Page 21 - MATINF Nr. 9-10

A generalization of Mollweide’s formula, rather Newton’s 21

Anticipating reducing the formula (3) to (1), rename elements of the figure as in Figure 2:

0
0
That is, angles α and γ have been sub-divided into α + α and γ + γ , and the segment labels
have been scrambled: a = |BC|, b = |AD|, c = |AB|, d = |CD|. Note that δ = π − β (as β and
δ are inscribed angles subtending opposing arcs). With these changes, the formula in question
becomes
0
1
0
1
c + d 1 sin (α + α + β) sin (α + α + β)
cot θ = 2 = 2 . (4)
1
1
0
0
a + b 2 cos (γ + γ − (π − β)) sin (γ + γ + β)
2 2
If we slide vertex D along the circle to coincide with C (so that E does as well), we find
0
0
that α and d shrink to nothing; γ and θ adjust to match β and γ, respectively; and β and δ
don’t change at all (see Figure 3).

0
0
Figure 3: If C = D = E, then α = 0, γ = θ and γ = β.

So, (4) becomes

1
1
c + 0 1 sin (α + 0 + β) sin (α + β)
2
2
cot γ = = . (5)
1
1
a + b 2 sin (γ + β + β) sin (γ + 2β)
2 2
Since α + β + γ = π, we have
α + β = π − γ, γ + 2β = π − (α − β)

whence (5) becomes

1
1
1
c cos γ sin π − γ cos γ
· 2 = 2 2 = 2
1
1
1
a + b sin γ sin π − (α − β) cos (α − β)
2 2 2 2
and we can write
1
a + b cos (α − β)
= 2 ,
1
c sin γ
2
which is Newton’s formula.