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Observe here again, sir, and you will find that your objection vanishes as soon
t6as it is examined. You say, I do not
tamb in these instances obtain the remainder
td-e +1-1+1-1+1-1, &c. True sir, but what if I obtain a remainder equal to it! Have you any objection to this? Now mark, in every subtraction, if it is But if the number of terms be infinite, truly made, the reinainder added to
viz. if the series be atotctdtetit, what is subtracted is equal to the sub
&c. ad infin. then this series mulupired trahend, by my second postulate, which by 1--1, will be = atbato-btd you say you admit. Consequently sir, -ctedtf-ety-f, &c." Q. E, D. io l-it1-1+1.&c. let it. Now, Sir, what becomes of your exul. 1--1, &c. be added, and the sum is i; tation; and how came you to be guilty and the like conclusion is true in the of so unpardonable an omission, as not other instances. But if this be the case even to mention this proposition? You -1+2-1+1-1, &c. is equal to 1–1+.
bave, however, been guilty of a greater 1- 1+1-1, &c. For if to 1-1+1-1
and more unpardonable omission than +1-i, &c. '1-1+1-1+1-1, &c.
For having granted that
For having gran be added, and the sum is 1; and if also the number of terms in an infinite series to 1-1-1-1+1-1, &c. -1+2-1
cannot be greater than and alsa +1-1, &c. be added, and the suin is also 1. I think you will not deny, Mr. that my method in proposition 3, of obSaint, that 1–1+1--1, &c. and 1+2 taining the last term of an infinite series -1+1-1, &c. must be equal to each is just; you have wholly neglected to other. Now, if it clearly appears from notice the necessary consequence of this all this, that such expressions as 1-1, concession, which is, the complete sub1-2+1, &c. are not equivalent to 0, version of the leading propositions in Dr, and yet are not quantities, is there any Wallis's Arithmetic of Infinites, as I absurdity in asserting that they are ana- have abundantly shown in the treatise Jogous to points at the extremitics of under discussion. Thus in the infinite se. lines, which are something belonging to, ries (+1+2+3+4, &c. the last or great, without being lines; and therefore that est term is 0+1+1+1+1, &c, and the those expressions are something belong. number of terms is 1+1+1+1+1, &c. ing to number, without being number and 0+1+1+1+1, &c. multiplied by
Why you exult so much at my having 1+1+1+1, &c. produces 0+1+2+3 by a very obvious deduction shown the +4+5, &c. Thus too in the series truth of my method of finding the last 0+1+4+9+16 +25, &c.; the last term of an infinite series, I cannot con. term is 0+1+3+5+7+9 +11, &c. and ceive. For in the cighth proposition, I the number of terms is 1+1+1+1, &c. have demonstrated the truth of this and the last terni multiplied by the namo universally, and I chose previously to her of terms is equal to 0+1+1+9+16, elucidate it by induction in the third &c. Thus again, in the series 0+1+8 proposition, from the facility with which +27 +64 +125,&c. ; the last term is 0+ such induction may be made. My 1+9+19+37 +61, &c, and the number eighth proposition, therefore, is as follows: of terms is 1+1+1+1, &c. and the last “ In every series of terms in arithmetical term multiplied by the number of terms, or geometrical progression, or in any produces 0+1+8+27 +64 +125, &c. progression in a hich the terms mutually And so in other instances which are enu. exceed each other, the last term is equal merated in prop. 3. Hence, as I infer to the first term, added to the second in corol. 4, to prop. 8.“ In every infinite term, diminished by the first; added to series whether fractional or integral, the the third term, diminished by the second; ternis of which have an uninterrupted added to the fourth term, diminished by continuity, the last term multiplied by the third; and so on. And if the num- the number of terms will be cqual to the ber of terins be inquire, the last terin sum of the series. Now if this, Sir, be is equal to the series multiplied by admitted to be true, and I defy you, or 1-1."
any mathematician, to show that it is Demonstration :
not, the following propositions of Dr. “Iet the terms, whatever the series Wallis, are evidently false. “In the may le, be represented by a, b, c, d, e, arithmetical series 0+1+2+3+4, &c. thiun atbWatc-bt-dm-cote-d=e. if the last terin be multiplied into the
number of terms, the product will be Is not the subtraction lawfully made acdouble che sum of all the series."
cording to the algebraic rule for sub« In the scries of squares 04144-49 traction? And I also add, is it not aca +16, &c. infinitely continued, the last tually made ? For there is no other terin being multiplied into the number way of actually subtracting 41 from 0, of terms, will be triple, to the sum of all than by changing the sign. At least this the series."
is acknowledged to be the case by all “ In the series of cubes 0-41 +8+27 modern writers on algebra. Now, Sir, +64+125, &c. infinitely continued, the if the expression 1-1 while it remains last terin being multiplied into the num- in this form, and no actual subtraction ber of terms will be quadruple the sum is made of 1 from 1, is an infinitesimal, of all the series."
