Newton's Interpolation Formulas. An unpublished Manuscript of Sir Isaac Newton

1927 ◽  
Vol 58 (1) ◽  
pp. 53-95
Author(s):  
Duncan C. Fraser

The mortality of Europeans in West Africa has on several occasions been the subject of discussion in this Institute or of note in the pages of the Journal. In 1886 Dr. T. B. Sprague read a paper on the mortality of Europeans in the Congo, based on the statistics from Stanley's work on the Pounding of the Free State: In 1892 Dr. T. G. Lyon contributed a paper on mortality in various regions of the world from data gathered from Government Reports of the Colonial Office. In 1897 Dr. A. E. Sprague read a paper on the experience of 971 Belgian Government officials on the Congo and 178 employees of a Dutch Trading Company. In the same year Mr. J. E. Hart gave the experience of Colonial Office officials on the Gold Coast and Sierra Leone. Lastly, in 1911, Mr. Hart furnished a copy of the Report on Vital Statistics of European Government officials in the employ of the Colonial Office for the previous year.

1897 ◽  
Vol 33 (4) ◽  
pp. 307-310
Author(s):  
J. R. Hart

A few statistics relating to this subject recently came into my hands, and although the results obtained from them cannot be considered to be of great weight, a short communication may be of interest. In the hope that useful information might be forthcoming, I made enquiry as to whether any record is kept of the dates of departure, death, or retirement of persons who go out in the employment of African merchants to the West Coast; and ascertained that it was unlikely that data could be supplied from that source. But I thought it worth while to make similar enquiry at the Colonial Office; and although the West African department could not officially furnish me with information, as they had none here of which they could vouch for the accuracy, I obtained, through the courtesy of one of the officials, the particulars given below. These were contained in a list of all the Europeans employed by the Governments of the four West African Colonies—the Gambia, Sierra Leone, the Gold Coast, and Lagos—during the ten years, 1 January 1881 to 31 December 1890, showing when their service began, and, if ended before 31 December 1890, when and why it ended.


Gentlemen, The time has again come round for my addressing you, and for ex­pressing my own gratitude, as well as yours, to your Council for their constant and zealous attention to the interests of the Royal Society. We have been compelled during several late years to have recourse to legal proceedings on the subject of the great tithes of Mablethorp, a portion of the Society’s property, and I rejoice to say with success. In my last address, I was required to give our thanks to Mr. Watt and to Mr. Dollond for the valuable busts which they had kindly presented to us. That of Mr. Dollond is placed at the commence­ment of the staircase leading to our apartments, and serves to indi­cate that his valuable improvements in the construction of our tele­scopes have been so many steps to the acquisition of higher and higher knowledge of the great universe of which this globe forms so insignificant a part. By the liberality of Mr. Watt we shall soon be furnished with handsome pedestals for the busts of his father and of Sir Isaac Newton, the two great lights of British mechanical genius and British philosophical science. Mr. Gilbert has kindly undertaken to furnish a similar pedestal for the bust of his father, and we have thought it right to provide one for that of Sir Joseph Banks. These will shortly form a conspicuous ornament of our place of meeting. The magnetical observatories are still carrying on their observa­tions, both in Her Majesty’s dominions and in foreign countries, and another naval officer, Lieut. Moore, has proceeded to the Antarctic Seas to complete a portion of the survey of Captain Sir James Ross, which was interrupted by stress of weather. That gallant and enter­ prising officer will, I hope, ere long give to us and to the public his own narrative of his important discoveries. Detailed accounts of the botany and zoology of the regions visited by him are preparing under the patronage of the Government, while Colonel Sabine is proceeding with the raagnetical observations, which were the more immediate objects of this, one of the most important voyages of discovery ever undertaken.


