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buch desirable objects, than for one of the cumulalarum Mole," the repeal of bad members for London, Westminster, laws is much more wanted than the Middlesex, or Southwark, to prepare and enactment of new; and is one of tha bring in a bill, having sent provisions, best proofs of legislative wisdom, and and there can be no doubt but it would of attention to the laws, the conslitution, be hailed as a salutary measure in and and the welfare of the people, out of parliament, and be carried into

CAPEL LOFFT. execution, attended by no sentiment besides that of gratitude to its projector, To the Editor of the Monthly Magazince and applause to him who reduced it to a

SIR, Jaw.

THE number of lives lost, and the CHELSEA. COMMON SENSE, 1 damage and inconvenience susa Nov. 17, 1810.

:tained, pariicularly in winter, when

torrents of water, and great quantities To the Editor of the Monthly Magazine.

of ice and snow force their way down

rivers, (at other seasons perbaps insige SIR

niticant) carrying away the bridges in T THINK you will do right to notice their course; baving frequently occupied

1 in the Monthly Magazine, a legisla- my mind, and believing I have devised a tive phænomenon which I believe has mode by which the same may be pre. not occurred before since the 9th Hen. vented, I have obtained a patent for a 3.; in the passing of two excellent acts new method of erecting bridges, &c. of parliament the same day. I mean without arches or sterlings; the advane chap. 51 and 2 of 50 Geo. III. This tages to be derived from which are, that took place the 9th of June, 1810, 545 they are not subject to be injured or de. years from the re-enactment of Magna stroyed by floods no kind of ground is

unsuitable for the foundation-ihey may The first of these repealed, the statute be erecied in the most difficult and ala of 7 James I. c. 4. by which a woman, most inaccessible places-roads may be among those of the most uneducated, ige continued over marshy grounds without norant, and neglected, class of society, the danger of being destroyed in winter, chargeable to a parish by her inconti- and are alike applicable to every situation nence, was to be kept to close imprison. whether public or private-are erected ment and labor for one whole vear. And on in a small space of time-and compara. the second offence was to find good sure tively inconsiderable expence. dies for her behaviour,and to be imprisoned As I hope and trust this will be found till found : which, the repealing act very of essential benefit to society, I beg the justly observes, might be imprisonment favour of you to give publicity to this, for life. It also provides that on signs by inserting it in the ensuing number. of reformation, the imprisonment, which Bristol, March, 1811. SARAH GUPPY, is to be not less than six weeks, nor more than one year, is mitigable, by warrant To the Editor of the Monthly Magazine, of the committing inagistates to discharge SIR, the prisoner at any time after the six N OT being a professional man, I weeks.

I was not aware, in the statement By the 2d the statute of 8 and 9 W. of facts respecting Admiral Patton, III. c. 30, s. 2. is repealed, which re- which I sent for insertion in the Monthly quired the poor to be badged with a Magazine, that the term ordinary seamen larye Roman P. and the initial of their was appropriated in the ship's books to parish, so as to be conspicuous; and landmen and inexpert sailors, who repunish them, on refusing or neglecting ceive inferior wages to those who are to wear this badge, by whipping.

rated uble; whereas the persons I meant "The first of these acts had lasted al to discriminale by that appellation, are most exactly two centuries, and the se. all included in the class of able seamen, cond above one. The first was fre. who draw the bigliest wages, upon an, quently, the second rarely, executed. equal footing with the most expert sailors, But it was high time for the repeal of meriting the appellation of prime seamen, both of them.

and greatly their superiors in professional Those repealing acts have among their merit. li is upon this distinction Adessential excellences, the saine merit of miral Patton's plan for the improvement conciseness.

and security of the navy rests; whicha In this " legum aliarum super alias may be understood by my stalement;

but

but it would prevent misconception, and under and opposite to the earth will be obviate caril upon this head, if the ex- highest in the perigeon and lowest in the pression inferior seamen were substituted apogeon. There will consequently be for ordinary seamen, wherever it occurs high water and low water once in about ju my statement.

