that they will not throw much lustre on the philosophy of the Stagirite, whatever proof of his deductive powers they may

afford.

In page 42, speaking of the translation, Mr. Taylor says that he has given, as nearly as possible, the literal meaning of every sentence, without paraphrasing what might be the sense. of the author, or expanding what might appear too concise. Superior to the fleeting and contemptible applause of the present day, he has translated and explained as Zeuxis painted; and he expects that posterity will award him his meed of never dying fame.

On the mention of the two Greek interpreters of Aristotle, Alexander Aphrodisiensis and Syrianus, the translator re-commences his attack on Dr. Gillies in very severe terms. In particular, he comments on an opinion of that gentleman concerning the real subject of Aristotle's Metaphysics; and he' endeavours to shew that so far was Aristotle from opposing the doctrines of the Polytheists, (or men who believed in the existence of divine natures, the immediate progeny of one first, cause,) that in the 8th chapter of the 12th book he demonstrates the existence of these divine natures. To the extract from Aristotle, he adds the authority of Maximus Tyrius, and then concludes with Dr. Gillies; asserting that he has been induced to be thus severe on the Doctor by no personal enmity whatever, but by a sincere love of truth.

Here ends the censure on Dr. Gillies, but here cease not the anger and hostility of Mr. Taylor: the critics, the merciless critics themselves, are next subjected to his lash. In deciding on a work, their situation, and the nature of their duties, must necessarily render them regardless of every consideration except that of its merit: but if there be critics who would not feel emotions of deep commiseration, when they hear that the present translation has been effected amidst the pressure of want, and the languor and weakness occasioned by continual disease,' we know none such: far from us, and far from our friends, be that indifference which views distressed learning without sympathy. This suffering in the present instance has been at length relieved; and we read with pleasure the translator's testimony of gratitude towards two gentlemen, his friends and patrons, Messrs. William and George Meredith.At the close of the Introduction, we are also informed that an English translation of Plato's works is soon to appear; and that for this the public will be partly indebted to the aid which nobleman of high rank' (the Duke of Norfolk, we understand,) has generously offered to Mr. Taylor.

6 a

Q3

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To proceed in due order, we should now enter on a discussion of the merits of Mr. Taylor's translation, and of his copious notes: but we have already given our opinion respecting the work itself, and the consequent nature of his undertaking; his powers as a translator are sufficiently known; and of his notes we can truly say that to us they have not made

"all things plain and clear." (Hudibras.)

We shall therefore refrain from any minute or elaborate criticism on this part of the present volume, and proceed to the subjoined original tract by Mr. T.-his Dissertation on nullities and diverging series. Here we hoped that, in a mathematical dissertation, the author would have imitated the prudence of mathematicians, by premising definitions from which deductions might be strictly made: but the cause of obscurity in the foregoing translation operates alike through this paper; and we have still to complain of the "verborum præstigia et incantationes."

Mr. Taylor begins by observing that mathematicians, unable to lay down any clear and intelligible doctrine with regard to nullities, have never, in their speculations concerning them, suspected that they are in reality infinitely small quantities, and that they have a subsistence prior to number and even to the monad itself. To prove this point is one object of the dissertation; and another is to shew the errors of mathematicians respecting neutral and diverging series. A neutral series is one which neither converges nor diverges; such is 1-1+1 -1, and this series Euler affirmed to be, because, said he, if we stop at -1, the series gives o, and if we finish by +1, it gives 1. But this is precisely what solves the difficulty, for, since we must go on to infinity, without stopping either at 1 or at +1, it is evident that the sum can be neither o nor 1, but that this result must be between these two, and therefore be="-a seemingly refined reason, but a very weak and unsatisfactory one; since, as Mr. T. justly observes, this neutral series is equally the result of the developement of

I

In the next observation of Mr. Taylor, and on which he seems to plume himself, we do not perceive much that is worthy of commendation. A mathematician asserting that I divided by +1 produced 1-1+1−1, &c. must mean at the same time to assert that 1—1+1—1, &c. multiplied by 1+1, was 1: for there is no independent method of prov

* Bacon de Augmentis Scientiarum, p. 155.

ing that divided by 1+1 produces the neutral series above mentioned; and the first proof adopted would be the multiplication of the quotient by the divisor. This proof, however, is unsatisfactory; or, to speak exactly, it is not founded on the original properties of number, but is merely a consequence of an operation performed after a certain rule.-The author's remark concerning 1-1, viz. that it is not the same as o, is true in a certain sense: but he would have spoken with greater accuracy if he had said, not always, or not in all situations. In all reasonings concerning real arithmetical equalities, 1-1, and o, are the same: but, when 1-1, or generally a-b, are employed as symbols, their significancy depends on the order and position in which they are placed.

