Axiom is from the Greek ἀξίωμα :
that which is thought fit, a requisite; that which a pupil is required to know beforehand; a self-evident principle.
Postulate is from the Latin postulo :
I ask, request, desire, demand. I pretend, claim.
See : Anthony Lo Bello, Origins of mathematical words, John Hopkins UP (2013), page 41 :
Euclid did not use the word axiom, instead calling his geometrical assumptions αἴτηματα, demands, postulates, and the nongeometrical ones koinai ennoiai, common notions. It was Proclus (fifth century A.D.), in his Commentary on the First Book of Euclid's Elements, who first called the latter statements axioms, and his decision has prevailed to this day.
The medieval Latin translators rendered αἴτηματα by petitiones, requests, and koinai ennoiai by scientia universaliter communis, knowledge common to everyone.
αἴτημα is from αἰτέω : I ask, request, demand.
Thus, it has the same meaning of the Latin : postulo, that is the correct Latinn translation of aitēma.
See also Aristotle, Post.An., I :
[76a32] I call principles in each genus those which it is not possible to prove to be.
[76a38] Of the things they use in the demonstrative sciences some are proper to each science and others common — but common by analogy, since things are useful in so far as they bear on the genus under the science. Proper: e.g. that a line is such and such, and straight so and so; common: e.g. that if equals are taken from equals, the remainders are equal.
Thus, we can sey that Euclid organizes his Elements according to Aristotle's view :
there are five common notions (later called: axioms), i.e. self-evident truths of a general nature, and five postulates (αἴτηματα), principles specific of the geometrical science.
Hypothesis is from Greek ὑπόθεσις from hupotíthēmi, “I set before, suggest”.
See also : Arpad Szabò, The Beginnings of Greek Mathematics, Reidel (1978), Ch.3.26 The difference between postulates and axioms, page 302 :
One of the facts which has emerged from our discussion [...] is that aitemata and axiomata were synonymous. Furthermore, the manner in which they were used by every ancient mathematician except Euclid suggests that these terms were interchangeable with a nuber of other words having to do with the foundations of
For example, Book I of Archimedes' On the Sphere and Cylinder begins with two lists of such principles. Tho assertions in the first list are called axiomata and are obviously definitions, whereas those in tho second list are called lambanomena and seems to be related to Euclid's postulates.