And from hence the following general rule for drawing tangents to curves is derived.

Find the fluxionary value of the abfciffe, from the equation expreffing the nature of the curve; multiply this fluxionary value by the ordinate, and divide this laft product by the fluxion of the fame ordinate. Or, which is the fame thing, in the rcom of the fluxionary value of the fubtangent, substitute the fluent itself, and the result will be the value of the subtangent in the terms of the equation first given.

Let it, for inftance, be required to draw a tangent to the point C in the parabola AC.

Put AB = x, BC=y, BV=t, and V the parameter 1, and because from the nature of the parabola x=yy, by taking the fluxions of the quantities on each fide of the equation, we fhall have A x=2yy, whence t=2yy; and fubftituting in the room of y, to which it is equal by hypothefis, we fhall have t=2x=VB; wherefore making AV =AB, or VB=2AB, we fhall have the point V in the produced axis of the parabola, to which if a line, as VC, be drawn, it will be a tangent to the curve in the point C, as was required.

B

C

And univerfally, if n reprefent the exponent of the power of BC, then 1x=y, will exprefs the nature of all parabolic curves; whence x = ny ÿ, and t = x×2=1× ny y j

y=ny", and putting & in the room of y", to which it is fuppofed equal, we have t=nx BV. Wherefore, universally, as ::::t, that is, as unity is to the index of the power of the ordinate, so is the abfciffe to the subtangent,

As we hope foon to fee Mr. Landen's treatife on the Refidual Analysis, we shall say nothing farther of it at present; but cannot conclude this article without expreffing our defire, that fo ingenious a gentleman may meet with the fuccefs his discovery deferves; a discovery that escaped the fagacity of his predeceffors, and which few, befides himself, are now capable of carrying to perfection. May it never be faid, that a mathematician of the firft clafs published propofals for printing a treatife on a new branch of the algebraic art, but did not meet with proper encouragement!

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The

The Hiftory of the Popes, from the foundation of the See of Rome, to the prefent Time. Vol. IV. By Archibald Bower, Efq; Heretofore Public Profeffor of Rhetoric, Hiftory, and Philofophy, in the Universities of Rome, Fermo, and Macerata, and in the latter place, Counsellor of the Inquifition. 4to. Is. 6d. in boards. Sandby.

MR

R. Bower defires that his Readers would excuse the smallnefs of this volume, (which contains only 350 pages) as he has published it at this time to fatisfy the world, as early as he could, that he is determined to purfue the work he has undertaken. Ample amends, he says, fhall be made, in the next volume, for what is wanting, as to fize, in this; and the whole be comprised, the Public may depend upon it, in three volumes more.

As Mr. Bower's abilities as an Hiftorian are already so well known, we shall only make one very obvious remark upon his Hiftory of the Popes; it is this, He is diffufe and tedious in thofe parts of his work which are dry and uninteresting; and if, according to his promife, he comprifes the whole in three volumes more, he muft neceffarily treat the most important and interefting parts of his fubject, in a very flight and fuperficial manner. A judicious Hiftorian would have comprehended in one volume what Mr. Bower has fpun out into four; but, indeed, the HISTORIAN OF THE POPES feems much better qualified for writing upon thofe fubjects where fiction and invention are required, than upon thofe where integrity, judgment, and impartiality are indifpenfibly neceflary.

His fourth volume contains the hiftory of the Popes from the year 757, to the year 867, and has little in it that is either important or entertaining. Thofe, indeed, who are fond of church hiftory, will find, in the pontificate of Hadrian, a pretty full account of the council of Nice, and of the various abfurdities and ridiculous notions, concerning the doctrine of image-worship. But we fhall, for the amusement of our Readers, close this article with fome extracts from Mr. Bower's account of Pope Joan.

• After Leo IV. and before Benedict III. is commonly placed the famous Pope Joan, by thofe who believe that fuch a Pope ever exifted. But before I enquire whether ever fuch a Pope exifted, or not, the reader will expect some account of the birth, of the education, of the various adventures, of fo extraordinary a woman, before, as well as after, fhe attained to the pontifical dignity, as it has been delivered down to us by the writers who speak of her as a real, and not as a fabulous perfon. She was,

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according to most of those writers, the daughter of an English miffionary; who, leaving his own country, went over to Germany, with great numbers of his countrymen, to inftruct the Saxons, whom Charlemagne had converted with his victorious army to the Chriftian religion. The miffionary carried over his wife with him, which obliged him, as fhe was big with child, to stop at Ingelheim; and there fhe was delivered of a daughter, whom fome call Joan, and others Agnes, Gerbert, Ifabel, Marguerite, Dorothy, and Jutt. As Joan (fo Tfhall call her, as fhe is most commonly known by that name) fhewed from her infancy a strong inclination to the ftudy of letters, and her father, who was a man of great learning, indulging that inclination, took upon him to inftruct her, he made under him fuch aftonishing progress in the different branches of literature, that The was looked upon by all as a prodigy. Her paffion for learning did not render her infenfible to a paffion of a different nature. As fhe was no lefs famous for her beauty and address, than for her genius and her learning, a young monk, of the monaftry of Fuld, in Germany, fell violently in love with her; and his flame kindling one no less violent in her breast, it was agreed between them, that, to enjoy more freely the company of each other, the fhould privately withdraw from her father's houfe, fhould difguife her fex, and, in that difguife, apply to the abbot to be admitted into the fame monaftery. She was then only twelve years old; but her paffion infpiring her with a refolution fuperior to her age as well as to her fex, the forfook her parents unaffected, and diffembling her fex, prefented herself to the abbot, and fo impofed upon him by an affumed modefly, and a pretended defire of confecrating herfelf from her tender years to God, and avoiding the temptations of the world, that might, in confederacy with her paffions when they grew stronger, rob her of her innocence, that he embraced her with great joy, and received her, as a most promifing youth, amongst his monks. And now the two lovers had, to their inexpreffible fatisfaction, opportunities every day of feeing one another, of converfing familiarly together, and expreffing to each other the violence of their paffion, undisturbed and unfufpected. However, they are faid to have kept, notwithstanding the violence. of their paffion, within bounds in indulging it; but within what bounds we are not told; and to keep any bounds in indulging a violent paffion, is a task to which few, if any at all, are equal. The lovers did not long continue in that happy ftate; but eloping together, for what reasons we are not informed, from the monaftery, they came privately over to England, the young monk being a native of this country. Here they pursued their ftudies together with uncommon application. From hence they went to France, from France to Italy, and from Italy to Greece;