twenty-eight of our days, or a periodical ROBERT Patton, month, which is not quite equal to a day Hampshire, February 2, 1811. and night in the moon. Besides this,

the attraction of the sun will also pro. To the Editor of the Monthly Magazine. duce another tide, (as it does with us), SIR,

returning to all parts of the moon livice I JAVING paid some attention to the in a lunation or synodical month, which II astronomical quere in your last, I is the lunar day and night. venture to submit the following ideas on The varying positions of the sun and the subject, by way of an attempt at a moon will produce either spring or neap solution.

tides, according as their accions concur Although the moon contains only a with, or counteract, each other, and the fortieth part of the quantity of matter greatest spring rides in the parts directly that the earth does, she is said to be the under, and opposite to, the earth, being largest secondary planet in the system, when the sun is in the moon's perigeon in proportion to its primary. The argue at the conjunction, or in her apogeon at mene inay therefore be most fairly stated the opposition; and the lowest neap as between these two, since it would tides when the moon is in her apogeon apply yet more strongly to the others. at a quadrature. In the circle, ninety The attraction of the moon upon the degrees from those points, or what we waters of the earth is just sufficient to call the moon's liinb; the contrary will raise a moderate and beneficial tide, take place. The fluctuations of the lu. which is met by the several places of the nar waters, arising from these causes, are earth iwice in each diurnal revolution: probably sufficient to preserve their sweetwhereas if we were its secondary planet, ness, and to answer other purposes of and the primary one forty times our bulk, convenience, as the tides do with us.. it would attract the waters with forty In the above theory it is taken for times the force that they are now subject granted that lunar seas exist, which I to; and consequently vast regions of the find is denied by some philosophers, and earth would twice in every day be inun- I observe that the one, who is possessed dated ; and a great additional inconve- of the most powerful apparatus for obnience would certainly result from the servation, speaks of the moon as if it increased rapidity in the ebbing and were decidedly not a terraqueous globe. flowing of the waters. It may indeed Others however are of a different way of be doubted whether they would be at all thinking, and there seem to be good arnavigable. But supposing that we (like guments for their opinions : but whichthe moon) always turned the same face ever way that question be decided, I aptowards our primary, then, although the prehend it is agreed on all hands that waters which were under and opposite the moon is furnished with an atmo. to it would be greatly raised by its at. sphere, and the reasoning above may be traction; yet as they would remain con- applied to that, although there should be stantly in the same state, (except such no seas. For I presume it to be indis. gentle variations as might result from putable that the earth and moon, by causes hereafter to be noticed) the effect their attraction, raise tides in each other's arising from the difference of bulk would atmosphere, and that the air in the pro. be the same as if there were no tide at tuberant parts must be thereby consiall; 80 that none of the above inconve- derably rarified, and in those remote niences would be perceived.

from them, as much condensed; both It is evident therefore that the moon which effects inust be abundantly greater can have no sensible tides resulting from in the moon than on the earth. It might the earth's attraction, except wbat arise therefore be a serious inconvenience to from the variation in the degree of that the former if so considerable an alterattraction, in consequence of the changes ation in the state of the air were to reof distance; and as her eccentricity is cur at all places successively at short considerable, and the earth so large a intervals of time, as the cides of the same body, it is probable that this change of nature do with us; for besides their ef. distance may have the effect of produc- fect (or influence as it is called) on the ing a gentle tide, whereby the waters minds and bodies of an unfortunate de.

