Acute is from the Latin word acus for needle, with derivatives generalizing to anything pointed or sharp. The root persists in the words acupuncture (to treat with needles) and acumen (mentally sharp). An acute angle then, is one which is sharp or pointed.
Angle comes from the Latin root angulus, a sharp bend. As with many g sounds the transfer from Latin to the German and English languages switched to a k spelling. The word ankle is from the same root.
Algebra comes from an Arabic book that revolutionized how mathematics was done in western cultures. “Al-jebr w’al-mugabalah” written by Abu Ja’far Ben Musa (about 825 AD) who was also known as al-Khowarazmi. The phrase Al-jebr at the start of the title became the word Algebra in western languages. The phrase means “the reunion of broken parts”.
Algorithm, as it is used in mathematics means a systematic procedure to solve a problem. The word is derived from the name of the Hindu mathematician, al-Khowarazmi (See algebra). The first use of the word I am aware of was by G W Liebniz in the late 1600. It remained a little known and little used term in western mathematics until the Russian mathematician Andrei Markov (1856-1922) introduced it. The term became very popular in the areas of math focused on computing and computation.
Analogy The word analogy comes from the early greek roots ana + logos . Logos was the early greek root for lots of related mental constructions such as word, speech, logic, and reason. An analogy refers to things which share a similar relation. Originally it was more of a mathematical term interchangeable with ratio or proportion; as in “2,4,8 is analogous to 3,6,12”. Later this idea of similar relations was extended to things which shared a logical relationship. Analog clocks and computers are so named because they operate off mechanical objects (gears and pulleys) that transform motions in proportional movements.
An ex-teacher by the name of Pat has a nice list of math terms and their etymology on the Internet. Love it! Here are some of the terms (take with a grain of salt and double check: he or she is wrong about “centurion,” anyway):