AXIOMS. 1. Magnitudes which are equal to the same are equal to one another. SCHOLIUM. This axiom is frequently employed in the Elements under the form, "Magnitudes which are equal to equals are equal to one another." 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Magnitudes which are double of the same are equal to one another. 7. Magnitudes which are halves of the same are equal to one another. SCHOLIUM. A similar extension may be given to the 6th and 7th axioms as that given above to the 1st by substituting "equals" for "the same." 8. Magnitudes which coincide with one another are equal to one another. SCHOLIUM. The converse of this axiom is sometimes made use of in the elements, namely, "Magnitudes which are equal coincide with one another when similarly placed." 9. The whole is greater than its part. 10. Two straight lines cannot enclose a space. SCHOLIUM. This axiom may be otherwise expressed, namely, "If two straight lines coincide in two points, they coincide when produced." 11. All right angles are equal to one another. SCHOLIUM. Angles being a species of magnitude, this axiom is a particular case of the 8th. 12. Through the same point two straight lines cannot be drawn parallel to the same straight line. SCHOLIUM. This axiom is substituted for that of Euclid, not only as being more self-evident, but because the latter being the converse of the 17th proposition, it was considered that it ought to be demonstrated, which has been done after the 29th proposition. The axioms are self-evident theoreins, that is, theorems which do not admit of being demonstrated, but the truth of which is nevertheless so apparent as to be instantly admitted. No theorem should be considered as an axiom simply because it is self-evident, but only when it will not admit of being demonstrated by means of arguments founded on still simpler theorems; for it is desirable that the number of axioms should be reduced as much as possible, for which reason the 20th proposition, and some others, although as self-evident as any of the axioms, are demonstrated at length. EXPLANATORY REMARKS. A proposition in geometry is something either proposed to be solved or proved; and is consequently divided into two kinds, a problem or a theorem. A problem proposes something to be done; while a theorem makes an assertion, of which it proposes to demonstrate the truth. The statement of the thing to be done, or of the assertion to be proved, is termed the enunciation of the proposition. In the following work the enunciation is distinguished by a bolder type. The enunciation of a problem may be divided into two parts, the data or things given, and the quæsita or things sought to be done. The former is distinguished by being printed in italics. In like manner the enunciation of a theorem may be divided into two parts, the hypothesis, and the consequence, which it is to be proved results from that hypothesis. The former is distinguished from the consequence by being in italics. The solution of a problem is the mode in which the quasita are found, or the thing proposed to be done is accomplished, and is always performed by means of some other problems the truth of which has been either admitted or proved previously. The construction of a theorem is certain things which may be required to be done by means of problems, in order to admit of the truth of the theorem being demonstrated. The demonstration either of a problem or theorem is a succession of arguments logically deduced from theorems already admitted or proved, by means of which the truth of the solution of a problem or the assertion of a theorem are undeniably established. A lemma is a proposition of no importance in itself, merely introduced for the purpose of demonstrating some other proposition. A corollary is a proposition the truth of which immediately follows from that to which it is affixed. A scholium is a note or remark appended to any proposition by way of explanation or elucidation. In the marginal references the following abbreviations are employed:- Post. 1, signifies the first postulate. I. 31, signifies the 31st proposition of the first book. PROPOSITION I. PROBLEM. To construct an equilateral triangle upon a given finite straight line (AB). SOLUTION. From the center A, at the distance AB, describe the circle BCD (a), and from the center B, at the distance BA, describe the circle ACE; and from the point C, in which the circles cut one another, draw the straight lines CA, CB to the points A, B (b); then ABC will be the triangle required. (a) Post. 3. DEMONSTRATION. It is evident that the triangle ABC is constructed on the line AB. And it is also equilateral: for, the lines AC and AB being both radii of the same circle, BCD are equal (c), and the lines AB and CB being both radii of the same circle, ACE are equal; then, because the lines AC and CB are both equal to the same line AB, therefore they are equal to each other (d); that is, the three sides AB, AC, and CB are equal, and the triangle ABC is therefore equilateral. SCHOLIUM. In this Prop. the following axiom is tacitly assumed by Euclid, viz. :-"That a circle whose center is in the circumference of another circle, must be partly within that circle, and partly without it, and therefore, that those circles must necessarily cut or intersect cach other." PROPOSITION II. PROBLEM.