wards, as their lodge was at my service, whenever I was pleased to do them the favour to call. This was civil, and I returned them the thanks they deserved.

Here dinner was brought in, and with thefe gentlemen I fat down to feveral 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 chearful glafs after dinner, and laughed a couple of hours away in a delightful manner. They were quite polite, friendly and obliging; and I foon found, in converfing with them, that they were men of great reading, and greater abilities. Philofophy had not faddened their tempers. They were as lively companions, as they were wife 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 Mr. Evelyn in his letter to Mr. Boyle *; but with this difference, that they have no chaplain, may rife when they please, go and come as they think fit, and are not obliged to cultivate every one his garden. Every member lays down a hundred pounds on the first day of the year, and out of that fund they live,

pay

* You will find this curious letter in Biograph. Britan.

332

pay their fervants, keep their horses, and purchase every thing the fociety requires. What is wanting at home, this ftock produces, and is to be expended only at Ulubra, for every thing neceffary and comfortable, except raiment and horses. When they are abroad, it is at a plus-expence.

I call these gentlemen philofophers, because, exclufive of their good morals, they devote the principal part of their time to natural philofophy and mathematics, and had, when I first faw them, made a great number of fine experiments and obfervations in the works of nature, tho' they had not been a fociety for more than four years. They make records of every thing extraordinary which come within their cognifance, and register every experiment and obfervation. I faw feveral fine things in their tranfactions, 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 greateft ufe, because fuch determinations exhibit the quadratures of curvilinear spaces, an easy method in doing the thing, is a promotion of The rule is geometry in the best manner.

this.

59. Suppose

determine

tangents of

59. Suppofe BDE the curve, BC the ab- A rule to cifax, CD the ordinate = y, AB the tan- expedigent linet, and the nature of the curve be tiously, the fuch, that the greatest power of y ordinate be curve lines. on one fide of the equation; then y3——x3— xxy + xyy-a3a ay—a a x + axx — ayy: but if the greatest power of y be wanting, the terms must be put = o.

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

and

The fraction, by affuming the terms wherein the abcilla x occurs, and retaining the figns, and if the quantity x be of one dimenfion, to prefix unity as above, &c. &c; and then it will be ~3x3—2xx y + xy y -a a x + 2 axx: xyy—a

Microfco

vations

made at

then diminish each of these by x, and the denominator will be-3xx-2xy +yy—a a xx—2xy+yy—aa +2ax.

This fraction is equal to AB, and therefore —3a3+2 aay➡2 aax+axx—ayy

t is =

-3xx-2xy+yy—aa+zàx

In this easy way may the tangents of all geometrical curves be exhibited; and I add, by the fame method, if you are fkilful, may the tangents of infinite mechanical curves be determined. Many other fine things, in the mathematical way, I looked over in the journal of thefe gentlemen. I likewife faw them perform feveral extraordinary experi

ments.

60. They make all the mathematical inpical obfer- ftruments they use, and have brought the microscope in particular, to greater perfection Ulubre. than I have elsewhere feen it. They have them of all kinds, of one and more hemifpherules, and from the invented fpherule 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 microfcope, much farther than Marfhal's is by Culpepper and Scarlet, and made feveral good alterations in the folar or camerà obfcura microscope; and in the catoptric microscope, which is made on the model of the Newtonian telescope.

4

61. In

fcope.

61. In one of their beft double reflecting Colours in optical inftruments, I had a better view of the microthe variety and true mixture of colours than ever I faw before. The origins and mixtures were finely visible. In a common green ribbon, the yellow, the light red, and a blue, appeared diftinct and very plain: the lively green was a yellow and blue: in a sea green, more blue than yellow: the yellow was a light red and a pellucid white: All the phanomena of colours were here to be found

out.

micro

62. In this inftrument, the finest point of a Works of needle appeared more blunt and unequal, and art in the more like a broken nail, than I had before fcope. feen it the fineft edge of a razor was like the back of a dog, with the hair up:

the finest paper, was great hairs, cavities, and inequalities and the fmootheft plate of glafs, was very rough, full of cracks, fiffures and inequalities. Very different, indeed, are the things finished by human art, from the things finished by the hand of nature. The points, the edges, the polish, the angles, every thing that nature produces, appear in the inftrument in a perfection that astonishes the beholder.

dom in a

63. In the views I here took of the vege- The vegetable world, with my eye thus armed, I faw table kingmany extraordinary things I had never ob- microferved before. I took notice, in particular, fcope.

that