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thufiafm of liberty, it alfo ached under the fevere humiliations of a constrained captivity.

In this diftreffing embarrassment, the Earl of Moira, whole mind and whofe conduct do honour to human nature, received intimation of the Duke's confinement; and, by his interference and friendfhip, Biron was liberated. But the pow. er of legal profecution had only augmented the enthufiafm of freedom; and he returned to Paris, to unite with the most popular leaders of the revolution.

There he renewed his friendship with the Duke of Orleans (who had affumed the title of Egalité); and, by his influence, was prevailed on to take the command of the army of La Vendée. Whether Biron felt the dreadful effects of anarchy, while he hourly received accounts of maffacres and horrors; or whether the fufferings of the ill-fated and perfecuted Marie Antoinette impreffed his fenfible and philanthropic mind, is not clearly afcertained; but he certainly evinced an inactivity of foul, which terminated in his deftruction. He was recalled to Paris, deprived of the rank which he held in the army; imprifoned; and executed!

Here let the fenfible reader beftow a tear, while reflection fhews the progrefs of Biron's fall from power to degradation; from the moft fplendid altitudes of fame and fortune, to the gloomy platform of the guillotine! and, while memory tranfcribes his many virtues, his gallant actions, his amiable fenfibility, and his romantic enthufiafm on the page of Time, let Pity efface with her fpontaneous tears, the frailties of human nature, and the laft fad clofe of his unfortunate deftiny.

honour to diftinguifh one who fo far outfhone his cotemporaries; and the merits of John Jebb were fufficiently acknowledged, by being the fecond in the lift. Waring took his first, or bachelor's degree, in 1757, and the Lucafian Profefforship became vacant before he was of fufficient ftanding for the next, or Master's degree, which is a neceffary qualification for that office. This defect was fupplied by a royal mandate, through which he became Mafter of Arts in 1765; and, fhortly after his admiffion to this degree, the Lucafian profeffor.

The

The royal mandate is too frequently a screen for indolence; and it is now become almoft a custom, that heads of colleges, who ought to fet the example in difcipline toothers are the chief violaters of it, by making their office a pretext for taking their Doctor's degree in Divinity, without performing those exercises which were defigned as proofs of their qualifications. Such indolence cannot be imputed to Waring; yet feveral circumftances previous to his election into the profefforial chair, difcovered that there was, at least, one person in the Univerfity who difapproved of the anticipation of degrees by external influence.Waring, before his election, gave a fmall specimen of his abilities, as proof of his qualifications for the officewhich he was then foliciting; and a controverfy on his merits enfued: Dr. Powell, the master of St. John's College, attacking, in two pamphlets, the Profeffor; and his friend, afterwards Judge Wilfon, defending. attack was fcarcely warranted by the errors in the fpecimen; and the abundant proofs of talents in the exercise of the profefforial office are the best answers to the M. R. farcafms which the learned divine amused himself in cafting on rifing merit. An office held by a Barrow, a Newton, a Whifton, a Cotes, and a Sanderfon, 'must excite an ingenuous mind to the greateft exertions; and the new profeffor, whatever may have been his fuccefs, did not fall behind any of his predeceffors, in either zeal for the fcience, or application of the powers of his mind to extend its boundaries. In 1762, he published his Mifcellanea Analytica, one of the most abftruse books written on the abstrusest parts of Algebra. This work extended his fame over all Europe. He was elected, without folicitation on his part, member of the focieties of Bononia and Göttingen; and received flattering marks of esteem from the most eminent mathematicians at home and abroad. The difficulty of this work may be prefumed from the writer's own

SOME ACCOUNT OF DOCTOR WARING,
THE LATE CELEBRATED MATHE-

MATICIAN.

EDWARD WARING, Lucasian Profeflor of Mathematics in the Univerfity of Cambridge, was the fon of a wealthy far mer, of the Old Heath, near Shrewsbury. The early part of his education he received at the free-fchool in Shrewsbury; whence he removed to Cambridge, and was admitted on the 24th of March 1753 a member of Magdalen College. Here his talents for abftrufe calculation foon developed themfelves, and, at the time of taking his degree, he was confidered as a prodigy in thofe fciences which take the fubject of the bachelor's examination. The name of Senior Wrangler or the first of the year was thought icarcely a fufficient

words,

words, "I cannot fay that I know any one who thought it worth while to read through the whole, and perhaps not the half of it."

