traordinary, and requested I would oblige them with some particulars that occurred. I did so immediately, and told them, among other things, of my reception at Burcot-Lodge, and of the skeleton of JOHN ORTON which I found in the cottage on the side of a woody hill. I let them know the goods and conveniences I saw there, and that I was so pleased with the beauties of the place, the little mansion, the once fine gardens, and the useful things on the premises, that I intended to return to it, and make it my summer retreat; that I had left a man there to that purpose, who was at work in the garden, and expected to be back in a month's time, with such things as were wanting to make it an agreeable and comfortable little country house..

The

The philosophers wondered not a little at what they heard. If they were surprized at seeing me as a traveller in such a place, they were much more astonished at my relation. They could not enough admire Mrs. BURCOT and Mrs. FLETCHER. history of the penitent ORTON, they thought very strange. They told me they were glad I had a thought of making Orton-Lodge a summer retreat, and hoped it would occasion my calling upon them many times; that I should always be heartily welcome to their house, and might with less difficulty go backwards and forwards, as their lodge was

at my service, whenever I was pleased to do them the favor to call. This was civil, and I returned them the thanks they deserved.

Here dinner was brought in, and with these gentlemen I sat down to several excellent dishes. There was the best of every kind of meat and drink, and it was served up in the most elegant manner : their wine in particular was old and generous, and they gave it freely. We took a cheerful glass after dinner, and laughed a couple of hours away in a delightful manner. They were quite polite, friendly and obliging; and I soon found in conversing with them, that they were men of great reading, and greater abilities. Philosophy had not saddened their tempers. They were as lively companions, as they were wise and learned men.

These gentlemen are twenty in number, men of fortune, who had agreed to live together, on the plan of a college described by Evelyn in his letter to the Hon. Robert Boyle; but, with this difference, that they have no chaplain, may rise when they please, go and come as they think fit, and every one is not obliged to cultivate his garden. Every member lays down a hundred pounds on the first day of the year, and out of that fund they live, pay their servants, keep their horses, and purchase every thing the society requires. What is wanting at home, this stock

produces, and is to be expended only at Ulubræ, for every thing necessary and comfortable, except raiment and horses. When they are abroad, it is at a plus-expence.

I call these gentlemen philosophers, because, exclusive of their good morals, they devote the principal part of their time to natural philosophy and mathematics, and had, when I first saw them, made a great number of fine experiments and observations in the works of nature, though they had not been a society for more than four years. They make records of every thing extraordinary which come within their cognizance, and register every experiment and observation. I saw several fine things in their transactions, and among them a most ingenious and new method of determining expeditiously the tangents of curve lines; which you know, mathematical reader, is a very prolix calculus, in the common way: and as the determination of the tangents of curves is of the greatest use, because such determinations exhibit the quadratures of curvilinear spaces, an easy method in doing the thing, is a promotion of geometry in the best manner. The rule is this.

Suppose B D E the curve, B C the abcissa = x, C D the ordinate y, A B the tangent line = t, and the nature of the curve be such, that the greatest

VOL. II.

с

power of y ordinate be on one side of the equation;

a a x + a x x — ay y: but if the greatest power of y

be wanting, the terms must be put

= 0.

Then make a fraction and numerator; the numerator, by taking all the terms, wherein the known quantity is, with all their signs; and if the known quantity be of one dimension, to prefix unity, and of two, 2, if of three, 3, and you will have 3 a3 +

2 aay

2a a x + a x x

ay y:

The fraction, by assuming the terms wherein the abscissar occurs, and retaining the signs, and if the quantity be of one dimension, to prefix unity, as above, &c, &c; and then it will be 3 x 3 x x y + x Y Y -a a x + 2 a xx. then diminish each of these by x, and the denominator will be

2

This fraction is equal to A B, and therefore t is = ·3a3+2aay - 2a a x + a xx-av u

- 3 x x − 2 x y +yy aa + 2 ax

In this easy way may the tangents of all geometrical curves be exhibited; and I add, by the same method, if you are skilful, may the tangents of infinite mechanical curves be determined. Many other fine things, in the mathematical way, I looked over in the journal of these gentlemen. I likewise saw them perform several extraordinary experiments.

They make all the mathematical instruments they use, and have brought the microscope in particular, to greater perfection than I have elsewhere seen it. They have them of all kinds, of one and more hemispherules, and from the invented spherule of Cardinal de Medicis, not exceeding the smallest pearl placed in a tube, to the largest that can be used. They had improved the double reflecting microscope, much farther than Marshal's is by Culpeper and Scarlet, and made several good alterations in the solar or camera obscura miscroscope; and in the catoptric microscope, which is made on the model of the Newtonian telescope.

In one of their best double reflecting optical instruments, I had a better view of the variety and true mixture of colours than ever I saw before. The origins and mixtures were finely visible. In a com