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- Source: Beryl May Dent
Beryl May Dent (10 May 1900 ā 9 August 1977) was an English mathematical physicist, technical librarian, and a programmer of early analogue and digital computers to solve electrical engineering problems. She was born in Chippenham, Wiltshire, the eldest daughter of schoolteachers. The family left Chippenham in 1901, after her father became head teacher of the then recently established Warminster County School. In 1923, she graduated from the University of Bristol with First Class Honours in applied mathematics. She was awarded the Ashworth Hallett scholarship by the university and was accepted as a postgraduate student at Newnham College, Cambridge.
She returned to Bristol in 1925, after being appointed a researcher in the Physics Department at the University of Bristol, with her salary being paid by the Department of Scientific and Industrial Research. In 1927, John Lennard-Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics. LennardāJones pioneered the theory of interatomic and intermolecular forces at Bristol and she became one of his first collaborators. They published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, that were to become the foundation of her master's thesis. Later work has shown that the results they obtained had direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.
In 1930, she joined Metropolitan-Vickers Electrical Company Ltd, Manchester, as a technical librarian for the scientific and technical staff of the research department. She became active in the Association of Special Libraries and Information Bureaux (ASLIB) and was honorary secretary to the founding committee for the Lancashire and Cheshire branch of the association. She served on various ASLIB committees and made conference presentations detailing different aspects of the company's library and information service. She continued to publish scientific papers, contributing numerical methods for solving differential equations by the use of the differential analyser that was built for the University of Manchester and Douglas Hartree. She was the first to develop a detailed reduced major axis method for the best fit of a series of data points.
Later in her career she became leader of the computation section at Metropolitan-Vickers, and then a supervisor in the research department for the section that was investigating semiconducting materials. She joined the Women's Engineering Society and published papers on the application of digital computers to electrical design. She retired in 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department. In 1962, she moved with her mother and sister to Sompting, West Sussex, and died there in 1977.
Early life
Beryl May was born on (1900-05-10)10 May 1900, at Penley Villa, Park Lane, Chippenham, Wiltshire, the eldest daughter of Agnes Dent (1869ā1967), nĆ©e Thornley, and Eustace Edward (1868ā1954). She was baptised at St Paul's, Chippenham, on 8 June 1900.:ā1ā They had married at St Mary's Church, Goosnargh, near Preston, Lancashire, on 27 July 1898. Her mother was educated at the Harris Institute, Preston, passing examinations in science and art. She was a teacher at Attercliffe School, in northeast Sheffield, before moving to Goosnargh School, near her hometown of Preston, where her elder brother and sister, John William and Mary Ann Thornley, were the head teachers. In March 1894, she had applied for the headship at Fairfield School, Cockermouth, making the shortlist, but the board decided to appoint a local candidate.
On 18 March 1889, Dent's father was appointed to a teaching assistant position at Portland Road School, in Halifax, West Yorkshire, after completing a teaching apprenticeship with the school board. In the same year, Florence Emily Dent, his elder sister, was appointed head teacher at West Vale girls' school, Stainland Road, Greetland, moving from the Higher Board School at Halifax. In August 1889, he obtained a first class pass in mathematics from the Halifax Mechanics' Institute. He enrolled on a degree course at University College, Aberystwyth, in the Education Day Training College. In January 1894, he was awarded a first by Aberystwyth, and a first in the external University of London examinations. His first teaching post was at Coopers' Company Grammar School, Bow, London, before moving to Chippenham, where he was a senior assistant teacher at the Chippenham County School.
In October 1901, Dent's father left Chippenham to become head teacher of the then recently established Warminster County School, that adjoined the Athenaeum Theatre in Warminster. The family moved to Boreham Road, Warminster, where houses were built in the early 19th century. In April 1907, they moved to 22 Portway, Warminster, situated a short distance from the County School and the Athenaeum.:ā264ā He was elected chair of the Warminster Urban District Council from 1920 to 1922, and remained as head teacher of the County School until his retirement in August 1929.
Dent's father was also a regular cast member of the Warminster Operatic Society at the Athenaeum and other venues. Dent and her younger sister, Florence Mary, would often appear with him on stage in such operettas as Snow White and the seven dwarfs and the Princess JuāJu (The Golden Amulet), a Japanese operetta in three acts by Clementine Ward. In Princess JuāJu, she played La La, one of the three maidens attendant on the princess, and sang the first act solo, She must be demure. In act two of the same musical, she performed in the fan dance, Spirits of the Night. She also acted in a scene from Tennyson's Princess at the County School's prize giving day on 16 December 1913.
