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William Emerson's father, Dudley Emerson, ran a school in Hurworth. He had two sons, the younger, named Dudley as his father, died when he was young. Dudley Emerson was skilled in mathematics and his school was successful. He educated his son William, the subject of this biography, teaching him reading, writing and mathematics, as well as a little Latin. William also studied languages with the local curate who lived in Dudley's house. At this stage William had little interest in learning, spending many hours looking for birds' nests. To complete his education, Dudley sent William to a school in Newcastle and then to one in York. At these schools he became fascinated by mathematics, which he also studied on his own by reading his father's mathematics books. After the death of his father, William took over running the school in Hurworth in 1730. This, however, did not prove successful for although Emerson was a highly intelligent and learned man, he had no patience as a teacher and frequently lost his temper with pupils who were unable to benefit from his high-powered teaching. The school was forced to close in 1733 as the pupils deserted it. Emerson's father had left him an estate at Castle Gate, near Eastgate in Weardale, and this brought in an income of between £70 and £80 a year. He decided that he could live on this, and from then on he devoted himself to studying mathematics.
Two years after the school in Hurworth closed, Emerson married. Chris Lloyd explains the events surrounding the marriage :-
In 1735, he married Elizabeth, the daughter of the Reverend Dr John Johnson, the rector of Hurworth. The rector had offered a £500 dowry with his daughter's hand, but he disapproved of her choice. He treated his scruffy son-in-law with contempt and refused to pay, so Emerson loaded all his wife's clothes in a barrow and wheeled it round to the parsonage, saying he refused "to be beholden to such a fellow for a single rag". He also vowed to prove the rector wrong.
We note at this point that William and Elizabeth Emerson had no children. The description of Emerson's behaviour towards his father-in-law looks eccentric and, indeed, we should explain that Emerson was a highly eccentric man. Again we quote from  for this colourful description:-
... vulgar language was typical of Emerson, as was the coarseness of his dress. His wife, Elizabeth, spun and bleached the linen which she then turned into his clothes on her elaborate sewing machine. He always buttoned the top and bottom of his coat but left the middle open and billowing. To keep his chest warm, he wore his shirt back to front. He was also renowned for inventing shin-covers: pieces of sacking tied above the knee with string. Shin-covers allowed him to sit in his favourite chair as close to the fire as possible without his protruding lower legs getting burnt.
In 1743 Emerson published his first mathematical textbook, The doctrine of fluxions. The Preface begins as follows:-
To say any thing in praise of the Method of Fluxions. or of its dignity and rank among the mathematical sciences, would be as needless as to describe the excellency of bright sunshine above the twinkling light of stars; since any one who is acquainted with the sciences will allow it to be a method of calculation incomparably superior to all other methods that ever were known or found out; and beyond which nothing further is to be hoped or expected. It lens its aid and assistance to all the other mathematical sciences, and that in their greatest wants and distresses: It opens and discovers to us the secrets and recesses of Nature, which have always before been locked up in obscurity and darkness. To this all the noble and valuable discoveries of the last and present age are entirely owing: And by this method Sir Isaac Newton, the worthy inventor, determined and settled the system of the whole visible World.
A second edition was published in 1757 and a third enlarged edition in 1768. In  a letter by Francis Holliday (written in September 1776) is given in which he described the opinions of William Jones and Abraham de Moivre concerning Emerson's book:-
[Jones and de Moivre] expressed their great approbation of Mr Emerson's Book of Fluxions, then lately printed, with regard to the method and neatness of the manner of his treating the subject, but were of the opinion he began his book too high for beginners, and that his 4, 5, etc Propositions were very elegant and useful in comparing Fluents, but too difficult and abstracted.
This does sound a little like the failed schoolmaster who found it impossible to come down to the level of his pupils. However, Holliday goes on in this letter to express his own views as to why Emerson spent so little time explaining the basics ideas:-
The reason I suppose why Mr Emerson began his subject high was designedly; for no doubt, he considered that all authors that wrote upon Fluxions, spent a great deal of words in explaining the first principles, and sometimes made them rather more obscure by a multitude of words laid the subject so low, and upon such incorrect principles, that the Science could not be defended against the Cavils of such as were professed Enemies to Science.