Mathematics did not, however, engrofs the whole of his attention. He could dedicate fome time to the study of his future profeffion; and in 1767, he was admitted to the degree of Doctor of Phyfic; but, whether from the incapacity of uniting together the employments of active life with abftrufe fpeculation, or from the natural diffidence of his temper, for which he was moft peculiarly remarkable; the degree which gave him the right of exercising his talents in medicine, was to him merely a barren title. Indeed, he was fo embarrassed in his manners before strangers, that he could not have made his way in a profeffion in which so much is done by addrefs; and it was fortunate that the eafe of his circumftances permitted him to devote the whole of his time to his favourite purfuit. His life paffed on, marked out by difcoveries, chiefly in abftract fcience; and by the publication of them in the Philofophical Tranfactions, or in feparate volumes, wider his own inspection. He lived fome years after taking his doctor's degree, at St. Is, in Huntingdonshire: while at Cambridge he married-quitted Cambridge with a view of living at Shrewsbury; but the air or fmoke of the town being injurious to Mrs. Waring's health, he removed to his own eftate at Plaisley, about eight miles from Shrewsbury, where he died, univerfally esteemed for inflexible integrity, modefty, plainnefs, and fimplicity of manners. They who knew the greatness of his mind from his writings, looked up to him with reverence every where; but he enjoyed himself in domeftic circles, with thofe chiefly among whom his purfuits could not be the object either of admiration or envy. The outward pomp which is affected frequently in the higher departments in academic life, was no gratification to one whofe habits were of a very oppofite nature; and he was too much occupied in science, to attend to the intrigues of the univerfity. There, in all questions of science, his word was the law; and at the annual examination of the candidates for the prize inftituted by Dr. Smith, be appeared to the greatest advantage. The candidates were generally three or four of the beft proficients in the mathematics at the previous annual examination for the bachelor's degree, who were employed from nine o'clock in the morning to ten at night, with the exception of two hours for dinner, and twenty minutes for tea, in an

fwering, vivâ voce, or writing down anfwers to the profeffor's questions, from the first rudiments of philofophy, to the deepeft parts of his own and Sir Ifaac Newton's works. Perhaps no part of Europe affords an inftance of fo feveré a procefs; and there was never any ground for fufpecting the profeffor of partiality. The zeal and judgment with which he performed this part of his office cannot be obliterated from the memory of thofe who paffed through his fiery ordeal.

Wishing to do ample juftice to the talents and virtue of the profeffor, we feel ourselves fomewhat at a lofs in (peaking of the writings by which alone he will be known to pofterity. He is the difcoverer, according to his own account, of nearly four hundred propofitions in the analytics, and the account is fcarcely exaggerated; yet we have reafon to believe, that the greater part of thefe difcoveries will fink into oblivion; and that pofterity will be as little attentive to them as his own cotemporaries. If, according to his own confeffion, "few thought it worth their while to read even half of his works," there must be some grounds for this neglect, either from the difficulty of the fubject, the unimportance of the difcoveries, or a defect in the communication of them to the public. The fubjects are certainly of a difficult nature, the calculaions are abstruse, yet Europe contained many perfons not to be deterred by the moft intricate theorems. then, that the difcoveries were unimportant? If this were really the cafe, the want of utility, would be a very fmall disparagement among thofe who cultivate fcience with a view chiefly to entertainment and the exercife of their rational powers. We are compelled then to attribute much of this neglect to a perplexity in style, manner, and language; the reader is stopped at every inftant, firft to make out the writer's meaning, then to fill up the chafm in the demonftration. He must invent anew every invention; for, after the enunciation of the theorem or problem, and the mention of a few steps, little affiftance is derived from the profeffor's powers of explanation. Indeed, an anonymous writer, certainly of very confiderable abilities, has aptly compared the works of Waring to the heavy appendages of a Gothic building, which add little of either beauty or ftability to the structure.

Shall we fay

A great part of the difcoveries relate to an affumption in Algebra, that equations may be generated by multiplying together others of inferior dimenfions. The roots

of

they,

I

or
2 a.

