PROPOSITION.
If certain Magnitudes at any Rate equally exceed
ing one another, and whoſe exceſs is equal to
the leaſt of them, be ſo diſpoſed in the Balance,
as that they hang at equal diſtances, to divide
the Center of Gravity of the whole Balance
ſo, that the part towards the leſſer Magnitudes
be double to the remainder.
ing one another, and whoſe exceſs is equal to
the leaſt of them, be ſo diſpoſed in the Balance,
as that they hang at equal diſtances, to divide
the Center of Gravity of the whole Balance
ſo, that the part towards the leſſer Magnitudes
be double to the remainder.
In the ^{*} Ballance A B, therefore, let there be ſuſpended at equal di-
ſtances any number of Magnitudes, as hath been ſaid, F, G, H, K,
N; of which let the leaſt be N, and let the points of the Suſpenſions
be A, C, D, E, B, and let the Center of Gravity of all the Magnitudes
ſo diſpoſed be X. It is to be proved that the part of the Ballance B X
towards the leſſer Magnitudes is double to the remaining part X A.
ſtances any number of Magnitudes, as hath been ſaid, F, G, H, K,
N; of which let the leaſt be N, and let the points of the Suſpenſions
be A, C, D, E, B, and let the Center of Gravity of all the Magnitudes
ſo diſpoſed be X. It is to be proved that the part of the Ballance B X
towards the leſſer Magnitudes is double to the remaining part X A.
* Or Beam.
Let the Ballance be divided in two equal parts in D, for it muſt ei
ther fall in ſome point of the Suſpenſions, or elſe in the middle point be
tween two of the points of the Suſpenſions: and let the remaining di
ſtances of the Suſpenſions which fall between A and D, be all divided
into halves by the Points M and I; and let all the Magnitudes be divi-
166[Figure 166]
ded into parts equal to
N: Now the parts of F
ſhall be ſo many in num
ber, as thoſe Magnitudes
be which are ſuſpended
at the Ballance, and the
parts of G one fewer,
and ſo of the reſt. Let
the parts of F therefore be N, O, R, S, T, and let thoſe of G be N, O,
R, S, thoſe of H alſo N, O, R, then let thoſe of K be N, O: and all the
Magnitudes in which are N ſhall be equal to F; and all the Magnitudes
in which are O ſhall be equal to G; and all the Magnitudes in which
are R ſhall be equal to H; and thoſe in which S ſhall be equal to K; and
the Magnitude T is equal to N. Becauſe therefore all the Magnitudes
in which are N are equal to one another, they ſhall equiponderate in
the point D, which divideth the Ballance into two equal parts; and for
the ſame cauſe all the Magnitudes in which are O do equiponderate in
I; and thoſe in which are R in C; and in which are S in M do equi
ponderate; and T is ſuſpended in A. Therefore in the Ballance A D at
the equal diſtances D, I, C, M, A, there are Magnitudes ſuſpended ex
ceeding one another equally, and whoſe exceſs is equal to the leaſt: and
the greateſt, which is compounded of all the N N hangeth at D, the
ther fall in ſome point of the Suſpenſions, or elſe in the middle point be
tween two of the points of the Suſpenſions: and let the remaining di
ſtances of the Suſpenſions which fall between A and D, be all divided
into halves by the Points M and I; and let all the Magnitudes be divi-
166[Figure 166]ded into parts equal to
N: Now the parts of F
ſhall be ſo many in num
ber, as thoſe Magnitudes
be which are ſuſpended
at the Ballance, and the
parts of G one fewer,
and ſo of the reſt. Let
the parts of F therefore be N, O, R, S, T, and let thoſe of G be N, O,
R, S, thoſe of H alſo N, O, R, then let thoſe of K be N, O: and all the
Magnitudes in which are N ſhall be equal to F; and all the Magnitudes
in which are O ſhall be equal to G; and all the Magnitudes in which
are R ſhall be equal to H; and thoſe in which S ſhall be equal to K; and
the Magnitude T is equal to N. Becauſe therefore all the Magnitudes
in which are N are equal to one another, they ſhall equiponderate in
the point D, which divideth the Ballance into two equal parts; and for
the ſame cauſe all the Magnitudes in which are O do equiponderate in
I; and thoſe in which are R in C; and in which are S in M do equi
ponderate; and T is ſuſpended in A. Therefore in the Ballance A D at
the equal diſtances D, I, C, M, A, there are Magnitudes ſuſpended ex
ceeding one another equally, and whoſe exceſs is equal to the leaſt: and
the greateſt, which is compounded of all the N N hangeth at D, the