CIRCLE. 
ratio of their diameters, and consequently 
of their radii. 
A circle is equal to a triangle, the base of 
■which is equal to the periphery, and its alti- 
tude to its radius : circles therefore are in a 
ratio compounded of the peripheries and 
the radii. 
To find the proportion of tlie diameter of 
a circle to its circumference. Find, by 
continual bisection, the sides of the incribed 
polygon, till you arrive at a side subtending 
any arch, however small ; this found, find 
likewise the side of a similar circumscribed 
polygon ; multiply each by the number of 
the sides of the polygon, by which you will 
have the perimeter of each polygon. The 
ratio of the diameter to the periphery of the 
circle vrill be greater than that of the same 
diameter to the perimeter of the circum- 
scribed polygon, but less than that of the in- 
scribed polygon. The difference of the two 
being known, the ratio of the diameter to 
the periphery is easily had in numbers very 
nearly, though not justly true. Tims Archi- 
medes fixed the proportion at 7 to 22. 
Wolfius finds it as 10000000000000000 to 
31415926535897932: and the learned Mr. 
Machin has carried it to one hundred places, 
as follows : if the diameter of a circle be 1, 
the circumference will be 3,14159, 26535, 
89793, 23846, 26433, 83279, 50288, 41971, 
69399, 37510, 58209, 74944, 59230, 78164, 
05286, 20899, 86280, 34825. 34211, 70679 
of the same parts. But the ratios generally 
used in practice are that of Archimedes, 
and the following ; as 106 to 333, as 113 to 
355, as 1702 to 5347, as 1815 to 5702, or as 
1 to 3.14159.- 
CiRCLE, the quadrature of the, or the 
manner of making a square, whose surface 
is perfectly and geometrically equal to that 
of a circle, is a problem that has employed 
the geometricians of all ages. 
Many maintain it to be impossible; Des 
Cartes, in particular, insists on it, that a 
right line and a circle being of different na- 
tures, there can be no strict proportion be- 
tween tliem : and in effect we are at a loss 
for the just proportion between the diameter 
and circumference of a circle. ' 
Archimedes is the person who has come 
nearest the trutli ; all the rest have made 
paralogisms. Charles V. offered a reward 
of 100,000 crowns to the person who should 
solve this celebrated problem ; and the States 
of Holland have proposed a reward for the 
same purpose. 
CincLE, great, of the sphere, that which 
having its centre in the centre of the sphere, 
divides it into two equal hemispheres ; such 
are the equator, ecliptic, horizon, the coin- 
res, and the azimuths, &c. See Equator, 
Ecliptic, &c. 
Circle, lesser, of the sphere, that which 
having its centre in the axis of the sphere, 
divides it into two unequal parts ; these are 
usually denominated from the great circles 
to which they are parallel, as parallels of tlie 
equator. 
CiRC/.E of curvature, a circle, the curva- 
ture of which is equal to that of a certain 
curve at a given point. 
Circle, horary, on the globe, a brazen 
circle fixed on every globe with an index, 
to shew how many hours, and consequently 
how many degrees any place is east or west 
of another. 
Circle of perpetual apparition, one of the 
lesser circles, parallel to the equator, de- 
scribed by any point touching the northern 
point of the horizon, and carried about with 
the diurnal motion : all the stars included 
within this circle are always visible above 
the horizon. 
Circle of perpetual occultation, another 
circle at a like distance from the equator, 
on the soutli, containing all those stars which 
never appear in our hemisphere. 
Circles, diurnal, are immoveable circles, 
supposed to be described by the several 
stars and other points of the heavens, in 
their diurnal rotation round the earth ; or 
rather, in the rotation of the earth round 
its axis. 
Circles of latitude, or secondaries of the 
ecliptic, are great circles perpendicular to 
the plane of the ecliptic, passing through the 
poles of it, and through every star and pla- 
net. Ttmy serve to measure the latitude of 
the stars, which is an arch of one of those 
circles intercepted between the star and the 
ecliptic. 
Circles of longitude are several lesser 
circles parallel to the ecliptic, still diminish- 
ing in proportion as they recede fi'om it; on 
these the longitude of the stars is reckoned. 
Circles of decimation, on the globe, are 
with some writers, the meridians on which 
the declination or distance of any star from 
tlie equinoctial is measured. 
Circles, horary, in dialling, are the lines 
which shew the hours on dials, though 
these be not drawn circular, but nearly 
straight. 
Circles, polar, are parallel to the equa- 
tor, and at the same distance from the 
poles that the tropics are from the equator. 
See Arctic. 
