A storm--in Shakespeare or a teapot
(7 letters) 
a tempest
In the acronym HOMES, it's the "H"
Huron
Idiomatically, you do this "at straws" in making a feeble argument
grasp
Details are sketchy on him; was he a 9th century B.C. guy? 8th? From Ionia? Did he really create the 2 epic poems he's credited with?
Homer
IR: This radiation just below visible light on the spectrum
Infrared
Doorway part brides have been carried over
(9 letters) 
threshold
Differing by a letter, these 2 rivers both rise in Switzerland; one is about 830 miles long & the other, more than 500
the Rhone & Rhine
This pencil product can also be used as an industrial lubricant
graphite
We have a theorem about this guy who was born in Ionia around 570 B.C.: a2 + b2 = c2
Pythagoras
G: This prefix meaning "billion"; it comes before "hertz" & "watts"
Giga
Math term for a digression
(7 letters) 
a tangent
You'll find the United Arab Emirates at the south end of this body of water
the Persian Gulf
It can mean to wrestle, or to cope with a difficult problem
grapple
The first great civilization to emerge around Greece flourished circa 2000 B.C. with the Minoans on this island
Crete
HTTP: Hypertext transfer this
Protocol
Boss Tweed's corrupt Hall
(7 letters) 
Tammany
Under an agreement first drawn up in 1903, the U.S. maintains a strategically important naval base on this Cuban bay
Guantanamo
The Vietnam Veterans Memorial wall is sculpted from this durable rock
granite
In the long run, you'll know General Miltiades lost only around 200 men & the Persians, 6,400, in the 5th century B.C. Battle of this
Marathon
PPM: Used to describe the concentration of small amounts of something in a solution
parts per million
A really fast heart rate
(11 letters) 
tachycardia
In ancient times the Dardanelles Strait was called this & was associated with the story of Hero & Leander
the Hellespont
A top music industry award gets its name from this old-time record player with a big horn
gramophone
This 400s B.C. physician had a school at Cos where he taught for fees; no word on if he had a loan program
Hippocrates
Pa: This unit of pressure named for a 17th century Frenchman
Pascal