cremental parts), and the velocities in the laft ratio's, as vanishing or ceafing to be j but from given fluxions to find the fluents ;and be ready in drawing tangents to curves; in the folution of problems de maximis & minimis, that is, the greatest or leaft poffible quantity attainable in any cafe; in the invention of points of inflection and retrogreffion; in finding the evoluta of a given curve; in finding the cauftic curves, by re flection and refraction, &c. &c. ---- this was amazing beyond any thing I had feen; or did ever fee fince, except Mrs. Benlow of Richmondshire, with whom I became ac quainted in 1739. (See Memoirs of feveral Ladies of Great Britain, Vol. I.) With aftonishment I beheld her. I was but a young beginner, or learner, in respect of her, though I had applied fo close to fluxions (after I had learned algebra), that my head was often ready to split with pain; nor had I the capacity, at that time, to comprehend thoroughly the procefs of feveral operations the performed with beauty, fimplicity, and charming elegance. Admirable Maria! No one have I ever feen that was her fuperior in this fcience: one equal only have I known, the lady a little before mentioned. And does not this demonftrate, that the faculties and imagination of women's minds, properly cultivated, may equal thofe of the greateft

men?

tion on the

of the

women.

men? And fince women have the fame im- A reflecprovable minds as the male part of the fpe- education cies, why should they not be cultivated by the fame method? Why should reafon be left to itself in one of the fexes, and be difciplined with fo much care in the other. Learning and knowledge are perfections in us not as we are men, but as we are rational creatures, in which order of beings the female world is upon the fame level with the male. We ought to confider in this particular, not what is the fex, but what is the fpecies they belong to. And if women of fortune were fo confidered, and educated accordingly, I am fure the world would foon be the better for it. It would be fo far from making them thofe ridiculous mortals Moliere has described under the character of learned ladies; that it would render them more agreeable and useful, and enable them by the acquifition of true fenfe and knowledge, to be fuperior to gayety and fpectacle, dress and diffipation. They would fee that the fovereign good can be placed in nothing else but in rectitude of conduct; as that is agreeable to our nature; conducive to well-being; accommodate to all places and times; durable, felf-derived, indeprivable; and of confequence, that on rational and mafculine religion only they can reft the foal of the foot, and the fooner they turn to it, the happier here and hereafter they fhall be.

Long

The au

thor's departure

Long before the power of fenfe, like the fet ting fun, is gradually forfaking them, (that power on which the pleasures of the world depend) they would, by their acquired understanding and knowledge, fee the folly of pleasure, and that they were born not only to virtue, friendship, honefly, and faith, but to religion, piety, adoration, and a generous furrender of their minds to the fupreme cause. They would be glorious creatures then. Every family would be happy.

But as to Mifs Spence, this knowledge, with a faultless perfon, and a modefty more graceful than her exquifite beauty, were not the things that principally charmed me: nor was it her converfation, than which nothing could be more lively and delightful: nor her fine fortune. It was her manners. She was a Christian Deift, and confidered Benevolence and Integrity as the effentials of her religion. She imitated the piety and devotion of Jefus Chrift, and worshipped his God and our God, his Father and our Father, as St. John expressly ftiles the God of Chriftians, xx. 17. She was extremely charitable to others, and confidered confcious virtue as the greatest ornament and moft valuable treafure of human nature. Excellent Maria!

§. 6. With this young lady, and her two fervants (her footman and her woman,) I

went

tor for

1731.

went up to London. We fet out from Cleator from Cleathe 31st day of July, and without meeting London, with any mischief in all that long way, came July 31. fafe to London. We were nine days on the road; and as the weather was fine, and our horfes excellent, we had a charming journey. My companion was fo agreeable, that had it been two thousand miles from Cleator to London, inftead of 272, I should still have thought it too short. Her conversation was fo various and fine, that no way could feem tiresome and tedious to him that travelled with her. Her notions and remarks were ever lively and inftructive. It was vaft plea fure to hear her, even on the driest and inoft abftrufe fubjects, on account of the admiration her difcourfe raised, and the fine knowledge it communicated, to one who underftood her. I will give an instance.

§. 7. In riding over the mountains the first day, we miffed the road in the evening, and inftead of getting to a very good inn, where we intended to reft, we were forced to stop at a poor little public house, and right glad to get in there, as the evening was tempeftuous and wet, dark and cold. Here we got fome bacon and fresh eggs for fupper, and the ale was good, which amufed us well enough till nine o'clock. We then proposed to play at cribbage for an hour, and called for a pack

of

A difcourfe on Auxions.

of cards; but they had none in the houfe, and we were obliged to divert ourselves with conversation, till it was time to retire. Mils Spence began in the following manner.

Was Newton, Sir, or Leibnitz, the author of that method of calculation, which lends its aid and affiftance to all the other mathe matical sciences, and that in their greatest wants and diftreffes? I have heard a foreigner affirm, that the German was the inventor of fluxions.

That cannot be (I replied). In the year 1696, Dr. Barrow received from Mr. New ton a demonftration of the rule of the диаdrature of curves, which the Doctor communicated to Mr. Collins; and as this is the foundation of fluxions, and the differential calculus, it is evident Mr. Newton had invented the method before that time.

In the beginning of the year 1673, Leibnitz was in England, again in October 1676; and the interval of this time he spent in France, during which he kept a correfpondence with Oldenburgh, and by his means with 7. Col lins; and fometimes alfo with Newton, from the last of whom he received a letter, dated June 18, 1676, wherein is taught the method of reducing quantities into infinite feries, that is, of exhibiting the increments of flowing quantities. This method was utterly

unknown