Beryl May Dent

Former Warminster County School, where Dent was educated and her father was head teacher.

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 Gathorne‑Hardy, son of Gathorne Gathorne-Hardy, 1st Earl of Cranbrook.[36][37][lower-alpha 10] 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 County School. In 1915, she passed the senior examination with second class honours and a distinction in French, and subsequently, her scholarship was renewed.[38][39][40] She then joined the sixth form and won the school prize for French in December 1916.[41][lower-alpha 11] In March 1918, she applied for a scholarship in mathematics from Somerville College, Oxford, one of the first two women's colleges in Oxford. She was highly commended but was not awarded a scholarship nor an exhibition.[42]

Paul Dirac, Dent's fellow student on the honours course in mathematics at Bristol.

In 1923, Dent was accepted as a postgraduate research student by Newnham College, Cambridge.

Dent was accepted on to the general Bachelor of Arts (BA) degree course at the University of Bristol and passed her intermediate degree examinations in June 1920.[lower-alpha 12] For the following academic year, she took the honours course in mathematics at Bristol.[44] After spending the summer of 1921 at her parent's home in Warminster,[45] she returned for the start of the 1921 to 1922 academic year to find that Paul Dirac had joined the mathematics course.[46][lower-alpha 13] Dent and Dirac were taught applied mathematics by Henry Ronald Hassé, the then head of the Mathematics Department, and pure mathematics by Peter Fraser. Both of them had come from Cambridge.[46] Fraser introduced them to mathematical rigour, projective geometry, and rigorous proofs in differential and integral calculus.[47][lower-alpha 14] Dent studied four courses in pure mathematics:

• Geometry of conics; Differential geometry of plane curves
• Algebra and trigonometry; Differential and integral calculus
• Analytical projective geometry of conics
• Differential equations; Solid geometry.[48]

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.[49][lower-alpha 15] Dent studied four courses in applied mathematics:

• Elementary dynamics of a particle and of rigid bodies
• Graphical and analytical statics; Hydrostatics
• Dynamics of a particle and of rigid bodies
• Elementary theory of potential with applications to electricity and magnetism.[48]

In June 1923, she graduated with Dirac, gaining a Bachelor of Science (BSc) degree in applied mathematics with First Class Honours.[50][46][lower-alpha 16] 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, Cambridge.[49][51][52][lower-alpha 17] Dent spent a year at Cambridge, leaving in 1924 without further academic qualification, prior to being appointed to a temporary position teaching science at Barnsley Girls' High School, Huddersfield Road, Barnsley.[52] Before 1948, Cambridge University denied women graduates a degree, although in the same year as Dent left Cambridge, Katharine Margaret Wilson was the first woman to be awarded a PhD at Cambridge.[54][lower-alpha 18]

H. H. Wills Physics Laboratory, University of Bristol, where Dent worked as a researcher.

John Edward Lennard‑Jones, Dent's advisor and co-author at Bristol in the 1920s.

In 1925, Dent was 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, the forerunner of the Science and Engineering Research Council (SERC).[56]^:107[52][lower-alpha 19] 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.[58][lower-alpha 20] From August 1925, John Lennard‑Jones, Trinity College, Cambridge, was elected Reader in Mathematical Physics.[60][58] In March 1927, Lennard‑Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics.[61][62]^:24[63][lower-alpha 21] Lennard‑Jones pioneered the theory of interatomic and intermolecular forces at Bristol and Dent became one of his first collaborators.[58][46]

Lennard‑Jones 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.[46][64] 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 Lennard‑Jones, Crystal parameters, and with Lennard‑Jones and Sydney Chapman, Structure of carbonate crystals (1927) and Part II. Structure of carbonate crystals (1927).[lower-alpha 22] On 28 June 1927, she was awarded a MSc degree for her thesis and research work.[65] 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.[lower-alpha 23] 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.[62]^:26[1][lower-alpha 24]