20

a-x

is equal to

a2x2

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for, fay

I

I

a + x

But when x is equal to a, the nu

merator and denominator of the fraction are both, in their language, equal

a-x

of thefe latter equations are frequently when xa, P is equal to
terms called negative, or impoffible; and
the relation of these terms to the coefficients
of the principal equation is a great ob-
ject of inquiry. In this art the profeffor, that is, when x is equal to a, P=
was very fuccefsful, though little affiftance"
is to be derived from his writings, in look-
ing for the real roots. We fhall not, per-
haps, be deemed to depreciate his merits,
if we place the feries for the fum of the
powers of the roots of any equation, among
the most ingenious of his difcoveries; yet
we cannot add, that it has very usefully
enlarged the bounds of science, or that
the algebraift will ever find occafion to in-
troduce it into practice. We may fay the
fame on many ingenious transformations
of equations, on the difcovery of impoffi-
ble roots, and fimilar exertions of undoubt-
edly great talents. They have carried the
affumption to its utmoft limits; and the
difficulty attending the fpeculation has
rendered perfons more anxious to afcer-
tain its real utility; yet they who reject it
may occafionally receive ufeful hints from
the Mifcellanea Analytica.

in

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2a.

to nothing. Therefore, nothing divided
by nothing, is equal to In the fame
manner
a1x which,
a3—x3 a2+ax+x1X; a-x
when x is equal to a, becomes

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I

I .There

3a2

fore, nothing divided by nothing is equal

[blocks in formation]

is abfurd. But we need only trace back our fteps to fee the fallacy in this mode of reafoning, For P is equal to fome num-ber multiplied into that is, when x is equal to a, P is equal to fome number multiplied into nothing, and divided by nothing; that is, P is, in that cafe, no number at all. For a-a cannot be divided by a-x when x is equal to a, fince, in that cafe, a-x is no number

at all.

The first time of Waring's appearing public as an author, was, we believe, in the latter end of the year 1759, when he published the first chapter of the Mifcellanea Analytica, as a fpecimen of his qualifications for the profeffor fhip; and this chapter he defended, in a reply to a pam. phet entitled, Obfervations on the First Chapter of a book called Mifcellanea Analytica. Here the profeffor was ftrangely profeffor could admit fuch paralegifins inIf, in the beginning of his career, the puzzled with the common paradox, that nothing divided by nothing may be equal the mathematicians, for nearly a century to his fpeculations; and the writings of to various finite quantities, and has rebefore him, may plead in his excufe; we courfe to unquestionable authorities in proof of this pofition. The names of Mac-fhould be built rather on the affumptions are not to be furprifed that his difcoveries laurin, Saunderson, De Moivre, Bernou- of others, than on any new principles of illi, Monmort, are ranged in favour of his. his own. opinion: but Dr. Powell was not fo eafly convinced, and returns to the charge, in the Defence of the Obfervations; to which the profeffor replied in a Letter to the Rev. Dr. Powell, Fellow of St. John's College, Cambridge, in antwer to his Obfervations, &c. In this controverfy, it is certain that the profeffor gave evident proofs of his abilities; though it is equally certain that he followed too implicitly the decifions of his predeceffors. No apparent advan tage, no authority, whatever, fhould in duce mathematicians to fwerve from the principles of right reafening, on which then icience is fuppofed to be peculiarly founded. According to Maclaurin, the Profeffor, and others, If P=" then,

ax

tion, that nothing could be divided by noAcquiefcing in the frange nothing, and produce a variety of numbers, he as easily adopted the pofition, that an equation has as many roots as it has dimenfions. Thus 2 and

- 4 are faid to be

roots of the equation x2-2x-8, though 4 can be the root only of the equation; 22x8, which differs fo materially from the preceding, that in one cafe ze is added, in the other cafe it is fubtracted from x2.

the principles, the deductions are, in geAllowances being made for this error in neral, legitimately made; and any one who can give himself the trouble of demonftrating the propofitions, may find fufficient employment in the profeffor's ana lytics. Perhaps it will be fufficient for a ftudent to devote his time to the fimpleft

cafe

cafe x+1=0; and when he has found a few thoufand roots of +1 and the publication of them may afford to pofterity 2 ftrong proof of the ingenuity of their predeceffors, and the application of the powers of their mind to ufeful and important truths. In this exercise may be confulted the method given by the profeffor, of finding a quantity, which, multiplied into a given irrational quantity, will produce a rational product, or confequently exterminate irrational quantities out of a given equation; but if an irratio al quantity cannot come into an equation, the utility of this invention will not be admitted without hesitation.