scription

SIR

scription of people, which I apprehend be to an accurate knowledge of the ana to be a certain fact, I make no doubt cient philosophy, or to an intimate ace but that, combined with other unknown quaintance with Pagan theology, your causes, it is of no small consequence in claims to the higher bonor of refuring producing alterations of the weather. Wallis or Newton have no foundation, It appears therefore to be wisely ordered except in the ebullitions of your own that in the inuon the effects on the earth's vanity. great attraction should be always nearly Now then, Sir, to the point: I am the same at each particular place; and ready to grant your three first postulates, it may probably be a principal cause of though I cannot help remarking, that, in the constant serenity which seems to a work abounding with so many pretene take place in the lunar atmosphere. sions to perfect accuracy, it would have

JOHN ANDREWS. accorded better with those pretensions, Alodbury, March 3, 1811.

if these postulates had been preceded by

definitions of the terms addition, suba For the Monthly Magazine, traction, division, &c. more particularly KEMARKS on the elements of the TRUE as you appear on some occasions to have

ARITHMETIC of INFINITES, by THOMAS used these terms in a sense differing from TAYLOR (the Platonist); in & LETTER that in which they are commonly reto the AUTHOR, by w. SAINT, ad. ceived. Your fourth postulate, however, dressed to THOMAS TAYLOR, ESQ. of I by no means so readily grant; it runs WALᎳᏫᎡᏘᏗ.

thus, " That to multiply one number, or · Norwich, March 4, 1811, one series of nuinbers, by anothier, is

the same thing as to add either of those V OUR“ Elements of the True Arith- numbers, or series of numbers, to itself,

I metic of Iufinites," baving acci- as often as there are units in the other." dentally fallen into my bands, I was Now, to say nothing of the absurdity of anxious to see in what manner you had calling this a postulute, which is, in retreated this subject, not only on account ality, a definition, I do not believe that of your professed admiration of the sci- it conveys even your own meaning, for entific accuracy of the ancient mathema. surely you will not say that 3 multiplied ticians, but froin the vaunting style in by 2 is the same as 3 added twice to it. whicb you seemed to exult over the mo. self-for 3 added once to itself makes 6, derns, even in the very tiile page of and if added twice to itself it will make your performance : I therefore eagerly 9; and I cannot think, Sir, that you meant applied to vour book with the confident to say that 3 multiplied by 2 is equal to expectation that I should be made ac. 9. Moreover, Sir, I beg to ask you what quainted with the true “ nature of infi. you can mean in this postulate by a nitesimals," and that I should find that « series of numbers," unless several or you had treated this curious branch of many numbers connected together by malliematics in the most unexception- the sign plus or minus ? And if so, I able manner. Judge then, Sir, of my will further ask you how the units in surprise when, instead of that divine ac- either series are to be ascertained, (for curacy, that logical precision, that luni. the purpose of knowing how many times nous arrangement, for which the wri- the other series is to be added to itself" tings of the ancients are so pre-eminently to produce the product), unless by an distinguished, I inet with nothing but actual summation of that series, that is absurd premises, confused reasoning, and by collecting its terms into one sum ac. false conclusions!

cording to their signs? Now if you had I can scarcely hope to convince you, to multiply the series 1+1+1+1, &c. Sir, that your performance abounds wiib ad infinitum by 1--1, since you have errors and absurdities; but, as you have asserted in the corollaries to your first evir.ced an almost unexampled degree proposition that 1–1 is that " which is of buldness, not to say arrogance, even neither quantity nor nothing, but which in the title page of your work, by decla- is something belonging to number withring therein that you have“ demonsirated out being number." You would thus hare all the propositions in Dr. Wallis's Arilho to add the infinite series 1+1+1+1, metic of Infinites, and also the principles &c. to itself, as many times as are deof the Doctrine of Fluxions, to be fulse," noted by that's which is neither quantity I think it but right to convince others, nor nothing, but which is something bee or at least to attempt to convince them, longing to number without being num that, however just your pretensions may ber," In like manner, Sir, to multiply

