Geometry · Wentworth · Plane and solid geometry

Wentworth's Plane and Solid Geometry

G. A. Wentworth's geometry for students about 13 to 16: definitions and axioms, then theorems and problems with full proofs, and hundreds of original exercises. 148 guides cover all nine Books, plane (I–V) and solid (VI–IX), each a 30 to 45 minute session, with the right page of the book beside it.

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How to use it

  • One session at a time. Each guide is about 30 to 45 minutes: a few definitions, two to four propositions, or a set of exercises. Three or four sessions a week covers plane geometry in a school year.
  • Draw every figure. Have a ruler, compasses and sharp pencil ready. The guides tell you exactly what to draw, and an accurate figure is half the proof.
  • Say the proof back. The author wanted students to prove propositions aloud, pointing to each line and angle. After you walk through a proof together, your student closes the book and gives it back.
  • Ask "Why?" Every step of a proof has a reason: an axiom, a definition, or an earlier section (§). You do not need to know the answer; ask the question and look it up together.
  • Choose among the exercises. The book has far more exercises than anyone works. The guides suggest a few each time; brief hints and answers are in each guide, but let your student try first.
  • Review each Book. At the end of a Book, go back over the propositions, drawing figures freehand and stating them aloud, as the author advises.

Book 1 Introduction

Solids, surfaces, lines and points; axioms, theorems and the general axioms (§§ 1–35). Lessons 1–2.

  1. 1Solids, surfaces, lines and pointsIntroduction, §§ 1–17✓
  2. 2Proofs, axioms, theorems and the general axiomsIntroduction, §§ 18–35✓

Book 2 Book I: Rectilinear Figures

Lines and angles, perpendiculars and parallels, triangles, loci, quadrilaterals, polygons and symmetry, with the methods of proof and the original exercises. Lessons 3–21.

  1. 3Lines, planes and the straight lineBook I, §§ 36–56✓
  2. 4Angles and how they are measuredBook I, §§ 57–80✓
  3. 5Straight angles, supplements and vertical anglesBook I, Props. I–IV✓
  4. 6Perpendiculars: equal obliques and the shortest lineBook I, Props. V–VII✓
  5. 7Lines drawn to a line: which is longer?Book I, Props. VIII–IX✓
  6. 8Parallel lines and the transversalBook I, Props. X–XIII✓
  7. 9More angle pairs of parallelsBook I, Props. XIV–XVII✓
  8. 10Triangles and the sum of their anglesBook I, §§ 117–137 (Prop. XVIII)✓
  9. 11Sides of a triangle, and two ways triangles are equalBook I, Props. XIX–XXII✓
  10. 12Isosceles triangles, three sides, and right trianglesBook I, Props. XXIII–XXV✓
  11. 13Unequal sides and unequal anglesBook I, Props. XXVI–XXIX✓
  12. 14Loci: the perpendicular bisector and the angle bisectorBook I, §§ 156–162 (Props. XXX–XXXI)✓
  13. 15Quadrilaterals and parallelogramsBook I, §§ 163–182 (Props. XXXII–XXXIV)✓
  14. 16More on parallelograms, and parallels cutting equal partsBook I, Props. XXXV–XXXVIII✓
  15. 17Polygons and the sum of their anglesBook I, §§ 191–207 (Props. XXXIX–XL)✓
  16. 18Symmetry, and a review of Book IBook I, §§ 208–213 (Props. XLI–XLII)✓
  17. 19Methods of proving theoremsBook I, §§ 214–215, Exs. 19–27✓
  18. 20Original exercises on angles and trianglesBook I, Exs. 28–58✓
  19. 21Original exercises on quadrilateralsBook I, Exs. 59–90✓

Book 3 Book II: The Circle

Arcs, chords and tangents, the theory of limits, the measure of angles, and the problems of construction. Lessons 22–39.

