*Last revision: Aug. 19, 2019*

**LAGRANGE MULTIPLIER.** Joseph-Louis
Lagrange states the general
principle for maximising a function of *n* variables when there are one
or more equations between the variables in his
*Théorie des Fonctions Analytiques*
(1797, p. 198): "il suffira d'ajouter à la function proposée les functions qui
doivent être nulles, multipliées chacune par une quantité indéterminée ...".
Lagrange originally applied the multiplier technique to problems in the calculus
of variations in his
*Mécanique Analytique*
(1788, pp. 46-7). (See H. H. Goldstine *A History of the Calculus of Variations from the 17 ^{th}
through the 19^{th} Century* (1980).)

Although "Lagrange multiplier" is the standard term today,
"undetermined multiplier" and "indeterminate multiplier"
were the usual terms in the 19^{th} century and for much of the 20^{th}.

*Lagrange’s method of multipliers* is found in 1845 in the title
“On the Theory and Application of Lagrange’s Method of Multipliers” by William Rutherford,
in *The Mathematician,* editors
Thomas Stephens Davies, William Rutherford, and Stephen Fenwick. [Google print search by James A. Landau]

The term "Lagrange’s method of
undetermined multipliers" appears in J. W. Mellor, *Higher
Mathematics for Students of Chemistry and Physics* (1912) [James
A. Landau].

The term "Lagrange multiplier rule" appears in "The Problem of Mayer
with Variable End Points," Gilbert Ames Bliss, *Transactions of the
American Mathematical Society,* Vol. 19, No. 3. (Jul., 1918).

*Lagrange multiplier* is found in "Necessary Conditions in the
Problems of Mayer in the Calculus of Variations," Gillie A. Larew,
*Transactions of the American Mathematical Society,* Vol. 20,
No. 1. (Jan., 1919): "The [lambda]’s appearing in this sum are the
functions of *x* sometimes called Lagrange multipliers."

The use of **Lagrangian** for the augmented function
dates from the 1960s, see e.g. Samuel Zahl "A Deformation Method for Quadratic Programming,"
*Journal of the Royal Statistical Society, B*, **26**, (1964), p.
153. (JSTOR search) *The Lagrangian function* or *the Lagrangian expression*
were once the popular terms.

**LAGRANGE MULTIPLIER TEST** in Statistics.
This test principle was introduced by S. D. Silvey "The
Lagrangian Multiplier Test," *Annals of Mathematical Statistics*,
**30**, (1959), 389-407. However, while Silvey’s derivation was new, the
test statistic was already in the literature as the "score test." Econometricians
tend to favour the Lagrange term and statisticians the score term.

See the entries SCORE TEST and WALD TEST.

**LAGRANGE’S THEOREM.** *Formule de Lagrange* appears in
*Traité élémentaire de calcul
différentiel et de calcul intégral* (1797-1800) by
Lacroix.

*Lagrange’s theorem* appears in 1818 in
*An Elementary Treatise on Physical Astronomy* by Robert Woodhouse:
“we require a formula assigning *v*
in terms of *nt.* Such a formula we may deduce from Lagrange’s Theorem.”
[Google print search by James A. Landau]

*Lagrange’s theorem* appears in *An Elementary Treatise on
Curves, Functions and Forms* (1846) by Benjamin Peirce: "The
theorem (650) under this form of application, has been often called
*Laplace’s Theorem*; but, regarding this change as obvious and
insignificant, we do not hesitate to discard the latter name, and
give the whole honor of the theorem to its true author, Lagrange."

*Lagrange’s formula for interpolation* appears in 1849 in *An
Introduction to the Differential and Integral Calculus,* 2nd ed.,
by James Thomson.

*Lagrange’s method of approximation* occurs in the third edition
of *An Elementary Treatise on the Theory of Equations* (1875) by
Isaac Todhunter.

**LAGRANGIAN FUNCTION** is found
in 1877 in E. J. Routh, *A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion.*
[Google print search by James A. Landau]

**LAGRANGIAN (as a noun)** occurs in Th. Muir, "Note on the
Lagrangian of a special unit determinant," *Transactions Royal Soc.
South Africa* (1929).

**LAPLACE’S COEFFICIENTS.** According to Todhunter (1873),
"the name Laplace’s coefficients appears to have been first used" by
William Whewell (1794-1866) [Chris Linton].

*Laplace’s coefficients* appears in 1830 in the *Encyclopædia Metropolitana,* 2nd ed.
[Google print search by James A. Landau]

**LAPLACE’S EQUATION** appears in 1812 in *Philosophical Transactions
of the Royal Society of London for the year MDCCCXII Part I.* [Google print search by James
A. Landau]

The **LAPLACE EXPANSION** of a determinant is generally traced to Laplace’s memoir
in *Histoire de l'Académie royale des sciences*
*1776* (Année 1772, 2e partie) pp. 267-376
(see pp. 294-304). Thomas Muir makes a detailed examination of the argument in
*The Theory of Determinants in the Historical Order of
Development* vol. 1, pp. 24-33: he concludes, “there can be no doubt
that if any *one* name is to be attached to the theorem it should be that of Laplace.” [John Aldrich]

*Laplace’s expansion-theorem*
is found in *Proceedings of the Royal Society of Edinburgh Volume VIII November 1872 to July 1875,*
in a paper read on Jan. 5, 1874. [Google print search by James A. Landau]

**LAPLACE’S FUNCTIONS** appears in English in 1833
in *Elementary principles of the theories of electricity, heat and molecular actions*
by Robert Murphy. [Google print search]

The term **LAPLACE’S OPERATOR** (for the differential operator
^{2}) was used in 1873 by James Clerk Maxwell in
*A Treatise on Electricity and Magnetism*
(p. 29): "...an operator occurring in all parts of Physics, which we may
refer to as Laplace’s Operator" (OED).

See VECTOR ANALYSIS and the Earliest Uses of Symbols of Calculus page

The term **LAPLACE TRANSFORM** was used by Boole and
Poincaré. According to the website of the University of St.
Andrews, Boole and Poincaré might otherwise have used the term
Petzval transform but they were influenced by a student of
Józeph Miksa Petzval (1807-1891) who, after a falling out with
his instructor, claimed incorrectly that Petzval had plagiarised
Laplace’s work.

**LAPLACIAN** is found in 1889 in *A Treatise on Statics with Applications to Physics Vol.
II,* 4th ed., by George Minchin. The preface
(which is dated February, 1886) has on page vi:

In this Chapter I have also ventured to introduce the term “Laplacian” with reference to those remarkable coefficients which occur in the development of the reciprocal of the distance between two points. The general term “Spherical Harmonic” is, of course, retained; but it seems to me that the name of Laplace ought to be explicitly connected with the branch of Mathematical Physics which he did so much to develop, and which has now become of such great importance. The pure mathematicians having their “jacobians,” “Hessians,” “Cayleyans,” &c., the term “Laplacian” is surely justified.

[Google print search by James A. Landau]

**LAPLACIAN** (as a noun, for the differential operator
^{2})
was used in 1935 by Pauling and Wilson in *Introd. Quantum Mech.* (OED).

See VECTOR ANALYSIS.

**LATENT VALUE** and **VECTOR.** See EIGENVALUE.

The term **LATIN SQUARE** was named by Euler (as quarré
latin) in 1782 in "Recherches sur une Nouvelle Espèces
de Quarrés Magique," *Verh. uitgegeven door het Zeeuwsch
Genootschap d. Wetensch. te Vlissingen*, **9**,
85-232.