Indeed this view is supported by reading Emerson's long Preface where he spends much time defending the method of fluxions against criticism. Two further books were published in 1749: The Projection of the Sphere, Orthographic, Stereographic and Gnomical and The Elements of Trigonometry "containing the properties, relations, and calculations of sines, tangents, secants, etc. The doctrine of the Sphere, and the Principles of Plain and Spherical Trigonometry all plainly and clearly demonstrated". The second mentioned work is in three volumes and we give a short extract from the Preface:-
In this treatise I have ventured to lay down the whole both theory and practice; and to take in all things of any consequence that any way belong to the subject. ... I won't say that I have quite exhausted the subject, but I am of the opinion that I have omitted little or nothing of any consequence.
In 1754 he published Principles of mechanics explaining and demonstration The general laws of motion, the laws of gravity, motion of descending bodies, projectiles, mechanic powers, pendulums, centres of gravity, strength and stress of timber, hydrostatics and construction of machines. This, the most popular of his books, ran to six editions, the last (which was revised and corrected) being published in 1836. Perhaps one of the most remarkable facts here is that Emerson constructed the many instruments illustrated in this work, including the spinning-wheel he constructed for his wife. In 1755 he published A Treatise of Navigation but an important event took place in 1763 when he was introduced to the publisher John Nourse by his friend Edward Montagu. Montagu was a mathematician of some note, an FRS and also a Member of Parliament.
William Bowe, who knew Emerson personally, suggests in  that Emerson's first publications were not popular and, had it not been for Nourse, nothing further may have been published. This is not totally convincing given that a second edition of The Doctrine of Fluxions was published in 1757, before he met Nourse. However, it is clear that Nourse strongly encouraged Emerson and from 1763 onwards all his books were published by Nourse. He returned to fluxions, publishing The Method of Increments herein the principles are demonstrated and the Practice thereof shown in the Solution of Problems in 1763. The Preface begins as follows:-
He that writes upon a new or strange subject, little treated on before; has something more to do, than he that writes only on common or known matters. For he not only has his general scheme to lay, and the several parts of his work to connect properly together; but he has numberless computations to go through, which never before have been attempted; and which, without trial, he is ignorant whether they will succeed or not; and like a traveller wandering in an unfrequented and desolate country, must often turn back and begin his progress anew.
Emerson agreed with John Nourse to write a course of mathematics and thirteen volumes were envisaged. Part of this course was a second edition of The Elements of Trigonometry and a new work of two volumes A treatise of algebra (1765). The Preface contains strong claims as to the importance of the subject:-
The subject of the following book is Algebra, a science of universal use in the mathematics. Its business and use is to solve difficult problems, to find out rules and theorems in any particular branch of science; to discover the properties of such quantities as are concerned in any subject we have a mind to consider. It properly follows these two fundamental branches, Arithmetic and Geometry, but is vastly superior in nature to both, as it can solve questions quite beyond the reach of either of them.
Emerson maintained a remarkable output of textbooks: The Elements of Optics (four books) (1768); A System of Astronomy (1769); The Laws of Centripetal and Centrifugal Force (1769); The Mathematical Principles of Geography (1770); and Tracts (1770). Also in 1770 appeared two works on Newton's fluxions. One was A short comment on Sir I Newton's Principia containing notes upon some difficult places of that excellent book. Here are some short extracts from the Preface:-
The Principia being a book which is universally read by all the world, that pretend to any degree of philosophical learning; it cannot be improper to explain such passages therein as seem obscure and difficult. For although it is written in as clear a style as can be done in so few words; yet, by reason of its conciseness, and the difficulty of the subjects treated on, many things occur which require some further explication, especially to young beginners. ... This little Treatise was written many years since; for when I studied the Principia, I was frequently at a stop, which obliged me to make calculations here and there, as I went on; and when I had done, I set them down as notes upon these places. Wherein I only meddled with these places, that appeared difficult to me. These notes collected together are the subject of the following Comment.
