INTRODUCTORY. 1. Little is known of Euclid beyond the fact that he lived about three centuries before Christ (325-285) at Alexandria, where he became famous as a writer and teacher of Mathematics. Among the works ascribed to him, the best known and most important is The Elements, written in Greek, and consisting of Thirteen Books. Of these it is now usual to read Books I.-IV. and VI. (which deal with Plane Geometry), together with parts of Books XI. and XII. (on the Geometry of Solids). The remaining Books deal with subjects which belong to the theory of Arithmetic. 2. Plane Geometry deals with the properties of all lines and figures that may be drawn upon a plane surface. Euclid in his first Six Books confines himself to the properties of straight lines, rectilineal figures, and circles. 3. The subject is divided into a number of separate discussions, called propositions. Propositions are of two kinds, Problems and Theorems. A Problem proposes to perform some geometrical construction, such as to draw some particular line, or to construct some required figure. A Theorem proposes to prove the truth of some geometrical statement. 4. A Proposition consists of the following parts : The General Enunciation, the Particular Enunciation, the Construction, and the Proof. (i) The General Enunciation is a preliminary statement, describing in general terms the purpose of the proposition. (ii) The Particular Enunciation repeats in special terms the statement already made, and refers it to a diagram, which enables the reader to follow the reasoning more easily. (iii) The Construction then directs the drawing of such straight lines and circles as may be required to effect the purpose of a problem, or to prove the truth of a theorem. (iv) The Proof shews that the object proposed in a problem has been accomplished, or that the property stated in a theorem is true. 5. Euclid's reasoning is said to be Deductive, because by a connected chain of argument it deduces new truths from truths already proved or admitted. Thus each proposition, though in one sense complete in itself, is derived from the Postulates, Axioms, or former propositions, and itself leads up to subsequent propositions. 6. The initial letters Q.E. F., placed at the end of a problem, stand for Quod erat Faciendum, which was to be done. The letters Q.E.D. are appended to a theorem, and stand for Quod erat Demonstrandum, which was to be proved. 7. A Corollary is a statement the truth of which follows readily from an established proposition; it is therefore appended to the proposition as an inference or deduction, which usually requires no further proof. 8. The attention of the beginner is drawn to the special use of the future tense in the Particular Enunciations of Euclid's propositions. The future is only used in a statement of which the truth is about to be proved. Thus: "The triangle ABC SHALL BE equilateral" means that the triangle has yet to be proved equilateral. While. "The triangle ABC is equilateral" means that the triangle has already been proved (or given) equilateral. 9. The following symbols and abbreviations may be employed in writing out the propositions of Book I., though their use is not recommended to beginners. and all obvious contractions of words, such as opp., adj., diag., etc., for opposite, adjacent, diagonal, etc. Let AB be the given straight line. It is required to describe an equilateral triangle on AB. Construction. With centre A, and radius AB, describe the circle BCD. Post. 3. With centre B, and radius BA, describe the circle ACE. Post. 3. From the point C at which the circles cut one another, draw the straight lines CA and CB to the points A and B. Then shall the triangle ABC be equilateral. Post. 1. Because A is the centre of the circle BCD, Proof. Def. 15. And because B is the centre of the circle ACE, therefore BC is equal to AB. Def. 15. Therefore AC and BC are each equal to AB. But things which are equal to the same thing are equal to one another. Therefore AC is equal to BC. Therefore AC, AB, BC are equal to one another. and it is described on the given straight line AB. Ax. 1. Q.E.F. PROPOSITION 2. PROBLEM. From a given point to draw a straight line equal to a given straight line. K H F B Let A be the given point, and BC the given straight line. It is required to draw from A a straight line equal to BC. Construction. Join AB; Post. 1. I. 1. and on AB describe an equilateral triangle DAB. With centre B, and radius BC, describe the circle CGH. Post. 3. Post. 2. Produce DB to meet the circle CGH at G. Proof. Because B is the centre of the circle CGH, And because D is the centre of the circle GKF, therefore BC is equal to BG. therefore DF is equal to DG. Post. 2. Def. 15. Def. 15. And DA, a part of DF, is equal to DB, a part of DG; Def. 24. therefore the remainder AF is equal to the remainder BG. Ax. 3. But BC has been proved equal to BG; therefore AF and BC are each equal to BG. And things which are equal to the same thing are equal to one another. Therefore AF is equal to BC; Ax. 1. Q.E.F. PROPOSITION 3. PROBLEM. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which AB is the greater. It is required to cut off from AB a part equal to C. Construction. From the point A draw the straight line I. 2. and with centre A and radius AD, describe the circle DEF, cutting AB at E. Post 3. Then AE shall be equal to C. Proof. Because A is the centre of the circle DEF, Def. 15. But C is equal to AD. Therefore AE and C are each equal to AD. and it has been cut off from the given straight line AB. Constr. Ax. 1. Q.E..F. |