12 MATHEMATICS. 
described in and around it, and their circumferences determined, that of 
the circle will be intermediate. In this way Archimedes determined 
the ratio of the diameter to the circumference as 7 : 22, and Ludolph, as 
1:314159. The latter number is employed and indicated by z, as the 
ratio to a diameter of 1 (or unity). Accordingly, since circles are to each 
other as their diameters, the circumference of any circle may be found by 
multiplying its diameter by 7 (= 3.1415926). 
Every circle may be regarded as a regular polygon of an infinite number 
of sides; hence, also, as a triangle whose base is equal to the circum- 
ference of the circle, and whose altitude is the radius. We consequently 
obtain the area of a circle by multiplying the circumference by half the 
radius, or according to the preceding proposition, by multiplying the 
second power of radius by 7. A sector is equal to a triangle whose base is 
the length of the arc, and whose altitude is equal to the radius (pil. 3, 
jig. 84). 
Allied to the circle are the symmetrically curved lines, the oval and the 
ovate: each one consisting of four elongated quadrants. In the former 
the quadrants are all equal; in the latter only the two lying on the same 
side of the short axis. The following are some constructions of ovals. In 
pl. 2, fig. 87, an isosceles triangle, CDE, is constructed upon the base CD, 
and under it another and equal arc, CDF. From C and JD, with any 
radius, CA—DB, describe arcs intersecting the equal sides produced of 
each triangle in G and J, H and K: and finally, connect these points by 
ares described with the radii FH=EK, from F and FE as centres. Fig. 
88 agrees with the preceding construction, except in that the two equal 
triangles employed, are equilateral. In fig. 88, the length of the oval, or of 
the major axis, AB, is given. Divide it at C and D, into three equal parts. 
From the points C and D, with radii equal to 4 AB, describe circles 
intersecting in HZ and F. From these points draw two diameters in each 
circle, GF, EI, FH, EK, and with one of these diameters as radius, from 
the points # and F, describe the ares JK and GH, completing the outline. 
In this construction the breadth of the oval is a little more than # of the 
length. An oval of less breadth, with the same length, AB, may be thus 
obtained (fig. 89). Divide AB into four equal parts, and from the points 
of division, C, D, E, with a radius equal to 1 of AB, describe three circles, 
intersecting ea@h other in F, G, H, and I. Through these points draw in 
the first and third circles the diameters MH, NI, F'K, and GL, projonging 
them until they intersect in O and P. From O and P, with radii OK= 
PM, describe the arcs KL and MN. In this construction the breadth of 
the oval is not quite 2 the length. In fig. 91, a half-oval of given length, 
AB, is constructed in the following manner. From the points A and B, in 
the line AB, any part, AH — BK, is taken, and with this distance as radius, 
ares are described cutting each other in J and L; with the distance 
between J and JZ as radius, describe, from these points as centres, arcs 
cutting each other beneath AB; finally, from D as centre, complete the 
circle by the are IL. 
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