which I have abundantly proved it is, What other reason, Mr. Saint, can be it most clearly follows that 1-1 while assigned for your omitting to notice this it remains in this form, and one unity is discovery of sine, than a conviction of not actually added to the other, differs the truth of it?
from the aggregate 2 by 1-1. You will find your next objection And now, Sir, I shall conclude witla answered in the 29th proposition of my thanking you for the opportunity yog treatise, if you read it with attention; have afforded me of vindicating my Arith. and I shall therefore proceed to your last metic of lofinites, and also for the coin: remark. You ask me “ how in taking pliment you have paid to my heart; but 1+1 from 2, I obtained the remainder it would have been better, if, in doing 1-1." It was as follows, Mr. Saint, it, you had not run your head against From
mine, as I am afraid it has injured Subtract
Manor Place, Walworth,
May 7th, 1811,
MEMOIRS AND REMAINS OF EMINENT PERSONS.
MEMOIRS OF THE LATE
of his eminence as a landscape painter, PAUL SANDBY, Esq. R. A. &c. &c, at least in the formation of his peca. THOUGH the subject of this Me- liar style, as, though he there sa w na
1 moir has left behind hien that, ture in her wildest form, the necessity which will, in time to cuine, distinguish under which he lay of attending ia him from the common dead; a few particular accuracy in filling up the facts, relating to an individual, whose plans, may be supposed to have formed long career and exertions of eminent in hiin that correct and faithful habit, talents, bave been a public good, will, I with wbich he after viewed and delia presume, be acceptable to the nuinerous neated her. readers of the Monthly Magazine.
It is now too late, (except perhaps Mr. Paul Sandby was born in Note from his intimate connexions,) to learn tingham, in the year 1726, and came how he passed his early days, or under to Loudon at the early age of six- whose superintendence he received his teen; was suon after placed in the education; but from the respectable drawing-room in the Tower, (instituted and ancient family from whom he for the purpose of instructing persons sprung, and his personal and mental in drawing military plans, &c.) and acquirements, it was evident that he had from thence he was selected to attend been carefully attended to. The cir. the survey of the Highlands of Scotland, cumstances that led to his professional (as draughtsman) then carrying on under excellence are more our iminediale ens Colonel David Watson, and, which quiry, and more interesting to others, took place soon after the rebellion in and to those especially who are to fole 1745.
low his pursuits in art. As circumstances are the great go- In the life of a painter little variety vernors of men, and pay in most in- is to be looked for; the next day being stances be said to be the makers of but a repetition of the last, and the them; perhaps, the destination of Mr, succeeding one varying only as be creeps Sandby to the Highlands was the source on towards perfection. Mr. Sandby,
however, did not pass his life unaccom- of a sheet of paper; yet in the year panied by interesting circumstances; 1791, being applied to by Sir Nigel for while in Edinburgh, though very Gresby to paint a roon the baronet young, bis talents were discovered, and had built at his seat at Drakelow duly appreciated by many eminent per- Blouse, near Burton-upon-Trent, Mr. Sons in that cits; and he was particu. Sandby, justly relying upon his powers, larly marked by the friendship of Al undertook the task; and notwithstandlan Ramsay, the celebrated poet, of ing the unusually large size of the the late ingenious Sir John Clerk, of room, he actually began and compleated Pennycuik, and many others, with within the short space of two months: whoin, and his brother officers in the which to those who see it there appears survey, he passed a pleasant period. the labour of years, being one con Though thus circumstanced, we find hiin tinued subject of a landscape round ivot ide or indifferent to the cultivation three sides of the rooin. Many of of his native taste and talents; for at the trees are nearly thirty feet in this time be mare many very accurate bright, and the ceiling has a beautiful vjets of Edinburgh and its vicinity; sky. At the same time he contrived to and becoming acquainted with Mr. Beil, make numerous sketches in the park av engraver in that city, he got sume and grounds. josight into his mode of etching, and Mr. Sandby's brother residing in Windbrimself etched a number of scenes sor Great Park (of which he was depuis in the neighbourhood, which were ranger,) shortly after he returned from done on the spot, upon the copper. Scotland, he went to live in that neighAt the same time also he took numerous bourhood; and those who are acsketches from nature, with surprising quainted with Mr. Sandby's style, will accuracy, and made many drawings of see that at this time he fixed his prinfigures, in the costume, and of the ha- ciples, from studying in the park and hits and employments, of the inhabi. foiest. While residing at Windsor, he tants of Edinburgh, that are peculiarly was noticed by the late Duke of Moninteresting, and which mark a fertility tague, then governor of the castle, for of genius, that had only to select its whom he made many fine drawings, path in art, to attain excellence. lle views of the castle and adjacent park; chose landscape-painting, and a few and his Grace continued ever after to facts will trace him through his progress be his warme friend and patron. In up to that height, which he confessedly the year 1768, he was appointed by attained.
