The surface of the Moon is not smooth, uniform, and precisely spherical as a great number of philosophers believe it to be, but is uneven, rough, and full of cavities and prominences, being not unlike the face of the Earth, relieved by chains of mountains and deep valleys.

If you could see the earth illuminated when you were in a place as dark as night, it would look to you more splendid than the moon.

We must say that there are as many squares as there are numbers.

And believe me, if I were again beginning my studies, I should follow the advice of Plato and start with the mathematical sciences, which proceed very cautiously and admit nothing as established until it has been rigorously demonstrated.

Nature is written in mathematical language.

When the moon is ninety degrees away from the sun it sees but half the earth illuminated (the western half). For the other (the eastern half) is enveloped in night. Hence the moon itself is illuminated less brightly from the earth, and as a result its secondary light appears fainter to us.

Philosophy itself cannot but benefit from our disputes, for if our conceptions prove true, new achievements will be made; if false, their refutation will further confirm the original doctrines.