1+1-1 rather than 11 have obtained TI , 1+i ought certainly lo have terminated with same with bta? I must therefore, sir, push this question, why did you adopt the fraction and it should have the position of the integer l'ajter the been the object of the demonstration to fraction rather than before it? ! prove that was equal to the infinite anticipate your answer in these words, series 1+1+1+1, &c. whereas, by ad d “ because this position, and this only, he would produce the result which I have ding to this part of the cnunciation the obtained." How necessary then was it, words “ which is equal to 1+1+1+1, &c. ad infinitum," the proposition t's sir, I again repeat, that you should have is rendered identical, and ineans neither Po previously proved, that, in the addition more nor less than that there cannot be of numbers, a particular regard to their position was essentially necessary to a greater number of terms in any infi. ? i obtain, I will not say their correct sum, nite series, than un infinite number Per for that I deny, but the conclusions which of terms! or that the number of " terms in an infinite series is infinite! U you have deduced in your propositions. Yo Now, as it would be the height of folly In your third proposition from among for one moment to dispute the truth of many, curious speciinens of your reason. ing, I will select the following. “ For this assertion, I will not dwell on the 1+1+1+1, &c. ad infin. is evidently demonstration which you have been equal to the last term of the series pleased to give to this notable truth, but will only ask you, by what method you " 1+2+3+4, &c. ad infin. For the sum " of two of these terms, beginning from the 1--1+1 obtained I first term, viz. 1+1 is equal to the se first term. viz. ti is equ 1-1 cond term of the series 1+2+3+4, &c. The suin of three of the terms, beginning for the sum of 1 and you must froin the first term, viz. 1+1+1, is equal to 3, or the third term of the se. : by first expres- ries. The sum of four of the terins is equal to 4, or the fourth term of the sing the sum of 1, and T 1 series, and so on; and therefore the sun i by 17i+l, of the infinite series 1+1+1+1, &c. and then by adding the product of the will be equal to the last term of the integer and denominator to the nume- series 1+2+3+4, &c." I mean not, sator, and placing their sum over the sir, to dispute the justness of this infere denominator, agreeably to the common ence, but can it be possible that you Tule for reducing a mixed number to an should have deduced such conclusions iniproper fraction ; but, suppose you had from such premises ! You, who only a written the integer 1 before the fraction few pages before, in your preface, were . I, thus it vaunting in these words, “I rejoice to , and reduced this find as the result of this discovery, that it mixed number as above, would you affords a most splendid instance of the absurdity which may attend reasoning by Lor1+1+1+1 induction from parts to wholes, or from wholes to parts, when the wholes are them:or 142+1+1. selves infinite?" flave you not here rea1-1 " soned by induction from “ parts to Ac? Will you say that these resulting wholes," when the wholes are theinselves series 1+1+1+1, &c. and 1+2+1+1, infinite! And may it not be perempto&c. are equal, are the same, are identical ? rily demanded of you, “first to cast out If not, ought you not to have proved by the beam which is in thine own eye, that way of Lemma, previously to your enter thou mayest see clearly to cast out the ing upon the “ Elements of the True mote of thy brother's eye." Arithmetic,” and as an indispensable In your corollary to this proposition requisite to understand even the first proposition of your work, that the num. you say, “ And - is less than ber denoted by a, adued to the number denoted by b, is not the saine as the 3 , 1-1 number denoted by b, added to the num- inni by J 1 ;" here again, sir, I ber denoted by a, or that a tob is not the will not stop to dispute your conclue MONTHLY Mag. No, 212. - 2 Ş. 1 "fl+1+1 induct not have obtained 1-1+1 1-1 &c. instead of 1+1-1 sione 1 sion, namely, that the difference be- ask you, in taking 1+1 from 2, did you 0-411 . 