Abbildungen der Seite
PDF
EPUB
[blocks in formation]

YOU

Norwich, March 4, 1811.

OUR "Elements of the True Arithmetic of Infinites," having accidentally fallen into my hands, I was anxious to see in what manner you had treated this subject, not only on account of your professed admiration of the sci'entific 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 your performance: I therefore eagerly applied to your book with the confident expectation that I should be made ac. quainted with the true nature of infinitesimals," and that I should find that you 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 all the propositions in Dr. Wallis's Arith metic of Infinites, and also the principles of the Doctrine of Fluxions, to be fulse," I think it but right to convince others, or at least to attempt to convince them, that, however just your pretensions may

1

be to an accurate knowledge of the an cient 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: I am ready to grant your three first postulates, though I cannot help remarking, that, in a work abounding with so many preten sions 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 from 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 re ality, 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 the 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 by 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 is by collecting its terms into one sum according to their signs? Now if you had to multiply the series 1+1+1+1, &c. ad infinitum by 1-1, since you have asserted in the corollaries to your first proposition that 1-1 is that “which is neither quantity nor nothing, but which is something belonging to number with out being number." You would thus have to add the infinite series 1+1+1+1, &c. to itself, as many times as are de noted by that" which is neither quantity nor nothing, but which is something be longing to number without being num ber." In like manner, 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" au infinite number of times; and this sum Leing 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 a) you would have an infinite number added to itself "neither quantity nor nothing" times, equal to "neither quantity nor nothing" added to itself an infinite number of tines! I know not, Mr. Taylore 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 in stance 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 at some conclusion which may be applicable to practical purposes, neglects that rigid accuracy of demonstration, which may called the impregnable fortress of the mathematical science, and for which the genius of ancient mathematicians was so pre-eminently distinguished."

be

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 specimen of the accuracy of your logic. "Proposition 1. "1-1 is an infinitesimal, or infinitely Sinall part of the fraction

1 1+1' infinite series of 1-1 is equal to

and an

1

1+1

In like manner, also, 1-2+1 is an infinitely small part of

1-1-1+2-1-1+2-1-1+2-1-1&c 1+1

ad infin, and an infinite series of 1-2+1 is equal to

1-1-1-2-1-1+2-1-1, &c. 1+1

And 1-2 is an infinitely small part 1-1-1-1-1, &c.

of

ad infin. and an 1+1 infinite series of 1-2 is equal to 1-1-1-1--1-1, &c.

From 1

Subtract 1-1+1-141-1+1—1, &c. Remainder+1−1+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 to 1. The series 1-1+1-1+1-1, &c. is therefore equal to and consequently 1-1 is an infinitesimal. For 1+1 it cannot be 0, since an infinite series of O, added to an infinite series of 0, can never be equal to 1.

"In like manner,

1

If from 1-1-1+2-1-1+2—1—1, &c. Subtract 1-2+1+1-2+1+1−2+1, &c. 1-2+1+1-2+1+1−2, &c.

Remaind.

and therefore 1-2+1 is an infinitesimal;

and so of the rest.

"Corol. 1. Hence such expressions quantities nor nothings, but they are as 1-1, 1-2+1, 1—2, &c. are neither something belonging to number, without being number; just as a point, which is longing to, without being a line. the extremity of a line, is something be

"Corol. 2.

Hence, likewise such

expressions when they are considered as parts of infinite series, are not to be they are expressed, viz. 1—1, for instance taken separate from the terms by which is not to be considered as a subtraction of 1 from 1; for, in this case, it woul be 0. Nor is 1-2 to be considered as a subtraction of 2 from 1, since it would 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 modern 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 part" which follow were designed to supply this deficiency. Your demonstration, I presume, begins at the word "From ;"-if so, let me ask you by what Thus, too, 1-3 means you obtained the remainder 1—2—2—2—2,&c. +1-1+1-1+1-1+1-1, &c.? is the infinitesimal of Your answer must certainly be, that you 1+1 actually subtracted the first term of the 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 term) of the first line or subtrahend, and that

1-4 of

others,

1+1

2

-"

you

1

[ocr errors]

you called the remainder 0 or nothing, or
rather dat or, to which you annexed
the other terms of your second line, or
number to be subtracted, with their
signs changed, agreeably to the common
rule 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-
sion, I would ask, why the first term of
your second line should be actually sub-
tracted from the subtrahend, rather than
put down after that subtrahend with its
sign changed, in like manner as all the
terms after the first in that line are put
down? in which case, instead of the dot
ог 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 re-
mainder, which I also readily admit; but
what, let me ask, would have been the
result of the second part of your demon-
stration, where you attempt to shew that
1-2+1 is an infinitesimal; if, instead
of actually subtracting the first term of
your second line from the same term of
your first, you had only put the former
down 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 this latter 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
:+1—1+1−1, &c. as may be seen on
inspection:
From
Subtract

1

1-1+1-1+1, &c.

Remainder is —1+2—1+1−1, &c.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

thus an infinite number of remainders might be obtained from the infinite variety of positions in which the subtrahend might be placed, and any one of these I will affirm to be as correctly the re mainder 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 is absolutely necessary that the subtrahend should be placed over the first 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 particular 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, false, and the superstruc ture which you have raised upon it in stantly falls to the ground; or I should rather have said the temple erected by Wallis and Newton, which you have 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 assert that the expres sions 1-1, 1-2+1, &c. 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? Surely these can only be the shadows of the ghosts of departed nothings!

