because, exclusive of their good morals, they devote the principal part of their time to natural philosophy and mathematicks, and had, when I first saw them, made a great number of fine experiments and observations in the works of nature, tho' they had not been a society for more than four years. They make records of every thing extraordinary which come within their cognisance, 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 power of y ordinate be on one side of the equation; then y 3 = − x 3 − xxy + xyy − a 3 + aay − aax + axx − ayy: but if the greatest power of y be wanting, the terms must be put = 0.

[illustration] [Graph of curve]

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 − 3a 3 + 2aay − 2aax + axx − ayy:
The fraction, by assuming the terms wherein the abscissa x occurs, and retaining the signs, and if the quantity x be of one dimension, to prefix unity, as above, etc, etc; and then it will be − 3x 3 − 2xxy + xyy − aax + 2axx:

then diminish each of these by x, and the denominator will be − 3xx − 2xy + yy − aa + 2ax.
This fraction is equal to A B, and therefore t is = − 3a 3 + 2aay − 2aax + axx − ayy/− 3xx − 2xy + yy − aa + 2ax 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