Latin square appears in English in 1890 in the title of a
paper by Arthur Cayley, "On Latin Squares" in *Messenger of Mathematics*.
Graeco-Latin square appears in H. F. MacNeish
"Euler Squares." *Ann. Math.* **23**, (1921-1922), 221-227.

*Graeco-Latin square* appears in 1898 in “A New Method in Combinatory Analysis, with application to Latin Squares and associated questions,”
by Major P. A. MacMahon,
in *Transactions of the Cambridge Philosophical Society Volume XVI.*
[Google print search by James A. Landau]

The term was introduced into statistics by R. A. Fisher, according
to Tankard (p. 112). Fisher used the term in 1925 in
*Statistical Methods
for Research Workers* p. 229 (OED).

See the entry EULER’s GRAECO-LATIN SQUARES CONJECTURE.

**LATITUDE** and **LONGITUDE.** Henry of Ghent used the
word *latitudo* in connection with the concept of latitude
of forms.

Nicole Oresme (1320-1382) used the terms latitude and longitude approximately in the sense of abscissa and ordinate.

**LATTICE** and **LATTICE THEORY** are found in 1899 in
“Memoir on the Theory of the Partitions of Numbers” by
Major Percy Alexander MacMahon, *Philosophical Transactions of the Royal Society of London Series A Vol.
192*: “There is also a
lattice theory connected with unipartite partitions on a line, for the
unit-graph of such a partition is nothing more than a number of units
and zeros placed at the points of a two-dimensional lattice....” [Google print search by James
A. Landau]

**LATTICE** and **LATTICE THEORY** in algebra. These terms were introduced by
Garrett Birkhoff in his
“On the combination of subalgebras,” *Proceedings of the Cambridge Philosophical Society*, **29**, (1933), 441–464
and became well-known through his book *Lattice Theory* (1940). See
*Enyclopedia of Mathematics* where earlier work by
Schröder and Dedekind and contemporary work by Ore are also described.

Man stele sich jetzt in der Ebene ein rechtwinkliges Coordinatesystem (x, y) und die ganze Ebene durch Parallelen mit den Axen in den Abständen = 1 von einander in lauter Quadrate von den Dimensionen = 1 getheilt vor.sollen alle Eckpuncte von Quadraten heifsen, welche nicht in den beiden Coordinaten – Axen liegen.Gitterpuncte

The term *Gitterpuncte* may well have been used earlier.

The earliest use of *lattice point* in English given by the *OED* is from
Cayley’s translation of Eistenstein, which is titled “Eisenstein’s
Geometrical Proof of the Fundamental Theorem for Quadratic Residues” and appears in the
*Quarterly Mathematical Journal,* vol. I. (1857), pp. 186-191:
“Imagine now in a plane, a rectangular
system of coordinates (*x, y*) and the whole plane divided by
lines parallel to the axes at distances = 1 from each other into
squares of the dimension = 1. And let the angles which do not lie on
the axes of coordinates be called *lattice points.*”

[James A. Landau]

The term **LATUS RECTUM** was used by Gilles Personne de Roberval
(1602-1675) in his lectures on Conic Sections. The lectures were
printed posthumously under the title *Propositum locum geometricum
ad aequationem analyticam revocare,...* in 1693 [Barnabas Hughes].

*Latus rectum* is found in English in *A Mathematical Dictionary*
by Joseph Raphson: “In a Parabola the Rectangle of the Diameter, and Latus Rectum, is equal to the Rectangle of the Segments of the double Ordinate.” [OED]

**LAURENT EXPANSION.** Pierre-Alphonse
Laurent announced this result in his "Extension du théorème de M. Cauchy relatif à la
convergence du développement d'une fonction suivant les puissances ascendantes
de la variable," *Comptes
rendus,* **17**, (1843) pp. 348-9. The full paper was published in
*Journal de l’Ecole Polytechnique*. **23**, (1863), 75-204. (Kline p. 641)

The term **LAW OF INTERTIA OF QUADRATIC FORMS** was introduced by
James Joseph Sylvester
in his “A demonstration of the theorem that every homogeneous
quadratic polynomial is reducible by real orthogonal substitutions to the form
of a sum of positive and negative squares,” *Philosophical
Magazine*, **4,** 138-142. It is sometimes called **Sylvester’s
law of inertia**. See Kline (p. 799) and the entry in *Encyclopedia
of Mathematics*. [John Aldrich]

**LAW OF LARGE NUMBERS, STRONG LAW, WEAK LAW.**
*La loi de grands nombres* appears in 1835 in Siméon-Denis Poisson (1781-1840),
"Recherches sur la Probabilité des Jugements, Principalement en Matiére
Criminelle,"
*Comptes Rendus Hebdomadaires des Séances
de l'Académie des Sciences,* 1, 473-494
also in his "La loi de grands nombres" in
ibid. 2, (1836) 377-382. (Porter p. 77).

By Poisson’s time there were several theorems that could be covered by this
phrase. The first, given by
Jacob Bernoulli in
*Ars Conjectandi* (1713), was about "Bernoulli
trials" (a 20^{th} century term). It was often called "Bernoulli’s theorem":
see e.g. Todhunter’s
*A History of the Mathematical Theory of Probability*
(1865, p. 71). In the course of the 19^{th} century analogous results
were found for other types of random variable.

*Law of large numbers* is found in 1863 in “On the Military Statistics of the United States of America”
by F. B. Elliot:
“....may be expressed by a very simple analytical
function (see Note in Appendix), first investigated by J. Bernoulli in
its relation to the probable distribution of errors of observation of a
single object; extended by Poisson, under the title the law of large
numbers, to the measurement of many objects, representative each of a
common type; and first applied by M. Quetelet to the physical
measurement of men.” [Google print search by James A. Landau]

In the 20^{th} century a new kind of convergence result was obtained,
based on almost-sure convergence, *not* convergence in probability, as in the Bernoulli tradition. The first of these
results (on Bernoulli trials) was given by E.
Borel in 1909 (see NORMAL NUMBER) and a
more general result was given by
F. Cantelli in 1917. In 1928
A. Y. Khintchine
introduced the term **strong law of large numbers** to distinguish these results from the
"ordinary" Bernoulli-like results: "Sur la loi forte des grands
nombres," *Comptes Rendus de l'Académie
des Sciences,* **186** page
286.

The obvious term for the Bernoulli results viz., the **weak
law of large numbers**, seems to have come later. A *JSTOR* search (restricted
to journals in English) produced
W. Feller’s 1945 "Note on the
Law of Large Numbers and "Fair"
Games," *Annals of Mathematical Statistics*, **16**, 301-304.

**LAW OF QUADRATIC RECIPROCITY.** Legendre used *loi de réciprocite* in 1808 in his *Essai Sur La Théorie des Nombres; Seconde
Édition*: “Démonstration du Théorème contenant la loi de réciprocite
qui existe entre deux nombres premiers quelconques.”

In English, *law of quadratic reciprocity* is found in *Report of the annual meeting of the
British Association for the Advancement of Science* in the Sept. 1859 meeting, by H. J. Stephen Smith.
[Google print search by James A. Landau].

**LAW OF SINES (Snell’s law).** In a letter to the editor of
the Edinburgh Review dated Jan. 3, 1837, and printed in *History of
the Inductive Sciences from the earliest to the present times Volume the
First* by John W. Parker, William Whewell wrote, “This letter
is a repetition of one written in October last, with an alteration of
the paragraph respecting Sir C. Bell and Mr. Mayo. .... Most of the
other corrections I reject. On the subject of the discovery of the law
of the sines in refraction, his criticism is erroneous. Snell’s
law *is* the law of sines; Descartes had seen Snell’s papers;
and from Descartes’s habits of mind, I am persuaded that he *did*
borrow the discovery thence.”