1–1 by the infinite series 1+1+1+1, . From 1 &c. would be to add that to iiselt which Subtract 1-1+1-171-1+1-1, &c. is “ neither quantity nor nothing" au Remainder +1-1+1-1+1-1+1, &c. infinite number of times; and this sum “ But by the second postulate the reteing equal to the former (unless indeed mainder added to what is subtracted is pou deny that 2 multiplied by 3 is the equal to the subtrahend Hence the sesame with 3 multiplied by 2, or, more ries 1--1-1-1+1-1, &c. added to generally, that a multiplied by b is the 1-1+1-141-1, &c. is equal to same with b inultiplied by ") you would 1. The series 1-1+1-1+1-1, &c. have an infinite number added to itself “neither quantity nor nothing" times,

p" times. is therefore equal to 1, and conseequal to " neither quantity nor nothing" quently 1-1 is an infinitesimal. For added to itself an infinite number of it cannot be 0, since an infinite series of tines! I know not, Mr, Taylory what 0, added to an infinite series of o, can you may think of this, but I will tell you never be equal to 1. most freely that I think it to be infinite « In like inanner, nonsense! And I was not a little asto- Jf from 1-1-1+2-1-1+2-1-1,&c. nished to meet with this “ splendid in Subtract 1-2+1+1-2+1+1-2+1, &c. stance of absurdity,” to use your own remaind: 19.I

Remaind. • 1—2+1+1^2+1+1=2, &c. language, in the very outset of a work

and therefore 1-2+1 is an infinitesimal; in which you most modestly observe that and so of the rest.

The rambling and precipitate genius of “ Corol. 1. Hence such expressions modern mathematicians, eager to arrive as 1-1, 142+1, 1-2, &c. are veither at some conclusion which may be applicable quantities nor nothings, but they are to practical purposes, neglects that rigide something belonging to number, without accurucy of demonstration, which may be being number; just as a point, which is called the impregnable fortress of the ma.

the extremity of a line, is something be

the thematical science, and for which the ge

longing to, without being a line.

lowing to w nius of ancient mathematicians was so

“ Corol. 2. Hence, likewise such pre-eminently distinguished.

expressions when they are considered as But I pass, Sir, from your postulates parts of infinite series, are not to be to your first proposition, the enunciation

inciation taken separate from the terms by which

taken and demonstration of which I will here they are expressed, viz. 141, for instance put down at length, as affording a fair

is not to be considered as a subtractiok specimen of the accuracy of your logic. Of

of 1 from 1; for, in this case, it woulu

from 7. For; “ Proposition 1.

be 0. Nor is 1-2 to be considered as “1_1 is an infinitesimal, or infinitely a subtraction of 2 from 1, since it would

1 ainall part of the fraction

then be -1. But these expressions ara, and an always to be considered in connexion

with the numbers by which they are infinite series of 1-1 is equal to 1, formed. In like manner, also, 1–2+1 is an in

“ Corol. 3. Hence, the series whicha

are called by modern mathematicians finitely small part of 1-1-1+4-1-1+2-1-1+2-1-1&c neutral and diverging series, are erro

neously so called, for they are in reality ad infin, and an infinite series of 1-2+1 converging series." is equal to :

In this proposition, Sir, you begin by

affirming that 1-1 is an infinitesimal, 1-1-1+2–1-172-1-1, &c.

without having previously defined what 1+1

constitutes an infinitesimal; And 1-2 is an infinitely small part of however, the qualifying words“ infinitely

perhaps, 1-1-1-1-1, &c.

ad infin and a small part" which follow were designed 11

to supply this deficiency. Your demone infinite series of 1–2 is equal to stration, I presume, begins at the word 1-1-1-1--1-1. &c. ..

