degree

from The American Heritage® Dictionary of the English Language, 5th Edition.

• noun One of a series of steps in a process, course, or progression; a stage.
• noun A step in a direct hereditary line of descent or ascent.
• noun Relative social or official rank, dignity, or position.
• noun Relative intensity or amount, as of a quality or attribute.
• noun The extent or measure of a state of being, an action, or a relation.
• noun A unit division of a temperature scale.
• noun Mathematics A planar unit of angular measure equal in magnitude to ¹/₃₆₀ of a complete revolution.
• noun A unit of latitude or longitude, equal to ¹/₃₆₀ of a great circle.
• noun The greatest sum of the exponents of the variables in a term of a polynomial or polynomial equation.
• noun The exponent of the derivative of highest order in a differential equation in standard form.
• noun An academic title given by a college or university to a student who has completed a course of study.
• noun A similar title conferred as an honorary distinction.
• noun Law A division or classification of a specific crime according to its seriousness.
• noun A classification of the severity of an injury, especially a burn.
• noun Grammar One of the forms used in the comparison of adjectives and adverbs. For example, tall is the positive degree, taller the comparative degree, and tallest the superlative degree of the adjective tall.
• noun One of the seven notes of a diatonic scale.
• noun A space or line of the staff.
• idiom (by degrees) Little by little; gradually.
• idiom (to a degree) To a small extent; in a limited way.

from The Century Dictionary.

• To advance by a step or steps.
• To place in a position or rank.
• noun In physical chemistry, the number of conditions of a thermodynamic system which can be changed independently of each other, without destroying the system by suppressing one of its phases. For example, a system composed of water existing in the two phases, liquid and solid, and depending for equilibrium on the two conditions, temperature and pressure, has one degree of freedom and only one: any desired temperature may be given to it within certain limits, but the pressure is thereby fixed; and any pressure may be established within certain limits, but the temperature is determined in so doing.
• noun A step, as of a stair; a stair, or set of steps.
• noun A step or single movement toward an end; one of a series of advances; a stage of progress; a phase of development, transformation, or progressive modification.
• noun Specifically In grammar, one of the three stages, namely, positive, comparative, and superlative, in the comparison of an adjective or an adverb. See comparison, 5.
• noun The point of advancement reached; relative position attained; grade; rank; station; order; quality.
• noun In universities and colleges, an academical rank conferred by a diploma, originally giving the right to teach.
• noun In geneal., a certain distance or remove in the line of descent, determining the proximity of blood: as, a relation in the third or fourth degree. See first extract, and forbidden degrees, below.
• noun In algebra, the rank of an equation, as determined by the highest power under which an unknown quantity appears in it.
• noun One of a number of subdivisions of something extended in space or time.
• noun In arithmetic, three figures taken together in numeration: thus, the number 270,360 consists of two degrees (more commonly called periods).
• noun In music: One of the lines or spaces of the staff, upon which notes are placed. Notes on the same degree, when affected by accidentals, may denote different tones, as D, D♮, and D♭; and, similarly, notes on different degrees, as D♭ and C♮, may denote identical tones, at least upon instruments of fixed intonation.
• noun The difference or step between a line and the adjacent space on the staff (or vice versa). Occasionally, through the use of accidentals, this difference is only apparent (see above).
• noun The difference, interval, or step between any tone of the scale and the tone next above or below it, as from do to re, from mi to fa. The interval may be a whole step or tone, a half step or semitone, or (in the minor scale) a step and a half, or augmented tone. See step, tone, interval, staff, scale. [To distinguish between degrees of the staff and degrees of the scale, the terms staff-degree and scale-degree are sometimes used.]
• noun Intensive quantity; the proportion in which any quality is possessed; measure; extent; grade.
• noun In criminal law: One of certain distinctions in the culpability of the different participants in a crime. The actual perpetrator is said to be a principal in the first degree, and one who is present aiding and abetting, a principal in the second degree.
• noun One of the phases of the same kind of crime, differing in gravity and in punishment.

from the GNU version of the Collaborative International Dictionary of English.

• noun obsolete A step, stair, or staircase.
• noun One of a series of progressive steps upward or downward, in quality, rank, acquirement, and the like; a stage in progression; grade; gradation
• noun The point or step of progression to which a person has arrived; rank or station in life; position.
• noun Measure of advancement; quality; extent.
• noun Grade or rank to which scholars are admitted by a college or university, in recognition of their attainments; also, (informal) the diploma provided by an educational institution attesting to the achievement of that rank.
• noun (Genealogy) A certain distance or remove in the line of descent, determining the proximity of blood; one remove in the chain of relationship.
• noun (Arith.) Three figures taken together in numeration; thus, 140 is one degree, 222,140 two degrees.
• noun (Algebra) State as indicated by sum of exponents; more particularly, the degree of a term is indicated by the sum of the exponents of its literal factors; thus, a2b3c is a term of the sixth degree. The degree of a power, or radical, is denoted by its index, that of an equation by the greatest sum of the exponents of the unknown quantities in any term; thus, ax4 + bx2 = c, and mx2y2 + nyx = p, are both equations of the fourth degree.
• noun (Trig.) A 360th part of the circumference of a circle, which part is taken as the principal unit of measure for arcs and angles. The degree is divided into 60 minutes and the minute into 60 seconds.
• noun A division, space, or interval, marked on a mathematical or other instrument, as on a thermometer.