1-1 obtain the remainder 1-1? You have tween 1 and j ismi, though, it carefully concealed the modus operandi, be in direct violation of common sense, for, if you had not, the absurdity of your but I will only ask, how you make the attempt at deinonstration would have been most glaring; since you could only fraction -- =1? for though I do not obtain the reinainder 1-1 by actually subtracting the first term of 1+1 from deny this equality, yet, if you admit it, 2, and by only denoting the subtraction I think you must likewise admit that they e of the lust term, by puiting it down with series 1-1+1 1+-1-1, &c. ad infin. the minus sign or - prefixed: now, upon is =1; for if = =I, its equal what principle could you, in subtracting 1- 1 a number which consists of two parts, or menibers, from another number, actually 1--1X - vide your 29th proposition) subtract the first part, or member, and only denote the subtraction of the other will also = 1; but = =1+1+1+1, part, or member, by connecting it with c . 1 the sign — or minus, with the result of &c. ad infin. therefore 1-1x the actual subtraction of the first mem. ber; when, in the very words of your 1--1x(1+1+1+1, &c. adinin.) that is proposition, you assert, that numbers =1-1+1-1+1--1, &c. ad infin, as connected together by a negative sien, thus appears : are different from the same numbers when 1+1+1+1, &c. ad infin. actually subtracted and expressed by oue 1--1 number? What a “ splendid instance," 1+1+1+1,&c and infin. have you here exbibited of the accuracy - -1-1--1. &c. ad infin. of your reasoning! What, sir, in future will be thought of Thomas Taylor, the 1-1+1-1+1-1+1-1,&c.ad infin. Platonist? Of Thomas Taylor, the transe therefore this series is also =1, but in lator of Proclus on Euclid? Of Thomas your first prop. you obtained this series Taylor, the admirer of Grecian geometry? 1 ; see therefore to what non Of Thomas Taylor, who boasts lumiself the vindicator of the very scientific ac. strous absurdities you are led by your curacy of the ancients? Of Thomas own accurate reasoning in the “ Ele. Taylor, who, in the Elements of his True ments of your True Arithmetic.” Arithmetic, reasons thus: “1+1 is not What I have said will, I think, be am. the same as 2; but sir Isaac Newton, in ply sufficient tu sher the fallacy of your all his researches, both mathematical reasoning, in this your work of boasted and philosophical, reasoned on the sup. accuracy. I cannot, however, refrain position that 1+1 was the same as 2; from adding a few remarks on your ninth therefore, the results of sir Isaac Newproposition, which is thus enunciated. ton's researches are a mass of errors and 6. Numbers connected together by an falsehoods; and Newton bimself, was affirmutive or negative sign, are dif. not only a man of a "rambling and preferent from the same numbers when ac cipitate genius, but a perpetual bluntually added together, or subtracted, ond dorer?". I am fully aware, that, in anerpressed by one number !” Had this swer to these questions, you will say, proposition been promulgated by any that you are perfecily indifferent to the ordinary person, I should doubtless have opinions of others, both with respect to considered it as the effect of folly or yourself and your works; for that you madness; bur, as procceding from one “ have long since learnt, from the school who holds a respectable rank in the re- of Pyrhagoras, that the praise or republic of letters, I would willingly ac- prehension of the stupid, is alike ridi. tribute it to some other cause. Sini. culous.” Highly as I applaud this truly gularly strange and ridiculous as this philosophical indifference, yet I must proposition must appear, its deinon- say, that, however regardless you may be stration, however, is, it possible, still of your own reputation, you ought at more absurd; it begins by stating that, least to possess some little respect for the “1+1 is not the saine as 2; for 1+1 reputation of those ancients, whom you subtracters froin 2 leaves the infini. so frequently and so ardently profess to mal 1-1:” llon, sir, allow ine to veneiate and admire. I clieat you, therefore, therefore, in the names of Jupiter and first principles, the accuracy of their Juno, of Bacchus and Venus, and of logic, and the truth of their conclusions; every other God and Goddess, whom but which I now find to abound in zidi. you may worship; nay inore, I conjure culous quirks, foolish conceits, und glaring you for the honour of the “sacred Mue absurdities.” Some such reflections as jesty of Truth,”* not again to prostitute these would most probably be made by the names of Euclid, Apollonius, and every one unacquainted with mathea Archimedes, as