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bat it would prevent misconception, and obviate cavil upon this head, if the expression inferior teamen were substituted tor ordinary seamen, wherever it occurs ju my statement.
Robert Patton. Hampshire, February 2, 1811.
To the Editor of the Monthly Magazine.
HAVING paid some attention to the astronomical quere in your last, I venture to submit the following ideas on the subject, by way of an attempt at a solution.
Although the moon contains only a fortieth part of the quantity of matter that the earth does, she is said to be the largest secondary planet in the system, in proportion to Us primary. The argument may therefore be most fairly stated as between these two, since it would apply yet more strongly to the others. The attraction of the moon upon the waters of the earth is just sufficient to raise a moderate and beneficial tide, which is met by the several places of the earth twice in each didmal revolution: whereas if we were us secondary planet, end the primary one forty times our bulk, it would attract the waters with forty times the force that they are now subject to; and consequently vast regions of the earth would twice in every day be inundated; and a great additional inconvenience would certainly result from the increased rapidity in the - ebbing and flowing of the waters. It may indeed be doubted whether they would he at all navigable. But supposing that we (like the moon) always turned the same face towards our primary, then, although the waters which were under and opposite to it would be greatly raised by its attraction; yet as they would remain constantly in the same state, (except such gentle variations as might result from causes hereafter to be noticed) the effect arising from the difference of bulk would be the same as if there were no tide at all ;"so that none of the above inconveniences would be perceived.
It is evident therefore that the moon can have no sensible tides resulting from the earth's attraction, except what arise from the variation in the degree of that attraction, in consequence ot the changes of distance; and as her eccentricity is considerable, and the earth so large a body, it is probable that this change of distance may have the effect of producing • gentle tide, whereby the waters
under and opposite to the earth will be highest in the perigeon and lowest in the apogepn. There will consequently be high water and low water once in about twenty-eight of our days, or a periodical month, which is not quite equal to a day and night in the moon. Besides this, the attraction of the sun will also produce another tide, (as it does with us), returning to all parts of the moon twice in a lunation or synodical month, which is the lunar day and night.
The varying positions of the sun and moon will produce either spring or neap tides, according as their actions concur with, or counteract, each other, and the greatest spring tides in the parts directly under, and opposite to, the earth, being when the siin is in the moon's perigeon at the conjunction, or in her apogeon at the opposition; and the lowest neap tides when the moon is in her apogeon at a quadrature. In the circle, ninetydegrees from those points, or what we call the moon's limb, the contrary will take place. The fluctuations of the lunar waters, arising from these causes, are probably sufficient to preserve their sweetness, and to answer other purposes of convenience, as the tides do with us.
In the above theory it is taken for granted that lunar seas exist, which I find is denied by sonic philosophers, and I observe that the one, who is possessed of the most powerful apparatus for observation, speaks of the moon as if it were decidedly not a terraqueous globe. Others however are of a different way of thinking, and there seem to be good arguments for their opinions: but whichever way that question be decided, I apprehend it is agreed on all hands that the moon is furnished with an atmosphere, and the reasoning above may be applied to that, although there should be no seas. For I presume it to be indisputable that the earth and moon, by their attraction, raise tides in each other's atmosphere, and that the air in the protuherant parts must be thereby considerably rarified, and in those remote from them, ne much condensed; both which effects must be abundantly greater in the moon than on the earth. It might therefore be a serious inconvenience to> the former if so considerable an alteration in the state of the air were to recur at all places successively at short intervals of time, as the tides of the same nature do with us; for besides their effect (or influence as it is called) on the minds and bodies of an unfortunate <ie
IcriptioB scriftion of people, which I apprehend to be a certain fact, I make no doubt but that, combined with other unknown causes, it is of no small consequence in producing alterations of the weather. It appears therefore to be wisely ordered that in the moon the effects on the earth's great attraction should be always nearly the same at each particular place; and it may probably be a principal cause of the constant serenity which seems to take place iu the lunar atmosphere.
John Andrews. Modbury, March 8, 1811.
For the Monthly Magazine. Iemarks on the Elemenis of the Trce
ARITHMETIC of INFINITES, by THOMAS
Taylor (the flatonist); in a Letter to the Author, by w. Saint, mldressed to THOMAS TAVLOR, ESQ. of WALWORTH.
