Dear Sir,

Having, as you know, bestowed a good deal of time and attention on the study of that part of physical astronomy which relates to the mutual disturbances of the planets, I have, naturally, been led to consider the various methods of resolving the formula (a2 + b2 − 2ab cos θ)2 into infinite series of the form A + B cos θ + C cos 2θ + &c. In the course of these investigations, a series for the rectification of the ellipsis occurred to me, remarkable for its simplicity, as well as its rapid convergency. As I believe it to be new, I send it you, inclosed, together with some remarks on the evolution of the formula just mentioned, which, if you think proper, you may submit to the consideration of the Royal Society.

The custom which this Society has established, of giving some account of the lives of its deceased members, is in every case gratifying to friendship, in many interesting to curiosity, but in those which serve to record the pursuits and occupations of men of letters, it is more strictly and properly an object coming within the views of a literary institution. The history of the authors is always in a great degree the history of the literature of a country; and even exclusive of an immediate relation to their works, the narrative of their private and domestic habits is often, in a moral point of view, useful and interesting to the scholar and the author. In both these respects, I may claim the attention of the Society to the following short account of the life and writings of our late worthy colleague, Mr William Tytler.

The mountain of Gibraltar is situated in 36° 9' north latitude, and in 5° east longitude from Greenwich. It is the promontory which, with that of Ceuta upon the opposite Coast of Barbary, forms the entrance of the Straits of Gibraltar from the Mediterranean; and Europa Point, which is the part of the mountain that advances most towards Africa, is generally regarded as the most southern promontory in Europe. The form of this mountain is oblong; its summit a sharp craggy ridge; its direction is nearly from north to south; and its greatest length, in that direction, falls very little short of three miles. Its breadth varies with the indentations of the shore, but it no where exceeds three quarters of mile. The line of its ridge is undulated, and the two extremes are somewhat higher than its centre.

The nature of those mathematical propositions, which were called Porisms by the ancient geometers, is now no longer a matter of uncertainty. The relation which they bear to other mathematical truths, the way in which they may at first have been observed, the kind of analysis to be employed in their investigation, their application to the solution of problems, have all been considered by some eminent mathematicians of the present age.

Doctor John Roebuck was born at Sheffield in Yorkshire, in the year 1718. His father was a considerable manufacturer and exporter of Sheffield goods, who, by his abilities and industry, had acquired a competent fortune. John, his eldest son, the subject of this memoir, was intended, by his father, for carrying on his own lucrative business at Sheffield; but was, from his early youth, irresistably attached to other pursuits, more calculated to gratify his ambition, and give fuller play to his powers. Notwithstanding this disappointment in his favourite object, his father had liberality enough to encourage his rising genius, and to give him all the advantages of a regular education.

In the second volume of the Asiatic Researches, an extract is given from the Surya Siddhanta, the ancient book which has been long, though obscurely, pointed out as the source of the astronomical knowledge of the Brahmins. The Surya Siddhanta is in the Sanscrit language: It is one of the Sastras, or inspired writings of the Hindoos, and is called the Jyotish, or Astronomical, Sastra. It professes, as we learn from Mr Davis, the ingenious translator, to be a revelation from heaven, communicated to Meya, a man of great fanctity, about four millions of years ago, toward the close of the Satya Jug, or of the Golden Age of the Indian mythologists; a period at which man is said to have been incomparably better than he is at present; when his stature exceeded twenty-one cubits, and his life extended to ten thousand years.

The life of which I am about to give some account to this Society cannot be called a literary one; of literary lives only it is perhaps the proper business of the Royal Society to record the particulars; but it has been in the practice of allowing a wider range to this customary notice of its deceased Members. Of the lives of such as were eminent in station or in usefulness, in abilities or in virtue, it has been accustomed to hear a narrative, which, though not important to learning, is interesting to humanity. Under this title, it will indulge me with a short account of the life of Lord Abercromby.

The mineral, of which I have the honour to lay an account before the Society, was brought to Edinburgh in considerable quantity about six years ago by a dealer in fossils, though indeed it had found its way, long before this period, into one or two collections.

By some it was mistaken for fluor. Its great specific gravity, its fibrous appearance, and its quality of forming an insoluble substance with sulphuric acid, made it generally be received as the native carbonate of barytes. From a few experiments, I was led at that time to entertain some doubt of its being any form of barytes ; and for several years, when I filled the chemical chair in the University of Glasgow, I used, when I exhibited the mineral itself, to mention in my lectures such of its properties as I had discovered, and which indicated that it did not belong to the barytic genus.

The Journal of the Weather, of which an abstract is here communicated, has been kept in a house in Windmill Street, on the south side of Edinburgh. The latitude of Edinburgh College, as deduced from a series of astronomical observations made at Hawkhill, is 55° · 57ʹ · 5″ nearly. Windmill Street is about 500 yards farther to the fouth.

The barometer used in these observations is a portable one, of the construction invented by Dr Lind, physician at Windsor; the mercury was boiled in the tube, and the scale is divided into the five-hundredth parts of an inch. The place where it stands is 265 feet above the level of the sea, or of the mean high-water mark at Leith. The height of it is marked every morning at 10 o'clock, as well as that of a thermometer, in the same room, which gives the temperature of the mercury.

As the Members of this Society afforded to M. Chevalier an early and warm encouragement and patronage, and readily gave his Tableau de la Plaine de Troye a place in their Transactions, as well as admitted the author to a seat among their number; and as that paper, since the time of its publication, has excited a good deal of interest and speculation, they will, no doubt, hear with pleasure, that it has now received, at least in all its material circumstances, a most ample and satisfactory confirmation.

In August last, I read to this Society the Description of an Air Thermometer intended to record the various degrees of heat at every instant; and mentioned my intention of constructing a barometer, which would, in the same manner, record the variations of the weight of the atmosphere: Both which I proposed to adapt to one piece of clock-work.

This piece of machinery appearing too complicated and expensive for general use, I contrived a thermometer, which marks the extreme points of heat and cold from any two times of observation. Of which instrument I also lately read an account, and produced the machine itself.