Education
= Warminster County School (1909ā1917)
=
From 1909, Dent was educated at Warminster County School, where her father was head teacher. At school, she was close friends with her neighbour at Portway, Evelyn Mary Day, the eldest daughter of Henry George Day, a former butler to Colonel Charles Gathorne GathorneāHardy, son of Gathorne Gathorne-Hardy, 1st Earl of Cranbrook. In August 1914, she passed the University of Oxford Junior Local Examination with First Class Honours, and on the strength of her examination result, she was awarded a scholarship by the school. In 1915, she passed the senior examination with second class honours and a distinction in French, and subsequently, her scholarship was renewed. She then joined the sixth form and won the school prize for French in December 1916. In March 1918, she applied for a scholarship in mathematics from Somerville College, one of the first two women's colleges in the University of Oxford. She was highly commended but was not awarded a scholarship nor an exhibition.
= University of Bristol (1919ā1923)
=
In 1918, Dent joined the Royal Aircraft Establishment (RAE) at Farnborough, Hampshire. The First World War opened new employment opportunities for women, and RAE was one of the first military establishments to recruit women into engineering, and mathematical and computational research.:ā116ā:ā10ā In the same period that Dent was at RAE, Lorna Swain, then mathematics tutor at Newnham College in the University of Cambridge, worked at the establishment on the problem of aircraft propeller vibration.:ā84ā The Treasury reduced RAE's funding after the end of the war, and consequently, the number of resources and staff available to support research fell significantly.:ā58ā In 1919, she left RAE after being accepted on to the general Bachelor of Arts (BA) degree course at the University of Bristol. In June 1920, she passed her intermediate examination in French with supplementary courses in Latin, history, and mathematics.:ā1ā
In the following academic year, Dent joined the honours course in mathematics and took an intermediate examination in physics.:ā1ā After spending the summer of 1921 at her parents' home in Warminster, she returned for the start of the 1921 to 1922 academic year to find that Paul Dirac had joined the mathematics course. The course of mathematics at Bristol University normally lasted three years, but because of Dirac's previous training, the Department of Mathematics had allowed him to join in the second year. They were taught applied mathematics by Henry Ronald HassĆ©, the then head of the Mathematics Department, and pure mathematics by Peter Fraser. Both of them had come from Cambridge; Fraser had been appointed in 1906 to the staff of the Bristol University College, soon to become the University of Bristol, and HassĆ© joined him in 1919 as professor of mathematics.:ā111ā Fraser introduced them to mathematical rigour, projective geometry, and rigorous proofs in differential and integral calculus. Dirac would later say that Peter Fraser was "the best teacher he had ever had."
Dent studied four courses in pure mathematics:
There was a choice of specialisation in the final year; applied or pure mathematics. As the only official, registered fee-paying student, Dent had the right to choose, and she settled on applied mathematics for the final year. The department could offer only one set of lectures so Dirac also had to follow the same course. Dent studied four courses in applied mathematics:
= Newnham College, University of Cambridge (1923ā1924)
=
In June 1923, Dent graduated with Dirac, gaining a Bachelor of Science (BSc) degree in applied mathematics with First Class Honours. On 7 July 1923, she was awarded the Ashworth Hallett scholarship by the University of Bristol and was accepted as a postgraduate student at Newnham College in the University of Cambridge. On her death in 1922, Lilias Sophia Ashworth Hallett left one thousand pounds each to the University of Bristol and Girton College, University of Cambridge, to found scholarships for women.:ā259,ā261ā The University of Bristol scholarship was open to women graduates of a recognised college or university, and worth Ā£45 at the time (equivalent to Ā£3,200 in 2023). She spent a year at Cambridge, leaving in 1924 without further academic qualification. Before 1948, the University of Cambridge denied women graduates a degree, although in the same year as she left Cambridge, Katharine Margaret Wilson was the first woman to be awarded a PhD by the university.
