sequel
- 01Opinions of the Work and Preface
- 02Typographical Errors Corrected in Project Gutenberg Edition
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