Five-minute digest
Elements, Book I
Summary
Book I of Euclid's Elements, composed at Alexandria around 300 BCE, is the opening of the most widely studied mathematics textbook in history. It builds the geometry of straight lines, angles, triangles and parallelograms from a short list of starting points, and it ends with the theorem now named for Pythagoras and its converse. The library text is Book I in the English edition of the Irish mathematician John Casey, The First Six Books of the Elements of Euclid (Dublin, 1885 and later editions), taken from Project Gutenberg. Casey keeps Euclid's order of propositions but rewords some definitions, adds notes, questions for examination and exercises, and rearranges the starting assumptions in the manner of nineteenth-century school editions. The diagrams are not reproduced, so readers should draw each figure as they go. The book opens with definitions: a point has position but no dimensions, a line is length without breadth, and so on through angles, triangles, polygons and the circle. Three postulates grant the right to draw a straight line between two points, to extend it, and to describe a circle with any centre and radius. Twelve axioms follow, from 'things which are equal to the same are equal to each other' to the statement that two lines making interior angles less than two right angles will meet. Casey places among these axioms the famous assumption about parallels that Euclid listed as his fifth postulate. Forty-eight propositions follow, each either a problem (something to construct) or a theorem (something to prove). Proposition 1 constructs an equilateral triangle on a given line using two circles. Propositions 4 and 8 give the side-angle-side and side-side-side tests for congruent triangles; Proposition 5 proves that the base angles of an isosceles triangle are equal, the step long nicknamed the pons asinorum. Propositions 13 to 26 treat angles at a point, the exterior angle, and the relations of sides and angles in a triangle, including Proposition 20, that any two sides of a triangle together exceed the third. Propositions 27 to 32 develop parallel lines, and Proposition 32 proves that the three angles of every triangle equal two right angles. Propositions 33 to 46 deal with parallelograms and with comparing areas, ending with the construction of a square. Proposition 47 proves that in a right-angled triangle the square on the hypotenuse equals the squares on the other two sides, and Proposition 48 proves the converse. Casey closes with questions on the whole book and more than fifty further exercises.
Top Lessons
Begin from what is granted
Book I asks the reader to accept only a few definitions, postulates and axioms and then proves everything else. Each proposition cites the earlier ones it uses, so the whole book becomes a single chain. The method teaches that a proof is only as strong as its starting points and that every step must be accounted for.
Theory, not only practice
Casey points out that Proposition 2, which copies a length to a new point, would be unnecessary if Euclid allowed the compass to carry distances. Euclid restricts himself to the ideal operations of drawing lines and circles because he is teaching theoretical geometry: the constructions must be justified, not merely performed.
The parallel assumption
The assumption that certain lines must meet (Casey's Axiom xii, Euclid's fifth postulate) is used for the first time in Proposition 29. Everything before it holds without it. Later mathematicians spent centuries trying to prove it from the other assumptions, and in the nineteenth century Lobachevsky, Bolyai and Riemann showed that consistent geometries can be built on different assumptions about parallels.
Proof by contradiction
Many propositions, such as 6, 14 and 29, are proved indirectly: Euclid supposes the opposite of what he wants and shows that it leads to something absurd, such as a part equal to the whole. Casey's questions ask students to name this method and to see when Euclid uses it.
Verified Quotes
“A point is that which has position but not dimensions.”
“A line is length without breadth.”
“Things which are equal to the same, or to equals, are equal to each other.”
“The whole is greater than its part.”
“Euclid’s object was to teach Theoretical and not Practical Geometry, and the only things he postulates are the drawing of right lines and the describing of circles.”
“the sum of the three internal angles is equal to two right angles.”
“In a right-angled triangle (ABC) the square on the hypotenuse (AB) is equal to the sum of the squares on the other two sides (AC, BC).”
Medical Lens
A work of pure geometry with no medical content. Its deductive form influenced later ideals of method in many sciences, including the systematic medical writing of Galen, who admired geometrical demonstration, but Book I itself says nothing about medicine.