Career
= High School for Girls, Barnsley (1924ā1925)
=
Dent spent the summer of 1924 at her parents' home in Warminster, playing mixed doubles tennis in a tournament organised by the local Women's Unionist Association. In September of the same year, she was appointed an assistant teacher in mathematics at the High School for Girls, in Barnsley, Huddersfield Road, on an annual salary of Ā£250 (equivalent to Ā£18,000 in 2023). Annie Rose Nuttall, the school's head teacher, was a former student at Newnham College. In the early 1920s, women who had studied university level mathematics faced limited employment prospects, as mathematics and engineering professions, other than perhaps school teaching, were dominated by men. Dent resigned her position on 31 August 1925 after being appointed a demonstrator (research assistant) in the Department of Physics at the University of Bristol, with her salary being paid by the Department of Scientific and Industrial Research, the forerunner of the Science and Engineering Research Council (SERC).:ā107ā
= Department of Physics, University of Bristol (1925ā1929)
=
In 1924, the University of Bristol Council had set aside a portion of a bequest from Henry Herbert Wills for the Department of Physics where Arthur Mannering Tyndall was building up a staff for teaching and research in the H. H. Wills Physics Laboratory, Royal Fort House Gardens. From August 1925, John Lennard-Jones, of Trinity College, University of Cambridge, was elected reader in mathematical physics. In March 1927, LennardāJones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics.:ā24ā LennardāJones pioneered the theory of interatomic and intermolecular forces at Bristol and Dent became one of his first collaborators.
LennardāJones and Dent published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, with the objective of calculating theoretically the properties of carbonate and nitrate crystals. Dent's thesis for her master's degree, Some theoretical determinations of crystal structure (1927), was the basis of the three papers that followed in 1927; with LennardāJones, "Some theoretical determinations of crystal parameters. CXVI", and with LennardāJones and Sydney Chapman, "Structure of carbonate crystals" and "Part II. Structure of carbonate crystals". On 28 June 1927, she was awarded a MSc degree for her thesis and research work. In 1927, the physics laboratory at Bristol had a surplus of funds, and so it was decided that the funds would be used to provide more technical help. Consequently, Dent was asked to combine her research duties with the post of part-time departmental librarian, the first appointment of librarian in the Department of Physics.:ā26ā
In 1928, LennardāJones and Dent published two papers, "Cohesion at a crystal surface", and with Sydney Chapman, "The change in lattice spacing at a crystal boundary", that studied the force fields on a thin crystal cleavage. Around this time, quantum mechanics was developed to become the standard formulation for atomic physics. LennardāJones left Bristol in 1929 to study the subject for a year as a Rockefeller Fellow at the University of Gƶttingen. She wrote one last paper before leaving the physics department at Bristol: "The effect of boundary distortion on the surface energy of a crystal" (1929) examined the effect of the polarisation of surface ions in decreasing the surface energy of alkali halides. In November 1929, she was appointed to the position of technical librarian for the scientific and technical staff in the research department at Metropolitan-Vickers, Trafford Park, Manchester.:ā14ā15ā
In December 1929, Dent resigned her position at Bristol and it was accepted with regret by the university council. Marjorie Josephine Littleton, the daughter of a local Bristol councillor and a graduate of Girton College, University of Cambridge, was appointed as her replacement on the 1 February 1930. Littleton was later Sir Neville Mott's co-author and research assistant in the physics department.:ā517ā In 1930, LennardāJones returned to Bristol, as Dean of the Faculty of Science, and introduced the new quantum theories to the Bristol group.
= Metropolitan-Vickers, Trafford Park (1930ā1960)
=
Metropolitan-Vickers was a British heavy industrial firm, well-known for industrial electrical equipment and generators, street lighting, electronics, steam turbines, and diesel locomotives. They built the Metrovick 950, the first commercial transistorised computer. In 1917, a Research and Education Department was established at the Trafford Park site, when the care of the library came within the remit of James George Pearce. He made the library the centre of a new "technical intelligence" section.:ā193ā In the 1920s, the post of librarian was held by Lucy Stubbs, a former assistant librarian at the University of Birmingham, and past member of the first standing committee of ASLIB.:ā228ā Stubbs did not possess scientific qualifications, maintaining that a librarian, if assisted by other technical staff, did not need to understand science or engineering.:ā193ā In 1929, James Steele Park Paton reorganised and expanded the section with Dent succeeding Stubbs as technical librarian on 6 January 1930.:ā15ā She joined the scientific and technical staff as was one of only two senior women in the research department,:ā314ā and in contrast to Stubbs, was employed principally for her technical skills.:ā193ā
Dent was honorary secretary to the founding committee for the ASLIB Lancashire and Cheshire branch from 1931 to 1936.:ā204ā205ā In 1932, the branch had twenty-six members and had organised four meetings, including one addressed by Sir Henry Tizard, the then President of ASLIB. After the war, it formed the basis for the Northern Branch of the association.:ā412ā Technical librarianship emerged as a new scientific career in interwar Britain and rapidly became one of the few types of professional industrial employment that was routinely open to both women and men.:ā301ā By 1933, Dent reported that the Metropolitan-Vickers library had three thousand engineering volumes and around the same number in pamphlets and patent specifications. Besides covering electrical subjects, the library covered accountancy, employment questions, and subjects of interest to the sales department. It also issued a weekly bulletin, scrutinised patents, handled patents taken out by research staff, and exchanged information with associated companies.
