Elements, Books I-VI

Summary

Euclid's Elements, composed in Alexandria around 300 BC, gathers the geometry and number theory of the Greek mathematicians before him into a single chain of reasoning. Its first six books are plane geometry: from a short list of definitions, postulates and axioms, Euclid proves, one proposition at a time, the properties of triangles, parallels, areas, circles, inscribed polygons and similar figures. Every later step rests explicitly on earlier ones, and nothing is assumed that has not been stated. For more than two thousand years these books were the standard way of teaching what a proof is. The edition attached here is John Casey's The First Six Books of the Elements of Euclid (third edition, Dublin, 1885), prepared for Irish colleges and schools. Casey prints Euclid's propositions in large type, using the older English 'right line' for straight line, and surrounds them with his own annotations, questions for examination, alternative proofs and more than eight hundred exercises. He also adds the first twenty-one propositions of Book XI and an appendix on solids. Readers should distinguish Euclid's text from Casey's commentary, and notice that in Book V Casey states Euclid's definitions but replaces his proofs with algebraic ones, remarking that Euclid's demonstrations there were no longer read. Book I builds from the construction of an equilateral triangle to the congruence theorems, the theory of parallels and the area of parallelograms, and closes with the Pythagorean theorem (I.47) and its converse. The parallel postulate, which Casey numbers as Axiom xii, is the one assumption that later mathematicians tried for centuries to prove from the others. Book II treats rectangles and squares in a geometric form of what we would now write as algebraic identities, including the division of a line so that the rectangle of the whole and one part equals the square on the other (II.11). Book III is the geometry of the circle: chords, tangents and the angle in a semicircle. Book IV inscribes and circumscribes figures, from triangles to the regular pentagon and fifteen-sided polygon. Book V gives the general theory of ratio and proportion traditionally credited to Eudoxus, and Book VI applies it to similar figures, ending with the extension of the Pythagorean theorem to any similar figures on the sides of a right triangle. In a classical Christian school the Elements serves both as mathematics and as training in logic and patience. Students learn to state exactly what is given, what is to be shown, and which earlier result justifies each step. Its beauty is the beauty of order: a few self-evident starting points, carefully kept, yield a whole world of necessary truths. The book also invites philosophical questions that Plato, Augustine and many later Christian thinkers raised: why the mind can grasp truths that no drawn figure perfectly shows, and what that suggests about the order of creation.

Plot or Argument

• Book I: Foundations, triangles and parallels

  Definitions of point, line, angle, figure and circle; three postulates of construction; Casey's twelve axioms, including the parallel postulate as Axiom xii. Propositions on equilateral triangles (I.1), isosceles triangles (I.5), congruence, parallel lines (I.27-I.31), the angle sum of a triangle (I.32), areas of parallelograms and triangles, and the Pythagorean theorem and its converse (I.47-I.48).

• Book II: Theory of rectangles

  Propositions in which every enunciation concerns a square or rectangle, equivalent to algebraic identities such as the square of a sum. It includes the cutting of a line so that the rectangle of the whole and one segment equals the square on the other (II.11), and generalizations of I.47 to obtuse and acute triangles.

• Book III: Theory of the circle

  Chords, centres, tangents and segments of circles; the equality of angles in the same segment; the right angle in a semicircle (III.31); and the relations between intersecting chords and secants.

• Book IV: Inscribed and circumscribed figures

  Problems of inscribing triangles, squares and regular polygons in circles and circles in them, culminating in the regular pentagon (IV.11) and the regular fifteen-sided polygon (IV.16).

• Book V: Theory of proportion

  Euclid's definitions of multiple, ratio and proportion, designed to handle incommensurable magnitudes. Casey keeps the definitions and Euclid's numbering but supplies algebraic proofs and an explanation of incommensurable ratios.

• Book VI: Application of proportion to figures

  Triangles of equal height are as their bases (VI.1); similar triangles and polygons; mean proportionals; division of a line in extreme and mean ratio (VI.30); and the theorem that similar figures on the sides of a right triangle obey the Pythagorean relation (VI.31).

• Casey's additions

  Questions for examination, alternative proofs and graded exercises after each book; the first twenty-one propositions of Book XI on lines and planes in space; and an appendix on prisms, pyramids, cylinders, spheres and cones.

Lessons

• What counts as a proof

  Each proposition in the Elements states what is given, constructs what is needed, and demonstrates the result using only definitions, postulates, axioms and propositions already proved. Students learn that a convincing diagram is not a proof; the reasons must be named. Casey's habit of citing [I. iv.] or Axiom i. at each step makes the chain visible.

  • In I.1, which postulate justifies drawing each circle, and which axiom proves the triangle equilateral?
  • Why does Euclid not simply measure the sides with a ruler?
  • Can you find a step in any proof that seems obvious but is not justified by anything stated?

• The fifth postulate

  Casey's Axiom xii states that two lines meeting a third with interior angles less than two right angles will meet on that side. It is less self-evident than the others, and Euclid avoids using it until I.29. Attempts to prove it led, in the nineteenth century, to consistent non-Euclidean geometries. The lesson is that axioms are chosen starting points whose status deserves examination.

  • Why might Euclid have delayed using this axiom for twenty-eight propositions?
  • Which later propositions in Book I depend on it, directly or indirectly?
  • What does it mean for a geometry to be consistent but different from Euclid's?

