32. It is evident from 22 and 23 that when one straight line meets another so as to form two angles, these angles are supplementary. 33. Two straight lines are said to be parallel when, lying in the same plane, and extended indefinitely both ways, they do not meet each other. 34. As we have before conceived a line (22) to move, so we may conceive one geometrical magnitude to be applied to another for the purpose of comparison. If they coincide, point for point, they are said to be equal. 35. Thus, if two angles can be so placed that their vertices coincide in position and their sides in direction, two and two, the angles must be equal. 36. Conversely, if two equal angles be conceived to be so placed, one upon the other, that their vertices and one pair of sides coincide respectively, then the other pair of sides must also coincide, otherwise one angle would be greater than the other. 37. Geometrical magnitudes are geometrical lines, angles, surfaces, and solids. 38. We shall have occasion to express the addition and subtraction of geometrical magnitudes, as well as the multiplication and division of these magnitudes by numbers. 39. For example, the sum of the two lines AB and CD is obtained by conceiving them to be placed so as to form one continuous straight line as HK. Similarly, the difference of two lines is obtained by cutting off from the larger a line equal to the smaller. Similarly, HKN H A 1 N -B -D K represents the sum of the two angles A and B. To multiply a line by a number is to add it to itself the required number of times. (See above.) To divide a line by a number is to conceive the line to be divided into the required number of equal parts. лл K I H The same is true of other geometrical magnitudes. (Illustrations of each should be given.) 40. An axiom is a truth that needs no argument; i.e. the mere statement of it makes it apparent, e.g.: GENERAL AXIOMS. (I. The whole of anything is greater than any one of its parts. II. The whole of anything is equal to the sum of all its parts. III. Quantities which are respectively equal to the same or equal quantities are equal to each other. IV. Quantities which are respectively halves of the same or equal quantities are equal to each other. V. Quantities which are respectively doubles of the same or equal quantities are equal to each other. VI. If equal quantities be added to equal quantities, the sums are equal. VII. If equal quantities be subtracted from equal quantities, the remaining quantities are equal, VIII. If equal quantities be multiplied by the same or equal quantities, the products are equal. I IX. If equal quantities be divided by the same or equal quantities, the quotients are equal. X. If equal quantities be either added to or subtracted from unequal quantities, the results will be unequal. XI. If equal quantities be either multiplied or divided by unequals, the results will be unequal. 41. The results obtained by the addition to, subtraction from, multiplication or division of, unequals by unequals are indeterminate with one exception. The pupil should ascertain for himself this exception. 42. Particular axioms. XII. Between two points only one straight line can be drawn; or if others are drawn, they must coincide. XIII. A straight line is the shortest of all possible lines. connecting two points. XIV. Conversely, the shortest line between two points is a straight line. XV. If two straight lines have two points in common, they will coincide however far extended. XVI. Two straight lines can intersect in only one point. point (as P) only one line A (as AB) can be drawn par allel to another line (as CD); C r if others are drawn, they must coincide, XIX. If a line makes B D P tersect the other if sufficiently extended. XX. The extension or shortening of the sides of an angle does not change the magnitude of the angle, 43. A theorem is a truth which is made apparent by a course of reasoning or argument. This argument is called a demonstration. Every theorem consists of two distinct parts, either expressed or implied; viz. the hypothesis and conclusion. The conclusion is the part to be proven, and the demonstration is undertaken only upon the ready granting of the conditions expressed in hypothesis; e.g.: Hyp. If two parallel lines be crossed by a transversal, 44. In demonstrating the theorems in this book the pupil should first analyze the theorem and write it after the above model. For instance, let us analyze the following theorem ; viz.: A perpendicular measures the shortest distance from a point to a straight line. Now this theorem, analyzed and written according to our model, would read as follows: Hyp. If from a given point to a given straight line a perpendicular and other lines be drawn, Con. the perpendicular will be the shortest one of those lines. 45. The converse of a theorem is another theorem in which the hypothesis becomes the conclusion, and conclusion the hypothesis. For example, the converse of the theorem mentioned in Sect. 43 would read as follows; viz.: Hyp. If two straight lines in the same plane be crossed by a transversal so as to make the alternate interior angles equal, Con. these two straight lines will be parallel. The converse of most of the theorems in this work will be left for the pupil to state. 46. A problem, in geometry, is the required construction of a geometrical figure from stated conditions or data; e.g.: It is required to construct the triangle which has for two of its sides AB and CD, and the angle H included between these two sides: A postulate is a self-evident problem, or a construction to the possibility of which assent may be demanded or challenged without argument or evidence. (Both theorems and problems are commonly designated as propositions.) A C H -B -D 47. Before we can accomplish the demonstration of a theorem in geometry, the following postulates must be granted; viz.: I. A straight line can be drawn from one point to any other point. II. A straight line can be extended to any length, or terminated at any point. III. A circle may be described about any point as a centre and with any radius. IV. Geometrical magnitudes of the same kind may be added, subtracted, multiplied, and divided. V. A geometrical figure may be conceived as moved at pleasure without changing its size or shape. 48. A postulate is also used to designate the first step in the demonstration whereby certain conditions or data are demanded as fulfilled, or admitted as true, for the basis of the argument, and which begins with "Let," etc., or "Let it be granted that," etc. It really demands assent to the general conditions implied in the hypothesis, with special reference to a particular representative diagram or figure; e.g. in beginning the demonstration of the theorem given in Sect. 43, we should say: "Let AB and HK H B be two parallel straight lines crossed by the transversal CD." |