... containing a full Explication of the Method by which the most celebrated Geometers of the present Age have made such vast Advances in Mechanical Philosophy, A work very useful for those who would know how to apply Mathematics to Nature.The book is the first English text on Newton's method of fluxions, or, to phrase it in more modern terms, the first English calculus text. The book is a very full treatise, about three times the size of de l'Hôpital's famous calculus book. It contains 315 closely printed folio pages on fluxions as well as a twelve page introduction to conic sections at the beginning of the book. The work was dedicated to Sir Dalby Thomas, Kt., General and Chief Director for the Royal African Company of England, on the Coast of Guinea in Africa. The Royal African Company played a large role in Hayes' life and we shall return to discuss his involvement in a moment. First, however, we quote from Hayes' Preface to the Reader:-
A Preface is expected from every Author, and when tending to inform the Readers of the motives that induced him to write: And of the means they must use to understand what is wrote, is very proper; the former to justify himself from the Imputation of Vanity, and the latter to quicken and forward their Industry.It is also interesting to give the first paragraph of the main part of the book on fluxions to illustrate Hayes' style:-
As to the first, 'Tis manifest, since the World has been convinced of the mischief of Dogmatizing, either in Philosophy or Mathematics, that by allowing themselves a freedom of Thought, and boldly venturing forwards; the Advances in each are equally wonderful, that it is a difficult matter to resolve, whether an easy Acquiescence in all the Ancient Discoveries more obstructed, or the more generous Essays of modern Hero's improved our Knowledge; that since Men resumed their Native privilege, and allowed themselves the Liberty of enquiring freely into things, they have extended their Dominions over all the Earth, and their Knowledge far above the clouds; that these being the undeniable results of Mathematical studies; it is further plain, that they create in us more awful Thoughts and juster Notions of the works of God; and the admirable Harmony which only they discover, both in things in Heaven, and in things in Earth, undeniably prove one and the same great Author; in a word, that by them we create our Riches, enlarge our Power, and improve our reason; these then being some of the great Advantages Mankind receive from Mathematical learning, Who would not incessantly aspire after such useful Knowledge? Who can be blamed for using these endeavours to propagate the same?
As to the ensuing Treatise, the Author has been well assured that there are in England as many Lovers of the Mathematics as in any part of the World; that the multitudes of excellent Judgements and natural Parts, merely for want of a competent Knowledge in other Languages, have hitherto been deprived of the Opportunities of improving them, to the great disadvantage of the most Flourishing Island in the World; that in other Nations the best pieces of Learning are written in their own mother Tongues, for the good of their country which we seem purposely to slight, seeking a little empty applause by writing in a Language not easily attained, as if the Knowledge of things and words had a necessary dependence on each other ...
Magnitude is divisible in 'infinitum', and the Parts after this infinite Division, being infinitely little, are what Analysts call 'Moments' or 'Differences'; And if we consider Magnitude as Indeterminate and perpetually Increasing or Decreasing, then the infinitely little Increment or Decrement is called the Fluxion of that Magnitude or Quantity: And whether they be called Moments, Differences or Fluxions, they are still supposed to have the same Proportion to their Whole's, as a Finite Number has to an Infinite; or as a finite Space has to an infinite Space. Now those infinitely little Parts being extended, are again infinitely Divisible; and these infinitely little Parts of infinitely little Parts of a given Quantity, are by Geometers called 'Infinitesimae Infinitesimarum' or 'Fluxions of Fluxions'. Again, one of those infinitely little Parts may be conceived to be Divided into an infinite Number of Parts which are called Third Fluxions, etc.One of the big problems of Hayes' time was the problem of finding a method to calculate longitude, particularly on board a ship. Hayes' contribution was the publication "New and easy Method to find out the Longitude" which appeared in 1710. He was not seeking the large prize offered by the government, for this prize was only announced four years after Hayes' publication. His next publication was Of the Moon: a Philosophical Dialogue (1723). I [EFR] find this quite a fascinating work. It is written as a discussion between two philosophers, one being a scientist, the other a man of religion. The argument they engage in is whether the moon has light of its own. Both philosophers accept that the Bible is the word of God, but the scientific philosopher claims that he can prove beyond doubt that the moon shines by reflected light from the sun while the religious philosopher argues that since the Bible says that "God made two great lights; the greater light to rule the day and the lesser light to rule the night ..." then the moon must shine with light of its own. The man of science says that God's purpose in the Biblical revelation was spiritual not scientific, so the science in the Bible simply reflected knowledge at the time it was written. The man of religion say, if this is so why would God lie? And if he lies about this how could one trust anything. Clearly Hayes is trying to present both sides of the argument, but he believes the moon must have its own light because of God's revealed truth. This book is interesting, partly because it reminds us of Galileo's argument with Cardinal Bellarmine, partly because three hundred years later, in the 21st century, similar arguments still take place.
