Final answer:
True, the moon is a sphere that is always half-illuminated by the Sun. The visible phases of the Moon from Earth depend on our view of the illuminated half and its orbital position. Our perception of the Moon's phases, like the full moon, results from the angle of sunlight and the Moon's synchronous rotation.
Step-by-step explanation:
The statement that the moon is a sphere that is always half-illuminated is true. Always half-illuminated refers to the fact that one hemisphere of the Moon is receiving sunlight at any given time, just like the Earth during the day. However, the phases of the Moon that we see from Earth are not because some part of the Moon is in darkness but because we are seeing different portions of the illuminated half. As the Moon orbits the Earth, we observe a cycle of phases, including new, first quarter, full, and third quarter, which represent different angles from which we view the illuminated half.
Certain early astronomers correctly inferred the shape and illumination of the Moon. They noted that the full moon occurs when the Moon and Sun are at opposite ends of the sky, and this suggested a spherical shape. As the Moon moves through its orbit, it is constantly half-lit by the Sun, and due to its synchronous rotation (it rotates on its axis in the same time it takes to orbit the Earth), the same side always faces us. This rotation is important because without it, we would be able to see different sides of the Moon.
The waxing and waning of the Moon's illumination, as seen from Earth, occur because as the Moon orbits, the amount and angle of the sunlight hitting the Moon's surface that we can see from our vantage point change. When a full moon is observed, the Sun fully illuminates the face of the Moon that is visible to us during the night. The misleading part in many diagrams, such as Figure 4.14 mentioned in the reference, is that they often do not to scale and represent the Moon's orbit as being much closer and more aligned with the Earth-Sun line than it actually is.