Dent was now living at Clifton Hill House, the university hall of residence for women in Clifton.[66][lower-alpha 25] She had been appointed honorary secretary of the Bristol Cheeloo Association by March 1926.[69] The association's aim was to raise sufficient funds to support a chair of chemistry at Cheeloo University.[66] 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.[70] In 1927, she was one of eleven people elected to the standing committee of the University of Bristol's Convocation, the university's alumni association, and later, represented the Manchester branch of the association.[71][72][67] In 1928, Lennard‑Jones and Dent published two papers, Cohesion at a crystal surface (1928), and with Sydney Chapman, The change in lattice spacing at a crystal boundary (1928), that studied the force fields on a thin crystal cleavage.[73][74] Around this time, quantum mechanics was developed to become the standard formulation for atomic physics.[lower-alpha 26] Lennard‑Jones left Bristol in 1929 to study the subject for a year as a Rockefeller Fellow at the University of Göttingen.[76]

With her collaborator and advisor in Germany, Dent 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.[77] In December 1929, she resigned her position and it was accepted with regret by the Council of the University of Bristol.[78] She left Bristol for Stretford, Manchester, to become the technical librarian for the scientific and technical staff in the research department at Metropolitan-Vickers.[79]^:232 In 1930, Lennard‑Jones returned to Bristol, as Dean of the Faculty of Science, and introduced the new quantum theories to the Bristol group.[76][58][lower-alpha 27] Marjorie Josephine Littleton was appointed as Dent's replacement on the 1 February 1930. She was the daughter of a local Bristol councillor and a graduate of Girton College, Cambridge. She was later Sir Neville Mott's co-author and research assistant in the Physics Department.[81][82]^:517

Metropolitan-Vickers was a British heavy industrial firm, based at Trafford Park, Manchester. They were 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.[83] 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, a former engineer in the Transformer Department.[lower-alpha 28] He made the library the centre of a new technical intelligence section. In 1929, the technical intelligence section was substantially reorganised and expanded, and placed under the directorship of James Steele Park Paton.[79]^:193[84] In January 1930, Dent joined the senior scientific and technical staff in the research at department Metropolitan-Vickers.[62]^:49[79]^:193 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.[85]^:301

By 1933, the Metropolitan-Vickers library had 3,000 engineering volumes and around the same number in pamphlets and patent specifications.[86] 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.[87] From 1931 to 1936, Dent was honorary secretary to the founding committee for the Lancashire and Cheshire branch of the ASLIB.[88]^:204–205 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.[89]^:412 She was also 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.[90][lower-alpha 29] She served on various ASLIB committees and made conference presentations detailing different aspects of the company's library and information service.[79]^:228[91]

Differential analyser designed by Douglas Hartree, at the Science and Industry Museum in Manchester.

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.[92][93] For Myers, Hartree, and Porter, in The Effect of Space-Charge on the Secondary Current in a Triode (1937), she provided the key numerical integrations for differential equations to aid in the calculation of the space charge limitation of secondary current using a differential analyser.[94] In 1946, she was promoted to section leader of the company's new computation section. Her knowledge of higher mathematics meant that she was often 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).[95][lower-alpha 30]

In the latter half of her career, Dent became the section leader for the women in the research department that were working on semiconducting materials.[79]^:233 She performed the Fourier analysis in Shorting and Field Corrections in Hall Measurements (1954), that developed a method for correcting the measured Hall effect in semiconductors for inhomogeneities in the applied magnetic field.[97] 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.[98] 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.[99]

Dent joined the Women's Engineering Society and published papers on the application of digital computers to electrical design.[100][101] With Brian Birtwistle, she wrote programs for the Manchester University Ferranti Mark 1, that demonstrated that high speed digital computers could provide considerable assistance to the electrical design engineer.[102] Birtwistle would later have an extensive career in the computer industry, working at, amongst others, Honeywell Information Systems and ADP Network Services.[103] 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.[104][105]^:243

An atomic force microscope on the left with controlling computer on the right. Dent's work had direct application to the development of atomic force microscopy.

In 1928, Lennard‑Jones and Dent published two papers, Cohesion at a crystal surface (1928) and The change in lattice spacing at a crystal boundary (1928), 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.[73][74] 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).[123]^:796–797 The expression for the potential in vacuum, ${\displaystyle \varphi _{0}\left(r\right)}$, at the point r = {x, y, z}, near the cubic lattice of point ions with different signs, the charge ${\displaystyle e_{k}}$, 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:[123]^:797

${\displaystyle \varphi _{0}\left(r\right)={\frac {2\pi }{a^{2}}}\sum _{l,m}\sum _{s}{\frac {\left(-1\right)^{s}\exp \left[-\left({\frac {2\pi }{a}}\right){\sqrt {l^{2}+m^{2}}}\left(z+z_{s}\right)\right]}{\left|k_{l,m}\right|}}\times \sum _{k}e_{k}\cos \left[k_{l,m}\left(r_{\parallel }-r_{k}\right)\right]}$

(1)

${\displaystyle r_{\parallel }=\left\{x,y\right\}}$ is the lateral vector that fixes the observation point coordinates in the sample plane.

