Citations:vinculum

sense: horizontal overline used to group terms in a calculation

 * 1897, The Elements of Mechanical Engineering, volume 1, Scranton: International Textbook Company, page 113:
 * The parentheses, brackets [], and braces {} have the same meaning, and signify that the quantities within them are to be subjected to the same operations. A fourth symbol of aggregation is the vinculum &#x2015;&#x2015;&#x2015;, which is simply a horizontal line drawn above the quantities affected. Thus, $$\overline{a+2}\times4$$ shows that $$a+2$$ is to be multiplied by $$4$$.
 * 1834, Mark Napier, "History of the Invention of Lograithms, &c.", in Memoirs of John Napier of Merchiston, Edinburgh: William Blackwood, page 500:
 * To Vieta is ascribed the vinculum in algebraic notation, which Girard changed to the parenthesis. The English algebraists, chiefly, use the vinculum, which is drawn above the compound thus, $$\sqrt{a+b}$$.
 * 1992, Frederick Mosteller, "Writing about Numbers", in, 1992, John C. Bailar III and Frederick Mosteller, editors, Medical Uses of Statistics, second edition, page 388:
 * In $$\sqrt{a+b}$$, the horizontal bar or vinculum is actually a parenthesis that groups $$a+b$$. Without it, $$\sqrt{}a+b$$ might mean $$b+\sqrt{a}$$ or it might mean $$\sqrt{a+b}$$. Nevertheless, the square-root sign without the vinculum is often used.

sense: thing used to group terms in a calculation

 * 1848 edition, Jeremiah Day, An Introduction to Algebra, pages 11–12:
 * When the several members of a compound quantity are to be subjected to the same operation, they are frequently connected by a line called a vinculum. Thus $$a-\overline{b+c}$$ shows that the sum of $$b$$ and $$c$$ is to be subtracted from $$a$$. The marks used for parentheses,, are often substituted instead of a line, for a vinculum.
 * 1855, Noble Heath, Treatise on Arithmetic, Philadelphia: T. Ellwood Chapman, pages 67–69:
 * A vinculum is a bar &#x2015;&#x2015;&#x2015;, or parenthesis, used to connect several numbers or quantities, so that the whole may, as one quantity, be subjected to any of the cardinal operations.
 * Wherefore, we must, in removing the vinculum, change the sign of the number 2 from plus to minus: thus $$8-\overline{4+2}=8-4-2$$.
 * , P. O'Connell, reprinted in, 1888, W. J. C. Miller, editor, Mathematical Questions and Solutions, volume 49, London: Francis Hodgson, page 96:
 * $$A^2=(A_{1-8})^2+2\{A_{1-4}\times A_{9-12}+A_{1-2}\times A_{13-14}+A_{5-6}\times A_{9-10}+A_1\times A_{15}+A_3\times A_{13}+A_5\times A_{11}+A_7\times A_9\}$$
 * first sum the series under the vinculum, add the result to itself, and finally add $$(A_{1\ 8})^2$$

unclear sense

 * 1801, John Colson, translator, Analytical Institutions, translation of, Maria Gaetana Agnesi, Instituzioni Analitiche, volume 1, page 29:
 * To multiply radicals among themselves, supposing them to be of the same denomination, or reduced to such, the quantities must be multiplied into each other which are under the radical signs, and to the product must be put the same radical vinculum, with such a sign, either positive or negative, as the common rule requires. Thus to multiple $$\sqrt{}bc$$ into $$\sqrt{}xy$$, the product will be $$\sqrt{}bcxy$$. To multiply $$\sqrt{}\frac{aa-xx}{x}$$ into $$-\sqrt{}\overline{aa+xx}$$, the product will be $$-\sqrt{}\frac{a^4-x^4}{x}$$.
 * 1814, Charles Butler, An Easy Introduction to the Mathematics, volume 1, Oxford: Bartlett and Newman, page 359:
 * Add $$3\sqrt{x+y}+4\sqrt{x+y}+5\sqrt{x+y}+\sqrt{x+y}$$ together.
 * Explanation. The quantities under the vinculum being alike in all respects, I merely add the coefficients 3, 4, 5, and 1, (understood,) together, and to their sum 13 subjoin the vinculum for the answer.

sense: horizontal overline for any purpose

 * 2001, Jerry Bobrow, CliffsQuickReview Algebra I, Wiley, ISBN 076456370X, page 26:
 * Infinite repeating decimals are usually represented by putting a line over (sometimes under) the shortest block of repeating decimals. This line is called a vinculum. So you would write
 * $$-2.1\overline{47}$$ to indicate $$-2.1474747$$...
 * Notice that only the digits under the vinculum are repeated.
 * 2003, J. T. Glover, Vedic Mathematics for Schools, book 2, Motilal Banarsidass, ISBN 81-208-1670-6, page 10:
 * The word 'vinculum' is a line placed above a number to indicate a negative quantity or a deficiency. Thus $$2\overline{3}$$ is two tens less three units, that is 17.  When a number has several successive vinculum digits it may be 'devinculated' using the All from nine and the last from ten sutra.
 * Devinculate $$4\overline{231}$$
 * $$4\overline{231}=3769$$

sense: horizontal line separating numerator from denominator in a fraction

 * 1850, James Elliot, An Elementary Course of Practical Mathematics, vol. 1, Simpkin, Marshall & Co., page 5:
 * The line between the terms of a fraction serves as a vinculum as, $$\frac{a+b}{a-b}$$.
 * 2004, Gloria Harris and Garda Turner, Queensland Targeting Maths Teaching Guide Year 7, Pascal Press, ISBN 1920728287, page 98:
 * Show how the vinculum stands for division.
 * 2006 edition, A. S. Kalra, Excel Essential Skills Year 6 Advanced Mathematics Study Guide, Pascal Press, ISBN 1864410469, page 57:
 * The oldest and the most common way of writing a fraction is when it is written as two numbers separated by a line. The line separating the top and the bottom parts is called the vinculum.
 * 1982 edition, John Walker and Russell Rea, Breakthrough Maths 5, Methuen Australia, page 35:
 * Prepare the children for exercise 4 below by explaining the vinculum in a fraction.