the Marquis of Granby chief drawingAfter returning to London from Edin naster in the Royal Military Academy. burgh, Mr. Sandby employed himself in at Woolwich, which office he continued etching and engraving several plates' to hold during twenty-six years. of various subjects, which were pub Mr. Sandby was one of the original lished by Boydell, Ryland, &c. and members of the Royal Academy. The when the late Sir W. W. Wynne went artists about the year 1750, associated to down to Wynnstay, to meet his tenants gether in a kind of academy, in St. Mar. upon coming of aye, he was accom- tin's in the Fields; the year afterwards panied by Mr. Sandby, who afterwards they formed a plan of exhibiting their mare several tours in Waies with the works, and by that means w a great degree baronet. He also travelled with Sir Jo. attracting public attention. In January, seph Banks, the late Dr. Solander, and 1765, they were incorporated by a royal Mr. Lightfoot, upon a tour to the Prin- charter; and in 1768, his Majesty, in cipality; and this journey he ever order to give dignity to the new estaafter remembered with the fondest de- blishment, instituted a royal academy Jiglie, baving experienced from Sir Joseph of painting, sculpture, and architecture. Banks an attention and kindness, which The principal artists with whom this called forth in hin the highest feeling scheme of pracuring an annual exhibi of respect and affection for his liberal tion originated, were Wilson, Heyman, patron and worthy friend. During his West, Sandby, Stubbs, Chambers, Keyfirst visit at Wynnstay, a theatre was nolds, &c. He was also a member of got up, all the scenes of which were the Turk's Head Club, composed of painted boy Mr. Sandhy; and although some of the inost eminent artists of for very many years afierwards he never the day. extended luis pictures beyond the size Before the extraordinary merit of How
garth was duly appreciated, Mr. Sandby tion remained for Mr. Sandby's indus: joined with his friend Churchill, in ittry, and it was in the endeavour to diculing hun. Churchill and Wilkes were complete a plate in Le Prince's method, alınuse the only real characters which (by sitting the rosin over the surface, were acbacked by the moral pencil of dc.) that be discovered a readier and Ilogarth, but, in this instance Hogarth more beautiful efect might be obtained completely jailed. Nothing could be by bringing the rosin into solucio!), and more coarse thall the anuse and satire floating it on copper, in which way he employed by the two combatants; and afterwards carried the art to an astothe failure of Hogarth is more extraor. nishing dugree or perfection, as may be dinary, as, at first, one should have been seun in the many tine works executed led to suppose that, i proportions as by him in that manner. This process his feelings were excited by personal for a long time was known to hinn alone: aninosity, so much the greater would but with a liberalisy the more praises have been the success of his exertions. worily from its rarity, hic coinmunicated It was, however, the contrary. Hogarth, luis discovery to Sikes, Robinson, Mat appears as a real saurist, as a personal ton, and others, who have practised it adversary; his performances were not with great success. worthy of the ia'ents he employed. Be Mr. George Alexander Stevens had sides inis, he had fallen into an error conceived an idea of a lecture on wiys, common to many men of genius; he and coinmunicated it to his friend Mr. had mistaken huis own powers. Con- Sandly, who suggested the adding blocks ceiving he had discovered the true to the wigs, which was accordingly line of beauty, he published his “ Ana. done; anil Mr. S. made the designs for lysis of Beauty," which was far from the ot iebraled Lecture on Heads. meeting the success, or producing It is the object of this memoir to the effect, he expected. It was then connect with it a few reinarks on the the opportunity for the adversaries of state of landscape-drawing, in this Hogarth, and the friends of Churchill, country. It has been remarked about to open upon him; and he was as- that time, that, in a country like this, sailed froin several quarters in bure so profusely adorned with the beauties lesque prints, satirising nis system; and of nature, it was extraordinary that we some of the best of these were from should have produced so few good painMr. Sandby, who, afterwards becom. ters of landscape ; and that as our poets ing better acquainted with the merit warmed their imaginations with sunny of Hogarth by the production of his hills and sighed after grottos, so our works, he was the first to express his painters draw rocks and castellaced regret at having endeavoured in any mountains, because Virgil gasped for way to depreciate the merit of so ex- breath at Naples, and Salvator wandered traordinary a genius; and every thing' amidst the Alps and Apemmines. That was done by Mr. Sandby to suppress our own country affords subjects emia his former publications; and no one nently suited to landscape-painting, no could afterwards be more forward in one who views the fine productions of expressing his unqualified adiniration our modern artists can doubt. But this of this artist, with whom, indeed, he is of any recent introduction; and it subsequently beca:ne acquainted.