  1. 22The circle: definitionsBook II, §§ 216–234✓
  2. 23Central angles, arcs and chordsBook II, Props. I–IV✓
  3. 24Chords and their distance from the centreBook II, Props. V–VIII✓
  4. 25Tangents, parallels, and the circle through three pointsBook II, Props. IX–XI✓
  5. 26Tangents from a point, and two circlesBook II, Props. XII–XIV✓
  6. 27Measurement, ratio and the theory of limitsBook II, §§ 266–285✓
  7. 28Finding a ratio, and angles measured by arcsBook II, Props. XV–XVI✓
  8. 29Inscribed angles and angles made by chords, tangents and secantsBook II, Props. XVII–XX✓
  9. 30Review of Book II and theorems on the circleBook II, Exs. 101–117✓
  10. 31More theorems on circlesBook II, Exs. 118–133✓
  11. 32First constructions: perpendiculars and bisectingBook II, Props. XXI–XXV✓
  12. 33Copying angles, parallels and dividing a lineBook II, Props. XXVI–XXX✓
  13. 34Constructing triangles and parallelogramsBook II, Props. XXXI–XXXIV✓
  14. 35Circles about and in triangles; tangentsBook II, Props. XXXV–XXXVIII✓
  15. 36How to solve a construction problemBook II, §§ 319–322, Exs. 147–148✓
  16. 37Loci and locating pointsBook II, Exs. 149–177✓
  17. 38Constructing triangles and squares from given partsBook II, Exs. 178–207✓
  18. 39Constructions with quadrilaterals and tangent circlesBook II, Exs. 208–245✓

Book 4 Book III: Proportion. Similar Polygons

The theory of proportion, similar triangles and polygons, the Pythagorean theorem and numerical properties of figures, with constructions and computations. Lessons 40–56.

  1. 40Proportion: first theoremsBook III, §§ 323–331 (Props. I–V)✓
  2. 41Proportion: composition, division and moreBook III, Props. VI–XII✓
  3. 42A line parallel to a side of a triangleBook III, Props. XIII–XIV✓
  4. 43The bisector of an angle of a triangleBook III, Props. XV–XVI✓
  5. 44Similar polygons and similar trianglesBook III, §§ 351–358 (Props. XVII–XIX)✓
  6. 45More similar triangles: parallel sides, altitudes, transversalsBook III, Props. XX–XXIII✓
  7. 46Perimeters, and polygons divided into trianglesBook III, Props. XXIV–XXVI✓
  8. 47Original theorems on similar figuresBook III, Exs. 256–265✓
  9. 48The right triangle and the Pythagorean theoremBook III, Props. XXVII–XXVIII✓
  10. 49Squares of sides in any triangle; mediansBook III, Props. XXIX–XXXI✓
  11. 50Chords, secants, tangents and bisectorsBook III, Props. XXXII–XXXV✓
  12. 51Original theorems on lines in circles and trianglesBook III, Exs. 267–285✓
  13. 52Constructing proportional linesBook III, Props. XXXVI–XXXVIII✓
  14. 53Mean proportional, extreme and mean ratio, similar polygonsBook III, Props. XXXIX–XLI✓
  15. 54Construction problems with proportionsBook III, Exs. 290–311✓
  16. 55Problems of computation (1)Book III, Exs. 312–330✓
  17. 56Problems of computation (2)Book III, Exs. 331–348✓

Book 5 Book IV: Areas of Polygons

Areas of rectangles, parallelograms, triangles and trapezoids, comparing areas, and constructing figures of given area. Lessons 57–66.

  1. 57Areas of rectanglesBook IV, §§ 392–399 (Props. I–III)✓
  2. 58Parallelograms, triangles and trapezoidsBook IV, Props. IV–VII✓
  3. 59Similar figures and the square on the hypotenuseBook IV, Props. VIII–X✓
  4. 60Original theorems on areasBook IV, Exs. 356–372✓
  5. 61Constructing squares and triangles of given areaBook IV, Props. XI–XIV✓
  6. 62Squares equal to parallelograms; two lines from sum or difference and productBook IV, Props. XV–XVII✓
  7. 63Similar polygons of given sizeBook IV, Props. XVIII–XX✓
  8. 64Transforming figuresBook IV, Exs. 387–403✓
  9. 65Problems of computation: areas (1)Book IV, Exs. 404–423✓
  10. 66Problems of computation: areas (2)Book IV, Exs. 424–437✓

Book 6 Book V: Regular Polygons and Circles

Regular polygons, the measurement of the circle and π, inscribed polygons, maxima and minima, and the miscellaneous exercises. Lessons 67–79.