*The law of sines* is found in 1847 in *A Text-Book on Natural
Philosophy for the use of schools and colleges* by John William
Draper. The book more frequently has “law of the sines,” but
“law of sines” appears as a page heading. Page 186 has
“...the general law of refraction is as follows:--- In each medium
the sine of the angle of incidence is in a constant ratio to the sine of
the angle of refraction.... To a beginner, this law of the constancy of
sines may be explained as follows:---....”

[Google print search by James A. Landau]

**LAW OF SINES, LAW OF COSINES, LAW OF TANGENTS.** The three terms are found in 1888 in
*Academic Trigonometry. Plane and Spherical* by
T. M. Blakslee.

“Law of Sines. In any plane triangle, the sides are proportional to the sines of the opposite angles.”

“Law of Cosines. The square of any side of a (pl.) triangle is equal to the sum of the squares of the other two sides, minus twice their product by the cosine of the included angle.”

“Law of Tangents. The sum of any two sides of a (pl.) triangle is to their difference as the tangent of one-half the sum of the opposite angles is to the tangent of one-half their difference.”

[Google print search by James A. Landau]

**LAW OF SMALL NUMBERS** is a
translation of the German phrase coined by L. von Bortkiewicz, and used by him
as a title of his book *Das Gesetz der kleinen Zahlen* (1898). (David (2001)).
A *JSTOR* search found the English phrase in a note, probably by Edgeworth,
in the *Economic Journal* (1904, p. 496) on an Italian publication
treating “the relation between statistics and the Calculus of Probabilities
with special reference to Prof. Bortschevitch’s ‘law of small numbers.’”

**LAW OF THE ITERATED LOGARITHM** is found in English in Philip Hartman and Aurel Wintner, "On
the law of the iterated logarithm," *Am. J. Math.* **63**, (1941), 69-176.

The German name appears in A. Kolmogorov’s Über das Gesetz
des iterierten Logarithmus, *Mathematische Annalen*,
**101**,
(1929), 126-135. Kolmogorov was extending
the original result (for coin tossing) due to A. Khintchine "Über einen
Satz der Wahrscheinlichkeitsrechnung," *Fundamamenta Mathematicae*,
**6**, (1924), 9-20. Khintchine’s work in turn rested on partial results
formulated in terms of number theory. [John Aldrich]

The term **LEAST ACTION** was used by Lagrange (*DSB*).

The *OED* says the term *principle of least action*
is from the French *principe de la moindre quantité d'action,* lit.
“principle of the least amount of action,”
and attributes this term to Pierre Louis Moreau de Maupertuis in 1748,
in *Hist. de l'Acad. R. des Sci. et des Belles Lettres de Berlin 1746,* 286.

The term is in use in a paper by de Maupertuis in 1744, in French and in English translation. [James A. Landau]

In 1751 the mathematician Samuel König started a priority dispute by claiming the least action principle had been invented by Leibniz in 1707. English language magazine articles in 1752 discussing this dispute used the English term “least action.” See here. [James A. Landau]

**LEAST COMMON MULTIPLE.**
*Common denominator* appears in
English in 1594 in *Exercises* by Blundevil: "Multiply the
Denominators the one into the other, and the Product thereof shall
bee a common Denominator to both fractions" (OED).

*Common divisor* was used in 1674 by Samuel Jeake in
*Arithmetick,* published in 1696: "Commensurable, called also
Symmetral, is when the given Numbers have a Common Divisor" (OED).

*Least common multiple* is found in 1734 in *A Treatise of
Fractions, in Two Parts* by Alexander Wright: “The *least
Common Multiple* to any Numbers is the least Number, which being
divided by each of the Numbers proposed, leaves nothing remaining. Thus
12 is a common Multiple to 2, 3, 4, 6, and it is also their least common
Multiple.” [Google print search by James A. Landau]

*Least common denominator* is found in 1751 in *A Plain and
Familiar METHOD For Attaining the Knowledge and Practice of COMMON
ARITHMETIC,* 18th ed., by Edmund Wingate: “When 2 or more
Fractions having unequal Denominators, are given to be reduced to other
equal Fractions having the least common Denominator possible....”
[Google print search by James A. Landau]

*Lowest common denominator* appears in 1777 in
*Arithmetic, Rational and Practical,* 3rd ed., by John Mair:
“The lower fractions are reduced, the more simple
and easy will any operation be 5 if, therefore, it be required to reduce
fractions to the lowest or least common denominator possible, it may be
done, as follows ....” [Google print search by James A. Landau]

*Least common dividend* appears in 1780 in *A Treatise of Algebra, in Two Books,*
Book I, 2nd ed., by W. Emerson. [Google print search by James A. Landau]

*Lowest common multiple* appears in 1830 in
*A Treatise on Algebra* by George Peacock. [Google print search by James A. Landau]

**LEAST SQUARES.** See METHOD OF LEAST SQUARES.

**LEBESGUE INTEGRAL.** In 1899-1901
Henri Lebesgue
published five short papers in the *Comptes Rendus*. These formed the basis of his doctoral
dissertation *Intégrale, longueur, aire*,
published in 1902 in the *Annali di
Matematica.* The fifth paper of the series “Sur une généralisation de
l’intégral défini,”
*Comptes rendus* 132, (1901) 1025-1028
announced Lebesgue’s generalisation of the Riemann Integral. (From T. Hawkins
*Lebesgue’s Theory of Integration: Its Origins and Development*. 1970)

The term *Lebesgue integral* soon appeared in English, in a paper by
William
H. Young, “On an extension of the Heine-Borel Theorem,” *Messenger of Mathematics* **33**
(1903-04), 120-132. The date received by
the editors is not given, but Ivor Grattan-Guinness [“Mathematical
bibliography for W. H. and G. C. Young,” *Historia
Mathematica* **2** (1975),
43-58] places this paper chronologically between papers with received dates of
29 October 1903 and 6 December 1903.

The theorem in question has, as far as I know, not hitherto been formulated, though it can be deduced without difficulty from a theorem in a recent memoir by M. Lebesgue, which states that the Lebesgue integral, as we may conveniently call it, of a sum of any two functions (as far as our present knowledge of functions goes) is the sum of their Lebesgue integrals. It has only to be shown that the new notion of the Lebesgue integral coincides in the case of semi-continuous functions with the well-known one of upper or lower integral.

(footnote on p. 129)

Behind Young’s reference to the Lebesgue integral is a
tale of lost priority. For when Young next refers to the integral he indicates
the resemblance between Lebesgue’s work and his own researches. William H.
Young, “On upper and lower integration,” *Proceedings of the London Mathematical Society (2)*
(1905), 52-66. [Received by the editors
on January 14, 1904. The following footnote appears on the first page of the
paper and is dated April 2, 1904.]

This paper was written simultaneously with the preceding memoir [Young’s "Open sets and the theory of content"], at a time when the writer was unacquainted with the work of M. Lebesgue. The result of Theorem 2 is in perfect accord with Lebesgue’s expression for his integral as the common limit of two difference summations (Annali di Matematica,1902, p. 253); in fact, it is easily shown that, in the case of an (upper) lower semi-continuous function, the Lebesgue integral coincides with the upper (lower) integral. It may be further remarked that, in the general case, the Lebesgue integral may itself be expressed in precisely my form.In accordance with the alterations made in the preceding memoir (cp. footnote, p. 16), I have made a few verbal alterations in the present paper; I have also elaborated the proof of the final theorem, which, in its original form, was too condensed.