Mrwikb, Matcbi, 1811. Sir,
YOUR "Elements of the True Arithmetic of Infinites," having accidentally fallen into my hands, I wn,s anxious to see in what manner you had treated this suhject, not only on account of your professed admiration of the scientific accuracy of the ancient mathematicians, but from the vaunting style in which you seemed to exult over the moderns, even in the very title page of jour performance: I therefore eagerly applied to your hook with the confident expectation that I should be made acquainted with the true " nature of infi. nitesimals," and that I should find that jou had treated this curious branch of mathematics in the most unexceptionable manner. Judge then, Sir, of my surprise when, instead of that divine accuracy, that logical precision, that luminous arrangement, for which the writings of the ancients are so pre-eminently distinguished, I met with nothing but absurd premises,confused reasoning, and false conclusions!
I can scarcely hope to convince you. Sir, that your performance abounds with errors and absurdities; but, as you have evinced an almost unexampled degree of boldness, not to say arrogance, even in the title page of your work, by declaring therein that you have " demonstrated ell the propositions in Dr. Watlis's Arithmetic of Infinites, and also the principles of the Doctrine of Fluxions, to befuise," I think it but right to convince others, or at least to attempt to convince them, that, however just your pretensions may
be tp an accurate knowledge of the ancient philosophy, or to an intimate acquaintance with Pagan theology, your claims to the higher honor of refuting Wallis or Newton have no foundation, except in the ebullitions of your own vanity.
Now then, Sir, to the point: lam ready to grant your three first postulate^, though I cannot help remarking, that, in a work abounding with so many pretensions to perfect accuracy, it would have accorded better with those pretensions, if these postulates had been preceded by definitions of the terms addition, subtraction, division, &c. more particularly as you appear on some occasions to have used these terms in a'sense differing front that in which they are commonly received. Your fourth postulate, however, I by no means so readily grant; it runs thus, "That to multiply one number, or one series of numbers, by another, is the same thing as to add either of those numbers, or series of numbers, to itself, as often as there are units in the other.'* Now, to say nothing of the absurdity of calling this a postulate, which is, in reality, a definition, I do not believe that it conveys even your own meaning, for surely you will not say that 3 multiplied by 2 is t he same as 3 added twice to itself—for 3 added once to itself makes 6, and if added twice to itself it will make 9; and I cannot think, Sir, that you meant to say that 3 multiplied hy 2 is equal to 9. Moreover, Sir, I beg to ask you what you can mean in this postulate by a "series of numbers," unless several or many numbers connected together by the sign plus or minus? And if so, I will further ask you how the units in either series are to be ascertained, (for the purpose of knowing how many times the other series is to be" added to itself" to produce the product), unless by an actual summation of that series, that it by collecting its terms into one sum according to their signs? Now if you had to multiply the series l+l+l+l, &c. ad infinitum hy 1—1, since you have asserted in the corollaries to your first proposition that 1—1 is that " which it neither quantity nor nothing, but which is something belonging to number without being number." You would thus have to add the infinite series 1 -J- 1 -V-3-f-:, &c. to itself, as many times as are denoted by that" which is neither quantity nor nothing, but which is something belonging to number without being number." In like maimer, Sir, to multiply 1—1 by the infinite series 1+1+1+1, &c. would be to add that to itself which is " neither quantity nor nothing" an infinite number of times; and this sum teing equal to the former (unless indeed you deny that 2 multiplied by 3 is the same with 3 multiplied by 2, or, more generally, that a multiplied by b is the same with b multiplied by «) you would have an infinite number added to itself •'neither quantity nor nothing" times, equnl to " neither quantity nor nothing" added to itself an infinite number of times!—I know not, Mr. Taylor* what you may think of this, but I will tell you most freely that I think it to be infinite nonsense! And I was not a little astonished to meet with this " splendid instance of absurdity," to use your own language, in the very outset of a work in which you most modestly observe that "The rambling and precipitate genius of modern mathematicians, eager to arrive tit some conclusion which may be applicable to practical purposes, neglects that rigid accuracy of demonstration, which may be called the impregnable fortress of the mathematical science, and Jot which the genius of ancient mathematicians was so pre-eminently distinguished."
But I pass, Sir, from your postulates to your first proposition, the enunciation and demonstration of which I will here put down at length, as affording a fair •pecimen of the accuracy of your logic. "Proposition 1.
"1—lis an infinitesimal, or infinitely
•mall part of the fraction ——, and an
infinite series of 1—1 is equal to 7-7-7.
