William Spence (born 31 July 1777 in

Greenock Greenock (; sco, Greenock; gd, Grianaig, ) is a town and administrative centre in the Inverclyde council areas of Scotland, council area in Scotland, United Kingdom and a former burgh of barony, burgh within the Counties of Scotland, historic ...

, Scotland – died 22 May 1815 in

Glasgow Glasgow ( ; sco, Glesca or ; gd, Glaschu ) is the most populous city in Scotland and the fourth-most populous city in the United Kingdom, as well as being the 27th largest city by population in Europe. In 2020, it had an estimated popul ...

, Scotland) was a Scottish mathematician who published works on the fields of

logarithmic functions In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

s and their relation to

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

and

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

respectively.

Early life, family, and personal life

Spence was the second son to Ninian Spence and his wife Sarah Townsend. Ninian Spence ran a

coppersmith A coppersmith, also known as a brazier, is a person who makes artifacts from copper and brass. Brass is an alloy of copper and zinc. The term "redsmith" is used for a tinsmith that uses tinsmithing tools and techniques to make copper items. Hi ...

business, and the Spence family were a prominent family in

at the time. From an early age, Spence was characterised as having a docile and reasonable nature, with him being mature for his age. At school he formed a life-long friendship with

John Galt John Galt () is a character in Ayn Rand's novel ''Atlas Shrugged'' (1957). Although he is not identified by name until the last third of the novel, he is the object of its often-repeated question "Who is John Galt?" and of the quest to discover ...

, who documented much of his life and his works posthumously. Despite having received a formal education until he was a teenager, Spence never attended university, instead he moved to

where he lodged with a friend of his fathers, learning the skills of a manufacturer. Two years after his father's death in 1795, Spence returned to

in 1797. With the support of Galt and others, he established a small

literary society A literary society is a group of people interested in literature. In the modern sense, this refers to a society that wants to promote one genre of writing or a specific author. Modern literary societies typically promote research, publish newsle ...

, wherein once a month they read a range of essays on varying subjects, this society met frequently until 1804. After this, Spence visited many places in England, he lived in London for a few months where, in 1809, he published his first work. In 1814, he published his second work, getting married in the same year – Spence intended to live in London, and began his journey back before becoming ill, having travelled as far as

, he died in his sleep due to illness. Spence held an interest in

musical composition Musical composition can refer to an original piece or work of music, either vocal or instrumental, the structure of a musical piece or to the process of creating or writing a new piece of music. People who create new compositions are called ...

, and played the flute.

Published works

Spence published ''An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series in 1809.'' Throughout his work, he displayed a familiarity with the work of

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaArbogast, which is notable since at the time very few were familiar with their works. In his preface he derived the

binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

and mainly focused on the properties and analytic applications of the series: :

\pm x/1^n - x^2/2^n \pm x^3/3^n - ...

which he denoted with

L_n(1\pm x)

. He went on further to derive nine general properties of this function in a table. Spence also wrote on presenting analytical mathematics without the need of demonstrating the practical applications of such work. Spence continues to write that the functions

L_n(1\pm x)

can be expressed as

iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

s of the previous n term: :

L_1(1\pm x)  = \int \pm dx/1\pm x

L_2(1\pm x) = \int (dx/x) L_1(1\pm x)

, . . . ,

L_n(1\pm x) = \int(dx/x) L_(1\pm x)

for all values of x. Spence goes on to calculate the values of: :

L_2(x) = -\int^x_0\ln (1-t/t) \operatorname\!t

to nine decimal places, in a table, for all integer values of

1 + x

from 1 to 100, the first ever of its kind. Spence dilogarithm table

Later on he also created a similar table for

\tan^x

. Spence published his last work, ''Outlines of a theory of Algebraical Equations, deduced from the principles of Harriott, and extended to the fluxional or differential calculus was published in 1814.'' In which he took a systematic approach to solving equations up to the fourth degree using symmetrical functions of the roots. After Spence's death,

John Herschel Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint and did botanical wor ...

edited ''Mathematical Essays by the late William Spence'', which was published in 1819, with John Galt writing a biography on Spence.

Legacy

Spence's work was noted to be remarkable at the time, with John Herschel, his acquaintance and one of Britain's leading mathematicians at the time, had referenced it in one of his later publications ''Consideration of various points of analysis,'' which prompted Herschel to edit Spence's manuscripts. Spence was held in such high regard by Galt, and later Herschel that they published a collection of his individual essays in 1819. Posthumously, his work was met with appreciation from his contemporaries, with a review in the ninety-fourth number of the

Quarterly Review The ''Quarterly Review'' was a literary and political periodical founded in March 1809 by London London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River ...

(reproduced in Galt's ''The Literary and Miscellanies of John Galt, Volume 1)'' that described his first work in 1809 as: ''" hefirst formal essay in our language on any distinct and considerable branch of the integral calculus, which has appeared since… Hellinsʼs papers on the ‘Rectification of the Conic Sections".''

References

{{DEFAULTSORT:Spence, William Scottish flautists 19th-century Scottish mathematicians Scientists from Glasgow People from Greenock 1777 births 1815 deaths