If ever there was a motto thought up by a cynical teacher then it is this one, in spe laboramus, which means: we work in hope. It is the motto of North London Collegiate School, a private day school for girls, which was founded in London in 1850 and now has associated schools in several other countries.
Spe is the ablative singular of spes, spei a fifth declension noun. The ablative singular is used to agree with the preposition in.
spes, spei f 5th declension noun - hope
The phrase "ex pede Herculem" is literally from the foot Hercules. The meaning is that from the foot we can deduce the size of Hercules. There is a Wikipedia article about this phrase.
The idea is attributed to the Greek philosopher Pythagoras (of right angled triangles fame). In the modern world, it is an essential tool in paleontology, where the size of long extinct animals is estimated from measuring a single bone.
ex pronoun with ablative from, out of
pede ablative singular of pes, pedis m noun a foot