The second of these works on fluxions was A Defence of Sir Isaac Newton against the objections that have been made to Several Parts of the Principia (1770). The text begins as follows:-
This incomparable Treatise being written in a concise style, and in the synthetic method, and being upon subjects quite new and untouched before, the generality of readers could make little of it. As it contained a new system of philosophy, built upon the most sublime geometry, the greatest mathematicians were obliged to study it with great care and attention, and few became masters of the subject; so that for a long time it was little read. But, at last, when the value of it became more known, it gained universal approbation, and the whole world stood amazed at the numberless new discoveries contained therein. And, upon account of its universal agreement with all the phenomena of nature, it was adopted as the true system by all, except some few that, through envy or ignorance, were bigoted to some other scheme.
In this work he answers objections made by Johann Bernoulli, Daniel Bernoulli and Leonhard Euler as well as defending Newton's right to be considered as the inventor of the "method of fluxions' rather than Leibniz. Emerson's final book was Miscellanies or a Miscellaneous Treatise containing several mathematical subjects (1776). The topics included are (I) Laws of chance, (II) Annuities, (III) Societies, (IV) Moon's Motion, (V) Construction of Arches, (VI) Precession of Equinoxes, (VII) Construction of logarithms, (VIII) Interpolation, (IX) The longitude, (X) Interest, (XI) Figure of sines, etc., (XII) Fortification, (XIII) Gunnery, (XIV) Architecture, (XV) Music, (XVI) Rules of Philosophy, (XVII) Optical lectures, (XIII) Problems.
We should not give the impression that all of Emerson's mathematical publications were his books. He also contributed to various publications such as the Ladies' Diary, The Palladium, Miscellanea Curiosa Mathematica, and the Gentleman's Magazine. Often he used a pseudonym, especially 'Merones,' 'Nichol Dixon,' and 'Philofluentimechanalgegeomastrolongo.' He defended his writings vigorously against criticisms, even in the Preface to some of his books. For example, two and a half pages of the Preface to Miscellanies consists of an attack on someone who had criticised his A System of Astronomy.
William Bowe describes Emerson with these words :-
He was singular and uncouth in his dress and manners, and hasty and impetuous in his temper; but whatever failing he had, they were overbalanced by his virtues. He had a great, firm, and independent mind, that could not be brought to submit to any thing mean, base or disingenuous, by any power on earth: a pure, genuine, ardent love of truth, and detestation of falsehood of whatever species.
Gow reinforces Bowe's opinion of Emerson's character writing that he :-
... was cantankerous, opinionated, careful with money and eccentric. Nonetheless, he was sincere, forthright and well respected for his wide knowledge, mainly acquired from his own reading and study.
We note that Emerson was offered a fellowship of the Royal Society but turned it down, saying:-
It is a damned hard thing that a man should burn so many farthing candles as I have done, and then have to pay so much a year for the honour of F.R.S. after his name. When a man becomes eminent, he has to pay quarterly for it. This is the way ingenuity is rewarded in England. Damn them and their F.R.S. too.
Emerson was a strong healthy man for much of his life although he did nothing for his health with much drinking and enjoying his favourite hobby of spending hours fishing in water up to his waist. By 1776, however, his health had begun to deteriorate. He wrote in a letter to his publisher Nourse in that year (see ):-
I have been much afflicted with the gout and gravel [gall stones] all this winter, the gout has left me, but the worst thing, the gravel, still continues.
He spent another six years in increasing pain from the stones which eventually led to his death. His gravestone was inscribed in Hebrew and Latin with words written by Emerson himself:-
Below are interred the mortal remains of William Emerson; a man whose merit and knowledge remained long unnoticed, although in him were united the virtues of simplicity and perfect integrity, with uncommon genius, he was a great mathematician. If you have read his works, this stone need not inform you; if not, read them and learn.
Article by: J J O'Connor and E F Robertson
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