24 MATHEMATICS. 
perpendicular to each other. If, however, a point in space is to be 
determined by its polar co-ordinates, a line and two angles are required. 
Every line, straight or curved, is in analytical geometry expressed by an 
equation from which all the peculiarities of the lime may be derived by 
calculation. If we suppose all co-ordinates to be expressed in numbers, and 
indicate the abscissa by z, and the ordinate by y, then for every line the 
dependence between abscissa and ordinate of one and the same point of the 
line may be expressed by an equation, which holds good for every point 
of one and the same line. Thus for the equation of the straight line we 
have y=azx +b), or ax +by+c=o. 
Curved lines, or curves, are divided into curved lines of simple curvature 
which le in one and the same plane, and into curved lines of double 
curvature which lie in different planes. The former, to which we here 
limit ourselves, are again subdivided into algebraic, which may be 
expressed by an algebraic equality ; and transcendental, whose equations 
are transcendental, that is, consist of an infinitely great number of terms. 
Algebraic curves are divided according to the degree of their equations, 
into lines of the first, second, third, &c., order. Since, however, the 
straight line alone is expressed by an equation of the first degree, and is 
consequently the only line of the first order, we term lines of the second 
order, also, curves or curved lines of the first class; lines of the third order, 
curves of the second class, &c. 
Every curved line may have a touching line or tangent, as well as the 
circle. By this is understood a straight line which has one point in 
common with the curve, and indicates the position of the curve with 
respect to that point. Thus in pl. 3, fig. 134, a tangent is drawn through 
the point m. The part of the axis of abscissas between the ordinate and 
the tangent of a point, is called the sub-tangent. If we erect a 
perpendicular to a tangent at the point of tangency, and prolong it to 
the axis of abscissas, the part of the perpendicular (mn in the figure) 
contained between the latter and the point of tangency, is called the 
normal; that part of the axis of abscissas (np in the figure) between normal 
and ordinate, the swb-normal. 
The most important curves, as well as those of most frequent occurrence, 
belong to the first class. These are the ellipse, parabola, and hyperbola. 
They are also called the conic sections, because they are produced by 
intersecting a cone by a plane in various directions. If the plane of 
intersection be parallel neither to the axis nor side of the cone, the outline 
of intersection is called an ellipse (pl. 1, fig. 55). This is a closed curve 
line, having the peculiarity that in one of its axes there are two points. 
termed the foci, so situated that the sum of the distances of any point of 
the curve from the foci, will be the same. The more the direction of the 
generating plane approaches a perpendicular to the axis of the cone, the 
more do the foci approach each other; and when the perpendicular is 
attained, the foci meet in the centre, and the ellipse becomes a circle. 
Every line passing through the centre of an ellipse, is called a diameter ; 
the longest diameter (called major axis) is that which passes through the 
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