In article 8th, Mr. T. mentions what he calls remarkable properties of nullities; viz. that I-1, continually involved into itself, produces 1-2+1, 1-3+3-1, &c. and that, when is divided by these nullities, the series 1+1+1, &c. 1+2+3+, &c., 1+3+6+, &c. are produced but these properties, or particular deductions from the binomial theorem, are surely no new discoveries: they are easily and generally proved, thus:

The 9th article, generally and simply stated, is this:

In the 10th, Mr. T., after having observed that

says;

I

2

-, &c. = ( ; ——, ) ′′ = { 1 + 1 + 1, &c. Z nj

It may here be necessary to observe that it is not possible to conceive more than three kinds of the actual infinite; viz. the infinite in power, in magnitude, and in number. The infinite in power is that which subsists in divinity; in magnitude the actual infinite has no subsistence whatever; and in number it has partly a subsistence, and partly not: for it does not subsist collectively, or at once, but according to a part, or, in other words, according to the power of receiving an additional number beyond any assignable number. Hence one infinite series may be greater than another, because the terms in the one are continually greater than the terms in the other. That is to say, the one has the power of continually supplying greater terms than the other: not that the aggregate of one infinite

Q 4

series is greater than that of another when the terms are actually infinite; for this is impossible, because there can be no numerical infinite with an aggregate subsistence; but when one series continued to infinity is greater than another, the terms in it are infinite only in capacity. Modern mathematicians, not attending to this distinction, have had no clear conception of the nature of the mathematical infinite, considered as having an actual subsistence.'

a-a -= a

In the 14th article, some inaccurate reasonings of Emerson concerning infinites are corrected. The doctrine of infinites of the first, second, third order, &c. is well known to the wrangling questionists of Cambridge.-In the 15th, it is said that nullities are infinitely small quantities: for, observes the author, let a represent any finite quantity: then, if a be divided by the infinite quantity, the quotient will be 1-1: but here Mr. T. paralogises: for, in order that a may be capable of being divided into parts, the divisor must not be ∞ or -; and if the divisor be not co, then 1-1 is not o, or may be called an infinitely small quantity: therefore, what is proved is in fact supposed in the premises. If 1-1 be exactly o, then it is impossible to prove, by any independent arguments, that a divided by gives 1—1.—: it gives 1—1 because a certain rule demonstrated for real quantities is followed. Hence 2-2, 3-3, cannot with any propriety of language be called parts of 2, 3, &c.

a

a

In article 16, the author says that nullities multiplied by nullities are diminished, which is a property directly contrary to the nature of numbers, and evinces that they are essentially

different from quantities:' but is not 1/x/=/11/30

I

I

generally 10

10" 102n

9

or

In article 21, it is observed that the quotient of any number divided by a nullity is different from the quotient of the same number when distributed into unities, and divided by the same nullity. This is true: but there is nothing either wonderful or paradoxical in it; since the quotient here means only a series of numbers produced by operating in a certain manner with numbers arranged in a certain order; which quotient must be different, if either the manner of operation or the arrangement of numbers be altered.

In chapter 2, we meet with many observations that are true in themselves, although not in the sense in which Mr. T. means them to be true: for instance, he observes that 1-3 is not the

I

same as 2 since 1-3=1+3+32+3', &c. which infinite series cannot ; and this is true, because 1-3 is here used symbolically, and its significancy as a symbol depends on the permanence of the arrangement of its cyphers. This observation, we think, is sufficient to render baseless the towering reasongs of Mr. Taylor concerning the nonquantative subsistence of infinitely small quantities, and the negations of infinite multitude.

Chapter 3. contains reasonings similar to those of the two former. Its special object is to shew that in continued quantity there are points, and that there is a threefold order of points, viz. linear, superficial, and solid: but we do not stop here to examine the deductions and conclusions of the author: since the first steps from which he sets out are neither selfevident nor proved, and he continually assumes, as a property proved on independent principles, that which is merely a result of calculation conducted according to a certain form. We shall be at any time ready to give attention to Mr. T.'s system concerning nullities, &c. when, independently of a given method and form for the division of algebraic quantities, he can prove that a divided by is a-a, or that a divided by

a

The dissertation ends with chapter 4; in which Mr. T. attempts to shew that infinitely small quantities are admirable images of the To v, or The One, of the Pythagoreans and Plato, concerning which so much is said by Aristotle in the 13th and 14th books of his Metaphysics; and that they beautifully illustrate some of the most profound dogmas of antient theology. Part of the proof is this: infinitely small quantities are negations of infinite multitude; and a negation of all multitude is that which characterises the one, as is evident from the first hypothesis of the Parmenides of Plato; as all finite quantities likewise may be considered as consisting of infinite series of infinitely small quantities, it follows that infinite negations of multitude may be said to constitute all finite quantity.' We are then presented with an extract from Proclus's commentary on the Parmenides; so beautiful, according to Mr. T., that no apology is needed for its length: but so obscure and unintelligible, according to the Monthly Reviewers, that no apology is required for its non-examination in their work. We do not, indeed, clearly understand the part of the proof which we have quoted: but the dealers in hard words and dark notions may apprehend what our sentiments are, when we applaud the

spirit