Dent continued to publish papers in applied mathematics and contribute to papers on emerging computational technologies. In "On observations of points connected by a linear relation" (1935), she developed a detailed reduced major axis method for line fitting that built on the work of Robert Adcock and Charles Kummell. In 1937, David Myers, then at the Engineering Laboratory at the University of Oxford, asked Douglas Hartree and Arthur Porter to calculate the space charge limitation of secondary current in a triode.:ā91ā:ā96ā The calculations relied on some initial numerical integrations that were carried out by Dent on a differential analyser. The results corresponded closely to those obtained experimentally by Myers at Oxford.:ā91ā:ā97ā Her knowledge of higher mathematics meant that she was asked to check the mathematics in papers for publication by engineers at Metropolitan-Vickers. For example, Cyril Frederick Gradwell, a graduate of Trinity College, Cambridge, asked her to scrutinise the algebraic part of his work in "The Solution of a problem in disk bending occurring in connexion with gas turbines" (1950). She would later analyse the problem of stress distribution in a thick disk based on a method devised by Philip Pollock, for Richard William Bailey, the former director of the mechanical, metallurgical, and chemical sections of the research department at Metropolitan-Vickers.:ā16ā
Dent was a delegate at the fourteenth International Conference on Documentation and was invited to the Government's conference dinner on 22 September 1938 at the Great Dining Hall of Christ Church, Oxford. In 1939, she was elected to the editing committee of the ASLIB book list. In 1944, she was put in charge of the women working in the research department laboratory at Metropolitan-Vickers, and in 1946, she was promoted to section leader of the new computation section. Her role would bring her into contact with Audrey Stuckes, a materials science researcher in the department, and a graduate of Newnham College, who would later head the physics department at the University of Salford. In 1953, they collaborated on an investigation into the heating effects that occur when a current is passed through a semiconductor that has no barrier layer. Dent suggested methods to solve the equations and computed the numerical integrations. In the following year, she developed the Fourier analysis in "Regenerative Deflection as a Parametrically Excited Resonance Phenomenon" (1954), that calculated the optimal radial oscillations to maintain cyclotron resonance in a synchrocyclotron. The causes of axial spreading of the charged particle beam during extraction were also analysed.
Dent joined the Women's Engineering Society and published papers on the application of digital computers to electrical design. With Brian Birtwistle, she wrote programs for the Ferranti Mark 1 (Mark 1) computer at the University of Manchester, that demonstrated that high-speed digital computers could provide considerable assistance to the electrical design engineer. Birtwistle would later have an extensive career in the computer industry, working at, amongst others, Honeywell Information Systems and ADP Network Services. In 1958, she carried out computer calculations for the mechanical engineering team at the Nuclear Power Group, Radbroke Hall. Their paper outlined a procedure for calculating the theoretical deflection (bending) of a circular grid of support girders for a graphite neutron moderator in a gas-cooled reactor. A general expression was derived from the central deflection of the grid and the maximum bending moment on the central cross-beam for a range of grid diameters.
In 1959, and a year from retirement, Dent modelled a proposed Zeta circuit on the Mark 1 computer, for Eric Hartill's paper on constructing a high-power pulse transformer and circuit. The cost of the computation was about two thousand pounds (equivalent to Ā£59,000 in 2023), corresponding to around eighty hours of machine time. She retired from Metropolitan-Vickers in May 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department.:ā232ā:ā243ā
Personal life
In the 1920s, Dent was living at Clifton Hill House, the university hall of residence for women in Clifton. May Christophera Staveley was her warden and tutor at Clifton Hill House, and Dent returned to Bristol on 22 December 1934 for Staveley's funeral. Dent was a member of the Clifton Hill House Old Students Association, and secretary and treasurer of the group of former Clifton Hill House students.:ā1, 9ā She would later write "I was very sorry indeed to leave Bristol and have many happy memories of my time there. I shall miss living at the [Clifton Hill House] Hall very much.":ā15ā
In 1926, Dent was elected treasurer of the University of Bristol's Convocation, the university's alumni association. In 1927, she was one of eleven people elected to the standing committee of the Convocation:ā62ā She later represented the Manchester branch of the association. Around 1926, Dent was appointed honorary secretary of the Bristol Cheeloo Association. The association's aim was to raise sufficient funds to support a chair of chemistry at Cheeloo University. In an effort to publicise the cause and raise money, she presented to the local branch of the Women's International League in October 1928.
