It will be assumed as self-evident that : Things which are equal to the same thing are equal to one another. The whole is equal to the sum of its parts. If equals be added to equals the wholes are equal. If equals be taken from equals the remainders are equal. If equals be added to unequals the wholes are unequal. If equals be taken from unequals the remainders are unequal. Things which are double of equal things are equal to one another. Things which are halves of equal things are equal to one another. Equimultiples of equal things are equal. Things, of which the equimultiples are equal, are themselves equal. If one magnitude be greater than another any multiple of the former is greater than the same multiple of the latter. If a multiple of one magnitude be greater than the same multiple of another then the former magnitude is greater than the latter. ERRATA. 12, 1. 7, for equal circles read Os having equal radii. ,, 107,,, ,, 129,,, 6, for divides read divide. I, add if possible, let them bisect each other; ,, IZI,,, 13, omit to it. 201. In the diagram the diagonal of FB is AC. PLANE GEOMETRY. BOOK I. INTRODUCTION. A point has no magnitude. It has position. A line has length but not breadth. The extremities of lines are points. A straight line is one which lies evenly between its extremities. It is taken for granted: That a straight line may be drawn from any one point to any other point; And also that a terminated straight line may be produced to any length in a straight line. A superficies has length and breadth but not thickThe extremities of superficies are lines. ness. A plane is a superficies in which any two points being taken the straight line joining them lies wholly in that An angle is formed by two lines drawn from a point. The two lines are called the arms, and the point the vertex of the angle. If the arms are in one plane the angle is called a plane angle, and if they are straight lines a plane rectilineal angle. Whenever the term angle is used a plane rectilineal angle is meant unless otherwise expressed. The size of an angle does not depend on the length of its arms but on their direction, and is said to be greater or less according as the difference of direction of its arms is greater or less. For if two straight lines ABC, ABH could have a com.. mon segment AB ; then the straight line ABC might be turned about its extremity A, towards the side on which BH is, so as to cut BH; and thus two straight lines would enclose a space, which is impossible. Hence it follows that If two straight lines pass through the same point they will coincide entirely or cut one another. For if not, if possible let them fall otherwise as AOB, POQ having a common point O. Then AOB might be turned about one extremity A towards the side on which OQ is, so as to cut OQ and also ОР. Thus two straight lines would enclose a space, which is impossible. It is evident that Magnitudes which may be made to coincide, i.e. exactly fill the same space, are equal to one another. If two angles can be so placed that their vertices coincide and the arms of one fall along the arms of the other, those angles are said to be equal to one another. C B If the angle BAC were taken up, reversed, and then applied to its former position, so that A fell on its former position and AC along that of AB; then would AB fall along that of AC. STRAIGHT LINES, ANGLES AND TRIANGLES. DEFINITION. A triangle is a plane figure contained by three straight lines. PROPOSITION I. If two sides and the included angle of one triangle be respectively equal to two sides and the included angle of another triangle, the triangles shall be equal in all respects. D B In the AS ABC, DEF, K E let the two sides AB, AC be respectively equal to DE, DF, ▲ BAC = ▲ EDF; and then shall the As ABC, DEF be equal in all respects. the ABC may be applied to the ▲ DEF so that A may fall on D, and the straight lines AB, AC along DE, DF; For if it fell otherwise, as EKF, then two straight lines. would enclose a space, which is impossible; SO .. BC is = EF. Also the AABC will coincide with and is ... = ▲ DEF, LACB is LDFE. Hence the As ABC, DEF are equal in all respects. |