HIGHER GEOMETRY. 27 
point 2’, lying half way between the two, called the vertex, is the furthest 
distance from the base, this distance being equal to a diameter of the gene- 
rating circle. When the generating point lies without the circle, the 
cycloid produced is called the curtate or contracted cycloid (fig. 136). If 
it be within, it becomes prolate, or elongated ( fig. 135°). 
If the circle with its generating point revolve, not on a straight line, but 
upon the circumference of another circle, fixed, and in the same plane, then 
the curve produced will be an epicycloid, if the revolution be on the outer 
side of the fixed circle, and a hypocycloid when on the inner. We have 
here, as in the cycloid, the same distinctions into ordinary or common 
epicycloid (figs. 137, 140); prolate or elongated (figs. 138, 141); and 
curtate or contracted ( figs. 139, 142). 
Another transcendental curve, or rather genus of curves, is the quadra- 
trix, a curve line, described on a common axis with any other given curve, 
and indicating by its ordinates the area of the latter curve, since their 
ordinates are as the areas answering to the corresponding abscissas, with 
the given line as axis of ordinates. The oldest quadratrix is that of Dinos- 
tratus (fig. 126): let ab be a diameter of a circle, and the triangle acb so 
constructed that the height cn: the base ab:: angle cab: a right angle, then 
c will be a point of the quadratrix, whose equation iszg >> —y’. 
Another construction of the quadratrix is given by Tschirnhausen (ig. 
127). Let adb be a semicircle, o the centre, and m a point in the circum- 
ference, furthermore 2, a point of the diameter which lies in such a manner 
that quadrant ad: arc am::ao:an, and draw through m and n to ao and do 
parallels meeting in p, then p will be a point of this quadratrix whose 
equation is y = sin. a Ee 
2a 
We have still to explain the meaning of the terms evolutes and involutes. 
Suppose that on the elevated side of a curved line, a perfectly flexible thread 
be laid. If, now, this thread be kept continually stretched, and unlapped by 
degrees from the curved line, its end will describe a new curve, which is 
called the involute of the old curve, this latter being the evolute of the former. 
Thus the parabola of Neil is the evolute of the common parabola. In pi. 3 
( fig. 128), the involute of the circle is represented, which is constructed as 
follows: Through any points, 6, c, d, of a circle, tangents are drawn, 
and on these the points, 0’, c', d’, so taken that the tangents, bb’, cc’, dd’, 
shall equal the length of arcs of circles contained between the points of 
tangency and a fixed point, a. The points 0’, c’, d’, will then be points of the 
involute of the circle, which is a transcendental curve. 
Among the solids produced by the higher curves, the spheroid is the most 
important, resembling the sphere, and like it having a centre in which every 
diameter is bisected, but differing in these diameters being of unequal 
length (pl. 2, fig.73). Among all the diameters of a spheroid, three, 
perpendicular to each other, called its axes, are best worthy of mention. 
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