The following is from p. 143 of Ivor Grattan-Guinness,
“A mathematical union: William Henry and Grace Chisholm Young,”
*Annals of Science* 29(2) (August 1972),
105-186:

By 1904 Will had, independently of Lebesgue, constructed by different means an equivalent theory of integration. It was his first really important idea in mathematics and the discovery that he had been anticipated would have cracked many a lesser man. But he took it magnanimously; when he heard of Lebesgue’s work he withdrew his major paper and rewrote parts of it to include considerations on what he named for ever as 'the Lebesgue integral'. There were enough technical differences between the two approaches for Young to present his own results in this paper, and sufficient applications of his approach to all branches of analysis to make it significant in its time. In fact, some of the succeeding workers on 'Lebesgue integration' have preferred to follow Young’s rather than Lebesgue’s approach.

[This entry was contributed by Dave L. Renfro.]

**LEG** for a side of a right triangle other than the hypotenuse
is found in English in 1659 in Joseph Moxon, *Globes* (OED).

*Leg* is used in the sense of one of the congruent sides of an
isosceles triangle in 1702 Ralphson’s *Math. Dict.*:
"*Isosceles* Triangle is a Triangle that has two equal Legs"
(OED).

**LEIBNIZ SERIES.** See GREGORY’s SERIES.

**LEMMA** appears in English in the 1570 translation by Sir Henry
Billingsley of Euclid’s *Elements* (OED). [The plural of
*lemma* can be written *lemmas* or *lemmata.*]

**LEMNISCATE.** Jacob Bernoulli named this curve the
*lemniscus* in *Acta Eruditorum* in 1694. He wrote,
"...formam refert jacentis notae octonarii [infinity symbol], seu
complicitae in nodum fasciae, sive lemnisci" (Smith vol. 2, page
329).

**LEMOINE POINT.** See SYMMEDIAN POINT.

**LEPTOKURTIC.** See KURTOSIS.

**LEVERAGE** in least squares estimation. The earliest *JSTOR* appearances are
from 1978 but the term was apparently already established for David F. Andrews and Daryl Pregibon
write that "observations with large effects" are *usually* called "leverage points".
"Finding the Outliers that Matter," *Journal of the Royal Statistical Society. Series
B*, **40**, (1978), 85-93.

**L'HOSPITAL’S RULE.** In his "De progressionibus transcendentibus"
Euler referred to the method as "a known rule" ("per regulam igitur cognitam quaeramus
valorem fractionis") [Stacy Langton].

In 1891 in *An Elementary Treatise of the Differential and Integral Calculus*
by George A. Osborne, the method is not named: "The Differential Calculus furnishes the following method
applicable to all cases."

In *Differential and Integral Calculus* (1902) by Virgil Snyder and John Irwin Hutchinson, the
procedure is termed "evaluation by differentiation." The same term is
used in *Elementary Textbook of the Calculus* (1912) by the same
authors.

*de l'Hospital’s theorem on indeterminate forms* is found in approximately 1904 in the E. R. Hedrick translation of volume I of
*A Course in Mathematical Analysis* by Edouard Goursat. The translation carries the date 1904, although a footnote
references a work dated 1905 [James A. Landau].

The 1906 edition of *A History of Mathematics* by Florian Cajori, referring to
L'Hopital’s 1696 treatise, has: "This contains for the first time the method of finding the limiting value of a fraction
whose two terms tend toward zero at the same time."

In *Differential and Integral Calculus* (1908) by Daniel
A. Murray, the procedure is shown but is not named.

James A. Landau has found in J. W. Mellor, *Higher Mathematics for
Students of Chemistry and Physics,* 4th ed. (1912), the sentence,
"This is the so-called rule of l'Hopital."

*L'Hopital’s rule* is dated 1944 in MWCD11.

The rule is named for Guillaume-Francois-Antoine de l'Hospital (1661-1704), although the rule was discovered by Johann Bernoulli. The rule and its proof appear in a 1694 letter from him to l'Hospital.

The family later changed the spelling of the name to l'Hôpital.

**LIAR PARADOX.** This paradox exists in several forms. One is attributed
to the philosopher Epimenides in the sixth century BC: "All Cretans are
liars...One of their own poets has said so." Another is attributed to Eubulides
of Miletus a leader of the Megarian school from the fourth century BC. In his
*Life of Euclides*
Diogenes Laertius wrote that Eubulides "handed down a great many arguments in dialectic."
The Eubulides form is "Is the man a liar who says that he tells
lies?" W. & M. Kneale *The Development of Logic* (1962) pp. 227-8 say
that new variants were devised in the Middle Ages and they speculate that the medieval
logicians may have rediscovered the paradox from considering Titus 1:12-13:
“One of themselves, a prophet of their own, hath said, The Cretans are always liars,
evil wild beasts, lazy gluttons. This witness is true. ...” Paul evidently did not
see the paradox in this echo of Epimenides.

*Paradox of the liar* is found in 1940 in *An Inquiry Into Meaning And Truth* by Bertrand Russell:
“The inference from the paradox of the liar is as follows.” [*OED*]

*Liar paradox* is found in 1959 in
*The Foundations of Mathematics: A Study in the Philosophy of Science*
by Evert Willem Beth: “The natural first reaction to the liar paradox is to ascribe the contradiction to the fact that the statement involved refers to itself.” [*OED*]

See PARADOX.

**LIE GROUP** (named for Sophus
Lie) appears in L. Autonne, "Sur une application des
groupes de M. Lie.," *C. R.* CXII. 570-573 (1891).

A second 1891 usage is found in “La théorie des goupes continus de transformations due à M. Sophus Lie. ...” [Google print search by James A. Landau]

*Lie group* appears in English in 1897 in
"Sophus Lie’s Transformation Groups: A Series of Elementary, Expository Articles" by
Edgar Odell Lovett, *The American Mathematical Monthly,* Vol. 4, No. 10. [*JSTOR* search]

**LIFE TABLE.** John Graunt set down a life table in Chapter XI of
Observations
Made upon the Bills of Mortality (1662) in the course of estimating
the number of "fighting men" in London.

Edmond Halley
presented a more solidly based table in
"An Estimate of the Degrees of the Mortality of Mankind,
Drawn from Curious Tables of the Births and Funerals at the City of Breslaw;
With an Attempt to Ascertain the Price of Annuities upon Lives"
*Philosophical Transactions (1683-1775)*, Vol. 17. (1693), 596-610.

*Life tables* appears in 1825 in *The Caledonian Mercury*: “The
life tables from which premiums have hitherto been calculated, are acknowledged to be no longer applicable.” [*OED*]

*Life table* appears in 1825 in *Tables of Life Contingencies* by Griffith Davies:
“From the Single Life Table extract the value of ₤1 annuity on each of the proposed lives taken separately, and from the sum deduct
that of a similar annuity on their joint existence, found as direct in the last Example.” [Google print search by James A. Landau]

See Hald (1990, chapters 7 and 9) and Life and Work of Statisticians for other presentations of Halley’s article.

See SURVIVAL FUNCTION.

**LIKELIHOOD**. The
term was first used in its modern sense in R. A. Fisher’s
"On the "Probable Error" of a
Coefficient of
Correlation Deduced from a Small Sample", Metron, 1, (1921), 3-32.

Formerly, likelihood was a synonym for probability,
as it still is in everyday English. In his paper
"On
the Mathematical Foundations of Theoretical Statistics"
(*Phil. Trans. Royal Soc.* Ser. A. **222**, (1922), p. 326). Fisher made
clear for the first time the distinction between the mathematical properties
of "likelihoods" and "probabilities" (DSB).