In July 1929, in Dent's last year at Bristol, she went on holiday to North Devon with friends that included Gertrude Roxbee, known as "Rox", who had graduated with Dent in 1923 with a BSc in botany.:ā12ā13ā After moving to Manchester in January 1930, Dent found shared lodgings at 10 Montrose Avenue, West Didsbury, in the same house as Roxbee who, at that time, was a teacher at Whalley Range High School.:ā15, 50ā At weekends, she would ramble to Hebden Bridge, and with Roxbee, learnt to figure skate at the Ice Palace, a former ice rink on Derby Street in Cheetham Hill.:ā54ā
In September 1930, she returned to Bristol for the ninety-eighth conference of the British Association for the Advancement of Science (British Association), meeting her friends at an alumnae association lunch.:ā93ā In the afternoon of the 4 September 1930, she toured Avonmouth Docks as a conference member,:ā94ā and in the evening, was invited to a reception held by Walter Bryant, the then lord mayor of Bristol, at the Bristol Museum & Art Gallery.:ā94ā. On the following day, she visited an aircraft manufacturer at Whitchurch Airport and attended a garden party at Wills Hall.:ā94ā On the Monday of the conference, Dent was in the audience to see Paul Dirac present his paper on the proton and the structure of matter.:ā94ā She would later comment::ā94ā
I heard a striking paper by Dirac, who was a student with me, who is now a very famous person, as I always knew he would be ... I now go about boasting that I was in the same class!
Dent's father died on (1954-06-24)24 June 1954, at their shared home, 529 King's Road, Stretford, with the funeral service taking place at St Matthew's Church, Stretford. She had close links to St Matthew's; from 1956 to 1962, she served as a school manager for St Matthew's Church of England Primary School at Poplar Road, Stretford.
Later life and death
In 1962, Dent and her mother moved from Stretford to 1 Cokeham Road, Sompting, a village in the coastal Adur District of West Sussex, between Lancing and Worthing. Her mother died on (1967-04-05)5 April 1967 and was cremated at the Downs Crematorium on 10 April 1967. Dent's sister, Florence Mary, also lived in the house until her death on 13 September 1986(1986-09-13) (aged 84). After a brief period as a teacher at a prep school in Malmesbury, Wiltshire, Florence worked as a secretary for a marine insurance firm attached to Lloyd's of London at 12 Leadenhall Street, commuting into London from Harrow each day.:ā57ā59ā
Dent considered herself to be an Anglican layperson who was neither high nor low church. In April 1970, she was elected treasurer of Lancing and Sompting Churches Fraternal (the parish fraternity organisation), and in March 1972, she was elected electoral officer for the parochial church council of St Mary's Church, Sompting. Her Christian faith is perhaps not unexpected, given her father's work for the church in Warminster, and the era she grew up in, where religion pervaded social and political life. However, it is notable that she remained a Christian while pursuing a scientific career.
In June 1974, Dent was hospitalised for seven weeks at Southlands Hospital, Shoreham-by-Sea, and after a long period of disablement, she died at Worthing Hospital on 9 August 1977(1977-08-09) (aged 77). The funeral service was held on 12 August 1977 at St Mary's, followed by cremation. Her ashes were interred at Worthing Crematorium, in the Gardens of Rest, towards the Spring Glades, and her entry in the book of remembrance at the crematorium states:
Beryl May Dent 1900 ā A real Christian loved by all ā 1977
The bishop's chair at St Mary's, situated close to the altar, bears a brass plaque with the following inscription:
In loving memory of BERYL DENT 1900 ā 1977
Legacy
An archive of Dent's papers, that relate to her life and work in the 1920s in the physics department at the University of Bristol, is held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol. Included in that archive is a series of letterbooks, written in the 1930s by members of the Clifton Hill House Old Students' Association, that include news and photographs of Dent, her family, and friends.