“ From;"-if'so, let me ask you by what Thus, too, I means you obtained the remainder 1+1

1-2-2-2-2.&c. :+1-1+1- 1+1-1+1-1,&c.? is the infinitesimalof

--, Your answer must certainly be, that you

actually subtracted the first term of the 1-4 of 1-3-3-3-3, &c; and so of second line, or number to be subtracted 1+1

from the first term (and here only térm) achers,

of the first line or subtrahend, and that

you

1

1+1

you called the remainder O or nothing, or thus an infinite number of remainders rather dat or ., to which you annexed might be obtained from the infinite the other terms of your second line, or variety of positions in which the subtra. number to be subtracted, with their bend might be placed, and any one of signs changed, agreeably to the common these I will affirm to be as correcily the rea rule for the subtraction of algebraic mainder as the one you have above given : quantities. Now, surely, as the author and what indifferent person would not of an Elementary Treatise, you ought to consider my affirmation as of equal weight have previously demonstrated the grounds with yours, till you have demonstrated of this method of subtraction: passing that to obtain the true remainder it is over, however, this unpardonable omise absolutely necessary that the subtrahend sion, I would ask, why the first terın of should be placed over the first term of your second line should be actually subo the series to be subtracteri. The retructed from the subtrabend, rather than mainuer in your second example might put down after that subtrahend with its be varied in a similar manner by putting sign changed, in like manner as all the the first term of the second line under terms after the first in that line are put the second, third, fourth, term, &c. of down? in which case, instead of the dot the first line; but you wouid not then or • in the remainder, you would have obtain for a reinainder a series whicb had 1-1; to this, perhaps, you will would be constituted of a repetition of answer that you would still have had the your infinitesimal 1-2+1; unless, series 1-1+1-1+1–1, &c. for a re therefore, you can demonstrate that the mainder, which I also readily admit; but true remainder can only be obtained by what, let me ask, would bave been the that purticular position in which you result of the second part of your demon- have thought proper to place the subtrastration, where you attempt to shew that hend and series to be subtracted, your 1-2+1 is an infinitesimal; if, instead fundamental proposition is, to use your of actually subtracting the first terın of own language, false, and the superstrucyour second line from the same term of ture which you have raised upon it ine your first, you had only put the former stantly falls to the ground; or I should down with its sign changed after the rather have said the temple erected by lutler ? Would you not, in this case, Wallis and Newton, which you have in instead of · 149+1+1–2 +1+1-2, vain attempted to demolish, still stands &c. have obtained 1-it1–2+1+1- firm, unshaken, and immutable, upon

firm, unshaken, and i 2+1+1-2, &c. for a remainder and the eternal and adamantine rock of how then, Sir, would you have shewn science and truth. that this latter series consisted of your In your corollaries to this proposition, boasted infinitesimal 1-2+1?-Again, you are pleased to assert that the expreso if in the first example, instead of placing sions 1-1, 1-2+1, &c. are “ neither the subtrahend i over the first ierin of quantities, nor nothings;" that they are the second line, you bad put it over any not quantities I am ready to allow, as of the succeeding terms in the same line, numbers are rather the incasures or reas in the following instances, you could presentatives of quantities, than quantinot have obtained the remainder ties themselves : but that they are not

+1-1+1-1, &c. as may be seen on nothing I deny, and I will defy you to inspection :

prove that they are something. The in· From

genivus Bishop Berkeley very shrewdly Subtract 1–1+1-1+1, &c. asked, " Whether evanescent incre| Remainder is – 1+2-1+1--1, &c.

menis might not be called the ghosts of

departed quantities :" what then, may I 1

ask, shall your non-quantities be called, Subtract 1—1+1–1, &c. which are something yet neither quantity Remainder is -1+1+1-1+1, &c. m i t nor nothing? Surely these can only be

the shudows of the ghosts of departed Froin

nothings! Subtract . 1-1+1-1+1, &c.

Your second proposition is thus enun. Remainder is itit1-1, &c. ciated : « There cannot be a greater Frum

number of terms in any infinite series Subtract 1-1+1-1+1, &c. than, which is equal to 1+1+1+1, Remainder is -1+1--1+2—1, &c. &c. ad infinitum.” This enunciation

· From

ought

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