The solution of the problems of calculating from a sample the parameters of the hypothetical population, which we have put forward in the method of maximum likelihood, consists, then, simply of choosing such values of these parameters as have the maximum likelihood. Formally, therefore, it resembles the calculation of the mode of an inverse frequency distribution. This resemblance is quite superficial: if the scale of measurement of the hypothetical quantity be altered, the mode must change its position, and can be brought to have any value, by an appropriate change of scale; but the optimum, as the position of maximum likelihood may be called, is entirely unchanged by any such transformation. Likelihood also differs from probability in that it is not a differential element, and is incapable of being integrated: it is assigned to a particular point of the range of variation, not to a particular element of it.Likelihood was first used in a Bayesian context by Harold Jeffreys in his "Probability and Scientific Method,"

This entry was contributed by John Aldrich, based on David (2001). See BAYES, MAXIMUM LIKELIHOOD, INVERSE PROBABILITY and POSTERIOR & PRIOR.

**LIKELIHOOD PRINCIPLE.** This expression burst into print in 1962, appearing in "Likelihood
Inference and Time Series" by G. A. Barnard, G. M. Jenkins, C. B. Winsten (*Journal of the Royal
Statistical Society A,* **125,** 321-372), "On the Foundations of Statistical Inference" by
A. Birnbaum (*Journal of the American Statistical Association,* **57,** 269-306),
and L. J. Savage et al, (1962) *The Foundations of Statistical Inference.* It must have been current
for some time because the Savage volume records a conference in 1959; the term appears in Savage’s
contribution so the expression may have been his coining.

The principle (without a name) can be traced back to R. A.
Fisher’s writings of the 1920s though its clearest earlier manifestation is
in Barnard’s 1949 "Statistical Inference" (*Journal of the Royal
Statistical Society. Series B,* **11,** 115-149). On these earlier outings
the principle attracted little attention. See
Likelihood
and Probability in R. A. Fisher’s *Statistical Methods for Research Workers*.

The **LIKELIHOOD RATIO** figured in the test theory of J. Neyman and E. S. Pearson from the
beginning, "On the Use of Certain Test Criteria for Purposes of Statistical Inference, Part I" *Biometrika,*
(1928), **20A,** 175-240. They usually referred to it as *the likelihood* although the phrase "likelihood
ratio" appears incidentally in their "Problem of *k* Samples," *Bulletin Académie Polonaise
des Sciences et Lettres, A,* (1931) 460-481. This phrase was more often used by others writing about Neyman
and Pearson’s work, e.g. Brandner "A Test of the Significance of the Difference of the Correlation Coefficients
in Normal Bivariate Samples," *Biometrika,* **25,** (1933), 102-109.

The standing of "likelihood ratio" was confirmed by S. S. Wilks’s "The Large-Sample Distribution of the
Likelihood Ratio for Testing Composite Hypotheses," *Annals of Mathematical Statistics,* **9,**
(1938), 60-620 [John Aldrich, based on David (2001)].

See the entry WALD TEST.

The term **LIMAÇON** was coined in 1650 by Gilles Persone de
Roberval (1602-1675) [*Encyclopaedia Britannica,* article:
"Geometry"].

*Limaçon* is found in English in 1852 in
*A Treatise on Higher Plane Curves* by G. Salmon:
“The origin is a double point, and the curve is of the fourth
class, and is the ‘limaçon de Pascal.’” [OED]

The curve is sometimes called *Pascal’s limaçon,*
for Étienne Pascal (1588?-1651), the first person to study it.
Boyer (page 395) writes that "on the suggestion of Roberval" the
curve is named for Pascal.

**LIMIT.** Gregory
of St. Vincent (1584-1667) used *terminus* to mean the limit of a progression,
according to Carl B. Boyer in *The History of the Calculus and its Conceptual
Development.*

Isaac
Newton wrote justifying limits in the Scholium to
Section
I of Book I of the *Principia*
(*Philosophiae Naturalis Principia Mathematica*
or *The Mathematical Principles of Natural Philosophy*) (first edition 1687)

Perhaps it may be objected, that there is no ultimate proportion, of evanescent qualities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate, velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical. (Translated by Andrew Motte 1729)

Katz (p. 471) comments, "A translation of Newton’s words into an algebraic statement would give a definition of limit close to, but not identical with, the modern one."

In 1821 Augustin-Louis
Cauchy defined limit as follows: "If the successive values attributed to
the same variables approach indefinitely a fixed value, such that they finally
differ from it by as little as one wishes, this latter is called the limit of
all the others." *Cours d'analyse*
(Oeuvres
II.3), p. 19. (Translation from Katz page 641) Cauchy introduced the modern
ε, δ way of arguing.

This entry was contributed by John Aldrich. See also **Limit** and **Delta and epsilon** on the
Earliest Use of Symbols of Calculus page.

**LIMIT POINT** is found in 1850 in *A Treatise on Conic Sections* by George Salmon:
“Angle ... Subtended at limit points of system of circles.” [*OED*]

Cantor used *Häufungspunkt* (accumulation
point) in an 1872 paper “Über die Ausdehnung eines Satzes der Theorie der
trigonometrischen Reihen,” which appeared in
*Mathematische
Annalen* **5**, pp. 123-132. [Roger Cooke]

*Point of accumulation* appears in English in 1921 in *The Theory of Functions of a Real Variable and the Theory of Fourier Series, v. I,*
by Ernest William Hobson: “A limiting point of *G,* defined as above, is also called a **point of accumulation.**
Another definition of a limiting point is that it is a point *P* such that *G* contains a sequence of points {Pn} for which the distance PPn is less than an arbitrarily chosen ... .”
[Google print search by James A. Landau]

**LINDLEY’S PARADOX.** "An example is produced to show that, if *H* is a simple hypothesis and *x*
the result of an experiment, the following two phenomena can occur
simultaneously: (i) a significance test for *H* reveals that *x* is significant at, say, the 5% level; (ii) the posterior
probability of *H* given *x*, is, for quite small prior probabilities of *H*, as high as 95%."
D. V. Lindley "A Statistical Paradox"
*Biometrika*, **44**, (1957), pp. 187-192. Lindley notes that "the paradox is not
in essentials new, although few statisticians are aware of it." (p. 190) His earliest
reference is to the discussion in Jeffreys’s *Theory of Probability* on which he comments, "Jeffreys is concerned to
emphasise the similarity between his tests and those due to Fisher and the
discrepancies are not emphasised." **Lindley’s Paradox** is the common name although some writers refer to the
**Jeffreys-Lindley Paradox**.

**LINE FUNCTION** was the term used for *functional* by Vito
Volterra (1860-1940), according to the DSB.

**LINE GRAPH** is found in 1916 in *Combinatorial Analysis* by
Percy A. MacMahon. [Google print search by James A. Landau]

The term **LINE INTEGRAL** was used in 1873 by James Clerk Maxwell in a
*Treatise on Electricity and Magnetism*,
p. 71 in the phrase "Line-Integral of Electric Force, or Electromotive
Force along an Art of a Curve" (OED). Earlier in the book (p. 12) Maxwell
explained how "Line-integration [is] appropriate to
forces, surface-integration to fluxes." The *concept* of a line integral
is much older: see the entry GREEN’S THEOREM.

The alternate term *curve integral* has been seen in
1965 textbook, but it may be much older.

The term **LINE OF EQUAL POWER** was coined by Steiner.

**LINEAR ALGEBRA.** The DSB seems to imply that the term
*algebra linearia* is used by Rafael Bombelli (1526-1572) in
Book IV of his *Algebra* to refer to the application of
geometrical methods to algebra.

