Illustration of Proposition 14 of the book On Sizes.
Proposition 14 of the book On Sizes states that the straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the [circumference] within the earth's shadow a ratio greater than that which 675 has to 1 (Heath 1913:403). This is grossly understated, as can be found from the equation above, where θ is the moon's angular size, and α the angular size of the earth's shadow. (The moon's distance BC is an inconsequential variable.) Note that CM is always perpendicular to BM. For the scale intended in On Sizes, move the sliders to θ = 2° and α = 4°. For true scale, move the sliders to θ = 0.5° and α = 1.4°.