+ J11 like manner, also, 1—2+1 is an infinitely small part of ■1—1—1 + 2—1—1+g—1—1+2—1—1 &c
ad infin. and an infinite series of 1—2+1 h equal to
And 1—2 is an infinitely small part of
• ad infin. and an
From 1 Subtract 1—1 + f—1+1—1+1—1, &c
Remainder -~+l—1+1—1+1—1 + 1, &c "But by the second postulate the remainder added to what is subtracted is equal to the subtrahend, Hence the series 1—1+1—1+1—1, &c. added to 1—1+1—1 + 1—1, &c. is equal tr> 1. The series 1—1+1—1+1—1, &c.
is therefore equal to J-tti al,(l consequently 1-r-l is an infinitesimal. Fix" it cannot be 0, since an infinite scries of 0, added to an infinite series of O, caa never be equal to 1. "In like manner, Iffrum 1—1—1+2—1—1+2—1—1, &c. Subtract 1—2 + 1 + 1—2+1 + 1—2+1, &c.
Remaind. • 1—2 + 1+1—y + 1 + 1—*, &c7 and therefore 1—2+1 is an infinitesimal; and so of the rest.
"Corol. 1. Hence such expression* as 1—1, 1—2+1, 1—2, &c. are neither quantities nor nothings, but they ana something belonging to number, without being number; just as a point, which is the extremity of a line, is something belonging to, without being a line.
"Corol. 2. Hence, likewise such expressions when they are considered as parts of infinite series, are not to be taken separate from the terms by which they are expressed, viz. 1—1, for instance is not to be considered as a subtraction of 1 from 1; for, in this case, it wouU be 0. Nor is 1—2 to be considered as a subtraction of 2 from J, since it wnulrl then be —1. But these expressions are, always to be considered in connexion with the numbers by which they are, formed. •
"Corol. 3. Hence, the series which are called by modem mathematicians neutral and diverging series, are erroneously so called, for they are in reality converging series."
In this proposition, Sir, you begin by affirming that 1—1 is an infinitesimal, without having previously defined what constitutes an infinitesimal; perhaps, however, the qualifying words" infinitely small purt" which follow were designed to supply this deficiency. Your demon, stration, I presume, begins at the word "From ;"—ifso, let me ask you by what means you obtained the remainder ■ +1 — 1+1 —1 + 1 —1+1—l,&c? Your answer must certainly be, that you actually subtracted the first term of the second line, or number to be subtracted from the fir-t term (and here only term) of the first line or subtrahend, and that
jou tailed the remainder 0 or nothing, i>r rather dot or ", to which you annexed the other terms of your second line, or number to be subtracted, with iheir signs changed, agreeably to the common xule for the subtraction of algebraic quantities. Now, surely, as the author of an Elementary Treatise, you ought to have previously demonstrated the grounds of this method of subtraction: passing over, however, this unpardonable omis»ion, I would ask, why thefirst term of your second line should be actually subtracted from the subtrahend, rather than put down after that subtrahend with its sign changed, in like manner as all the terms after thefirst in that line are put down? in which case, instead of the dot or • in the remainder, you would have had 1—1; to this, perhaps, you will answer that you would still have had the series 1—1+1 —1+1—1,' &c. for a remainder, which I also readily admit; but whnt, let me ask, would have been the result of the second part of your demonstration, where you attempt to shew that 1—2+1 is an infinitesimal; if, instead of actually subtracting the first term of jour second line from the same term of jour first, you had only put the former «fown with its sign changed after the latter? Would you not, in this case, instead of • 1—2+1+1—2+1+1—2, &c. have obtained 1—1+1—2+1 + 1— 2+1+1—2, &c. for a remainder? and how then, Sir, would you have shewn that thiYlatter series consisted of your
boasted infinitesimal 1—2+1? Again
if in the first example, instead of placing the subtrahend 1 over the first term of the second line, you had put it over any of the succeeding terms in the same line, as in the following instances, you could not have obtained the remainder *+l—1+1—1, &c. as may be seen on inspection;
• From i
Subtract 1—1+1—1 + 1, &c.
Remainder is ■—1+2—1 + 1—i, &c.
thus an infinite number of remainder* might be obtained from the infinite variety of positions in which the subtrahend might he placed, and any one of these I will affirm to he as correctly the lemainder as the one you have above given: and what indifferent person would not consider my affirmation as of equal weight with yours, till you have demonstrated that to obtain the true remainder it i» absolutely necessary that the subtrahend should be placed over thefirst term of the series to be subtracted. The remainder in your second example might be varied in a similar manner by putting the first term of the second line under the second, third, fourth, term, &c. of the first line; but you would not then obtain for a remainder a series which would be constituted of a repetition of your infinitesimal 1—2+1; unless, therefore, you can demonstrate that the true remainder can only be obtained by that purticulur position in which you have thought proper to place the subtrahend, and series to be subtracted, your fundamental proposition is, to use your own language,/afce, and the superstructure which you have raised upon it instantly falls to the ground'; or I should rather have said the temple erected by Wallis and Newton, which you haTe in vain attempted to demolish, still stands firm, unshaken, and immutable, upon the eternal and adamantine rock of science and truth.
