DOC.

14

EINSTEIN AND BESSO MANUSCRIPT

421

[p.

30]

(Einstein)

[134]At

the

head

of

[p.

30],

above

the first

horizontal

line,

a

value for the

Newtonian

gravitational

constant

K

is

computed

by

considering

the orbit of

the

Earth around

the

sun as

a

circle and

setting

the

gravitational

force

on

the Earth

equal

to

the

centripetal

force. What

is

interesting

about the

calculation

is

that

in

the

course

of

it,

the mistake of

a

factor

10

in

the value used

on [p.

26]

and

[pp.

28-29]

for the

mass

of

the

sun

is

discovered. The calculation

proceeds as

follows.

[Eq.

184]

for K

is

obtained from

the

condition that the

gravitational

and

centripetal

forces

on

the Earth

balance each other

(on

[p.

35],

[eq.

214], and

[p.

42]

[see

note

195],

the

same equation

is

used

to

determine

K). In

the three lines below

[eq.

184],

the various factors

in this

expression

for

K

are

evaluated. For the

interpretation

of

[eq.

185]

and the numbers

to

the

right

of

[eq.

184], see

note 120.

An

error

occurred

in

computing

r3: 4.8

•

1039

should be 3.2

•

1039.

The results

are

inserted into

[eq. 184].

When the value for M

is not

corrected,

one

finds

K

=

9.6

•

10-9

m3/gs2,

which

is

a

factor

in

the order of

10 too

small. This

may

have

prompted

the correction of

the

erroneous

factor 10 in the

value for

M.

On

the

other hand, if

obtaining

a

wrong

value for

K

made

Einstein realize there had

to be

an error

somewhere,

it is

puzzling

that

the

remaining discrepancy

of

some

50% due

to

the

error

in r3

does

not

seem to

have bothered him. The

value is

used

in

the

calculation farther down

on

the

page (see

note

135).

[135]In the middle of

[p. 30],

between the

two

horizontal

lines,

another

attempt is

made

to

find

the

perihelion

advance of

Mercury

in

the

field

of

a

static

sun.

The end result of

this calculation,

in

fractions of

tt

per

half

a

revolution, is 1.65

•

10-8. The

calculation,

which contains several

errors, proceeds as

follows. The left-hand

side

of

[eq. 186]

is

equivalent

to

1/4(Ac0/F)2

(see

[eqs.

177-179]

on [p. 26];

C

=

F). In

[eq. 187],

equivalent to

[eq. 178],

C is computed.

Note that

the

eccentricity

e

of

Mercury's

orbit

is neglected,

and

that the

semi-major

axis

a

is

computed

via

a

=

(rm/re)re (see

note

122).

The

value

found for

C,

2.75

•

1019 cm2/s,

is too

high. Using

the numbers

given on

[p.

26]

and

taking

the

orbit's

eccentricity into

account,

one

arrives

at

C

=

2.67

•

1019

cm2/s.

The

erroneous

value for

C is

used

together

with

the

values

for

K and

M

in

[eqs.

184-185] to

compute

KM/cC,

giving

2.3

•

10-4. The

right-hand

side of

[eq. 186]

is

found

by

squaring

this

number.

Using

the numbers

given

on

[p. 26],

correcting

the

error

of

a

factor

10

in

the

mass

of the

sun,

one

obtains the value

1.65

•

10-4

rather than 2.3

•

10-4

for

KM/cC. In

[eq.

188],

[eq. 186]

is

multiplied

by

5/16.

In

fact,

in order to find the

perihelion

advance,

the

equation

should

be

multiplied

by

an

extra

factor

4

(see

[p.

26],

[eq. 177]; [p. 39],

[eq.

253]).

When this

is done,

using

the corrected value for

KM/cC,

the end result becomes

3.4.

10-8.

[136]After

discovering

the

error

in the value for the

mass

of the

sun

in

[eq. 185],

Einstein

presumably expected

to find

a

perihelion

advance

that

would

be

a

factor

100

smaller than the

advance he

had found

on [p.

28].

In fractions of

n

per

half

a

revolution, this

would be 3.4

•

10-8

(see note

128),

which

translates into

18" per century.

Presumably,

this

is

why

Einstein

wrote

this

number

immediately

below

[eq.

188].

When the various

errors

in

the calculation

on

[p. 30]

are

corrected,

this

is

indeed the result

one

finds

(see note

135).