CHAPTER I.
THE SITUATION OF A POINT AND LINE IN A PLANE.
1. P being the given point, draw from it,to YO andYO, and parallel to the same, thelines Pm and Pn. As there can be bnt onePm and one Pn, thus drawn from the point P,the position of this point is evidently deter-mined when Pm and Pn are known, XO andYO being supposed to be given in position.
So also any point is similarly determined whenin like manner, its situation, relative to twolines given in position, is known.
The two different kinds of lines whichhave here been used, in the determination of a point, and whichare of like importance in all calculations on this subject, havereceived peculiar names. XO and YO, or as extended, XX and
Y Y, are termed axes, as they arc the fixed lines to which all otherlines are referred. Pm and Pn are called the coordinates of thepoint P ; those parallel to one axis, XX, are designated by theletter x, and those parallel to Y Y, by the letter y. In like man-ner the axis XX is termed the axis of x, the coordinates x beingin that direction ; and for a like reason, the axis YY" is denomi-nated the axis of y. The axes bisect each other in the point O,which is termed the centre or origin. If the point P were movedtowards m, the line Pm would decrease, and finally would becomeequal to 0 ; and then, if continued, would increase on the oppositeside of the axis. If, therefore, the sign + is applied to Pm, thesign — should be applied to P'"m ; and so generally, if lines above
Y Y are -f-, those below will be —: so also, if lines to the rightof XX' are — those to the left will be +. It follows, hence, thatn O, or the distance of any point, as n, in the axis of x, from O,may equal ± a ; and ??iO = ± b, designating nO and mO, by a andb ; that is, the point n may be at equal distances either above orbelow the origin and the point m, either to the right or left of it.