may appear singular when it is said, Mr. Sandby was-honoured by the in, that the first person, who, by his works timacy of the late Mr. Charles Gre- familiarized us with our own scenery, ville, so well known as a collector and died but the last year. But such is the man of science; and through a como fact; for before the drawings of Mr. munication of Mr. Greville's, Mr. S. Sandby, I believe there were few of any
was enabled to inake some very im- merit representing English scenery, and · portant discoreries in working on cope it would have been curious through per, in a way which is now called aqua- such a man to have traced the art in its tinta,
- gradual, though rapid, progress; as the Upon Mr. Greville's return from Italy, had witnessed its dawn, and lived with where he had purchased the secret of unimpaired faculties to enjoy its splen. · Le Prince's method, he made it known dour; as no one (perhaps) had had inore • so Mi, Sandby; but it had been so ime opportunities, (from his extensive aci perfectly coinmunicated to Mr. Gre- quaintance with all classes of adınirers, • ville, that much research and investiga. patrons, and professors, than himself,)
or or could have better known their feel. considered, it should be remembered, ings and opinions, as they increased in that all his endeavours were to give to taste and intelligence, with the vast ex. his drawings a similar appearance to that ertions of the artists to reach their pre- seen in a camera-obscura, and when sent perfection.
i looked at with this impression, their It should be observed that for many beauty becomes very conspicuous; the years after Mr. Sandhy cuminenced land- truth in the reflected lights, the clearness scape drawing, uo colours were in general in the shadows, the aërial tint and keepo use except such as were peculiarly adapted ing in the distances, and skies, will be for the staining of maps and plans; and found to bave been generally go: up to jodeed it was himself who first set Mid. reach the artist's intention. As he never dleton the colour-maker.co prepare them appears to have introduced, or depended in somewhat like their present state, at all upon, vjoieni contrast for effect. and which are now brought to so great Ilis drawings will ever be esteemed by the perfection by Reeves, Newinan, and judicious, for their portrait-like resem. others.
blance to nature, and as bearing the In viewing the works of Mr. Sandby minutest inspection. Kesiding in the it is bardly possible to trace any other country where Mr. Sandby studied, and than nature for his guide : he looked having several Views in ihe neighboure alone to ber, and his style appears hood drawn by him, I have enjoyed over to have been compleatly formed in and over again, the pleasure which my the three places where he had chiefly ride, or walk, had afforded me, upan studied, Scotland, Wales, and Windsor. coming home and seeing in my roon so The Forest and Park of Windsor seem close a copy of every ibing I wished to to have preponderated, and from the recollect in scenes I had just been cone studies made in those places that it is templating. There is among them a evident it early appeared to him, ihat the drawing of the cottage in which I lise, becoming a draughtsman was no easy with many figures, and animals about it; matier of attainment; and whatever the and in which my own and other infant force of genius may be, without the children, instantly recognize, and call by mnost accurate and faithful minuteness, it name all the persons, and even the dogs is impossible to arrive at real excellence; and cattle. and accordingly his studies were prose. Whoever has seen the exquisite drau. cured with unremitting ardour.
ings of Wilson, will take a lesson from In his careful sketches and highly that great man's method (as well as from finished drawings, great precision is to be the similar one of Mr. Sandby) in the found in the outline. The foliage and ra- value of adhering to fact in their imita. mifications of the trees, the inanagement tion of nature; and yet it is to be la. of the perspective, in foreshortening the mented, that there are some (not proJimbs as they advance or recede together fessional men though, thank God!) who, with the fine feeling and exquisite taste while they are presumptuously advancing with which the extremities are touched, strong claims to connoisseurship, do is surprising; his pencil-sketches from great harm, by affecting to doubt its inNature have seldom been seen but by his portance, and prefer an undefined wild particular friends, by whom it has been rumble-cumble, (or any thing else you admitted, that those of about forty years please) of penciling, to a just represene back, have not been exceeded by any tation; which work they call bold, and one; and this all will allow to be no small sprited sketching; and aptly is it named, admission, when it is considered who are för bold inust be the doers, bolder the now living.
admirers. Ouiline was Mr. Sandby's peculiar fort; Not long before Mr. Sandby's death, he drew with amazing facility and the I repeated to him the remarks of a gene" greatest correctness,whatever might be the tleman who had just acquired a taste by subject, or however complicated its parts, commission. “Aye, (said be) these genHe sat down without the slightest embar- tlemen, when they attempt to fly their ressment, and drew buildings, figures, kites, little suspect how soon you discattle, or landscape, with equal ease, and cover the length of the string." free from all triling. His Views of Few people had a more varied mode Windsor Castle shew a thorough know. of execution, or possessed more knowledge ledge of perspective.
respecting his art, than Mr. Sandby; When the works of this master are some of his best works, I have always