they gave it freely. We took a chearful glais 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. Thefe 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 their fervants, keep their horfes, and purchase every thing the fociety requires. What is wanting at home, this stock produces, and is to be expended only at Ulubræ, for every thing neceffary and comfortable, except raiment and horses. * You will find this curious letter in Biograph. Britan. When When they are abroad, it is at a plus-expence. I call these gentlemen philofophers, becaufe, exclufive of their good morals, they devote the principal part of their time to natural philofophy and mathematics, and had, when I firft 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 cognisance, and register every experiment and obfervation. I saw several fine things in their tranfactions, and among them a moft 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, becaufe fuch determinations exhibit the quadratures of curvilinear spaces, an eafy method in doing the thing, is a promotion of geometry in the best manner. The rule is this. 59. Suppofe BDE the curve, BC the abciffax, CD the ordinatey, A B the tangent linet, and the nature of the curve be fuch, that the greatest E 4 A rule to determine expe-. ditiously, the tangents of curve lines. power power of y ordinate be on one fide of the equation; then y3x3xxy+xyy a 3 + a ay —a a x + a x x but if the greatest power of y be wanting, the terms must be put = 0. ayy: 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 dimenfion, to prefix unity, and of two, 2, if of three, 3, and you will have 3a3 + 2 a ay 2 a a x + a x x yy: a The fraction, by affuming the terms wherein the abfciffa 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 2xxy +xyy - a ax + 2 axx: then diminish each each of these by x, and the denominator will be 3xx-2xy+yy—aa+2ax. This fraction is equal to A B, and there- foret is= In this easy way may the tangents of all geo metrical 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 faw them perform feveral extraordinary experiments. Microscopical obfervations made at Ulubræ. 60. They make all the mathematical inftruments they use, and have brought the microfcope in particular, to greater perfection 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 Marfbal's is by Culpepper and Scarlet, and made feveral good alterations in the folar or camera obfcura microfcope; and in the catoptric microfcope, which is made on the model of the Newtonian te lescope. Colours in the 61. In one of their best microscope. double reflecting optical inftruments, I had a better view of the 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 fea green, more blue than yellow: the yellow was a light red and a pellucid white: All the phænomena of colours were here to be found out. Works of art Scope. 62. In this inftrument, the in the micro- fineft point of a needle appeared more blunt and unequal, and more like a broken nail, than I had before 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 smootheft plate of glafs, was very rough, full of cracks, fiffures and inequalities. Very different, indeed, are the things finished by human art, from the hings finished by the hand of nature. The points, the edges, the polish, the angles, every thing that na ture |