  1. 67Regular polygons and their circlesBook V, §§ 429–444 (Props. I–III)✓
  2. 68Similar regular polygons, and the apothem as a limitBook V, Props. IV–VI✓
  3. 69The circumference as a limit, and πBook V, Props. VII–IX✓
  4. 70Area of a regular polygon and of a circleBook V, Props. X–XII✓
  5. 71Inscribing the square, hexagon, decagon and pentedecagonBook V, Props. XIII–XVI✓
  6. 72Similar inscribed polygons, doubling the sides, and computing πBook V, Props. XVII–XIX✓
  7. 73Maxima and minima: triangles and polygonsBook V, §§ 482–486 (Props. XX–XXII)✓
  8. 74The regular polygon is the greatestBook V, Props. XXIII–XXVI✓
  9. 75Theorems on regular polygons and circlesBook V, Exs. 452–477✓
  10. 76Constructions with circles and regular polygonsBook V, Exs. 478–502✓
  11. 77Computing with circles and regular polygonsBook V, Exs. 503–530✓
  12. 78Miscellaneous theoremsBook V, Exs. 531–556✓
  13. 79Miscellaneous loci and problemsBook V, Exs. 557–603✓

Book 7 Book VI: Lines and Planes in Space

Planes and their determination, perpendicular and parallel lines and planes, dihedral angles, projections and polyhedral angles, with the exercises (§§ 492–583). Lessons 80–92.

  1. 80Planes, and what determines a planeBook VI, §§ 492–506 (Prop. I)✓
  2. 81A line perpendicular to a planeBook VI, Props. II–IV✓
  3. 82Oblique lines, loci in space, and parallel perpendicularsBook VI, Props. V–VII✓
  4. 83Lines parallel to planes; planes perpendicular to one lineBook VI, Props. VIII–X✓
  5. 84Parallel planesBook VI, Props. XI–XIII✓
  6. 85Angles with parallel sides; lines cut by parallel planesBook VI, Props. XIV–XV✓
  7. 86Dihedral anglesBook VI, §§ 536–548 (Prop. XVI)✓
  8. 87Measuring dihedral angles; perpendicular planesBook VI, Props. XVII–XIX✓
  9. 88Planes perpendicular to a third plane; bisecting planesBook VI, Props. XX–XXII✓
  10. 89Projections, the angle of a line with a plane, and the common perpendicularBook VI, Props. XXIII–XXV✓
  11. 90Polyhedral anglesBook VI, §§ 569–580 (Prop. XXVI)✓
  12. 91Face angles of a polyhedral angle; equal trihedral anglesBook VI, Props. XXVII–XXVIII✓
  13. 92Exercises on lines and planesBook VI, Exs. 616–629✓

Book 8 Book VII: Polyhedrons, Cylinders, and Cones

Prisms and parallelopipeds, pyramids and frustums, Euler's theorem, similar and regular polyhedrons, cylinders, cones and the prismatoid formula, with the problems of computation. Lessons 93–117.

  1. 93Polyhedrons, prisms and parallelopipedsBook VII, §§ 584–604✓
  2. 94Sections of a prism, lateral area, and equal prismsBook VII, Props. I–III✓
  3. 95Oblique prisms and parallelopipedsBook VII, Props. IV–VI✓
  4. 96Comparing rectangular parallelopipedsBook VII, Props. VII–IX✓
  5. 97Volumes of parallelopipeds and prismsBook VII, Props. X–XIII✓
  6. 98Problems of computation: prismsBook VII, Exs. 643–656✓
  7. 99Pyramids and their lateral areaBook VII, §§ 630–644 (Prop. XIV)✓
  8. 100Parallel sections of a pyramid; the pyramid as a limitBook VII, Props. XV–XVI✓
  9. 101The volume of a pyramidBook VII, Props. XVII–XIX✓
  10. 102Problems of computation: pyramidsBook VII, Exs. 662–675✓
  11. 103The frustum of a pyramidBook VII, Props. XX–XXI✓
  12. 104Pyramids with a common trihedral angle; the truncated prismBook VII, Props. XXII–XXIII✓
  13. 105Euler's theorem; the sum of the face anglesBook VII, Props. XXIV–XXV✓
  14. 106Similar polyhedronsBook VII, §§ 664–672 (Props. XXVI–XXVIII)✓
  15. 107The five regular polyhedronsBook VII, §§ 673–675 (Prop. XXIX)✓
  16. 108CylindersBook VII, §§ 676–694 (Props. XXX–XXXI)✓
  17. 109Area and volume of a cylinderBook VII, Props. XXXII–XXXV✓
  18. 110Problems of computation: cylindersBook VII, Exs. 678–691✓
  19. 111ConesBook VII, §§ 702–721 (Props. XXXVI–XXXVIII)✓
  20. 112Area and volume of a coneBook VII, Props. XXXIX–XLI✓
  21. 113The frustum of a coneBook VII, Props. XLII–XLIII✓
  22. 114The prismatoid formulaBook VII, §§ 731–736 (Prop. XLIV)✓
  23. 115Frustums of pyramids and of conesBook VII, Exs. 702–711✓
  24. 116Equivalent solidsBook VII, Exs. 712–722✓
  25. 117Similar solidsBook VII, Exs. 723–735✓

