Their greatest achievements were made in Mathematics and Medicine/engineering.
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Euclid mathematics, Hippocrates medicine.
Hippocrates and Euclid made their greatest achievements in the fields of Mathematics, Medicine, and Engineering. Found in "Halt Social Studies: World History" in Chapter 9, Section 4 - Greek Achievements on page 282.
Euclid's greatest accomplishment was making a compilation of all the mathematics known at the time when he lived, about 2400 years ago. He also developed proofs to previously unproven hypothesis and collated these along with theories developed by other ancient Greek and Egyptian mathematicians and philosophers. These were published in a 13 volume manuscript now known as Euclid's Elements, which forms the basis of the geometry taught in schools to this day.
Euclid is best known for being the Father of geometry. He also created the oldest mathematical school teaching book called 'Euclid's Elements' which is still being used in school's world wide.
He made his 'discoveries' by studying and compiling the works of other mathematicians who lived many years before him.