Classical Geometry

Return to QRV-321 Classical Geometry I In this lesson, we will study Euclid, Book I, Proposition 2. I. ENUNCIATION The proposition we are now to contemplate is stated by Euclid in these words: “To place at a given point (as an extremity) a straight line equal to a given straight line.” Read the words slowly. We are given two things: a point somewhere, and a straight line somewhere else. We are required to produce a straight line which has the given point for one of its ends, and which is equal to the given straight line. The parenthesis, “as an … Continue

Return to QRV-321 Classical Geometry I QRV-321 Lesson 003 – Book I, Common Notions I. INTRODUCTION This lesson studies the five Common Notions placed at the head of Book I of Euclid’s Elements of Geometry. Every science is built out of two kinds of statements: those that are proved, and those from which the proofs are made. The statements that are proved are called propositions, theorems and problems. The statements from which proofs are made are called principles. A science cannot prove everything, because every proof must start from something already granted; if nothing were granted, nothing could ever be … Continue

Return to QRV-321 Classical Geometry I QRV-321 Lesson 002 – Euclid Elements, Book I, Postulates I. INTRODUCTION This lesson studies the five postulates set down at the beginning of the first book of Euclid’s Elements. Geometry is the science of continuous magnitude — of lines, angles, surfaces and solids considered according to their measure, shape and position. Like every science, geometry proves its conclusions. But nothing can be proved out of nothing. Every proof rests on something already granted. The postulates are among the things granted at the beginning of geometry, before any proposition is proved, so that everything afterward … Continue

Return to QRV-321 Classical Geometry I QRV-321 Lesson 001 – Euclid Elements, Book I, Definitions I. INTRODUCTION This lesson begins the study of geometry, and it begins where every science must begin: with definitions. Geometry is the science of continuous magnitude — that is, of extension, of what has size and shape. Arithmetic studies multitude, which is number, and number is discrete: it is made of units that are separate from one another. Geometry studies magnitude, which is continuous: its parts hold together and are not separated by anything. Lengths, surfaces, and solids are continuous magnitudes, and geometry is the … Continue

Return to QRV-321 Classical Geometry I In this lesson, we will study Euclid, Book I, Proposition 1. I. ENUNCIATION The proposition we are now to contemplate is the first of the Elements. Euclid states it in these words: On a given finite straight line to construct an equilateral triangle. Let us first understand the words themselves, before we ask why we should care to know this. A straight line, in Euclid’s usage, is not what we today call a “line” stretching endlessly. It is what we would call a segment: a length with two ends. He calls it finite to … Continue