There are a number of references in the scientific literature to a burning mirror designed by Sir Isaac Newton (1). Together, they record that it was made from seven separate concave glasses, each about a foot in diameter, that Newton demonstrated its effects at several meetings of the Royal Society and that he presented it to the Society. Nonetheless, neither the earliest published list of instruments possessed by the Royal Society nor the most recent one mentions the burning mirror; the latest compiler does not even include it amongst those items, once owned, now lost. No reference to the instrument apparently survives in the Society’s main records. It is not listed by the author of the recent compendium on Newton’s life and work (2). There is, however, some contemporary information still extant (Appendix 1). Notes of the principles of its design and some of its effects are to be found in the Society’s Journal Book for 1704; some of the dimensions and the arrangement of the mirrors are given in a Lexicon published by John Harris which he donated to the Royal Society at the same meeting, 12 July 1704, at which Newton gave the Society the speculum. The last reference in the Journal Book is dated 15 November that year, when Mr Halley, the then secretary to the Society, was desired to draw up an account of the speculum and its effects (3). No such account appears to have been presented to the Royal Society. There is no reference in Newton’s published papers and letters of his chasing Halley to complete the task, nor is there any mention of it in the general references to Halley. The latter was, of course, quite accustomed to performing odd jobs for Newton; that same year he was to help the Opticks through the press. The only other contemporary reference to the burning mirror, though only hearsay evidence since Flamsteed was not present at the meeting, is in a letter the latter wrote to James Pound; this confirms that there were seven mirrors and that the aperture of each was near a foot in diameter (4). Because John Harris gave his Dictionary to the Royal Society in Newton’s presence, it is reasonable to assume that his description is accurate. As Newton would hardly have left an inaccurate one unchallenged, then, belatedly, the account desired of Mr Halley can be presented. In some respects, the delay is advantageous, since the subject of radiant heat and its effects, although already by Newton’s period an ancient one, is today rather better understood. On the other hand, some data has to be inferred, that could have been measured, and some assumptions made about Newton’s procedures and understanding that could have been checked (5).


The theory of the figures of the planets involves two questions perfectly distinct from each other; first, the figure which a mass of matter would assume by the mutual attraction of its particles, combined with a centrifugal force, arising from rotatory motion; and secondly, the force with which a body so formed will attract a particle occupying any proposed situation. The latter is the subject of the present inquiry; and it is also limited to the consideration of homogeneous bodies bounded by finite surfaces of the second order. This subject was first partially treated of by Sir Isaac Newton, who, in determining the attraction of spherical bodies, has also treated of other solids, formed by the rotation of curves round an axis, and of the attractions they exert upon bodies placed in the line of their axes. MacLaurin was the first who determined the attractions that such spheroids of revolutions exert on particles placed anywhere, either in or within their surfaces.


The Professor observes, that Sir Isaac Newton was the first mathematician who endeavoured to estimate the quantity of the precession from the attractive influence of the sun and moon on the spheroidal figure of the earth. His investigations relating to this subject evince the same transcendent abilities that are displayed in other parts of his Principia; but it is admitted, that, from a mistake in his process, his conclusion is erroneous. The investigations of other mathematicians in attempting the solu­tion of the same problem are arranged by the author under three general heads. The first arrive at wrong conclusions, in consequence of mistake in some part of their proceedings; the second obtain just conclusions, but rendered so by balance of opposite errors; the third approach as near the truth as the nature of the subject will admit, but, in the author’s estimation, are liable to the charge of obscurity and perplexity.


1977 ◽  
Vol 1 (4) ◽  
pp. 419-436 ◽  
Author(s):  
Kenneth F. Kiple ◽  
Virginia H. Kiple

West Africa’s disease environment of the eighteenth and nineteenth centuries was decidedly hazardous to the health of Europeans who ventured there. Comtemporary observers reported die-offs of white troops reaching the 80 percent mark annually, while the loss of one half of a ship’s company on the coast was not all that unusual. Philip Curtin has calculated that on the average England’s loss of white troops ranged between 300-700 per 1,000 mean strength per annum with his most recent word on the subject placing the overall white death toll at about half of the white soldiers, government officials, and civilian personnel who reached West African shores. K. G. Davies, on the other hand, would have the “risks of the African station” even higher with an individual facing “three chances in five of being dead within a year.”West African natives by contrast positively thrived amidst European death. Again referring to Curtin’s data, the biggest killer of whites by far was “fever.” The fevers of Sierra Leone, for example, dispatched white troops at the rate of 410.2 per thousand per annum during the years 1819-1836, yet caused the death of only 2.5 African troops per thousand per annum. A similar differential experience with fevers occurred throughout West Africa—an experience which Professor Curtin has suggested constituted a crucial reason for the Atlantic slave traded Europeans would have preferred to locate plantations in tropical Africa close to a seemingly inexhaustible source of cheap labor, but they were persuaded by the lethal nature of West African fevers to locate those plantations instead in the more salubrious New World. Put plainly, they found the expense of transporting African workers across the Atlantic eminently preferable to challenging the odds against their own survival in Africa.


