1. Mother: Maternal :: Stepmother: ?

Annotated solutions.

If there is more than one word that fits the analogy, we list the best
word first.  Goodness of fit considers many factors, such as parallel
spelling, pronunciation or etymology.  In general, a word that occurs
in Merriam-Webster's Third New International Dictionary is superior to
one that does not.  If we are unsure of the answer, we mark it with
a question mark.

Most of these answers are drawn from Herbert M. Baus, _The Master
Crossword Puzzle Dictionary_, Doubleday, New York, 1981.  The notation
in parentheses refers to the heading and subheading, if any, in Baus.

1. Mother: Maternal :: Stepmother: Novercal (STEPMOTHER, pert.)
2. Club: Axe :: Claviform: Dolabriform, Securiform (AXE, -shaped)
    "Claviform" is from Latin "clava" for "club"; "securiform" is from
    Latin "secura" for "axe"; "dolabriform" is from Latin "dolabra" for "to
    hit with an axe."  Thus "securiform" has the more parallel etymology.
    However, only "dolabriform" occurs in Merriam-Webster's Third New
    International Dictionary.
3. Cook Food: Pressure Cooker :: Kill Germs: Autoclave (PRESSURE, cooker)
4. Water: Air :: Hydraulic: Pneumatic (AIR, pert.)
5. Prediction: Dirac :: Proof: Anderson (POSITRON, discoverer)
6. Raised: Sunken :: Cameo: Intaglio (GEM, carved)
7. 1: 14 :: Pound: Stone (ENGLAND, weight)
8. Malay: Amok :: Eskimo Women: Piblokto (ESKIMO, hysteria)
9. Sexual Intercourse: A Virgin :: Bearing Children: A Nullipara
10. Jaundice, Vomiting, Hemorrhages: Syndrome :: Jaundice: Symptom (EVIDENCE)
11. Guitar: Cello :: Segovia: Casals (SPAIN, cellist)
12. Bars: Leaves :: Eagle: Stars (INSIGNIA)
13. Roll: Aileron :: Yaw: Rudder (AIRCRAFT, part)
14. 100: Century :: 10,000: Myriad, Banzai? (NUMBER)
     "Century" usually refers to one hundred years, while "myriad" refers
     to 10,000 things, but "century" can also mean 100 things.  "Banzai"
     is Japanese for 10,000 years.
15. Surface: Figure :: Mobius: Klein
16. Logic: Philosophy ::
    To Know Without Conscious Reasoning: Theosophy (MYSTICISM)
     There are many schools of philosophy that tout the possibility of
     knowledge without conscious reasoning (e.g., intuitionism).
     "Theosophy" is closest in form to the word "philosophy."
17. Alive: Parasite :: Dead: Saprophyte (SCAVENGER)
18. Sea: Land :: Strait: Isthmus (CONNECTION)
19. Moses: Fluvial :: Noah: Diluvial (FLOOD, pert.)
20. Remnant: Whole :: Meteorite: Meteoroid? (METEOR)
     A meteorite is the remains of a meteoroid after it has
     partially burned up in the atmosphere.  The original meteoroid
     may have come from an asteroid, comet, dust cloud, dark matter,
     supernova, interstellar collision or other sources as yet unknown.
21. Opossum, Kangaroo, Wombat: Marsupial ::
    Salmon, Sturgeon, Shad: Andromous (SALMON)
22. Twain/Clemens: Allonym :: White House/President: Metonym (FIGURE, of speech)
23. Sculptor: Judoka :: Fine: Martial (SELF, -defense)
24. Dependent: Independent :: Plankton: Nekton (ANIMAL, free-swimming)
25. Matthew, Mark, Luke, John: Gospels ::
    Joshua-Malachi: Nebiim (HEBREW, bible books)
26. Luminous Flux: Lumen :: Sound Absorption: Sabin (SOUND, absorption unit)
27. 2: 3 :: He: Li (ELEMENT)
28. Growth: Temperature :: Pituitary Gland: Hypothalamus (BRAIN, part)
29. Spider: Arachnoidism :: Snake: Ophidism, Ophidiasis, Ophiotoxemia
     None of these words is in Webster's Third.
30. Epigram: Anthology :: Foreign Passages: Chrestomathy, Delectus (COLLECTION)
     These words are equally good answers.
31. Pathogen: Thermometer :: Lethal Wave: Dosimeter? (X-RAY, measurement)
     What does "lethal wave" refer to?  If it is radiation, then
     a dosimeter measures the dose, not the effect, as does a thermometer.
32. Russia: Balalaika :: India: Sitar, Sarod (INDIA, musical instrument)
     Both are guitar-like instruments (lutes) native to India.
33. Involuntary: Sternutatory :: Voluntary: Expectorant? (SPIT)