ftop.

ftopping wherever they found mafters or profeffors capable of improving them in the knowlege they had already acquired. In Greece they chose Athens for the place of their abode, to perfect themselves there in the knowlege of the Greek tongue. They had not been long at Athens, when the monk was taken ill, and died in a few days, in fpite of all the care that could poffibly be used to fave his life. How deeply the furviving lover was affected with fo fatal a blow, no words can exprefs. Not able to bear the fight of any thing or place fhe had ever feen with him, fhe refolved, in the fame difguife, to repair to Rome; not to vifit the holy places there, but to divert her mind from dwelling too intenfely upon the irreparable loss she had sustained, and alleviate her grief with the fight of fo many great objects as would offer themselves there to her view. She had no occafion to repent of that refolution: her extraordinary talents made her foon known in that metropolis; and her modefty, her addrefs, her engaging behaviour, gained her the efteem as well as the affection of all who knew her. To difplay her talents, The opened a school; and had the fatisfaction of feeing it frequented by perfons of the first rank and distinction, by the most learned men at that time in Rome; nay, and by the public profeffors themselves, not afhamed, nor thinking it any fort of dif paragement for them to become her difciples. Thus fhe continued gaining daily new reputation and credit, not by her knowlege and learning alone, but by a conduct, in appearance, quite blameless, and an outward fhew of extraordinary fanctity, being ever the foremost in all public exercises of piety and devotion.

In the mean time died Pope Leo IV. and though men of extraordinary merit were not then wanting in Rome, yet was a woman preferred to them all, and as of all the best qualified for fo high a ftation, raised with one voice by the people and clergy to the pontifical throne. Thus did the world behold a woman fitting in the chair of St. Peter, and the keys, with the power of loofening and binding, fallen to the distaff. How long fhe was fuffered thus to impose on the Christian world, is not agreed amongst authors; but in this all agree, that neither the people nor the clergy had occafion, till fhe was difcovered, to repent of their choice; for fhe was difcovered in the end, and the difcovery of her fex was owing to the fame paffion that firft prompted her to difguife it. Had he been as chafte as many other women, who are faid to have disguised their fex before her time, as well as after it, the might have continued undifcovered, as well as they, to the hour of her death; but chastity was a virtue that he had been an utter ftranger to ever fince her infancy, and opportunities now offering daily to gra

tify an inclination that she never had the refolution to withftand, fhe yielded to it at all adventures, difcovered herself to one of her domeftics, on whofe fecrecy fhe knew she could rely, and disclosing to him all her fecrets, took him in the room of her former lover. He was true to his truft; and to none was their intimacy known, till the confequences naturally attending it, betrayed it to the world. Her holiness proved with child; and we are told, that having prefumed, in that condition, to exorcife a demoniac, and command the devil to tell her when he was to quit the body be poffeffed, the evil spirit anfwered, "Tell me firft, you who are Pope, and the father of fathers, when a fhe-pope is to be brought to bed, and I will then tell you when I am to quit the body I poffefs." That anfwer was understood, by those who heard it, as importing no more, than that the devil never would depart from that body; and no notice was therefore taken of it.

In the mean time her holiness advanced in her pregnancy; but not thinking herself so near her time as the really was, the unluckily ventured to affift at a proceffion, the annual proceffion of the rogation-week. In that week, the week preceding Whitfuntide, extraordinary devotions were performed to preferve the fruits of the earth, yet tender and liable to be blafted; and the Pope walked in folemn proceffion, with all the clergy, from the Vatican Bafilic to the Lateran. She might have excufed herself; and a woman of her art and address could not be at a lofs to find pretences to excuse herself from attending fo long and fo fatiguing a ceremony: but the chose to attend it, not apprehending that she was so near her time, say some Writers; while others gravely tell us, that, touched with remorse, the fincerely repented of her wickedness; and that an angel being thereupon fent from heaven, to offer her the alternative, to be either eternally damned in the other world, or endure in this the confufion that was due to her fins, fhe chofe of the two evils, the leaft. However that be, fhe fet out in proceffion from the Vatican, attended, according to cuftom, by the clergy in a body, by the senate, and immenfe crowds of people, and walked with great eafe till fhe came to the ftreet between the church of St. Clement and the Amphitheatre. There he was fuddenly feized with the pains incident to women in her condition; fell, overcome by, the violence of those pains, to the ground: and, while all about her were ftriving to help her up, and afford her fome relief, not knowing what had befallen her, fhe was, in the public ftreet, and in the prefence of the whole multitude, delivered of a fon, or, as a monkish poet expreffes it, of a little Pope. Some fay, that both the mother and the child died on the spot; and others, that the child died; but that the mo

ther