In your corollaries to this proposition, you are pleased to osser* that the expressions 1—1, 1—2+1, etc. are " neither quantities, nor nothings;" that they are not quantities I am ready to allow, as numbers are rather the measures or representatives of quantities, than quantities themselves: but that they are not nothing I deny, and I will defy you to prove that they are something. The ingenious Bishop Berkeley very shrewdly asked, " Whether evanescent increments might not be called the ghosts of departed quantities:" what then, may I ask, shall your non-quantities be called, which are something yet neither quantity nor nothing i Surely these can only he the shadows of the ghosts of departed nothings!
Your second proposition is thus enunciated: "There cannot be a greater number of terms in any infinite series
&C than ——which is equal to 1+1+1+1,
ought certainly fo have terminated with
the fraction , and it should have
been the object of the demonstration to
prove that — was equal to the infinite
series 1 + i-f-i+l, &c. whereas, by adding to this part of the enunciation the words "which is equal to 1+1+1 + 1, &c. ad infinitum," the proposition is rendered identical, and means neither more nor less than that there cannot be a greater number of terms in any infinite series, than an infinite number of terms! or that the number of terms in an infinite series is infinite! Now, as it would be the height of folly for one moment to dispute the truth of this assertion, I will not dwelt on the demonstration which you have been pleased to give to this notable truth, but will only ask you, by what method you
. . . 1+1—1 . . 1—14-1 obtained —-—:— rather than ■
for the sum of 1 and ?you must
have obtained — by first expressing the sum of 1, and -—— by lr\j
and then by adding the product of the integer and denominator to the numerator, and placing their sum over the denominator, agreeably to the common rule for reducing a mixed number to an improper fraction ; but, suppose you had written the integer 1 before the fraction
■ ., thus IJ , and reduced this
mixed number as above, would you
. . , 1—14-1 not have obtained !_0rl + l+i+i
&c. instead of "** — or 14/2+ 1+1,
&c? Will you say that these resulting series 1+1 + 1+1, &c. and 1+2+1 + 1, &c.are equal, are the same, are identical f If not, ought you not to have proved by way of Lemma, previously to your entering upon the "Elements of the True Arithmetic," and as an indispensable requisite to understand even the first proposition of your work, that the number denoted by «, added to the number denoted by b, is not the same as the number demited by b, added to the number denoted by a, or that a+6 is not the Monthly liUo. No. 212.,
same with i+a? I must therefore, sir, push this question, why did you adopt the position of the integer 1 after the
fraction rather than befo-e. it? I
1—1 , J
anticipate your answer in these words,
"because this position, and this only,
would produce the result which I have
obtained.'' How necessary then wns it,
sir, I again repeat, that you should have
previously proved, that, in the addition
of numbers, a particular regard to their
position was essentially necessary to
obtain, I will not say their cot'ect sum,
for that I deny, but the conclusions which
you have deduced iw your propositions.
In your third proposition from among many curious specimens of your reason, ing, I will select the following. "For 1+1+1+1, &c. ad infin. is evidently equal to the last term of the series 1+2+3+4, &c. ad infin. For the sum of two of these terms, beginning from tiie first term, viz. 1+1 is equal to the second term of the series 1+2+3+4, &x. The sum of three of the terms, beginning from the first term, viz. 1+l-J-i, j* equal to 3, or the third term of the series. The sum of four of the terms is equal to 4, or the fourth term of the series, and so on; and therefore the sum 'of the infinite series 1 +1+1+1, &c. will be equal to the last term of the series 1+2+3+4, &c." 1 mean not, sir, to dispute the justness of this infer* nice, but can it be possible that you should have deduced such conclusions from such premises} You, who only a few pages before, in your preface, were vaunting in these words, "I rejoice to find as the result of this discovery, that it affords a most splendid instance of tht absurdity which may attend reasoning by induction from parts to wholes, orjrom wholes to parts, when the wholes are thtmselves infinite?" Have .you not here reasoned by induction from "parts to wholes," when the wholes are themselves infinite > And may it not be peremptorily demanded of you, "first to cast out the beam which is in thine own eye, that thou mayest see clearly to cast out the mote of thy brother's eye."
In your corollary to this proposition . 0+1 you say, "And —— is less than
TUT by TT^1'" Iie,e a6ain> sir» I will not stop to dispute your conclu. 2 $ siou,