orthogonal

Etymology

It is borrowed from Middle French orthogonal (“having a right angle”),^[2][3][4][5] recorded in French from 1520,^[2] formed from orthogone and the suffix -al.^[2][3] The French noun goes back to Late Latin orthogōnius (“right-angled”),^[3][4][6] from Ancient Greek ὀρθογώνιος (orthogṓnios, “right-angled”),^[3][4][6][7] a compound of ὀρθός (orthós, “straight, right”) and γωνία (gōnía, “angle, corner”),^[6][3][7] which Etymonline derives from Proto-Indo-European *genu- (“knee; angle”).^[3] The OED compares Medieval Latin orthogonalis, recorded in British sources from about 1109,^[2] and the Shorter Oxford takes the English word directly from that form.^[8] Johnson (1755) derived it from French orthogonel.^[9]

Earliest recorded meaning

Having or forming a right angle; right-angled, rectangular: perhaps before 1560, in L. Digges's Pantometria, printed in 1571.^[2] Merriam-Webster Unabridged dates the word to about 1560,^[4] Etymonline to the 1570s^[3] and Merriam-Webster's Collegiate to 1612.^[10]

Semantic evolution

The word began in geometry, for a right angle or a figure with one, a sense now rare.^[2] From 1694 it was used more generally of lines and planes at right angles to each other.^[2] Johnson (1755) glossed it simply as "rectangular".^[9] In the 19th century mathematicians extended it to transformations that keep lengths and angles unchanged (1852), then to matrices (1891) and groups (1898), and in 1913 to vectors and functions whose inner product is zero, by analogy with perpendicular lines.^[2] Statisticians used it from 1933 for independent variates.^[2] From these technical senses comes a looser use for problems or aspects that are independent of each other and can be dealt with separately.^[1]

Timeline

• c. 1560 – having or designating a right angle; right-angled (now rare).^[2][4]
• 1694 – relating to right angles; at right angles to something else.^[2]
• 1852 – mathematics, of a linear transformation that preserves lengths and angles.^[2]
• 1891 – of a square matrix representing such a transformation.^[2]
• 1898 – of the group of all orthogonal matrices of a given order.^[2]
• 1913 – of vectors or functions, having an inner product of zero.^[2]
• 1933 – statistics, of variates, statistically independent.^[2]

Meanings

Adjective

1. Lying or meeting at right angles; perpendicular: two orthogonal walls.^{[11][4][7][2]}
2. (mathematics) Of two vectors or functions, having an inner product equal to zero.^[11][4][2]
3. (mathematics) Of a transformation or matrix, preserving lengths and angles.^[2]
4. Of curves, having perpendicular tangents where they cross.^[4]
5. (statistics) Statistically independent.^[2][1]
6. Of two problems or aspects, independent of each other and able to be treated separately.^[1]

Noun

1. In oceanography, an imaginary line drawn at right angles to the crests of waves.^[4]

Word family

• orthogonally
• orthogonality
• orthogonalize

Related words

• orthonormal
• ortho-
• -gon
• diagonal
• polygon
• orthodox
• knee

References

1. ^ a b c d e Wiktionary, the free dictionary, https://en.wiktionary.org
2. ^ a b c d e f g h i j k l m n o p q r s t Oxford English Dictionary Online (accessed March 2026), Oxford: Oxford University Press
3. ^ a b c d e f g Harper, Douglas. Online Etymology Dictionary, https://www.etymonline.com
4. ^ a b c d e f g h i Merriam-Webster Unabridged Dictionary (2024), online ed., Springfield, Mass.: Merriam-Webster
5. ^ Stevenson, Angus, ed. (2010), Oxford Dictionary of English, 3rd ed., Oxford: Oxford University Press
6. ^ a b c Klein, Ernest. A Comprehensive Etymological Dictionary of the English Language. 2 vols. Amsterdam: Elsevier, 1966–1967.
7. ^ a b c The American Heritage Dictionary of the English Language (online 2024), 5th ed., Boston: Houghton Mifflin Harcourt
8. ^ Shorter Oxford English Dictionary on Historical Principles (2002), 5th ed., ed. William R. Trumble and Angus Stevenson, Oxford: Oxford University Press
9. ^ a b Johnson, Samuel (1755), A Dictionary of the English Language, London; 4th ed. 1773; Johnson's Dictionary Online, ed. Beth Rapp Young et al.
10. ^ Merriam-Webster's Collegiate Dictionary (2003), 11th ed., Springfield, Mass.: Merriam-Webster
11. ^ a b Collins English Dictionary (2023), Glasgow: HarperCollins