${\displaystyle k_{l,m}}$ 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 layers of the crystal. Lennard‑Jones and Dent showed that this expression forms a rapidly convergent Fourier series.[123]^:797 Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959,[124]^:481 had suggested that the results obtained in Cohesion at a crystal surface (1928), 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.[125] However, later work by Cleveland, Radmacher, and Hansma, in Atomic Scale Force Mapping with the Atomic Force Microscope (1994), has shown that the Lennard‑Jones and Dent paper had direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.[126]

Richard J. Smith has stated that Dent, in On observations of points connected by a linear relation (1935), was the first to develop a reduced major axis (RMA) regressiom 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).[92][93] 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. RMA regression is specifically formulated to handle errors in both the x and y variables.[127]^:1 If the estimate of the ratio of the error variance of y to the error variance of x is denoted by 𝜆, then the reduced major axis method assumes that 𝜆 can be approximated by the ratio of the total variances of y and x.[128] 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.[127]^:2

Linear regression attempts to model the relationship between two variables by fitting a linear equation (straight line) to observed data.

Maurice Kendall and Alan Stuart showed that the maximum likelihood estimator of a likelihood function, depending on a parameter ${\displaystyle \theta }$, satisfies the following quadratic equation:[129]^:387

${\displaystyle \theta ^{2}x^{T}y+\left[\lambda x^{T}x-y^{T}y\right]\theta -\lambda x^{T}y=0}$

(2)

where x and y are the ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ vectors in a covariance matrix giving the covariance between each pair of x and y variables.

Using the quadratic formula to solve for the positive root (or zero) of (2):[130]

${\displaystyle \theta _{ML}\equiv {\frac {y^{T}y-\lambda x^{T}x+{\sqrt {(y^{T}y-\lambda x^{T}x)^{2}+4\lambda (x^{T}y)^{2}}}}{2x^{T}y}}}$

(3)

Inspection of (3) shows that as 𝜆 tends to zero, the positive root tends to ${\displaystyle \theta _{y}}$ equal to ${\displaystyle {y^{T}y}/{x^{T}y}}$, and as 𝜆 tends to plus infinity, the root tends to ${\displaystyle \theta _{x}}$ equal to ${\displaystyle {x^{T}y}/{x^{T}x}}$.[130] Dent had solved the maximum likelihood estimator in the case where the covariance matrix is not known, that is, when the variances in the x and y variables are unknown.[131]^:1049 Dent's maximum likelihood estimator is the geometric mean of ${\displaystyle \theta _{x}}$ and ${\displaystyle \theta _{y}}$, equal to:[130]

${\displaystyle {\sqrt {\left({y^{T}y}/{x^{T}x}\right)}}}$, where ${\displaystyle x^{T}y}$ is positive.

Dennis Lindley repeated Dent's analysis and stated that Dent's geometric mean estimator is not a consistent estimator for the likelihood function, [132]^{:235–236, 241} and that the gradient of the estimate will have a bias, and this remains true even if the number of observations tends to infinity.[133]^:15 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.[134] Furthermore, Albert Madansky noted that Lindley took the wrong root for the quadratic in (2) for the case where ${\displaystyle x^{T}y}$ is negative.[135]^:201–203 Subsequently, Theodore Anderson pointed out that the likelihood function has no maximum in this case, and therefore, there is no maximum likelihood estimator.[136]^:3

Although Dent's 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.[133]^:3[128]

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 (1956), to the Convention on Digital Computer Techniques at the Institution of Electrical Engineers.[137] The paper was intended to show, by describing three relatively simple applications, that the digital computer could be a very useful aid to the electrical design engineer. The three example problems were:

• Impulse voltage distribution on transformer windings
• Supply frequency ripple on transductor performance
• Starting torque of a synchronous motor.[138]

Illustration of transformer windings.

The Manchester University Ferranti Mark 1 computer 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:[139]

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.[102] Considerable 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.[140] 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.[141]