We noted above that Hayes dedicated his calculus text to the Director of the Royal African Company. This chartered company was formed in 1672 for the development of new trade with Africa. From its foundation until 1698 it held the English monopoly for slave trading. Although its monopoly was removed it continued trading in slaves until 1731, when it gave up trading in slaves but continued trading in ivory and gold dust. Hayes had an excellent reputation as a geographer and had made a voyage to Africa, spending some time there before returning to England. Following this he managed the Royal African Company, being elected as Sub-governor or Deputy-governor in an annual election for these positions.
Already an outstanding linguist, having expertise in classical and modern languages, he learnt Hebrew so that he could attack what was the main scholarly work of his life, namely an attempt to produce a chronology of world history using the religious writings :-
Following Muslim scholars who sought to lay down sound chronologies of the very ancient world, Hayes realised that for this it was necessary to establish relationships and links between lists of rulers in different parts of the world.His first publications in this area were A Vindication of the History of the Septuagint (1736), A Critical Examination of the Holy Gospels according to St Matthew and St Luke, with regard to the History of the Birth and Infancy of our Lord Jesus Christ (1738), and Dissertation on the Chronology of the Septuagint with an Appendix showing that the Chaldean and Egyptian Antiquities, hitherto esteemed fabulous, are perfectly consistent with the Computations of that most ancient Version of the Holy Scriptures (1741). He published a supplement of this last mentioned work in 1747 Series of Kings of Argos and of Emperors of China from Fohi to Jesus Christ. Again his aim was to show that the chronology of these other nations agreed with the chronology given in the Septuagint.
In 1752 the Royal African Company was dissolved; its successor was the African Company of Merchants. Hayes was seventy-four years old when the Company was dissolved and he retired to a new home he had purchased in Downe, Kent. William Bowyer suggests that Hayes, despite working in a managerial role for the Royal African Company all his life, had never been entirely happy so when the Company was no more:-
... Hayes found himself exonerated from that burden, which, though he had long supported it, yet was not altogether suitable to his inclinations.Looking at Hayes with 21st century eyes, we might find it strange to understand how a deeply devout Christian could spend his life running a company dealing in slaves. However, it is worth noting that during Hayes' time the practice was not seen as immoral, indeed the Royal African Company played a significant role in creating wealth for England which Hayes would have seen as highly laudable. At his home in Downe he began work on a major chronology Chronographia Asiatica et Aegyptiaca :-
He devoted the last eight years of his life to a far larger project, a Chronology of Asia and Egypt which he did not finish. In this work he argued that Josephus and the Seventy had had access to texts left out of the Old Testament canon but available to them in the Temple. On the basis of this it was possible to give credence to the testimony of the Letter, and hence also to what was contained in the Septuagint.By 1758 his health was deteriorating and he left his home in Kent and went to live in of Gray's Inn, in London, which provided office accommodation for barristers and was home to many important barristers and politicians. In  a quote from friend of Hayes' is given:-
If I differed in anything from him, as I could not espouse all his opinions, and never made a compliment of my own, he would reply in so mild a manner, that I never left him without admiring his great fund of learning, the clear method in which he explained his mind, and his sedate and serene temper.
Article by: J J O'Connor and E F Robertson