1740.

human race the barrows; and has the amazing impiety to offer human victims to heaven in her Autos de fe. Such a church cannot with juftice be called Chriftian. It is blacker than the blackest institution of paganism; because it speculats and acts more bafely; and its fpeculations and Actions are under the light of the gofpel (K).

With thefe ladys I left Mrs. Benlow, the latter end of September 1740, and she writ me word foon after, that they made her life compleatly happy. Their good fense and behaviour delighted her, and their management of affairs, without and within doors, fo intirely freed her from every trouble, that she could give all her time to books, mufic, and painting: That being fo circumstanced, fhe fat down once more to study the doctrine of fluxions, in which her father had taken great pains to inftruct her, and was now fatisfyed, the could form clear and distinct conceptions of the principles on which the arithmetic of infinites is founded; though my A remark- lord of Cloyne has been pleafed to declare to able letter the world that fuch a thing was impoffible; Mrs. Ben- for the doctrine of fluxions is defective. I low, in re- find, (continues this lady in her letter) that fluxions, the doctrine is so far from being defective, that it is easy to deduce the method of it from a author of few felf-evident truths, in the ftrict manner the Analyft. of the ancients; and to prove that, Sir Isaac Newton

written by

lation to

and the

right revd.

Newton has fairly and truly determined the fluxion of a rectangle under indeterminat quantitys. Nor is this all. It came into my head, that it was poffible to demonftrat, without the affiftance of time, velocity, or motion; or any confiderations of infinity; and I am now producing a little fpecimen of the practice of fluxions, intirely independent of these things. What then could cause this prelate, to abuse the mathematicians, and mifreprefent fluxions, in the manner he has done? What could tempt a man of his understanding to affirm, in the 9th section of his Analyft, that the true increment of the rectangle a b, is a B+bA+a B; when it must be a B+bA; as Sir Ifaac found it? and to say that, the indirect way used by Sir Ifaac is illegiti mat; when, in reality, it is rigorously geometrical, and the way my Lord of Cloyne would have proceeded, if he could have found out, that the velocity is that which the flowing rectangle has the very instant of time that is AB; and not the velocity that the rectangle has, while it is greater or less than AB?Shall we afcribe this procedure to his lordship's ignorance in the New Analyfis, after all his boafting that no man understands it better? Or fhall we fay that vanity, and a zeal for orthodoxy, incited him to write fo abufively and fally? In my opinion, the latter was the cafe.

on bishop

Berkley's

Remarks This is admirable, fewks; but as you have made but little progrefs yet in this fine part conduct, in of the mathematics, I must give you a little the mathe-explication of the bishop's behaviour in this maticians, article, that you may the better comprehend doctrine of Mrs. Benlow's letter.

respect of

and the

fluxions.

Doctor Berkley, bishop of Cloyne, published a thing called the Analyst, in the year 1734, and under pretence of fome abuses committed by mathematicians, in virtue of the authority they derive from their profeffion, does, in the Libel aforementioned, declare them infidels, makers of infidels, and feducers of mankind in matters of the highest concernment. These men (continues his lordship) affume an authority in things foreign to their profeffion, and undertake to decide in matters whereof their knowledge can by no means qualify them to be competent judges. They treat the principles and Mysterys of religion with freedom. Sir Ifaac Newton has prefumed to interpofe in prophecys and revelations. He decides in affairs of religion. And yet, it appears upon an enquiry, that the object, the principles, and method of demonftration, admitted by the mathematicians, are difficult to conceive, or imagine diftinctly. Their notions are most abftracted incomprehenfible mataphyfics, not to be admitted for the foundations of clear and accurat science. Their principles are obscure,

obfcure, repugnant, precarious. Their arguments are fallacious, indirect, illogical; and their inferences and conclufions falfe and unjuft. The inventor of fluxions, Sir Ifaac Newton, is dark and false in his reasonings on the fubject; and to make his false reasoning pafs upon his followers, he uses deceitful artifices. He obtains his ratios by means illegitimat; nor was it in his power to give a confiftent account of his principles. This, and a great deal more to the fame purpose, this right reverend author fays in his Analyst, and defence of it; and by blackening Sir Isaac Newton, and leffening the reputation of mathematicians, attempts to ferve chriflianity.

of a prac

written by

How far the credulous and injudicious A fpecimen may become infected by this uncommon way tice of of treating mathematics and mathematicians, fluxions, is not easy to say: but from a fhort account Mrs. Benof the nature of fluxions, and of the objects low. about which the method is converfant, it will appear that this author did not underftand the metaphyfics he would refute; and that it is not difficult to defend the principles and their demonftrations, from any imputations of fallacy or repugnancy. And if, after this, we review the other writings of this prelate, we shall find that, however orthodox the bishop was in his religion, and transcendental in his philofophy, yet he was far from being an able writer. For these things fee F 2

note

note L, at the end of this Letter. I fhall fay no more of him here than to observe, that in Mrs. Benlow's specimen of the practice of fluxions, independent of time, velocity, motion, &c. (the little MS. fhe mentions in her letter, and now in my poffeffion) fhe intirely knocks up this mighty analyft, and makes a reply impoffible. For, if time, velocity, and motion, in fluxions, really were what the bishop calls them, a reproach to mathematics, (tho in truth they are not); yet, if we can demonftrat without them, all my lord of Cloyne's reafonings are nothing in a moment. His call is anfwered at once (a).

In this useful manner, Mrs. Benlow was employed for several months after I left her, and the spent a great part of the next year, 1741. to wit, 41, in writing a thing fhe calls, A

(a) I call (fays this right reverend doctor) on the celebrated mathematicians of the prefent age, to clear up thofe obfcure analytics, and concur in giving to the public fome confiftent and intelligent account of the principles of their great mafter; which, if they do not, I believe the world will take it for granted that they cannot. Defence of free-thinking in the mathematics. You will afk me perhaps, if Mrs. Benlow was the only difcoverer of this method of demonftrating? I anfwer, No. The late ingenious Mr. Roger Paman, who was out with lord Anfon, and dyed lately at Jamaica, found out the fame method. But as his Harmony was not published till 45, Mrs. Benlow could not poffibly have a notion of his invention in 40. Nor did fhe hear of him, or fee his book, till laft year, 53, that I fent her one.