-From a given point (A) to draw a straight line equal to a given finite straight line (BC). SOLUTION. From the given point A draw a straight line to either extremity B of the given line (a). Upon AB construct an equilateral triangle (b). From the center B, at the distance BC, describe the circle CGH (c); and produce the straight line DB until it meets the circumference in G (d). From the center D, at the distance DG, describe the circle GKL (c); and produce the straight line DA until it meets the circumference in L (d). The straight line AL is equal to the given line BC. DEMONSTRATION. For the lines DG and DL being both radii of the same circle, GKL are equal (e), and if the equals DB (a) Post. 1. Def. 13 and 16. and DA (f) be taken from each respectively,. the remainders BG and AL are equal (g); but the lines BG and BC being both radii of the same circle CGII, are equal (e): therefore the lines BC and AL, being both equal to the same line BG, are equal to each other (h). Therefore, from a given point A, a straight line has been drawn equal to a given straight line BC. SCHOLIA. 1. The construction of this problem will somewhat vary according to the relative positions of the point A and the line BC. Def. 13 and 16. f) Construction. (g) Ax. 3. (h) Ax. 1. 2. In practice this problem will be solved by measuring the length of the given line BC with a pair of compasses; and then, applying one leg of the compasses to the point A, the other leg will mark the length of the line required to be drawn from A. In geometry, however, such use of the compasses is not permitted. The only way in which they may be employed is that allowed in the third postulate, viz. to describe a circle whose circumference shall pass through a given point about some other given point as a center. The compasses must be supposed to close of themselves whenever removed from the paper, so that no distance can be carried by means of them. This restricted use of the compasses being borne in mind will enable the student to see the necessity of the first three problems in this book. PROPOSITION III. PROBLEM.-From the greater of two given straight lines (AB and C) to cut off a part equal to the less. SOLUTION. From either extremity A of the greater given line draw a straight line AD equal to the lesser given line C (a). From the center A, at the distance AD, describe the circle DEF (b), which shall cut off AE equal to the lesser line C. DEMONSTRATION. For the lines AD and AE being both radii of the same circle DEF are equal (c). But AD and C are equal (d); therefore, because AE and C are both equal to the same line AD, they are equal to each other (e); and from AB the greater of two given lines, a part AE has been cut off, equal to C the less. G (a) I. 2. SCHOLIUM. By a similar operation the lesser line could be extended to equal the greater; thus from either extremity of the lesser line let a line be drawn equal to the greater, then about this same extremity as a center describe a circle with a radius equal to the greater line; extend the lesser line to meet the circumference of this circle, and it will equal the greater line. PROPOSITION IV. THEOREM. If two triangles (ABC and DEF) have two sides of the one respectively equal to two sides of the other (DE and DF to AB and AC), and the angles formed by those sides also equal to one another (D to A); [1] their bases or third sides (EF and BC) will be equal; [2] and the angles at the bases, which are opposite to the equal sides, will be equal (E to B and F to C); [3] and the triangles themselves will be equal. C E DEMONSTRATION. For, if the triangle ABC be applied to DEF, so that the point A may be on the point D, the point B on the straight line DE, and that AC and DF may lie on the same side; then AB must lie wholly on DE, for otherwise two straight lines would enclose a space (a); and because AB is equal to DE, the point B must coincide with the point E (6). Further, because the angles A and D are equal, the side AC must fall upon the side DF; and because AC is equal to DF, the point C must coincide with the point F. (a) Ax. 10. [1.] Therefore, as the points B and C coincide with the points E and F, the base BC must coincide with the base EF, and be equal to it (c); for otherwise two straight lines would enclose a space (a). [2.] And as the sides which form the augles B and C coincide with the sides which form the angles E and F, those angles themselves must coincide, and therefore must be equal (c). [3.] And as the straight lines which contain the triangle ABC coincide with and are equal to the straight lines which contain the_triangle DEF, therefore the triangles themselves must coincide, and must therefore be equal (c). SCHOLIA. 1. In the above demonstration (at b) an axiom is assumed the converse of the eighth axiom, namely, "Magnitudes which are equal coincide with one another when similarly placed." 2. In every triangle there are six quantities or magnitudes, namely, the three sides and the three angles; and (except in two particular cases) when any three of these are given, the other three can be found and the triangle determined. If, therefore, two triangles are found to agree in any three of those quantities by which the triangles are determined, it is evident that those triangles must be equal. The following are the only six cases which can occur: 1. The three angles. 2. The three sides. 3. Two sides and the angle between them. 4. Two sides and the angle opposite to one of them. 5. Two angles and the side between them. 6. Two angles and the side opposite to one of them. |