Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. One of a series of steps in a process, course, or progression; a stage: proceeded to the next degree of difficulty.
 n. A step in a direct hereditary line of descent or ascent: First cousins are two degrees from their common ancestor.
 n. Relative social or official rank, dignity, or position.
 n. Relative intensity or amount, as of a quality or attribute: a high degree of accuracy.
 n. The extent or measure of a state of being, an action, or a relation: modernized their facilities to a large degree.
 n. A unit division of a temperature scale.
 n. Mathematics A planar unit of angular measure equal in magnitude to 1/360 of a complete revolution.
 n. A unit of latitude or longitude, equal to 1/360 of a great circle.
 n. Mathematics The greatest sum of the exponents of the variables in a term of a polynomial or polynomial equation.
 n. Mathematics The exponent of the derivative of highest order in a differential equation in standard form.
 n. An academic title given by a college or university to a student who has completed a course of study: received the Bachelor of Arts degree at commencement.
 n. A similar title conferred as an honorary distinction.
 n. Law A division or classification of a specific crime according to its seriousness: murder in the second degree.
 n. A classification of the severity of an injury, especially a burn: a thirddegree burn.
 n. 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.
 n. Music One of the seven notes of a diatonic scale.
 n. Music 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: doesn't like spicy food, but can eat a little pepper to a degree.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The number of edges that a vertex takes part in; a valency.
 n. The curvature of a circular arc, expressed as the angle subtended by a fixed length of arc or chord.
from the GNU version of the Collaborative International Dictionary of English
 n. A step, stair, or staircase.
 n. One of a series of progressive steps upward or downward, in quality, rank, acquirement, and the like; a stage in progression; grade; gradation
 n. The point or step of progression to which a person has arrived; rank or station in life; position.
 n. Measure of advancement; quality; extent.
 n. 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.
 n. A certain distance or remove in the line of descent, determining the proximity of blood; one remove in the chain of relationship.
 n. Three figures taken together in numeration; thus, 140 is one degree, 222,140 two degrees.
 n. 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.
 n. 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.
 n. A division, space, or interval, marked on a mathematical or other instrument, as on a thermometer.
 n. A line or space of the staff.
from The Century Dictionary and Cyclopedia
 To advance by a step or steps.
 To place in a position or rank.
 n. A step, as of a stair; a stair, or set of steps.
 n. 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.
 n. 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.
 n. The point of advancement reached; relative position attained; grade; rank; station; order; quality.
 n. In universities and colleges, an academical rank conferred by a diploma, originally giving the right to teach.
 n. 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.
 n. In algebra, the rank of an equation, as determined by the highest power under which an unknown quantity appears in it.
 n. One of a number of subdivisions of something extended in space or time.
 n. In arithmetic, three figures taken together in numeration: thus, the number 270,360 consists of two degrees (more commonly called periods).
 n. 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.
 n. 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).
 n. 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 staffdegree and scaledegree are sometimes used.]
 n. Intensive quantity; the proportion in which any quality is possessed; measure; extent; grade.
 n. 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.
 n. One of the phases of the same kind of crime, differing in gravity and in punishment.
 n. 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.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a position on a scale of intensity or amount or quality
 n. a specific identifiable position in a continuum or series or especially in a process
 n. an award conferred by a college or university signifying that the recipient has satisfactorily completed a course of study
 n. the highest power of a term or variable
 n. a unit of temperature on a specified scale
 n. a measure for arcs and angles
 n. the seriousness of something (e.g., a burn or crime)
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

No man is so wicked _but_ (conjunctive adverb) he loves virtue = No man is wicked _to that degree in which_ he loves _not_ virtue (_so_ = _to that degree_, _but_ = _in which not_).
Higher Lessons in English A work on english grammar and composition

If it seems to be rolled to a width beyond the normal degree, it should be classified as if rolled only to the normal degree_.

In whatsoever degree more of the _man_ and less of the _mechanics_ appear, _in that degree_ is the result a work of art.

What distinguishes Political Correction, to a certain degree, is the context in which it pursues this partisan role.
Political Correction: Media Matters ReLaunches Sister Site, Then Slams Issa

For example, one might argue that if the degree is all that matters, why require attendance?

Your degree is a commodity: You are just one student in a sea of hundreds of thousands of business students who will graduate this year.
Series – Letters to a Marketing Student. Part One: The Most Valuable 10 Hours of Your Degree

I absolutely agree that having a degree is a valuable asset.

That said, obtaining a degree is a valuable asset.

In most cases, the degree is a must to get most jobs which is the main reason for school.
The Education of Educators, Bryan Caplan  EconLog  Library of Economics and Liberty

Government regulation to a degree is acceptable in a capitalist economy.
chained_bear commented on the word degree
"When the French Academy of Sciences took up the task of making an accurate map of France, they were helped by Galileo's pioneering work that had turned the moons of Jupiter into a celestial timepiece through calculations involving triangulating from a baseline on earth. To achieve stability for the baseline for the French map, workers cut varnished wooden rods for the sevenmile distance between Paris and Fontainebleau. It took them two years to determine that a degree ran for 69.1 miles, a figure that still stands. Their findings led to surprising results with some cities relocated 100 miles away from their old cartographical positions. France, it seemed, had shrunk, causing Louis XIV, the project's funder to exclaim that the effort had cost him 'a major portion of my realm.' Accuracy had its price."
Joyce Appleby, Shores of Knowledge: New World Discoveries and the Scientific Imagination (New York and London: W.W. Norton & Co., 2013), p. 171
December 28, 2016