31(19)
A SECOND
SUPPLEMENT,
BEING
Monſ. DE FERMAT’S Treatiſe on
Spherical Tangencies.
SUPPLEMENT,
BEING
Monſ. DE FERMAT’S Treatiſe on
Spherical Tangencies.
PROBLEM I.
HAVING four points N, O, M, F, given, to deſcribe a ſphere which
ſhall paſs through them all.
ſhall paſs through them all.
Taking any three of them N, O, M, ad libitum, they will form a triangle,
about which a circle ANOM may be circumſcribed, which will be given in
magnitude and poſition. That this circle is in the ſurface of the ſphere
ſought appears from hence; becauſe if a ſphere be cut by any plane, the
ſection will be a circle; but only one circle can be drawn to paſs through the
three given points N, O, M; therefore this circle muſt be in the ſurface of
the ſphere. Let the center of this circle be C, from whence let CB be
erected perpendicular to it’s plane; it is evident that the center of the ſphere
ſought will be in this line CB. From the fourth given point F let FB be
drawn perpendicular to CB, which FB will be alſo given in magnitude and
poſition. Through C draw ACD parallel to FB, and this line will be
about which a circle ANOM may be circumſcribed, which will be given in
magnitude and poſition. That this circle is in the ſurface of the ſphere
ſought appears from hence; becauſe if a ſphere be cut by any plane, the
ſection will be a circle; but only one circle can be drawn to paſs through the
three given points N, O, M; therefore this circle muſt be in the ſurface of
the ſphere. Let the center of this circle be C, from whence let CB be
erected perpendicular to it’s plane; it is evident that the center of the ſphere
ſought will be in this line CB. From the fourth given point F let FB be
drawn perpendicular to CB, which FB will be alſo given in magnitude and
poſition. Through C draw ACD parallel to FB, and this line will be