= Atomic force microscopy
=
In 1928, LennardāJones and Dent published two papers, "Cohesion at a crystal surface" and "The change in lattice spacing at a crystal boundary", that for the first time, outlined a calculation of the potential of the electric field in a vacuum, produced by a thin sodium chloride crystal surface. They gave an expression for the electric potential produced by a system of point charges in vacuum (although not a real cubic sodium chloride ionic lattice).:ā796ā797ā The expression for the potential in vacuum,
Ļ
0
(
r
)
{\displaystyle \varphi _{0}\left(r\right)}
, at the point r = {x, y, z}, near the cubic lattice of point ions with different signs, the charge
e
k
{\displaystyle e_{k}}
, and the period a (a crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion), can be represented in the form::ā797ā
r
ā„
=
{
x
,
y
}
{\displaystyle r_{\parallel }=\left\{x,y\right\}}
is the lateral vector that fixes the observation point coordinates in the sample plane.
k
l
,
m
{\displaystyle k_{l,m}}
is the reciprocal lattice vector.
s is the number of planes to be calculated inside the crystal; s set to zero would calculate the surface plane.
The expression sums the set of potential static charges for the surface and lower planes of the crystal lattice. LennardāJones and Dent showed that this expression forms a rapidly convergent Fourier series.:ā797ā Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959,:ā481ā had suggested that their results should be treated with caution. For example, the contraction a crystal plane would suffer under the conditions prescribed would not be the same as that of a similar plane with a solid mass of crystal behind it. Another difficulty arises because calculation of crystal surface field force fields are so great that simplifying assumptions have to be made to render them capable of a solution.
Michael Jaycock and Geoffrey Parfitt, then respectively senior lecturer in surface and colloid chemistry at Loughborough University of Technology and professor of chemical engineering at Carnegie Mellon University, concurred with Buckley, noting that "an ideal crystal, in which the ionic positions at the surface were identical to those achieved in the bulk crystal ... is obviously extremely improbable." However, they acknowledged that the LennardāJones and Dent model was singularly elegant, and like most researchers working before the advent of modern computers, they were limited in what could be attempted computationally. Nonetheless, LennardāJones and Dent demonstrated that the force exerted on a single ion, by a surface with evenly distributed positive and negative ions, decreases very rapidly with increasing distance.:ā14ā Later work by Jason Cleveland, Manfred Radmacher, and Paul Hansma, has shown that this result has direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.:ā543ā
= Reduced major axis regression
=
The theoretical underpinnings of standard least squares regression analysis are based on the assumption that the independent variable (often labelled as x) is measured without error as a design variable. The dependent variable (labeled y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Therefore, the underlying least squares regression assumptions can be violated. Reduced major axis (RMA) regression is specifically formulated to handle errors in both the x and y variables.:ā1ā If the estimate of the ratio of the error variance of y to the error variance of x is denoted by š, then the reduced major axis method assumes that š can be approximated by the ratio of the total variances of x and y. RMA minimizes both vertical and horizontal distances of the data points from the predicted line (by summing areas) rather than the least squares sum of squared vertical (y-axis) distances.:ā2ā
In Dent's 1935 paper on linear regression, entitled "On observations of points connected by a linear relation", she admitted that when the variances in the x and y variables are unknown, "we cannot hope to find the true positions of the observed points, but only their most probable positions." However, by treating the probability of the errors in terms of Gaussian error functions, she contended that this expression may be regarded as "a function of the unknown quantities", or the likelihood function of the data distribution.:ā106ā Furthermore, she argued that maximising this function to obtain the maximum likelihood estimation,:ā5ā subject to the condition that the points are collinear, will give the parameters for the line of best fit. She then deduced formulae for the errors in estimating the centroid and the line inclination when the data consists of a single (unrepeated) observation.:ā106ā
Maurice Kendall and Alan Stuart showed that the maximum likelihood estimator of a likelihood function, depending on a parameter
Īø
{\displaystyle \theta }
, satisfies the following quadratic equation:
where
x
{\displaystyle x}
and
y
{\displaystyle y}
are the
X
{\displaystyle \mathbf {X} }
and
Y
{\displaystyle \mathbf {Y} }
vectors in a covariance matrix giving the covariance between each pair of x and y variables. The superscript
T
{\displaystyle T}
indicates the transpose of the matrix.