• Pythagoras by areas

  I.47 proves that the square on the hypotenuse equals the squares on the other two sides by showing that two rectangles inside the large square equal the two smaller squares. The result is ancient, but Euclid's demonstration uses only Book I. VI.31 later extends it to any similar figures, showing how the same truth reappears at a higher level.

  • Trace which earlier propositions I.47 depends on.
  • Why does Euclid follow it immediately with its converse, I.48?
  • How does VI.31 generalize I.47, and why does it need the theory of proportion first?

• Ratio and the incommensurable

  The side and diagonal of a square have no common measure, a discovery that troubled Greek mathematics. Book V's definition of proportion lets Euclid reason about such magnitudes without whole numbers. Casey's notes explain incommensurables and replace Euclid's Book V proofs with algebra, which shows how methods change while results endure.

  • Why can the ratio of side to diagonal of a square not be written as a ratio of whole numbers?
  • What is gained and lost when Casey replaces Euclid's proofs with algebraic ones?
  • Why does Book VI need Book V before it can treat similar figures?

• Necessary truth and the mind

  No drawn line lacks breadth, as Casey notes on the postulates, yet the reasoning concerns perfect lines and circles. Plato used geometry to argue that the mind knows realities beyond the senses, and Christian thinkers such as Augustine saw in unchanging mathematical truth a sign of an ordered creation and a divine mind. Students can weigh these claims while doing the mathematics itself.

  • Casey says a drawn line is never a true geometrical line. What, then, is a geometry proof about?
  • Are Euclid's theorems discovered or invented? Give reasons.
  • How might Wisdom 11:20, 'thou hast ordered all things in measure and number and weight', bear on the study of geometry?

Context

Euclid worked in Alexandria under the early Ptolemies, probably around 300 BC, in the scholarly world that grew up around the Museum and Library. Almost nothing is known of his life. The Elements organizes earlier discoveries, including work associated with the Pythagoreans, Hippocrates of Chios, Theaetetus and Eudoxus, into a single deductive system. Its thirteen books also cover number theory, incommensurable magnitudes and solid geometry, ending with the five regular solids.

Composition

The Greek text survives in manuscripts descending largely from the revision of Theon of Alexandria (fourth century AD), with an earlier tradition preserved in a Vatican manuscript. The work passed through Arabic translations to the Latin West, notably the twelfth-century version associated with Adelard of Bath, and was among the first mathematical books printed (Venice, 1482). Robert Simson's eighteenth-century edition became the basis of British school texts. John Casey (1820-1891), professor of mathematics at the Catholic University of Ireland, published his annotated edition of Books I-VI in 1881-82; the attached third edition is dated 1885.

Reception

The Elements was a core school text from antiquity to the early twentieth century, studied in Byzantium, the Islamic world and medieval and modern Europe. Its axiomatic method shaped philosophy and science, from Spinoza's Ethics, set out in geometrical order, to Newton's Principia. The long effort to prove the parallel postulate ended in the non-Euclidean geometries of Lobachevsky, Bolyai and Riemann, and David Hilbert's 1899 axioms filled gaps that modern readers find in Euclid's foundations. Casey's edition was widely adopted in Irish and British schools.

Comparisons

• Elements, Book I

  The separate record for Book I covers the foundations alone; this record follows the argument through circles, polygons and proportion to similar figures, in Casey's annotated school edition.

• Introduction to Arithmetic

  Nicomachus treats number through classification and Pythagorean symbolism with few proofs; Euclid's geometry proves every claim, a contrast that shows two ancient ideas of mathematical knowledge.

• Meno

  In the Meno Socrates leads a slave boy to double a square, the same kind of truth Euclid demonstrates; Plato uses it to argue for recollection, while Euclid simply builds the proof.

• Ethics

  Spinoza's Ethics borrows Euclid's form of definitions, axioms and propositions for metaphysics and morals, raising the question whether that method suits subjects other than geometry.

• The Geometry

  Descartes translates geometric problems into algebraic equations, the approach Casey already uses in his notes on Book V; reading both shows how the methods of the Elements were transformed.

• Flatland: A Romance of Many Dimensions

  Abbott's Victorian satire, narrated by a Square, plays with Euclidean figures and dimensions and assumes a reader trained in school geometry of exactly the kind Casey's edition provided.

Quotes

Creative Prompts

• Classroom

  Using only a compass and straightedge, construct I.1 and I.47 on paper, then write each step of the demonstration beside the figure, citing the postulate, axiom or earlier proposition that justifies it.

• Essay

  Is the parallel postulate self-evident? Write an essay that explains Euclid's statement, why later mathematicians doubted it, and what their doubts teach about the nature of axioms.

• Dialogue

  Write a dialogue between Euclid and John Casey in which Casey defends replacing the Book V proofs with algebra and Euclid replies.

• Fiction

  Tell the story of a student in Alexandria who first understands the proof that the side and diagonal of a square have no common measure.

• Poetry

  Write a short poem whose structure follows a Euclidean proof: a given, a construction, a demonstration and a final line that states what was to be shown.

Further Reading

• The Thirteen Books of Euclid's Elements, Thomas L. Heath (translator and commentator)

  The standard English translation of all thirteen books, with extensive historical commentary.

• Meno, Plato

  The slave-boy geometry lesson that uses a theorem of the Euclidean kind to argue about knowledge.

• Euclid and His Modern Rivals, Lewis Carroll (Charles L. Dodgson)

  A witty 1879 defence of Euclid as a school text against Victorian replacements.

• A History of Greek Mathematics, Thomas L. Heath

  Places Euclid among his predecessors and successors.