you have already done matics, into whose bands your book in the “ Elements of your True Arith- might fall, and it was to prevent the inmetic," by asserting, that you “have jurious consequences to such froin a total vindicated the very scientific accuracy neglect of the most accurate of all of the ancient mathematicians." What sciences, which might be the result of opinion, sir, let me ask you, would any such reflections, that I resolved to poing one be led to entertain of the mathe- out a few of the leading absurdities of matical works of the ancients, who had your “ Elements of the True Arivunetic never seen them; from this “vindication of Infinites," which I have here done. of their very scientific accuracy," as you trusting that you will believe me, when I have modestly termed it? I fear, sir, assure you, that I entertain the liigliest that disgusted with the absurdity, non. esteem for your character in every other sense, and falsehood, of your “ True Arith point of vieve than as a writer on the metic," yet supposing this greatise of liui hematical Sciences, but in none inore your's, from your frequent assertions to than for those inestimable qualilies of ihat effect, to be a full and clear illus- heart for which I have so frequently heard tration of the writings of the ancient you admired, and which render you, mathematicians, such a one would, though an avowed professor of Pavanism, on reading your book, imbibe an eternal a bright example worthy the imitation of disgust, not only to all mathematical the followers of bim who went about works, both ancient and modern, but doing good.” W. SAINT. even to the very, names themselves of every branch of the mathematical sci- To the Editor of the Monthly Mugazine. ences. For would he not say, “ Ifthese SIR, be the · Elements of the True Ariel D EING engaged in a “Life” of Almetic, if this book, which to me appears D gernon Sydney, I shall be obliged so dark and empty, contains a clear and to any of your correspondents, who, full explanation of the writings of the through the inedion of your Magazine, ancient mathematicians, which are ac- will inform me of any rare sources of inknowledged by all. who are acquainted formation, whether in manuscript or in with them, to contain the purest spe- print. The materials contained in the cimens of right reasoning; what hope have Sydney papers, and in Dalrymple, as I of ever being able to understand even well as those in Ludlow and Burnet, are the elements of the mathematical already before me; but it seems pro. sciences?" bable, that many references to my hero “ If this book, in which it is maintained may be concealed in contemporary pub. that 1+1 is not equal to 2; that 1-1 is lications, either scarce or little known; not equal to 0, or nothing; that a frac- as, in a work of this description, I lately tion, with nothing added to it, is less by met with a very honourable notice of his unity, or 1, than the same fraction without conduct at Marston Moor, the addition of nothing; that an infinite Feb. 12, 1811. series of numbers, with nothing added to it, is infinitely less than the same series To the Editor of the Monthly Magazine. without the addition of nothing; if this SIR, book, I say, while it maintains such I SHOULD esteem it a very great manifest absurdities and self-contradic. I favour, if some of your numerous and tions, be a “Tindication of the accuracy ingenious correspondents would oblige of the ancient mathematicians," I must me, through the channel of your enter. for ever bid adieu to the prospect of obe taining Magazine, with a list of such taining any knowledge of mathematics, shrubs, herbs, and flowers, as emit the · which I had been taught to believe were inost salubrious and nutritious air, and distinguished for the simplicity of their in this I should wish wild, or hedge, berbs to be included. Also, if not going too * The Deity to whom Mr. Taylor de. far, I should be glad of any observations dicated his Proclus, that would tend to shew, whether the air ennitted SIR, emitted from the growing plant bas, or oxidated state, will be estimated much might be expected to have, the same or less than it really should be. If any of a similar effect on the constitution, as your correspondents would give a more the same when taken interually. accurate mode of analysis, through the March 27, 1811. medium of your Magazine, it would much oblige, N. To the Editor of the Monthly