The Proprietates Algebraicarum Curvarum, published in 1772, neceffarily labour under the fame defects with the Mif. cellanea Analytica, the Meditationes Algebraica, published in 1770, and the Meditationes Analytica, which were in the prefs during the years 1773, 1774, 1775, and 1776. Thefe were the chief and the moft laborious works edited by the profelior; and in the Philofophical Tranfactions is to be found a variety of papers, which alone would be fufficient to place him in the first rank in the mathematical world. The nature of them may be feen from the following catalogue.

lution to go through them, will not only add much to his own knowledge, but be ufefully employed in dilating on thofe articles for the benefit of the more general reader. We might add in this place a work written on morals and metaphyfics in the English language: but as a few copies only were prefented to his friends, and it was the profeffor's with that they fhould not have a more extenfive circulation, we fhall not here enlarge upon its

contents.

In the mathematical world the life of

Waring may be confidered as a distin
guifhed æra. The strictness of demonstra-
tion required by the ancients had gra-
dually fallen into difufe, and a more com-
modious though almoft mechanical mode
by Algebra and Fluxions took its place,
and was carried to the utmost limit by the
profeffor. Hence many new demonftra-
tions may be attributed to him, but four
hundred difcoveries can scarcely fall to the
lot of a human being. If we examine
thoroughly thofe which our profeffor
would diftinguish by fuch names, we fhall,
find many to be mere deductions, others,
as in the folution of biquadratics, anti-
cipated by former writers. But if we can-
not allow to him the merit of fo inventive
a genius, we must applaud his affiduity;
and, diftinguifhed as he was in the fcien-
tific world, the purity of his life, the fim-
plicity of his manners, and the zeal which
he always manifefted for the truths of the
Gofpel, will intitle him to the refpect of
all who do not efteem the good qualities
of the heart inferior to thofe of the head.
F.
London, Nov. 1799.

THOR OF THE "MICROSCOPE MADE
EASY," TO DR. DODDRIDGE.

Vol. LIII. page 294, Mathematical Problems.-LIV. 193, New Properties in Conics.-LV. 143, Two, Theorems in Mathematics. LXIX. Problems concerning Interpolations-86, A general Refolution of Algebraical Equations.LXXVI. 81, On Infinite Series. LXXVII. 71, On finding the Values of Algebraical Quantities by converging ferieles, and demonftrating and extending ORIGINAL LETTER OF H. BAKER, AUpropofitions given by Pappus and others.LXXVIII. 67, On Centripetal Forces. ib. 588, On fome Properties of the Sum of the Divifion of Numbers.-LXXIX. 166, On the Method of correfpondent Values, &c. ib. 185, On the Refolution of attractive Powers.-LXXXI. 146, On infinite Seriefes.-LXXXIV. 385-415, On the Summation of those Seriefes whole general term is a determinate function of z, the diftance of the term of the Series.

For thefe papers, the profeffor was, in 1784,defervedly honoured by the Royal Society with Sir GodfreyCopley's medal; and most of them afford very strong proofs of the powers of his mind, both in abftract fcience, and the application of it to philofophy; though they labour in common with his other works under the difadvantage of being clothed in a very unattractive form. The mathematician who has refoMONTHLY MAG. NO. 55.