*Linear algebra* is found in English in 1870 in
the *American Journal of Mathematics* (1881)4/107:
“An algebra in which every expression is reducible to
the form of an algebraic sum of terms, each of which consists of a single
letter with a quantitative coefficient, is called a linear algebra.”
[OED]

*Linear algebra* occurs in 1875 in the title, "On the uses and
transformations of linear algebra" by Benjamin Peirce, published in
*American Acad. Proc.* 2 [James A. Landau].

Pierce meant what today we would call a "finite dimensional algebra over a field," not the theory of vector spaces and linear transformations. [Fernando Q. Gouvea]

In 1898,
Alfred North Whitehead wrote in a footnote in *A Treatise on Universal Algebra with applications*:
“The type of multiplication is then called by Grassman (cf. *Audehanungelebre vom* 1862, §50)
‘linear.” But this
nomenclature clashes with the generally accepted meaning of a ‘linear algebra’
as defined by B. Peirce in his paper
on Linear Associative Algebra, *American Journal of Mathematics,*
vol. IV (1881). The theorem of subsection (2) is due to Grassman, cf. *loc. cit.*”
[Google print search by James A. Landau]

**LINEAR COMBINATION** is found in 1854 in
“On a Theory of the conjugate relations of two rational integral functions, comprising an application to the Theory of Sturm’s
Functions, and that of the greatest Algebraical Common Measure’ by J. J. Sylvester, in
*Abstracts of the Papers communicated the the Royal Society of London from 1850 to 1854 inclusive.*
[Google print search by James A. Landau]

**LINEAR DEPENDENCE** appears in the title
“The Theory of Linear Dependence” by Maxime Bôcher
published in 1900 in the *Annals of Mathematics*
[James A. Landau].

**LINEAR DIFFERENTIAL EQUATION** appears in J. L. Lagrange,
"Recherches sur les suites récurrentes don't les termed
varient de plusieurs manières différentes, ou sur
l'intégration des équations linéaires aux
différences finies et partielles; et sur l'usage de ces
équations dans la théorie des hasards," *Nouv.
Mém. Acad. R. Sci. Berlin* 6 (1777) [James A. Landau].

**LINEAR EQUATION** appears in English the 1816 translation of
*Lacroix’s Differential and Integral Calculus* (OED).

**LINEAR FUNCTION** is found in 1829 in an English translation of
*Mécanique Céleste* by Laplace. [Google print search by James A. Landau]

**LINEAR INDEPENDENCE** is found in 1846 in
“Researches
respecting Quaternions.
First Series,” by Sir William Rowan Hamilton in *Transactions of the Royal Irish Academy.*
[Google print search by James A. Landau].

**LINEAR OPERATOR.** *Linear operation* appears in 1837 in
Robert Murphy, "First Memoir on the Theory of Analytic Operations,"
*Philosophical Transactions of the Royal Society of London,* **127,** 179-210.
Murphy used "linear operation" in the sense of the modern term "linear operator"
[Robert Emmett Bradley].

**LINEAR PRODUCT.** This term was used by Hermann Grassman in his
*Ausdehnungslehre* (1844).

**LINEAR PROGRAMMING.** See PROGRAMMING.

**LINEAR TRANSFORMATION** appears in 1843 in the title
“Exposition of a general theory of linear transformations, Part II”
by George Boole in *Camb. Math. Jour. t. III.* 1843, pp. 1-20. [James A. Landau]

**LINEARLY DEPENDENT** was used in 1893 in "A Doubly Infinite
System of Simple Groups" by Eliakim Hastings Moore. The paper was
read in 1893 and published in 1896 [James A. Landau].

**LINEARLY INDEPENDENT** is found in 1841 in
“On a linear Method of Eliminating between double, treble, and other Systems of Algebraic Equations,” by J. J. Sylvester in
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Vol. XVIII.*
[Google print search by James A. Landau]

**LINK FUNCTION.** One of the components of a GENERALIZED LINEAR MODEL
is "the linking function, *θ* = *f*(*Y*) connecting the parameter
of the distribution of *z* [the
dependent variable] with the *Y*’s
of the linear model." Nelder & Wedderburn *Journal
of the Royal Statistical Society, A*, **135**, (1972), p. 372.
The term **link function** was introduced in J. A. Nelder’s "Log Linear Models for Contingency Tables: A Generalization of Classical
Least Squares," *Applied Statistics*, **23**, (1974),
pp. 323-329. (Based on David (1998))

The term **LITUUS** (Latin for the curved wand used by
the Roman pagan priests known as augurs) was chosen by Roger Cotes
(1682-1716) for the locus of a point moving such that the area of a
circular sector remains constant, and it appears in his *Harmonia
Mensurarum,* published posthumously in Cambridge, 1722 [Julio
González Cabillón].

The term **LOCAL PROBABILITY** is due to Morgan W. Crofton
(1826-1915) (Cajori 1919, page 379).

The term is found in 1865 in James Joseph Sylvester,
“On a Special Class of Questions on the Theory of Probabilities,” *Birmingham
British Association Report* 35:
“After referring to the nature of geometrical or local probability in general,
the author of the paper drew attention to a particular class of questions
partaking of that character in which the condition whose probability is to be ascertained
is one of pure form. The chance of three points within a circle or sphere being apices
of an acute or obtuse-angled triangle, or of the quadrilateral formed by joining four
convex quadrilateral, will serve as types of the class of questions in view.”

The term appears in “Note on Local Probability” by Crofton in
*Mathematical Questions, With Their Solutions,* Vol. 7, from January to July 1867:
“When, therefore, in questions on Local Probability, we speak of a point
or line taken at random, *i.e.* according to no law, there is no obscurity
in the idea; but it cannot be made a matter of arithmetical calculation till we have a definite
conception of the nature of the assemblage of points or lines one (or more) of which we take.”
[Herb Acree]

**LOCATION and SCALE.** The *location* and *scaling*
of frequency curves is discussed in §9 of R. A. Fisher’s
"On the Mathematical Foundations of Theoretical
Statistics" (*Phil. Trans. R. Soc.* 1922, p. 338). In §9 of
Two New Properties of Mathematical Likelihood
(*Proc.R. Soc., *A, 1934 p. 303) Fisher changed his terminology to *the
estimation of location and scaling*. The terms
*location parameter* and *scale parameter* were used by E. J. G. Pitman
in "Tests of Hypotheses Concerning Location and Scale Parameters," *Biometrika*,
**31**, (1939), 200-215.

David (2001)

**LOCUS** is a Latin translation of the Greek word *topos.*
Both words mean "place."

According to Pappus, Aristaeus (c. 370 to c. 300 BC) wrote a work
called *On Solid Loci* (Topwn sterewn).

Pappus also mentions Euclid in connection with locus problems.

Apollonius mentioned the "locus for three and four lines" ("...ton
epi treis kai tessaras grammas topon...") in the extant letter
opening Book I of the *Conica.* Apollonius said in the first
book that the third book contains propositions (III.54-56) relevant
to the 3 and 4 line locus problem (and, since these propositions are
new, Apollonius claimed Euclid could not have solved the problem
completely--a claim that caused Pappus to call Apollonius a braggard
(alazonikos). In Book III itself there is no mention of the locus
problem [Michael N. Fried].

*Locus* appears in the title of a 1636 paper by Fermat, "Ad
Locos Planos et Solidos Isagoge" ("Introduction to Plane and Solid
Loci").

In English, *locus* is found in 1727-41 in Chambers
*Cyclopedia*: "A *locus* is a line, any point of which may
equally solve an indeterminate problem. ... All *loci* of the
second degree are conic sections" (OED).