Book 9 Book VIII: The Sphere

Plane sections and tangent planes, spherical angles, polygons and polar triangles, the measurement of spherical surfaces and volumes, and the numerical and miscellaneous exercises. Lessons 118–132.

  1. 118The sphere and its circlesBook VIII, §§ 737–757 (Prop. I)✓
  2. 119Poles and polar distance; measuring a material sphereBook VIII, Props. II–IV✓
  3. 120Tangent planes; inscribed and circumscribed spheresBook VIII, Props. V–VIII✓
  4. 121Spherical angles and spherical polygonsBook VIII, §§ 778–790 (Props. IX–XII)✓
  5. 122Polar triangles and the angle sum of a spherical triangleBook VIII, §§ 791–798 (Props. XIII–XV)✓
  6. 123Bi-rectangular and symmetrical spherical trianglesBook VIII, Props. XVI–XVII✓
  7. 124Equal and symmetrical spherical trianglesBook VIII, Props. XVIII–XXII✓
  8. 125Unequal sides and angles; the shortest path on a sphereBook VIII, Props. XXIII–XXV✓
  9. 126Zones and the surface of a sphereBook VIII, §§ 817–828 (Props. XXVI–XXVII)✓
  10. 127Areas of lunes, spherical triangles and polygonsBook VIII, Props. XXVIII–XXX✓
  11. 128Volume of a sphere, sector and segmentBook VIII, §§ 838–849 (Props. XXXI–XXXII)✓
  12. 129Numerical exercises: surfacesBook VIII, Exs. 768–790✓
  13. 130Numerical exercises: volumesBook VIII, Exs. 791–810✓
  14. 131Miscellaneous exercises (1)Book VIII, Exs. 811–835✓
  15. 132Miscellaneous exercises (2)Book VIII, Exs. 836–865✓

Book 10 Book IX: Conic Sections

The parabola, the ellipse and the hyperbola: their construction, tangents and chief properties, and how each is cut from a cone; with a closing review of the table of formulas. Lessons 133–148.

  1. 133The parabolaBook IX, §§ 850–861 (Prop. I)✓
  2. 134Ordinates and tangents of the parabolaBook IX, Props. II–IV✓
  3. 135Tangents from a point; diameters of the parabolaBook IX, Props. V–VIII✓
  4. 136The area of a parabolic segment; the parabola as a section of a coneBook IX, Props. IX–X✓
  5. 137Exercises on the parabolaBook IX, Exs. 866–884✓
  6. 138The ellipseBook IX, §§ 896–910 (Prop. XI)✓
  7. 139Symmetry, focal radii and the auxiliary circleBook IX, Props. XII–XV✓
  8. 140Inside and outside; tangents of the ellipseBook IX, Props. XVI–XIX✓
  9. 141Tangents at corresponding points; conjugate diametersBook IX, Props. XX–XXII✓
  10. 142Area of an ellipse; the ellipse as a section of a coneBook IX, Props. XXIII–XXIV✓
  11. 143Exercises on the ellipseBook IX, Exs. 885–910✓
  12. 144The hyperbolaBook IX, §§ 945–959 (Prop. XXV)✓
  13. 145Symmetry and ordinates of the hyperbolaBook IX, Props. XXVI–XXIX✓
  14. 146Tangents of the hyperbolaBook IX, Props. XXX–XXXII✓
  15. 147Asymptotes; the hyperbola as a section of a coneBook IX, Props. XXXIII–XXXIV✓
  16. 148Review: the table of formulasTable of Formulas, pp. 461–464✓

The book itself

Each guide covers a few of the book's numbered sections (§) or propositions, shown beside the title. Proposition numbers start again at I in each Book. The guides are written for this site; the book is in the public domain.

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