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Stewart writes :

After graduating he prepared an elementary textbook in which he explored the basis of arithmetical notation and the principal arithmetical processes as functions of that notation, explaining these without resort to algebraic or geometrical techniques. He published this in 1707 as "Arithmetica", jointly with a further set of studies entitled "Miscellanea mathematica" ... and indicated that mathematics had been his primary interest for three years.

Berkeley was working on these mathematics texts, whose full title is Arithmetica absque Algebra aut Euclide demonstrata (Arithmetic demonstrated without algebra or Euclid), waiting for the chance to compete for a fellowship. In 1706 a College Fellowship became available and, after taking some extremely demanding competitive examinations, he became a Junior Fellow of Trinity College, Dublin on 9 June 1707. Later in that year, on 19 November, he read his article Of infinites to the Dublin Philosophical Society, but this mathematical and philosophical work was only published after his death. This work, as the previous mathematical ones, clearly shows that Berkeley was much influenced by Locke. As well as carrying out some light tutoring duties, he studied divinity and was ordained deacon in February 1709 and ordained a priest in the following year. During this time he published two important works, An Essay towards a New Theory of Vision in 1709 and A Treatise concerning the Principles of Human Knowledge in 1710. The purpose of the first, as stated by Berkeley, is:

... to show the manner wherein we perceive by sight the distance, magnitude, and situation of objects. Also to consider the difference there is between the ideas of sight and touch, and whether there is any idea common to both senses.

In the second work he examines abstract ideas. He states in the introduction the notions he discusses in depth:

It is agreed on all hands, that the qualities or modes of things do never really exist each of them apart by itself, and separated from all others, but are mixed, as it were, and blended together, several in the same object. But we are told, the mind being able to consider each quality singly, or abstracted from those other qualities with which it is united, does by that means frame to itself abstract ideas. ... Not that it is possible for colour or motion to exist without extension: but only that the mind can frame to itself by abstraction the idea of colour exclusive of extension, and of motion exclusive of both colour and extension.

Berkeley sent A Treatise concerning the Principles of Human Knowledge to Samuel Clarke and William Whiston .

See this link for Whiston's comments.

Although Berkeley continued to hold his fellowship at Trinity College, Dublin, until 1724, he spent most of the period from 1713 to 1724 away from Dublin. He went to London in January 1713 where he arranged publication of some of his works. In November of that year he set off for Italy as chaplain to Lord Peterborough. He wrote:

Green fields and groves, flowery meadows and purling streams, are no where in such perfection as in England: but if you would know lightsome days, warm suns, and blue skies, you must come to Italy, and to enable a man to describe rocks and precipices, it is absolutely necessary that he pass the Alps.

He returned to England in August 1714 and towards the end of the year he had a fever which Arbuthnot described with a joke aimed at Berkeley's philosophy:

19 October 1714 : ... poor philosopher Berkeley has now 'the idea of health', which was very hard to produce in him; for he had 'an idea' of a strange fever on him so strong, that it was very hard to destroy it by introducing a contrary one.

Berkeley returned to Italy in 1716 with George Ashe, son of the Trinity College provost, and he spent four years there. He gives a vivid description of the eruption of Vesuvius in 1717:

April 17, 1717 : ... with much difficulty I reached the top of Mount Vesuvius, in which I saw a vast aperture full of smoke, which hindered the seeing its depth and figure. I heard within that horrid gulf certain odd sounds, which seemed to proceed from the belly of the mountain; a sort of murmuring, sighing, throbbing, churning ... June 5, after a horrid noise, the mountain was seen at Naples to spew a little out of the crater. The same continued the 6th. The 7th, nothing was observed till within two hours of night, when it began a hideous bellowing, which continued all that night and the next day till noon, causing the windows, and, as some affirm, the very houses in Naples to shake. From that time it spewed vast quantities of molten stuff to the South, which streamed down the side of the mountain like a great pot boiling over.

On the journey back to England Berkeley spent some time in Lyon where he wrote the essay De motu (On motion) which he submitted for the Grand Prix of the Academy of Sciences . He was not successful. He reached London early in 1721 and by the start of the 1721-22 academic year he was in Dublin where he was now a Senior Fellow (a position to which he had been appointed in 1717 while in Italy). He took on additional duties such as :

... librarian, junior dean, Greek lecturer, divinity lecturer, senior proctor, and Hebrew lecturer.

