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Computing Gravitational Constant G from Cavendish's data

I use this formula (From Wikipedia)

\begin{equation*} G = \frac{2\; \pi^2 \;L \;r^2 \;\theta}{M \;T^2} \end{equation*}

G = Gravitational constant

L = Length of torsion balance (the distance between the centers of balls)

r = The distance of attraction (between weights and balls)

θ = Deflection of the arm from its rest position due to gravitational attraction

M = Mass of attracting lead weight

T = Natural period of oscillation of the balance

I take \(\theta\) and \(N\) from the 4th experiment, the rest are constants,

L = 1.862 m

r = 0.2248 m

\(r^2\) = 0.05053 \(m^2\)

\(\theta\) = 0.00806788 radians

M = 158.04 kg

T = 421 s

\(T^2\) = 177241 \(s^2\)

Cavendish gives the displacement of the arm due to attraction in terms of \(B\) which is the number of scale divisions. The scale is 38.3 inches away from the center. Each scale division is 0.050 inches. Therefore, each degree subtends an angle of 1/766 = 0.0013054 radians at the center. For this experiment,

\(B\) = 6.18 and

\(\theta\) = 6.18 * 1/766 = 0.008067 radians
Substituuting in the numbers,

\begin{equation*} G = \frac{2 \times \pi^2 \times 1.836 \times 0.05053504 \times 0.00806788}{158.04 \times 177241} = 5.27501\times 10^{-10} \end{equation*}

This value of G computed from Cavendish's experimental data is 8 times bigger than the accepted value of \(G = 6.67430 \times 10^{-11}\)

Notes

References are from Cavendish's article of 1798.
On page 509, Cavendish gives the distance of the ivory scale from the center of motion: "But the ivory scale at the end of the arm is 38.3 inches from the center of motion." On the same page he says that each division of the ivory scale is 1/20 of an inch, that is, 0.050 inch. By the radian rule, \(\theta = \frac{l}{r} = \frac{0.050}{38.3}= 0.0013054\).
The distance between centers of ball (the length of the arm) is given on page 509 as 73.3 inches, which is, 1.862 meters.

Table of values computed from Cavendish's data

On the last column I divided the value of G computed from Cavendish's data (G_cav) to the accepted value of G (G_true).
First 4 experiments were done with a soft wire. These give a G about 4 times greater than the modern value.
The rest of the experiments are done with a stiffer wire. These give a G about 8 times greater than the accepted value.
But the density of the earth computed from soft wire and the stiffer wire give the same 5.5 value.
So it seems that G is tied to the stiffness of the wire but not the density of the earth.

Exqeriment	B	B in Radian	N (sec)	G_Cav	G_Cav/ G_True
#1	14.1	0.0184	895	2.7E-10	4.0
#2	15.45	0.0201	882	3.0E-10	4.5
#3	15.22	0.0198	879	3.0-10	4.5
#3	14.50	0.0189	894	2.7E-10	4.1
#4	6.18	0.0080	421	5.3E-10	7.9
#4	5.92	0.0077	423	5.0E-10	7.5
#5	5.90	0.0077	425	5.0E-10	7.4
#5	5.98	0.0078	425	5.07E-10	7.5
#7	3.15	0.0041	424	2.68E-10	3.9
#9	6.32	0.0082	418	5.54E-10	8.2
#10	6.15	0.0080	419	5.37E-10	7.9
#11	6.07	0.0079	421	5.25E-10	7.7
#12	6.09	0.0079	423	5.22E-10	7.7
#13	6.12	0.0079	426	5.17E-10	7.6
#13	5.97	0.0077	427	5.02E-10	7.4
#14	6.27	0.0081	426	5.30E-10	7.8
#14	6.13	0.0080	426	5.18E-10	7.6
#15	6.34	0.0082	427	5.33E-10	7.9
#16	6.10	0.0079	436	4.92E-10	7.3
#17	5.78	0.0075	422	4.97E-10	7.3
#17	5.64	0.0073	423	4.83E-10	7.1