Magasine. Almondbury, March 18, 1811. . SIR, A N amatcur in philosophy, who has To the Editor of the Monthly Magazine, A not the opportunity of inaking ex. periments, wishes to propose the follow. I JAPPENING to run over a cata. ing question, Suppose a polished iron 11 logue of books published or sold at ball were suspended, in the manner of a the Leipzig fairs, of 1809, which I acci. pendulum, between two magnets; thé dentally found amongst a parcel of size of the ball, the length of the pen- foreign publications, I observed the fol. dulum, (baving as little friction as pos- lowing title, Arfixov TES 'Avoldogsxrac zt36 T* sible) and the form and distance of the ομιλουμένης τωρινης των Ελληνων γλοσσας, and magnets, being adjusted with the utinost wish to enquire of you, or any of your accuracy; and this apparatus contained correspondents, if this so called sredotegiten in a glass receiver, in which the atmo- ghoron is the language known by us at spherical air was previously as much ra- present as the modern Greek, or inerely refied as possible. a dialect, or union of dialects, in use The question is, whether the attractive among the ancients? If the adjective - power of the two magnets would operate Powgrym be a compound of spa, it probably reciprocally upon the polished-iron ball, signifies present, and the Arfixor here so as to keep it in a continued uniform spoken of, is perhaps a modern Greek oscillation between them, by the fresh Dictionary, impulse given to the ball at its near ap- In two Dictionaries, which I had an proximation (short of contact) to each opportunity of inspecting a few years magnet, in consequence of being drawn ago, I remember fiuding the titles of the somewhat beyond its otherwise natural new Greek very different from that here extent. alluded to. One was published in Leip. March 21, 1811. zig, if I mistake not, in 1796, and to the best of my recollection was called artwor To the Editor of the Monthly Magazine. dw dogwu cite or gespen Tinoy Xar itanixer, and the other, according to my common place ITIE following is the mode of an- book, bore the name of voor asfixov italie 1 alysis which cliemists in general ypaixixor. employ, in order to ascertain the quantity Now, sir, as I am not aware of any voof animal and vegetable matter in soils. cabulary of the modern Greek tongue After the finely divided matter of the existing in our language, I cannot but soil has been submitted to the action of think that it would be interesting to many, the muriatic acid, which takes up the not only to be informed, as I suspect, Jime, magnesia, and part of the oxide of that all the Dictionaries here quoted em. iron; the residuum, which generally con- brace the same language, but also to sists of silica, aluinina, and bighly oxidated receive some further account of a lan. iron, is exposed to a high degree of heat, guage, the real state of which appears in order to burn out the animal and ve. at present to be but little known; and getable matter, and the loss of weight it which, from its august descent, might be sustaius in this process, is supposed to be expected to have awakened the curiosity equal to the quantity of those kinds of of pbilologists at an earlier period. As matter contained in the soil. That this your Miscellany may doubtless fall into method is incorrect, seems obvious, from the hands of one or another, whose forthis consideration. That part of the tune and taste have enabled him to visit oxide of iron which is lelt, together the territories of ancient Greece, I trust with the vegetable matter after the action the perusal of this short potice will pre. of the muriatic acid upon the soil, ab- vail on any gentleman of this descripsorbs oxygen during calcination, and of tion, to give in his opinion of the language course obtains an increase of weight. at present in use among the descendants Hence the quantity of vegetable matter, of Plato and Demosthenes. calculated in this manner, and especially Kentish-Town, P&DAGOGUS. 