Your second proposition is thus enun ciated: There cannot be a greater number of terms in any infinite series.

[blocks in formation]

1-1

ought certainly to 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+1+1+1, &c. whereas, by ad-
ding 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 infi-
nite 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 dwell on the
demonstration which you have been
pleased to give to this notable truth, but
will only ask you, by what method you
obtained
1+1-1
1-1+1,

rather than

1-1 for the sum of 1 and

?

have obtained

1+1-1 1-1 sing the sum of 1, and

1-1

1-1

you must

1

by 금

[blocks in formation]

fraction rather than before it? I anticipate your answer in these words, "because this position, and this only, would produce the result which I have obtained." How necessary then was 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 correct sum, for that I deny, but the conclusions which you have deduced in your propositions.

"For

In your third proposition from among many curious specimens of your reason. ing, I will select the following. 1+1+1+1, &c. ad infin. is evidently equal to the last term of the series of two of these terms, beginning from the 1+2+3+4, &c. ad infin. For the sum first term, viz. 1+1 is equal to the second term of the series 1+2+3+4, &c. The sum of three of the terms, beginning from the first term, viz. 1+1+1, is equal to 3, or the third term of the seby first expres- ries. The sum of four of the terms is equal to 4, or the fourth term of the series, and so on; and therefore the sun will be equal to the last term of the of the infinite series 1+1+1+1, &c. series 1+2+3+4, &c." I mean not, sir, to dispute the justness of this infer should have deduced such conclusions ence, but can it be possible that you from such premises? You, who only a few pages before, in your preface, were find as the result of this discovery, that it vaunting in these words, "I rejoice to absurdity which may attend reasoning by affords a most splendid instance of the induction from parts to wholes, or from wholes to parts, when the wholes are themselves infinite?" Have you not here reawholes," when the wholes are themselves soned by induction from " infinite? And may it not be perempto rily 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."

and then by adding the product of the integer and denominator to the nume rator, 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

1 thus 1+

1-1'

and reduced this

mixed number as above, would you
1-1+1
For1+1+1+1

not have obtained

[merged small][ocr errors]

1-1 1+1-1 1-1

or 1+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? If not, ought you not to have proved by way of Lemma, previously to your enter ing upon the Elements of the True Arithmetic," and as an indispensable requisite to understand even the first proposition of work, that the num- you say, your ber denoted by a, added to the number denoted by b, is not the same as the 1 number denoted by b, added to the number denoted by a, or that a+b is not the MONTHLY MAG. No, 212.,

[blocks in formation]
[blocks in formation]

1

= ; see therefore to what mon1+1 strous absurdities you are led by your own accurate reasoning in the "Elements of your True Arithmetic."

What I have said will, I think, be amply sufficient to shew the fallacy of your reasoning, in this your work of boasted accuracy. I cannot, however, refrain from adding a few remarks on your ninth proposition, which is thus enunciated. "Numbers .connected together by an affirmative or negative sign, are dif. ferent from the same numbers when actually added together, or subtracted, and expressed by one number!" Had this proposition been promulgated by any ordinary person, I should doubtless have considered it as the effect of folly or madness; but, as procceding from one who holds a respectable rank in the republic of letters, I would willingly attribute it to some other cause. Sin. gularly strange and ridiculous as this proposition must appear, its demonstration, however, is, if possible, still more absurd; it begins by stating that, "1+1 is not the same as 2; for 1+1 subtracted from 2 leaves the infini

unal 1-1." How, sir, allow me to

ask you, in taking 1+1 from 2, did you obtain the remainder 1-1? You have carefully concealed the modus operandi, for, if you had not, the absurdity of your attempt at demonstration would have been most glaring; since you could only obtain the remainder 1-1 by actually subtracting the first term of 1+1 from 2, and by only denoting the subtraction of the lust term, by putting it down with the minus sign or prefixed: now, upon what principle could you, in subtracting a number which consists of two parts, or members, from another number, actually

you

subtract the first part, or member, and only denote the subtraction of the other part, or member, by connecting it with the sign -or minus, with the result of the actual subtraction of the first member; when, in the very words of your proposition, you assert, that nunibers connected together by a negative sign, are different from the same numbers when actually subtracted and expressed by one number? What a “splendid instance,” have here exhibited of the accuracy of your reasoning! What, sir, in future will be thought of Thomas Taylor, the Platonist? Of Thomas Taylor, the translator of Proclus on Euclid? Of Thomas Taylor, the admirer of Grecian geometry? Of Thomas Taylor, who boasts himself the vindicator of the very scientific accuracy of the ancients? Of Thomas Taylor, who, in the Elements of his True Arithmetic, reasons thus: "1+1 is not the same as 2; but sir Isaac Newton, in all his researches, both mathematical and philosophical, reasoned on the supposition that 1+1 was the same as 2; therefore, the results of sir Isaac Newton's researches are a mass of errors and fulsehoods; and Newton himself, was not only a man of a “ rambling and precipitate genius, but a perpetual blunderer?" I am fully aware, that, in answer to these questions, you will say, that you are perfectly indifferent to the opinions of others, both with respect to yourself and your works; for that you "have long since learnt, from the school of Pythagoras, that the praise or reprehension of the stupid, is alike ridiculous." Highly as I applaud this truly philosophical indifference, yet I must say, that, however regardless you may be of your own reputation, you ought at least to possess some little respect for the reputation of those ancients, whom you so frequently and so ardently profess to venerate and admire. I cutreat you, therefore

« ZurückWeiter »