1958 ◽  
Vol 51 (2) ◽  
pp. 124-127

Isaac Newton has received so many honors and has been the subject of so many biographies both during his lifetime and in the two hundred and thirty years since his death that such a note as we can write here seems hardly necessary.


1812 ◽  
Vol 102 ◽  
pp. 1-45 ◽  

In every physical inquiry the fundamental conditions should be such as are supplied by observation. Were it possible to observe this rule in every case, theory would always comprehend in its determinations a true account of the phenomena of nature. Applying the maxim we have just mentioned to the question concerning the figure of the planets, the mathematician would have to investigate the figure which a fluid, covering a solid body of any given shape, and composed of parts that vary in their densities according to a given law, would assume by the joint effect of the attraction on every particle and a centrifugal force produced by a rotatory motion about an axis. The circumstances here enumerated are all that observation fully warrants us to adopt as the foundation of this inquiry: for, with regard to the earth we know little more than that it consists of a solid nucleus, or central part, covered with the sea; and with regard to the other planets, all our knowledge is derived from analogy which leads us to think that they are bodies resembling the earth. There is one consideration, however, by which the general research may be modified without hurting the strictest rules of philosophizing; and that is, the near approach to the spherical figure which is observed in all the celestial bodies : and it is fortunate that this circumstance contributes much to lessen the great difficulties that occur in the investigation. But, even with the advantage derived from this limitation, the inquiry is extremely difficult, and leads to calculations of the most abstruse and complicated nature; and, when viewed in the general manner we have mentioned, it far surpassed the power of the mathematical and mechanical sciences as they were known in the days of Sir Isaac Newton, who first considered the physical causes of the figure of the planets. That great man was therefore forced to take a more confined view of the subject and to admit such suppositions as seemed best adapted to simplify the investigation. He supposed in effect that the earth and planets at their creation were entirely fluid, and that they now preserve the same figures which they assumed in their primitive condition; a hypothesis by which the inquiry was reduced to determine the figure necessary for the equilibrium of a fluid mass. The mathematicians, who have followed in the same tract of inquiry, have seldom ventured to go beyond the limited supposition proposed by Newton. They have succeeded in shewing that a mass revolving about an axis, and composed of one fluid of a uniform density, or of different fluids of different densities, will be in equilibrium, and will for ever preserve its figure when it has the form of an elliptical spheroid of revolution oblate at the poles. It has likewise been proved that the same form is the only one capable of fulfilling the required conditions ; which completes the solution of the problem in so far as it regards a mass entirely fluid.


1927 ◽  
Vol 58 (1) ◽  
pp. 21-52 ◽  
Author(s):  
H. E. Raynes

In two previous papers (vol. li, p. 77 and p. 211) an account has been given, of the published work of Newton on the subject of Interpolation by means of formulas of Finite Differences, and references have been made to the important letter which Newton sent on 24 October 1676, to Oldenburg, the Secretary of the Royal Society, with the intention that it should be communicated to Leibnitz. The second paper closed with a quotation from this letter, in which Newton mentions “a method [for the “construction of tables by interpolation] which I had almost “decided to describe here for the use of computers.” It has been my good fortune to discover that Newton had prepared an account of his method for inclusion in the letter and that the draft is preserved in the University Library at Cambridge among the Portsmouth Collection of Books and Papers written by or belonging to Sir Isaac Newton. By the freely granted permission of the Syndicate of the Library, I am able to give a reproduction and transcription of the manuscript, and I have added a translation.


1851 ◽  
Vol 2 ◽  
pp. 304-306
Author(s):  
D. R. Hay

The author stated in some prefatory remarks, that a belief in the operation of the laws of numerical harmonic ratio in the constitution of beautiful forms had long existed, although those laws had not been systematised so as to render them applicable in the formative arts. In proof of this, Mr Hay quoted a correspondence upon the subject of harmonic ratio, between Sir John Harrington and Sir Isaac Newton, in which the latter expresses his belief in such laws in the following words: “I am inclined to believe some general laws of the Creator prevailed with respect to the agreeable or unpleasing affections of all our senses; at least the supposition does not derogate from the power or wisdom of God, and seems highly consonant to the simplicity of the macrocosm in general.” The belief of this great philosopher, the author trusted, would form some apology to men of science for the repeated attempts he has made to establish the fact. These attempts he had hitherto made with reference to architecture, to ornamental design, and latterly to the human head and countenance; but on the present occasion he intended to shew the operation of these laws in constituting the symmetrical beauty of the entire human figure.


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