Euclid's Elements

John Casey, a professor of higher mathematics at the Catholic University of Ireland, presents this edition of the first six books of Euclid's Elements, along with twenty-one propositions from Book XI and an appendix on solid geometry. Casey aims to preserve the ancient Greek geometer's original text while incorporating modern geometric developments, addressing a need felt by teachers for a work combining classical content with contemporary methods. The work includes Casey's annotations, alternative proofs, and over eight hundred exercises, many drawn from French mathematical works by Catalan, Rouché, and de Comberousse, as well as contributions from the late Reverend Professor Townsend of Trinity College Dublin.

The Introduction defines geometry as the science of figured space, distinguishing among solids (three dimensions), surfaces (two dimensions), and lines (one dimension). A point has position but no dimensions. Plane geometry, the study of points, lines, and circles on a flat surface, is the subject of the Elements and of Casey's volume. Casey establishes notational conventions, including the congruence symbol (≡), which he credits to the mathematician Carl Friedrich Gauss.

Book I opens with definitions of the fundamental objects: a straight (or right) line lies evenly between its endpoints; a plane is a surface in which any straight line connecting two of its points lies entirely within it. Angles are classified as right (equal to their supplement), acute, or obtuse. Triangles are classified by sides as scalene, isosceles, or equilateral, and by angles as right-angled, obtuse-angled, or acute-angled. A circle is a plane figure defined such that all straight lines from its centre to the bounding curve are equal. Three postulates permit the drawing of a straight line between any two points, the extension of any line segment, and the description of a circle with any centre and radius. Twelve axioms establish the logical foundations, including the parallel postulate: If two lines meet a third so that the interior angles on one side sum to less than two right angles, the lines will meet on that side.

The propositions of Book I develop triangle congruence, the condition of two triangles being identical in shape and size. Proposition I constructs an equilateral triangle on a given line using intersecting circles, establishing the pattern of construction followed by demonstration. Propositions IV, VIII, and XXVI prove the three congruence cases: Side-Angle-Side, Side-Side-Side, and Angle-Side-Angle. Other propositions prove that the base angles of an isosceles triangle are equal, that adjacent angles on a straight line sum to two right angles, and that the sum of any two sides of a triangle exceeds the third.

The theory of parallel lines begins with Proposition XXVII, which proves that if a transversal (a line cutting across two other lines) makes alternate angles (angles on opposite sides of the transversal) equal, the lines are parallel. Proposition XXIX, the first to invoke the parallel postulate, proves the converse. Proposition XXXII proves that the three interior angles of any triangle sum to two right angles, with corollaries extending this to polygons: the angle sum of an n-sided polygon is 2(n − 2) right angles. The remaining propositions develop the theory of area for parallelograms and triangles, showing that figures on equal bases and between the same parallels have equal areas. Proposition XLVII, the Pythagorean theorem, proves that in a right-angled triangle the square on the hypotenuse (the side opposite the right angle) equals the sum of the squares on the other two sides. Proposition XLVIII proves the converse.

Book II translates algebraic identities into geometric form. Casey explains that rectangular areas are measured by the product of their sides. Key propositions prove the geometric equivalents of algebraic formulas, including the identity (a + b)² = a² + 2ab + b². Proposition XI divides a line in "extreme and mean ratio" (the golden ratio), where the whole is to the greater segment as the greater is to the lesser. Casey proves that the two parts are incommensurable, meaning no common unit measures both exactly. Propositions XII and XIII generalize the Pythagorean theorem to obtuse and acute triangles. Proposition XIV constructs a square equal in area to any given rectilinear figure, a shape bounded entirely by straight lines.

Book III develops properties of chords (line segments joining two points on a circle), tangents (lines touching but not cutting a circle), and angles in circles. The central angle theorem (Proposition XX) proves that the angle at the centre is double the angle at the circumference standing on the same arc, or portion of the circumference. Proposition XXII proves that opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) sum to two right angles. The alternate segment theorem (Proposition XXXII) proves that the angle between a tangent and a chord equals the angle in the segment on the opposite side of the chord. Propositions XXXV and XXXVI establish the power of a point: for any two chords through a fixed point, the products of their segments are equal.

Book IV consists of construction problems for inscribing and circumscribing triangles, squares, and pentagons in and about circles. Casey's annotations introduce the inscribed circle, the three escribed circles (each touching one side of a triangle externally), and the nine-points circle, which passes through the midpoints of the sides, the feet of the altitudes (perpendiculars from vertices to opposite sides), and the midpoints of the segments from vertices to the orthocentre (the intersection of the altitudes). Casey notes that Gauss proved in 1801 that regular polygons of 2ⁿ + 1 sides are constructible with ruler and compass when 2ⁿ + 1 is prime.

Book V develops the theory of ratios and proportional magnitudes. Casey replaces Euclid's original proofs with algebraic demonstrations, defining commensurable magnitudes as those sharing a common unit of measure and incommensurable magnitudes as those lacking one. He proves Euclid's Fifth Definition, the equimultiple criterion for proportionality, as a theorem. The propositions establish the standard transformations of proportions: invertendo, alternando, componendo, dividendo, and ex aequali.

Book VI applies proportion to similar figures. Proposition I proves that triangles of equal altitude have areas proportional to their bases. Propositions IV through VII establish conditions for triangle similarity: equiangular triangles (those with all corresponding angles equal) have proportional sides, and conversely. Casey notes that for triangles alone, either condition implies the other. Proposition XIX proves that similar triangles have areas in the duplicate ratio (the square of the ratio) of their homologous (corresponding) sides, and Proposition XX extends this to similar polygons. Casey develops the theory of centres of similitude, points from which one figure can be scaled to produce another. Proposition XXXI generalizes the Pythagorean theorem to any similar figures described on the sides of a right-angled triangle.

Book XI extends geometry to three dimensions, defining dihedral angles (angles between two planes), solid angles (formed by three or more plane angles meeting at a point), and normals to planes. The propositions establish that two intersecting lines determine a plane, that lines normal to the same plane are parallel, and that the sum of face angles of any convex solid angle is less than four right angles.

The Appendix treats prisms, pyramids, cylinders, spheres, and cones. Casey proves that a prism's volume equals base area times altitude and a pyramid's volume is one-third of this. Using a theorem on solids of revolution associated with the mathematician Guldinus, he derives volume formulas for the cone (πr²h/3) and sphere (4πr³/3), and proves that a sphere's surface area equals 4πr², equivalent to the area of four great circles (circles formed by planes passing through the sphere's centre).

The Notes address celebrated problems of classical and modern geometry. Casey presents the modern theory of parallels using the projective concept of the line at infinity, the theoretical line where parallel lines are said to meet. He offers proofs of the angle sum theorem independent of the parallel postulate by the mathematicians Adrien-Marie Legendre and William Rowan Hamilton, gives a construction for the regular seventeen-sided polygon by the mathematician André-Marie Ampère, discusses mechanical methods for finding mean proportionals (intermediate lines in continued proportion between two given lines) and trisecting an angle, and explains the quadrature of the circle, the classical problem of constructing a square equal in area to a given circle. Casey notes that π is irrational and presents an iterative method yielding π ≈ 3.1415926.