MATHEMATICS-GEOMETBY. 



CHAPTER I. 

 PLANE GEOMETRY. THE ELEMENTS OP EUCLID. 



INTRODUCTION. 



IN order to acquire clear conceptions of a point, a line, 

 and a surface, with the definitions of which Euclid sets 

 out, it will be best for the learner to consider them, at 

 first, in connection with a solid ; that is, with something 

 that has length, breadth, and thickness. All external 

 objects things that can be seen and felt are solids; 

 and however small the solid may be, it must have some 

 length, some breadth, and some thickness : using these 

 terms in their ordinary acceptation that is, in the sense 

 in which they are used in common discourse. 



The boundaries of a solid are called the surface, or the 

 superficies, of the solid ; and, as a boundary, cannot have 

 thickness ; because, if it had, it would be a part of the 

 solid, and not a side, face, or boundary of it : it follows 

 that a surface can have length and breadth only. 



Again the surface itself has its boundaries, or limits: 

 these boundaries are called lines : and as a boundary of a 

 surface cannot have either breadth or thickness, since it 

 is no part of the surface, much less a part of the solid, 

 it follows that a Hue has length only. 



Lastly, a line has its limits a beginning and a ter- 



mination : we speak of these, in common language, as 

 the ends of the line. Euclid calls them points ; and it is 

 plain that a point, being no part of the line, cannot have 

 length ; and as breadth and thickness are excluded even 

 from the line itself, it follows that a point has no dimen- 

 sions or magnitude. It merely indicates position, ; the 

 position, namely, of the commencement or termination 

 of a line. 



Euclid frequently speaks of taking a point in a line, 

 without meaning an extremity of the line ; but we may 

 conceive a line to be crossed, or cut by other lines, and 

 thus to be divided into shorter portions. Each portion 

 has its extremities ; so that we may conceive as many 

 points in the line as we please. We cannot represent 

 length only to the eye ; it is necessary, therefore, in con- 

 templating the black marks on paper, by which lines are 

 represented, entirely to disregard the breadth of them ; 

 and to fix the attention upon the length atone : the eye 

 may see breadth and thickness ; the mind takes note of 

 length only. It is usual to commence the study of 

 Geometry with definitions, which are as follows. 



ELEMENTS OF EUCLID. BOOK L 



ranmnom. 



A point has no magnitude : it has position only. 



ir. 

 A line has length only. 



in. 



A straight line is that which lies evenly between its 

 extreme points. 



It 1> distinguished by uniformity of direction between iti extremities. 



IV. 



A surface, or superficies, is that which has length and 

 breadth only. 



v. 

 Therefore the boundaries of a surface are lines. 



VI. 



A plane surface, or simply a plane, is that in which, 

 whatever two points be taken, the straight line, having 

 these points for extremities, lies wholly in that surface. 



VII. 



A plane rectilineal angle is the opening between two 

 straight lines, which meet together, but which do not 

 unite so as to form one continued straight line. 



That two straight linen A B, C B, meet each other in the point B, 

 forming an opening, of which B A and 

 B C are the boundaries, or limit* : this 

 opening Is called a plane rectilineal an- 

 fflf, or simply an angle. 

 The two straight lines A B, C B, in the 



1) 



forming a continued straight line, A C. 



second representation, also meet 

 each other; but they form no 

 opening or angle ; they unite in 



An angle Is sometimes referred to by simply naming the letter, placed 

 at the point In 

 which the lines * 

 forming the angle 

 meet: in this way 

 reference is made 

 to the angle B, or 

 the angle F, mean- 

 ing the opening be- 

 tween B A, B C, or that be- 

 tween F D, F E. But if there 

 be two or more openings, or 

 angles, at the same point, this 

 mode of reference will not do : 

 the rides, or boundaries, of the 

 particular one of those angles 

 meant, must also be pointed 

 out. Thus, if we wish to refer 

 to the angle whose sides are 

 B G, B H, attention must, in 

 some way, be distinctly di- 

 rected to these sides ; because, as there are several angles at 

 B, a reference to this point alone would be insufficient. The 

 plan adopted is this: not only the letter (B) at the vertex of 

 the angle, as this point is called, is used, but the other two letters 

 (G, H) which mark the sides of the angle the former letter being 

 always placed between these two. Thus GBU, or HBG, means the 

 angle whose vertex Is B, and whose sides are BO, B II. In like 

 manner A B H, or H B A , refers to the angle whose vertex is B, and 

 whose sides are B A, B H : but there is no necessity to use three 

 letters to denote an angle when there are no other angles at the 

 same point or vertex. 



VIII. 



When a straight line (A B) 

 standing on another straight line 



!C D) makes the adjacent angles 

 A B C, A B D) equal to one an- 

 other, each of the angles is called 

 a right angle ; and the straight 

 lines are said to be perpendicular 

 to each other. 



Thua the two lines, A B, C D, are perpendicular to each other. 