*Locus geometricus* is an entry in the 1771 *Encyclopaedia
Britannica.*

**LOGARITHM.** Before he coined the term *logarithmus* Napier called these
numbers *numeri artificiales*, and the arguments of his logarithmic function
were *numeri naturales* [Heinz Lueneburg].

*Logarithmus* was coined (in Latin) by John
Napier (1550-1617) and appears in 1614 in his *Mirifici
Logarithmorum Canonis descriptio.*

According to the OED, "Napier does not explain his view of the
literal meaning of *logarithmus.* It is commonly taken to mean
'ratio-number', and as thus interpreted it is not inappropriate,
though its fitness is not obvious without explanation. Perhaps,
however, Napier may have used logos merely in the sense of
'reckoning', 'calculation.'"

According to Briggs in *Arithmetica logarithmica* (1624), Napier
used the term because logarithms exhibit numbers which preserve
always the same ratio to one another.

According to Hacker (1970):

It undoubtedly was Napier’s observation that logarithms of proportionals are "equidifferent" that led him to coin the name "logarithm," which occurs throughout theDescriptiobut only in the title of theConstructio,which clearly was drafted first although published later. The many-meaning Greek wordlogosis therefore used in the sense ofratio.But there is an amusing play on words to which we might call attention since it does not seem to have been noticed. It is interesting that the Greeks also employedlogosto distinguishreckoning,or that is to say mere calculation, fromarithmos,which was generally reserved by them to indicate the use ofnumberin the higher context of what today we call the theory of numbers. Napier’s "logarithms" have indeed served both purposes.

*Logarithm* appears in English in 1616 in E. Wright’s English
translation of the *Descriptio*: "This new course of Logarithmes
doth cleane take away all the difficultye that heretofore hath beene
in mathematicall calculations. [...] The Logarithmes of proportionall
numbers are equally differing."

In the *Constructio,* which was drafted before the
*Descriptio,* the term "artificial number" is used, rather than
"logarithm." Napier adopted the term *logarithmus* before his
discovery was announced.

Jobst Bürgi called the logarithm *Die Rothe Zahl* since the
logarithms were printed in red and the antilogarithms in black in his
Progress Tabulen, published in 1620 but conceived some years earlier
(Smith vol. 2, page 523).

[Older English-language dictionaries pronounce *logarithm* with
an unvoiced *th,* as in *thick* and *arithmetic.*]

See also BRIGGSIAN LOGARITHM, COMMON LOGARITHM, NAPIERIAN LOGARITHM, NATURAL LOGARITHM.

**LOGARITHMIC CURVE.** Huygens proposed the terms *hemihyperbola*
and *linea logarithmica sive Neperiana.*

Christiaan Huygens used *logarithmica* when he wrote in Latin and *logarithmique*
when he wrote in French.

Johann Bernoulli used a phrase which is translated "logarithmic
curve" in 1691/92 in *Opera omnia* (Struik, page 328).

*Logarithmic curve* is found in English in 1715
in *The Elements of Astronomy, Physical and Geometrical*
by David Gregory and Edmond Halley:
"But this is the Property of the Logarithmic Curve very well known to
Geometricians; therefore the Curve *ACX* is a Logarithmic Curve,
whose Asymptote is the Right line *BZ.*" [Google print search]

**LOGARITHMIC FUNCTION.** Lacroix used *fonctions
logarithmiques* in *Traité élémentaire de
calcul différentiel et de calcul intégral*
(1797-1800).

*Logarithmic function* appears in 1831 in the second edition of
*Elements of the Differential Calculus* (1836) by John Radford
Young: "Thus, *a ^{x}*,

The term **LOGARITHMIC POTENTIAL** was coined by Carl Gottfried
Neumann (1832-1925) (DSB).

The term **LOGARITHMIC SPIRAL** was introduced by Pierre Varignon
(1654-1722) in a paper he presented to the Paris Academy in 1704 and
published in 1722 (Cajori 1919, page 156).

Another term for this curve is *equiangular spiral.*

Jakob Bernoulli called the curve *spira mirabilis* (marvelous
spiral).

**LOGIC.** The term *logikê* (knowledge of the functions of *logos* or reason)
was used by the Stoics but it covered many philosophical topics that
are not part of the modern subject. According to W. & M. Kneale *The Development of Logic*
(1962) pp. 7 & 23, the word
"logic" first appeared in its modern sense in the commentaries of
Alexander of Aphrodisias who wrote in the
third century AD. Until the late 19^{th} century the scope of the study
was determined by the contents of
Aristotle’s
(384-322 BC) writings on reasoning.
These were assembled by his pupils after his death and became collectively known
as the *Organon* or instrument of science. See Robin Smith’s
Aristotle’s Logic.

In the Middle Ages logic was one of the three sciences composing
the ‘trivium’, the former of the two divisions of the seven ‘liberal arts’.
The other constituents were rhetoric and grammar. The higher division, the ‘quadrivium’
consisted of arithmetic, geometry, astronomy and music. For the English word
*logic* the *OED*’s earliest quotation is from 1362, the second from
the Prologue to Chaucer’s *Canterbury Tales*: "A Clerk ther was of
Oxenford also, That unto logik hadde longe ygo." (c.1386.) See QUADRIVIUM.

Although the "Bibliography of Symbolic Logic" published in the first volume of the
*Journal of Symbolic Logic* (December 1936, pp. 121-216) starts in 1666 with Leibniz, the modern era in
logic begins in the 19^{th} century with the work of
Augustus de Morgan (1806-1871) and
George Boole (1815-1864).
By the end of the century many new terms had been coined, including names for the subject, for
the author’s particular take on it and for its various sub-divisions. Some of
the names are still in use, although their meaning has often shifted. Some authors
kept the new terms distinct, others would use them indifferently, e.g. Bertrand
Russell treated "symbolic logic" and "mathematical logic"
as interchangeable in his "Mathematical Logic as Based on
the Theory of Types," *American Journal of Mathematics*, **30**,
(1908), 222-262.

**Formal logic** and **symbolic logic** were
used as book-titles by De Morgan (1847) and J. Venn (1881) respectively. G.
Peano used **mathematical logic** as the name of his new subject. Its concerns
were not those of traditional logic, as he explained to Felix Klein in 1894:
"the aim of Mathematical logic is to analyse the ideas and reasoning which
feature especially in the mathematical sciences." (quoted on p. 243 of
Grattan-Guinness (2000). E. Schröder used the phrase **algebra of logic**
in the title of his main work, *Vorlesungen über die Algebra der Logik* (volume
1, 1890). **Logistic** (French *logistique*) was used by Couturat and
other speakers at the International Congress of Philosophy in 1904 and was popular
for a few decades. The terms **deductive logic** and **inductive logic**
originated in the 19^{th} century: W. S. Jevons called one of his books
*Studies in Deductive Logic* (1880) and Venn one of his, *The Principles
of Empirical or Inductive Logic* (1889). **Informal logic** is a new term,
having been in use only since the 1970s; see Leo Groarke’s
*Informal logic*.

This entry was contributed by John Aldrich.
See MATHEMATICAL LOGIC. A complete list of the set theory and logic terms on this web site is
here.
For the symbols of logic see *Earliest Use of Symbols*.

**LOGICISM** is the doctrine that mathematics is in some significant sense reducible to logic.
It is associated with the
*Principia Mathematica *(1910-1913) of
A. N. Whitehead and Bertrand Russell. According to Grattan-Guinness (2000, pp.
479 & 501), the word *Logizismus* was introduced by A. A. H. Fraenkel
*Einleitung in der Mengenlehre* (1928) and R. Carnap *Abriss der Logistik*
(1929). A *JSTOR* search found the English word in H. Reichenbach "Logical Empiricism in Germany and the Present State
of its Problems," *Journal of Philosophy*, **33**, (1936), p. 143.