In May 1724 Berkeley resigned his position at Trinity College to become Anglican Dean of Londonderry, but he never resided in the city spending most of the next four years in London. Over these years he planned to establish a College in Bermuda to train the sons of colonists and Native Americans whom he wished to convert to:

... religion, morality, and civil life.

Funds of 10000 pounds for Berkeley's project were promised after the House of Commons in London voted:

That an humble address be presented to his majesty, that out of the lands in St Christopher's, yielded by France to Great Britain by the treaty of Utrecht, his majesty would be graciously pleased to make such grant for the use of the president and fellows of the college of St Paul, in Bermuda, as his majesty shall think proper.

On 1 August 1728 he married Anne Foster, and soon after they set sail for America. They reached Newport, Rhode Island, and bought a farm. Their first two children Henry and George were born while the family lived near Newport, Rhode Island. There he waited for the 10000 pound grant to be paid which would enable him to build the planned College. However, by the middle of 1731 it became obvious that he would not receive the grant, and he returned to London in October. He wrote a number of articles during his time in America which he published in the two or three years after his return.

In January 1734 Berkeley was appointed Bishop of Cloyne and was consecrated in St Paul's Church, Dublin, on 19 May 1734. In this office he devoted himself to the social and economic plight of Ireland, doing his best as an Anglican bishop to help the conditions of all in the predominantly Roman Catholic country. In 1745, at the time of the Jacobite rebellion, Berkeley addressed Roman Catholics in his diocese, and in 1749 he addressed to the Roman Catholic Clergy A Word to the Wise. He received a reply in the Dublin Journal of 18 November 1749 in which the Catholic Clergy expressed:

... their sincere and hearty thanks to the worthy author, assuring him that they are determined to comply with every particular recommended in his address to the utmost of their power. ... in every page [Berkeley's address] contains a proof of the author's extensive charity; his views are only towards the public good; the means he prescribes are easily complied with; and his manner of treating persons in their circumstances so very singular, that they plainly show the good man, the polite gentleman, and the true patriot.

He was devoted to Cloyne and he always purchased his clothes locally even though this meant that they were not of top quality, but he wanted to support employment. He set up a school to teach spinning to children, and he wanted to make possible the manufacture of linen. He wrote:

The children begin to find a pleasure in being paid in hard money; which I understand they will not give to their parents, but keep to buy clothes for themselves ... I am building a workhouse for sturdy vagrants, and design to raise about two acres of hemp for employing them.

Berkeley is best known in the world of mathematics for his attack on the logical foundation of the calculus as developed by Newton . In his tract The analyst: or a discourse addressed to an infidel mathematician, published in 1734, he tried to argue that although the calculus led to true results its foundations were no more secure than those of religion. He declared that the calculus involved a logical fallacy of a shift in the hypothesis. He described derivatives as follows:

And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?

Berkeley's criticisms were well founded and important in that they focused the attention of mathematicians on a logical clarification of the calculus. He developed an ingenious theory to explain the correct results obtained, claiming that it was the result of two compensating errors. Ren writes in :

By reviewing Berkeley's lifetime and the content of the "Analysts", we conclude that his critique was correct and that it impelled the improvement of the foundations of calculus objectively. It is helpful for the normal development of mathematics to accept various forms of critique positively.

Many of the other references which we give also discuss Berkeley's attack on the calculus; see , , , , , , and . De Moivre , Taylor , Maclaurin , Lagrange , Jacob Bernoulli and Johann Bernoulli all made attempts to bring the rigorous arguments of the Greeks into the calculus. Maclaurin in Treatise on fluxions gave the best response to Berkeley.

By the late 1740s Berkeley's health was deteriorating. His youngest son William died in 1751 which added to his decline. Despite this he travelled with his wife to Oxford in July 1752 to see his second son George begin his studies at Christ Church. In fact his intention was to remain at Oxford for the rest of his life, which he anticipated would not be very long. He died of a heart attack on the evening of Sunday 14 January 1753, sitting with his family listening to his wife reading. He died so peacefully that it is said that the event went unnoticed, the family thinking he had fallen asleep. He left instructions that he was not to be buried for at least five days and he was buried at Christ Church, Oxford on 20 January.

Source:School of Mathematics and Statistics University of St Andrews, Scotland