11 the soil contain much iron in a highly Murch 19, 1811, For SIR, . For the Monthly Magazine. able. “ Pleasant is the joy of grief, it is like the shower of spring, when it sotiens the branch of the oak, and the young LETTER VII. leaf lifis its green bead :" but when clie AFTER some time spent in viewing this heart too surely feels that all the airy A grand scene, we recrossed the bridge, phantons of felicity are vanished, iis and resolved to gratify that prevailing feels hopes crushed by the rude hand of ading iraplanted in the breast of mankind in versity, and its feelings lacerated by ungeneral, namely, curiosity, even at the risk kindness and ingratitude, then is the of a tuinble into the stream, we scramp. moment of bicter retrospection; then bled down the rocky bank, and, stepping memory proves a torturing fienil, blaston the large stones that lay in all direc- ing cvery present enjoyment, and tions in the channel, reached the depth “ turning all the past to pain." But below the second fall, without accident, away with such reflections, for the preor the thousand difficulties and perils sent let us " leave dull care behind," and our guide to the cataract assured us must return to the beauties of the Array, and be the consequence of our undertaking. the sweet banks of Ullswater, to which Arrived at the spot we wished to gain, we hastened after quitting Gowbarrow, we placed ourselves in front of the se- and again beheld a picture, in which cond cascade, which, though neither so there appeared not a single object to astonishing nor grand as the one above, hurt the eye, or lead the beholder to is well worthy of a visit, and forms a yet imagine it would have been better omitmore pleasing subject for the pencil, than ted. All was beautiful, sweet, and dnes the celebrated fali we had seen. lovely, as fancy could pourtray; while in This is a sheet of water of about thirty the distant view into the gorge of Patterfeet in height, falling in one unbroken dale, at the extremity of the lake, there cataract, enclosed by dark brown pre is a grandeur and sublimity scarce to eipices, rendered yet more gloomy in be equalled in any part of Britain. their appearance by the profusion of Place Fell still claimed the pre-eminence trees that overhang the stream, uniting over its neighbours, on the opposite cheir lealy branches from either side of shore; while, on the side on which we the dell, (in that part scarcely thirty travelled, thick-wooded knolls, and awyards in width,) and forming an almost ful precipices, hung almost over the impenetrable shade, beneath which the road, and obscured the view of the bridge with some of the towering cliffs stupendous back-ground; the line of that bound one side of the principal fall, which is often broken by romantic appear with the most picturesque effect, woody glens, through which considerable and form one of the most roinantic, ad- streains of water pour, amidst rocks and mirable scenes I have ever beheld. Stones, roaring and sounding through From hence we again descended by the the shade, « and falling fast from gra. margin of the stream, will we reached the dual slope to slope, with wild infracted bottoin of the third fali, which is nearly course," hasten to the lake, and, inin. the same height as the one above; but gling with its placid flood, steal silently the scene of the rustic bridge, and its « Along the mazes of the quiet vale."" accompaniments, being more in the As you approach the entrance into Pata bird's-eye style, it is less interesting than terdale, the road winds sometimes close the former. Long did we linger amidst to the bases of the mountains, at others, this delightful scenery, now passing passing through rows of thick, umbraacross the stream upon the stones that geous trees, at length reaches the clean, rose far above the water, now seated on comfortable inn, at the mouth of that a jutting point of rock, listening to the sequestered vale, whiere, it being our inte rushing of the different falls, as their tention to remain a day or two), we pro. sounds swept along the dell, and min- cured apartments, and, after partaking gling with the gentle whispers of the of an excellent dinner, we strolled around wind, produced an effect that scarce the environs; from a rock behind the could fail in any bosom to awaken emo- inn, enjoying one of the finest views tions of the most pleasing nature. In imaginable, of the scenery which had mine, they produced a melancholy, perle recently afforded us some cause for adsive calm, while busy thought reverted to mii ation: Gowbarrow, forming the many à long-past incident and happy back ground of the picture, Place Feil circumstance; and recalling to memory the boundary on the rulit, and a chain many an absent or departed friend, my of lofty mountains that upon the left. heart swelled with emotions indescrib- This is a delighiful scene, and we belield |