This entry was contributed by John Aldrich. See also FORMALISM and INTUITIONISM.

The term **LOGISTIC CURVE** is attributed to Edward Wright (ca.
1558-1615) (Thompson 1992, page 145), although Wright used the term to refer to the logarithmic curve.

Pierre Francois Verhulst (1804-1849) introduced the term
*logistique* as applied to the sigmoid curve [Julio
González Cabillón]. David (1995) gives the citation
P. F. Verhulst (1845), "La Loi d' Accroissement de la Population,"
*Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles,* 18, 1-59.
Presumably the term refers to the "log-like" qualities of the curve.

The term **logistic regression** appears in D.
R. Cox "The Regression Analysis of Binary Sequences," *Journal of the Royal
Statistical Society. Series B (Methodological)*, **20**, (1958), 215-242.

**LOGIT** first appeared in Joseph Berkson’s "Application to the Logistic Function to Bio-Assay,"
*Journal of the American Statistical Association*, **39**, (1944), p. 361: "Instead of the observations
*q** _{i}* we deal
with their logits

See PROBIT.

**LOGNORMAL.** *Logarithmic-normal* was used in 1919 by S. Nydell in "The
Mean Errors of the Characteristics in Logarithmic-Normal Distributions,"
*Skandinavisk Aktuarietidskrift,* 2, 134-144 (David, 1995).

*Lognormal* was used by J. H. Gaddun in *Nature* on Oct. 20, 1945: "It
is proposed to call the distribution of *x* 'lognormal' when the distribution
of log *x* is normal" (OED).

The lognormal distribution was apparently first studied when
Donald McAlister answered a question put by Francis Galton: to what "law
of error" does the geometric mean bear the relationship that the arithmetic
mean bears to the normal distribution? See Galton’s
The
Geometric Mean, in Vital and Social Statistics
*Proceedings of the Royal Society of London*, **29**, (1879), 365-367,
which prefaced McAlister’s paper
"The
Law of the Geometric Mean." They
did not give a name to the new distribution. However, according to E. T. Whittaker
& G. Robinson *Calculus of Observations* (1924, p. 218), Seidel had
asked and answered the same question in 1863.

See also ARITHMETIC MEAN and GEOMETRIC MEAN.

**LONG DIVISION** is found in 1744 in *The Schoolmaster’s Assistant*
by Thomas Dilworth: “It very seldom happens that the Divisor consists
of more than one Denomination; yet because such Divisors may sometimes offer themselves,
I will give a few for the reader’s satisfaction, which musst be wrought
after the manner of Long Division, and may serve also as proofs to some of the foregoing
examples in Multiplication.” [OED]

**LORENZ ATTRACTOR.** This object was first described by the
meteorologist Edward Norton
Lorenz in his paper “Deterministic non-periodic flow,”
*J. Atmos. Sci.,* **20** : 2 (1963) pp. 130–141. The term
“Lorenz attractor” came into use in the 1970s when his work
began to be noticed. See MathWorld
and the Encyclopedia of
Mathematics.

**LORENZ CURVE.** This
diagram was introduced by
Max O. Lorenz in his “Methods of Measuring the Concentration of
Wealth,” *Publications of the American
Statistical Association*, **9**,
(Jun., 1905), pp. 209-219. The term *Lorenz
curve* quickly entered circulation—see e.g. W. M. Persons “The
Measurement of Concentration of Wealth,” *Quarterly
Journal of Economics*, **24**,
(Nov., 1909), p. 172. According to M. J. Bowman “A
Graphical Analysis of Personal Income Distribution in the United States,” *American Economic Review*, **35**, (1945), p. 617n, “The same idea was
introduced almost simultaneously by Gini, Chatelain and Séailles.” The Séailles
reference is to his 1910 book *La répartition
des fortunes en France*.

**LOSS** and **LOSS FUNCTION **in statistical
decision theory. In the paper establishing the subject ("Contributions to the
Theory of Statistical Estimation and Testing Hypotheses," *Annals of Mathematical
Statistics,* **10**, 299-326) Wald referred to "loss" but used
"weight function" for the (modern) loss function. He continued
to use weight function, for instance in his book *Statistical Decision Functions*
(1950), while others adopted loss function. Arrow, Blackwell & Girshick’s
"Bayes and Minimax Solutions of Sequential Decision Problems" (*Econometrica,
* **17**, (1949) 213-244) wrote *L* rather than *W* for the
function and called it the loss function. A paper by Hodges & Lehmann ("Some
Problems in Minimax Point Estimation," *Annals of Mathematical Statistics,*
**21**, (1950), 182-197) used loss function more freely but retained Wald’s
*W*.

This entry was contributed by John Aldrich, based on David (2001) and *JSTOR*.
See DECISION THEORY.

The term **LOWER SEMICONTINUITY** was used by René-Louis Baire (1874-1932), according
to Kramer (p. 575), who implies he coined the term. The term appears in Baire’s thesis,
“Sur les fonctions de variables réelles,” *Annali di Matematica Pura ed Applicata* (3) 3 (1899), 1-123,
and in his “Sur la théorie des fonctions discontinues,
*Comptes rendus,* **129**, (1899) 1010-1013.

The phrase **LOWEST TERMS** appears in about 1675 in Cocker’s
*Arithmetic,* written by Edward Cocker (1631-1676): "Reduce a
fraction to its lowest terms at the first Work" (OED). (There is
some dispute about whether Cocker in fact was the author of the
work.)

**LOXODROME.** Pedro Nunez (Pedro Nonius) (1492-1577) announced
his discovery and analysis of the curve in *De arte navigandi.*
He called the curve the *rumbus* (*Catholic Encyclopedia*).

The term *loxodrome* is due to Willebrord Snell van Roijen
(1581-1626) and was coined in 1624 (Smith and DSB, article: "Nunez
Salaciense).

**LUCAS-LEHMER TEST** occurs in the title, "The Lucas-Lehmer test for Mersenne
numbers," by S. Kravitz in the *Fibonacci
Quarterly* 8, 1-3 (1970). The test is named for
Edouard Lucas and
Dick Lehmer.

The term *Lucas’s test* was used in 1932 by A. E. Western in "On
Lucas’s and Pepin’s tests for the primeness of Mersenne’s numbers,"
*J. London Math. Soc.* 7 (1932), and in 1935 by D. H. Lehmer
in "On Lucas’s test for the primality of Mersenne’s numbers," *J.
London Math. Soc.* 10 (1935).

The term **LUCAS PSEUDOPRIME** occurs in the title "Lucas
Pseudoprimes" by Robert Baillie and Samuel S. Wagstaff Jr. in
*Math. Comput.* 35, 1391-1417 (1980): "If n is composite, but
(1) still holds, then we call n a Lucas pseudoprime with parameters P
and Q ..." [Paul Pollack].

**LUDOLPHIAN NUMBER.** The number 3.14159... was often called the
*Ludolphische Zahl* in Germany, for
Ludolph
van Ceulen.

In English, *Ludolphian number* is found in 1886 in G. S. Carr,
*Synopsis Pure & Applied Math* (OED).

In English, *Ludolph’s number* is found in 1894 in *History
of Mathematics* by Florian Cajori (OED).

**LUNE.** *Lunula* appears in *A Geometricall Practise named Pantometria* by Thomas Digges
(1571): "Ye last figure called a Lunula" (OED).

*Lune* appears in English in 1704 in *Lexicon technicum, or an
universal English dictionary of arts and sciences* by John Harris
(OED).