From BiblePortal Wikipedia

King James Dictionary [1]

Degree n.

1. A step a distinct portion of space of indefinite extent a space in progression as, the army gained the hill by degrees a balloon rises or descends by slow degrees and figuratively, we advance in knowledge by slow degrees. Men are yet in the first degree of improvement. It should be their aim to attain to the furthest degree, or the highest degree. There are degrees of vice and virtue. 2. A step or portion of progression, in elevation, quality, dignity or rank as a man of great degree.

We speak of men of high degree, or of low degree of superior or inferior degree. It is supposed there are different degrees or orders of angels.

They purchase to themselves a good degree. 1Tim.  3.

3. In genealogy, a certain distance or remove in the line of descent, determining the proximity of blood as a relation in the third or fourth degree. 4. Measure extent. The light is intense to a degree that is intolerable. We suffer an extreme degree of heat or cold. 5. In geometry, a division of a circle, including a three hundred and sixtieth part of its circumference. Hence a degree of latitude is the 360th part of the earths surface north or south of the equator, and a degree of longitude, the same part of the surface east or west of any given meridian. 6. In music, an interval of sound, marked by a line on the scale. 7. In arithmetic, a degree consists of three figures thus, 270,360, compose two degrees. 8. A division, space or interval, marked on a mathematical or other instrument as on a thermometer, or barometer. 9. In colleges and universities, a mark of distinction conferred on students, as a testimony of their proficiency in arts and sciences giving them a kind of rank, and entitling them to certain privileges. This is usually evidenced by a diploma. Degrees are conferred pro meritis on the alumni of a college or they are honorary tokens of respect, conferred on strangers of distinguished reputation. The first degree is that of Bachelor of Arts the second, that of Master of Arts. Honorary degrees are those of Doctor of Divinity, Doctor of Laws, &c. Physicians also receive the degree of Doctor of Medicine.

By degrees, step by step gradually by little and little by moderate advances. Frequent drinking forms by degrees a confirmed habit of intemperance.

Webster's Dictionary [2]

(1): ( n.) Grade or rank to which scholars are admitted by a college or university, in recognition of their attainments; as, the degree of bachelor of arts, master, doctor, etc.

(2): ( n.) A certain distance or remove in the line of descent, determining the proximity of blood; one remove in the chain of relationship; as, a relation in the third or fourth degree.

(3): ( n.) A line or space of the staff.

(4): ( n.) A step, stair, or staircase.

(5): ( n.) One of a series of progressive steps upward or downward, in quality, rank, acquirement, and the like; a stage in progression; grade; gradation; as, degrees of vice and virtue; to advance by slow degrees; degree of comparison.

(6): ( n.) The point or step of progression to which a person has arrived; rank or station in life; position.

(7): ( n.) Measure of advancement; quality; extent; as, tastes differ in kind as well as in degree.

(8): ( n.) A division, space, or interval, marked on a mathematical or other instrument, as on a thermometer.

(9): ( n.) Three figures taken together in numeration; thus, 140 is one degree, 222,140 two degrees.

(10): ( 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.

(11): ( 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.

Vine's Expository Dictionary of NT Words [3]

1: Βαθμός (Strong'S #898 — Noun Masculine — bathmos — bath-mos' )

denotes "a step," primarily of a threshold or stair, and is akin to baino, "to go;" figuratively, "a standing, a stage in a career, position, degree,"  1—Timothy 3:13 , of faithful deacons.

 Luke 1:52 James 1:9

International Standard Bible Encyclopedia [4]

dē̇ - grē ´ (מעלה , ma‛ălāh , "a going up" or "ascent," hence, a staircase or flight of steps; "rank": ταπεινός , tapeinós , "low"): By derivation it should mean "a step down" (Latin, de , down, gradus , step). It is used, however, of any step, up or down; then of grade or rank, whether high or low. (1) In its literal sense of step (as of a stair), it is used in the plural to translate Hebrew ma‛ălōth ("steps"), in the parallel passages  2 Kings 20:9-11 the King James Version (5 t);   Isaiah 38:8 the King James Version (3 t), where we read of the "degrees" (the Revised Version (British and American) "steps") on the "dial of Ahaz" (Hebrew "steps of Ahaz"). See Dial Of Ahaz . It seems to mean steps or progressive movements of the body toward a certain place in the phrase "A Song of Degrees" (the Revised Version (British and American) "Ascents"), which forms the title of each of the Psalms 120 through 134, probably because they were sung on the way up to the great feasts at Jerusalem. See Psalms (2) The secondary (but now the more usual) sense of rank, order, grade is found in the following passages: ( a )  1 Chronicles 15:18 , "their brethren of the second (degree)," literally, "of the seconds" (Hebrew mishnı̄m  ; compare  2 Chronicles 28:7 , "Elkanah that was next to the king," Hebrew, "the king's second," i.e. in rank); ( b )  1 Chronicles 17:17 , "a man of high degree" (Hebrew ma‛ălāh , "step"); ( c )  Psalm 62:9 , "men of low degree ... men of high degree," a paraphrase of Hebrew "sons of man ... sons of man," the first "man" being Hebrew 'ādhām ("common humanity"; compare Greek ánthrōpos , Latin homo , Welsh dyn ), and the second Hebrew 'ı̄sh (man in a superior sense; compare Greek anḗr , Latin vir, Welsh gwr); ( d ) "of low degree" for Greek tapeinos in Sirach 11:1;  Luke 1:52;  James 1:9; (e) In  1 Timothy 3:13 the King James Version "a good degree" (Greek bathmós kalós , the Revised Version (British and American) "a good standing") is assured to those who have "served well as deacons." Some take this to mean promotion to a higher official position in the church; but it probably means simply a position of moral weight and influence in the church gained by faithfulness in service (so Hort).

Cyclopedia of Biblical, Theological and Ecclesiastical Literature [5]

is the rendering in the A. V. of one Heb. and one, Gr. term (besides being employed as an adjunct in the phrases "man of high [or low] degree," where it has no [other] correspondent in the original) מִעֲלָה , Maalah' ( 2 Kings 20:9-11  Isaiah 38:8, referring to the graduated scale of the dial [q.v.] of Ahaz, and in the titles of the Psalms entitled "Song of Degrees" [see below]; a step, as elsewhere generally rendered); Βαθμός (only  1 Timothy 3:13, graduation or promotion, etc. of a deacon [q.v.] to a higher office; or perhaps rather a spiritual stand-point or condition, see Alford, in loc.), a step (as of a staircase or door,  Sirach 6:36). (See Stairs).