Using the quadratic formula to solve for the positive root (or zero) of (2)::ā183ā
Inspection of (3) shows that as š tends to plus infinity, the positive root tends to::ā183ā
Correspondingly, as š tends to zero, the root tends to::ā183ā
Dent had solved the maximum likelihood estimator in the case where the covariance matrix is not known.:ā1049ā Dent's maximum likelihood estimator is the geometric mean of
Īø
x
{\displaystyle \theta _{x}}
and
Īø
y
{\displaystyle \theta _{y}}
, equivalent to::ā184ā
Dennis Lindley repeated Dent's analysis and stated that Dent's geometric mean estimator is not a consistent estimator for the likelihood function,:ā235ā236,ā241ā and that the gradient of the estimate will have a bias, and this remains true even if the number of observations tends to infinity.:ā15ā Subsequently, Theodore Anderson pointed out that the likelihood function has no maximum in this case, and therefore, there is no maximum likelihood estimator.:ā3ā Kenneth Alva Norton, a former consulting engineer with the then National Bureau of Standards, responded to Lindley, stating Lindley's own methods and assumptions lead to a biased prediction. Furthermore, Albert Madansky, late H. G. B. Alexander professor of business administration at University of Chicago Booth School of Business, noted that Lindley took the wrong root for the quadratic in (2) for the case where
x
T
y
{\displaystyle x^{T}y}
is negative.:ā201ā203ā
Richard J. Smith has stated that Dent was the first to develop a RMA regression method for line fitting that built on the work of Robert Adcock in "A Problem in Least Squares" (1878) and Charles Kummell in "Reduction of observation equations which contain more than one observed quantity" (1879). It is now believed that she was the first to propose what is often called the geometric mean functional relationship estimator of slope, and that her essential arguments can be generalised to any number of variables.:ā106ā Moreover, although her solution has its theoretical limitations, it is of practical importance, as it likely represents the best a priori estimate if nothing is known about the true error distribution in the model. It is generally much less reasonable to assume that all the error, or residual scatter, is attributable to one of the variables.:ā3ā
= Electrical design using digital computers
=
In the 1950s, British electrical engineers would rarely use a digital computer, and if they did, it would be to solve some complicated equation outside the scope of analogue computers. To a certain extent, engineers were deterred by the difficulty and the time taken to program a particular problem. Furthermore, the varied and often unique problems that arise in electrical design practice, together with the degree of uncertainty of the numerical data of many problems, accentuated this tendency. On 10 April 1956, Dent and Brian Birtwistle presented their paper, "The digital computer as an aid to the electrical design engineer", to the Convention on Digital Computer Techniques at the Institution of Electrical Engineers. The paper was intended to show, by describing three relatively simple applications, that the digital computer could be a useful aid to the electrical design engineer. The three example problems were:
The Ferranti Mark 1 computer at the University of Manchester was used for the calculations in the three problems. Dent was allowed to use the university's library of subroutines, from which the following were taken and incorporated into the programs:
The first problem of calculating the impulse voltage distribution on transformer windings took about five hours of machine time. Conversely, a hand calculation, using a method described by Thomas John Lewis in "The Transient Behaviour of Ladder Networks of the Type Representing Transformer and Machine Windings" (1954), took around three months.:ā486ā The use of a computer in the second problem allowed for a more accurate solution as it was possible to include nonlinear magnetic characteristics in the calculation. In the last problem, the torque and speed curves for the synchronous motors were calculated in around fifteen minutes.:ā486ā Their paper was one of the first to recognise that high-speed digital computers could provide considerable assistance to the electrical design engineer by carrying out automatically the optimum design of products.
Significant research had been devoted to determining a transformer's internal transient voltage distribution. Early attempts were hampered by computational limitations encountered when solving large numbers of coupled differential equations with analogue computers. It was not until Dent, with Hartill and Miles, in "A method of analysis of transformer impulse voltage distribution using a digital computer" (1958), recognised the limitations of the analogue models and developed a digital computer model, and associated program, where non-uniformity in the transformer windings could be introduced and any input voltage applied.
Publications
= Selected papers and academic articles
=
= Publications detail
=
See also
Footnotes
References
Bibliography
Adcock, Robert James (March 1878). Hendricks, Joel Evans (ed.). "A Problem in Least Squares". The Analyst. 5 (2). Princeton: Princeton University: 53ā54. doi:10.2307/2635758. hdl:2027/njp.32101040986372. ISSN 0741-7918. JSTOR 2635758. OCLC 277641453.