     A better word would be an agent that tends to cause snorting or
     exsufflation, which is the voluntary, rapid expulsion of air from
     the lungs.
34. Unusual Hunger: Bulimia ::
    Hunger for the Unusual: Allotriophagy, Pica (HUNGER, unusual)
    These words are synonyms.
35. Blind: Stag :: Tiresias: Actaeon (STAG, changed to)
36. River: Fluvial :: Rain: Pluvial (RAIN, part.)
37. Country: City :: Tariff: Octroi (TAX, kind)
38. $/Dollar: Logogram :: 3, 5, 14, 20/Cent: Cryptogram (CODE)
39. Lung Capacity: Spirometer ::
    Arterial Pressure: Sphygmomanometer (PULSE, measurer)
40. Gold: Ductile :: Ceramic: Fictile (CLAY, made of)
41. 7: 8 :: Uranium: Neptunium (ELEMENT, chemical)
42. Judaism: Messiah :: Islam: Mahdi (MOHAMMEDAN, messiah)
43. Sight: Amaurosis :: Smell: Anosmia, Anosphresia (SMELL, loss)
     These words are synonyms.
44. Oceans: Cousteau :: Close Encounters of the Third Kind: Spielburg, Truffaut
     Steven Spielburg was the person most responsible for the movie;
     Francois Truffaut was a French person appearing in the movie.
45. Diamond/Kimberlite: Perimorph ::
    Fungus/Oak: Endophyte, Endoparasite (PARASITE, plant)
     An endoparasite is parasitic, while an endophyte may not be.  Which
     answer is best depends upon the kind of fungus.
46. Compulsion to Pull One's Hair: Trichotillomania :: 
    Imagine Oneself As a Beast: Zoanthropy, Lycanthropy
     Neither word is exactly right: "zoanthropy" means imagining oneself
     to be an animal, while "lycanthropy" means imagining oneself to be
     a wolf.
47. Cross: Neutralism :: Hexagram: Zionism (ISRAEL, doctrine)
48. Wing: Tail :: Fuselage: Empennage, Engines, Waist? (TAIL, kind)
     "Empennage" means the tail assemply of an aircraft, which is more a
     synonym for "tail" than "wing" is for "fuselage."  The four primary
     forces on an airplane are: lift from the wings, negative lift from
     the tail, drag from the fuselage, and thrust from the engines.  The
     narrow part at the end of the fuselage is called the "waist."
49. Bell: Loud :: Speak: Hear, Stentorian?
     The Sanskrit root of "bell" means "he talks" or "he speaks"; the
     Sanskrit root of "loud" means "he hears".  A bell that makes a lot
     of noise is loud; a speaker who makes a lot of noise is stentorian.  
50. Benevolence: Beg :: Philanthropist: Mendicant, Mendicate?
     If the analogy is attribute: attribute :: noun: noun, the answer
     is "mendicant";  if the analogy is noun: verb :: noun: verb the
     answer is "mendicate."
51. 10: Decimal :: 20: Vigesimal (TWENTY, pert.)
52. Five-sided Polyhedron: Pentahedron ::
    Faces of Parallelepiped Bounded by a Square: ?
     Does this mean a parallelepiped all of whose faces are bounded by
     a square (and what does "bounded" mean), or does it mean all six
     parallelograms that form the faces of a parallelepiped drawn in a
     plane inside of a square?
53. Motor: Helicopter :: Airflow: Autogiro (HELICOPTER)
54. Man: Ant :: Barter: Trophallaxis
55. United States: Soviet Union :: Cubism: ? (ART, style)
     If the emphasis is on opposition and collapse, there were several
     movements that opposed Cubism and that died out (e.g., Purism,
     Suprematism, Constructivism).  If the emphasis is on freedom of
     perspective versus constraint, there were several movements that
     emphasized exact conformance with nature (e.g., Naturalism, Realism,
     Photo-Realism).  If the emphasis is on dominating the art
     scene, the only movement that was contemporary with Cubism and
     of the same popularity as Cubism was Surrealism.  A better answer
     would be an art movement named "Turkey-ism", since the Soviet Union
     offered to exchange missiles in Cuba for missiles in Turkey during
     the Cuban Missile Crisis.
56. State: Stipend :: Church: Prebend (STIPEND)
57. Motorcycle: Bicycle :: Motordrome: Velodrome (CYCLE, track)
58. Transparent: Porous :: Obsidian: Pumice (GLASS, volcanic)
59. pi*r^2*h: 1/3*pi*r^2*h :: Cylinder: Cone