DEAR SIR,

Your laft favour came to London whilft I was at Ditton with his Grace the Duke of Montague, where and with whom I conftantly spend a week or ten days every Christmas and Eater, and I have no other holidays during the whole year. At these times feveral noblemen and gentlemen meet there to enjoy a perfect freedom, and throw off that ceremony and restraint which their rank fubjects them to at London. Our company this Chriftmas was his Grace, the Earls of Pembroke and Cardigan, the Lords Tyrawley, Edgcumbe, and Herbert, the Marquis Nicolini, our worthy prefident of the Royal Society, and feven or eight gentlemen of didination. The rules of the duke's houfe are, for every body to go to bed and rife at his own time, and amufe himself in

G

what

what way he likes beft; a liberty that renders the whole company perfectly eafy and fatisfied with each other, which people of different ranks in life can never be without it. In a morning every man orders what he pleases for his own breakfast, and waits for nobody. At dinner we fit promifcuously and lay afide all form, and our evenings are usually Ipent together in making experiments till ten o'clock, when we are called to fupper. But this

Christmas we had a choice band of muficians from London, who entertained us most evenings with felect and grand pieces of mufic from the beit Operas and Oratorios,accompanied by trumpets and kettle drums. For my own part, as his Grace has a fine library, I ipend much of my time there, and endeavour not to be idle though I am from bulineis. But I have made a tedious digreffion, when I only intended to produce a reafon for not anfwering your's before, and haften now to affure you of my hearty concern for your health,which I pray God to restore and confirm to you. I am pleafed to think any communication of mine can prove agreeable to your ingenious fociety, whofe profperity I fincerely with. The rules you were pleafed to fend me are, I make no doubt, well adapted to meetings in a country town, of which we here are very improper judges: and, therefore, inftead of pretending to advile in a matter they all understand much better than I can do, I intreat the favour of them to accept my best compliments, and believe me their humble fervant.

I am much obliged for your kind advice not to let my method of teaching [curing defects of speech] be loft with me; and can affure you I have taken good care to preferve it by means of my papers, if fo be any one can be found ftocked with attention and patience fufficient for it. My fon feems not to want a common fhare of both, but this bufinefs requires an ability of keeping the mind at full ftretch for an hour together, and a fagacity to difcover the ideas as they arile in the mind of the Jearner, and to give them words: to do all which nobody, I believe, at his years is capable.

I am forry any friend of your's needs affiitance in my way, and have often wifhed it were poffible in the nature of things to make the terms of fuch affiftance more eafy, and at the fame time provide for my family as well by this employment as I could do by fome other: but as each fcholar must be taught alone, I am able to undertake fo few, that I am obliged to keep to the original terms, not that I made

indeed, but that were offered me when I began this bufinefs, and knew much less of the matter than I do at prefent.

The inclosed letter to me will, I prefume, give you full satisfaction as to the man with two heads. My very intimate and moft valuable friend our prefident of the Royal Society, on my hinting how much I fhould be obliged to be able to give it you under his own hand, was fo good as to fend it me the next day.

We had lately at the Royal Society one Margaret Cutting of Wickham market, in Suffolk, a woman of about 30 years of age, whofe tongue at four years old was entirely eradicated by a cancer, notwithftanding which the continued to iptak plainly, according to an account given by me to the faid fociety, and printed in the Transactions fome years ago. We could depend on the authorities we had before of this extraordinary cafe, but found it more strange than we imagined upon having her before us; for on the strictest examination of skilful anatomifts, she has not the leaft remains of a tongue, and yet fpeaks fo plain you would not imagine fhe wanted any part of it, and fings very prettily.

I have often with pleasure and amazement examined the antennæ of butterflies and other infects: and have in fome taken notice of the ftructure you mention, which is not much unlike that of the beard of the wild oat, which is thereby capable of making an excellent hygrometer, as Dr. Hook has well oblerved.

My fon defires your acceptance of his most humble refpects, together with his beft thanks for your most friendly concern for him, which he hopes to become more deferving of: he likewife begs you will make his thanks and acknowledgments agreeable to the gentlemen of your fociety, for the great honour they do him by taking notice of any thing he can be capable of doing. I am very glad you are acquainted with my ingenious and worthy friend Dr. Miles, whofe almoft weekly correspondence I have been happy in for fome years, though in winter I fee him but feldom.

This letter has been writ at 3 or 4 fittings, and has nothing to recommend it but a hearty good will, accompanied with the molt fincere efteem and refpect for you, which therefore I affure myfelf you will accept without any ceremony, from

Dear Sir, Your most affectionate,
And obedient humble Servant,
H. BAKER.

London, Jan. 16, 1747-8.

P. S. My fervice to Mr. Shipley.

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