Cleveland, Jason Paul; Radmacher, Manfred; Hansma, Paul Kenneth (September 1994). "Atomic Scale Force Mapping with the Atomic Force Microscope". Applied Sciences. NATO Advanced Study Institute: Series E. 286. Office of Naval Research. Santa Barbara: University of California, Santa Barbara Physics Department: 1ā7. ISSN 0304-9930. OCLC 200640681. Technical Report No. 13. Defense Technical Information Center ADA284940. Retrieved 23 August 2021.
Excell, Phyl; Excell, Stanley (September 1979). "Monumental Inscriptions at Sompting Parish Church". Sussex Family Historian. September 1979 to December 1981. Vol. 4, no. 3. Brighton: Sussex Family History Group. pp. 84ā85. ISSN 0260-4175. OCLC 3114152.
Kummell, Charles Hugo (July 1879). Hendricks, Joel Evans (ed.). "Reduction of observation equations which contain more than one observed quantity". The Analyst. 6 (4). Princeton: Princeton University: 97ā105. doi:10.2307/2635646. hdl:2027/hvd.32044102938057. ISSN 0741-7918. JSTOR 2635646. OCLC 277641453.
Lewis, Thomas John (October 1954). "The Transient Behaviour of Ladder Networks of the Type Representing Transformer and Machine Windings". Proceedings of the Institution of Electrical Engineers. Part II: Power Engineering. 101 (83). London: Institution of Electrical Engineers: 541ā553. doi:10.1049/pi-2.1954.0114. ISSN 0369-8939. Paper No. 1691 Supply Section M. The paper was first received 24 February 1954, and in revised form 21 April 1954.
Murray, Janet Horowitz; Stark, Myra, eds. (1985). The Englishwoman's Review of Social and Industrial Questions. The complete run reproduced in facsimile in 41 volumes. January to October 1905. Vol. 36. New York: Garland Publishing. pp. 251ā252. hdl:2027/umn.31951002443042e. ISBN 978-0-8240-3761-1. OCLC 13062569.
Pollock, Philip John (August 1955). "Discussion on 'The Design of HighāSpeed SalientāPole A.C. Generators for Water Power Plants'". Proceedings of the Institution of Electrical Engineers. Part A: Power Engineering. 102 (4). London: Institution of Electrical Engineers: 476ā482. doi:10.1049/pi-2.1954.0114. ISSN 0369-8882. Retrieved 29 August 2021. NorthāWestern Centre, at Manchester, 6 May 1952.
Further reading
Byers, Nina; Williams, Gary (2006). Out of the shadows: contributions of twentieth-century women to physics. Cambridge: Cambridge University Press. ISBN 978-0-521-82197-1. OCLC 1050066680. Retrieved 10 December 2022.
Dummelow, John (1949). A history of the Metropolitan-Vickers Electrical Company Limited 1899ā1949. Manchester: Metropolitan-Vickers Electrical Company Limited. OCLC 5381038. With illustrations. Produced to commemorate the golden jubilee of the Metropolitan-Vickers Electrical Company Limited. A digital edition, digitised by Jim Lawton in 2008, is available at the Sydney Electric Train Society (SETS) website.
Fleming, Arthur Percy Morris; Churcher, Brian Andrew Graham; Davies, Leonard John (28 February 1952). "The research laboratories of Associated Electrical Industries Ltd". Proceedings of the Royal Society. B ā Biological Sciences. 139 (895). London: Royal Society: 208ā235. Bibcode:1952RSPSB.139..208F. doi:10.1098/rspb.1952.0008. PMID 14911826.
Napper, Brian (2003). "The Ferranti Mark 1". Computer 50: The University of Manchester Celebrates the Birth of the Modern Computer. Manchester: University of Manchester. Archived from the original on 10 February 2020. Retrieved 4 September 2020.
Swinton, Jonathan (4 March 2019). "Women At The Console". Alan Turing's Manchester. Manchester: Deodands. Archived from the original on 27 August 2020. Retrieved 27 August 2020.
External links
Beryl Dent's archive held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol.
Photograph of the staff working in the research department of Metropolitan-Vickers in 1954 in the collections of the Science and Industry Museum. Dent (front, fifth left) is one of only two women in the department. Arthur Fleming is also pictured (front, centre).
The Manchester transistor computer at the Computer History Museum.
Beryl May Dent at zbMATH.
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