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Thus, when we say a+x=b, we mean that a+r is the sum

of two numbers which is denoted by b: but, when

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1 -X

&c. then is only the mark of

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the series. We apprehend that it would be better to give to the sign its appropriate meaning, and to retain it strictly; which may be done, by simply adding to, or subtracting from the series an indefinite term, which is to be known from the

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On the binomial theorem, our author first lays down the rudiments of his own analysis; and, having given the history of its origin, he will startle many who have hitherto acquiesced in the usual demonstrations of this famous rule,

The precise object of the binomial theorem having been misunderstood, it is no matter of wonder, that the demonstrations given of it are inaccurate. The demonstrations alluded to, are conducted by combinations, by increments, by the multinomial theorem of De Moivre, by vanishing fractions, (Landen's method) and by the method of fluxions. All which demonstrations, without intending any disrespect to their learned or ingenious authors, I affirm to be imperfect. I stop not here to make good the assertion, it being my duty less to detect error, than to propose what may find reception as truth.'

P. 34. The author's proof is as follows:-D, D2, D3, D4, are notes of operations performed on other quantities. Thus Dam represents mam-1, D denoting the operation on the term a”, to produce the term following it in the binomial theorem, independently of r; namely, a is to be raised to a power less by unity, and then it is to be multiplied into them, the index of a. DD a signifies m Da-1, the first D denoting the preceding operation on am; and now, for the term m Da”—1, the same process is to be followed, when it will be seen that DD a represent m. m—1. am—2.

Let therefore, (a+r)"a"+ma"-1x+px2+9x3+ &c. (1),
or, =a”+D a”.x+px2+qx3+ &c. (2).
Increase by z, then (a+x+x)”=((a+z)+x)"=

(a+(x+2))"
)" and to expand ((a+2)+x)", in series (2) for a put
a+%, and in p, q, r, for a, put a+z, then since p, q, s, &c. are of
the form Na3, N, a numeral coefficient, when a is increased by z, the
expansion of (a‡z)' will be a'+5a=12+8.z2 +, &c. or a3 + Da
•+s‚2+xc. and consequently, p, q, r, &c. will become of the form
p+ Dp.z + &c. q + Dqz + &c.

bence,

((a+x)+x)" = (a+x)”+D(a+x)".x+(p+Dp .z+ &c.) x2 + &c. or a" + Da"%+p x2+qx3 + &c.

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+ Da" .x+D3a” „xz+Dp.x1z

+ &c. + &c.

To expand (a+(x+x))", in series (2) for x, x2, &c. put (x+2), (x+x)2, &c.

then (a+(x+x))"a"+Da” (x+x) + p(x2+2xx+ &c.) +q(x3+3x2%+ &c.)

Now, since the same operation is made on a+x+z, whatever is the operation that m designates, the two expansions or series must be identical, and two series are identical, when the coefficients of the terms involving the same powers of the arbitrary quantities are the

same.

hence, comparing 2pxz and D2a".xz, 2p must D3a"

; comparing 3 q.xx and Dp.xz, 3q=Dp=

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and if the coefficients of the terms affected with x, x" in the expansion of (a+x)" are N(-1) N(2)9

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D N(n-1),

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D-lam m(m-1) (m-2)... (m-n+2)

1.2.3..- 1

1.2.3...n- -1

m(m-1) (m-2) (m−3) ... (m−n+1)

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m. (m −1)

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1.2

or, expanding the coefficients by the

denote

(a+x)" = a+ma"-.x +
which theorem is known by the name of the binomial theorem.'

P. 24.

This proof is very ingenious; but, from the novelty of the terms, it may elude for a time even a careful mathematician: for which reason, we could have wished the author had gradually led his reader to the use of the notation, and shown him, from the instance a+x)2=a2+D ax2+pr2, how he may carry on this method with certainty to the higher powers. We may observe also, that, in comparing 2 px,

D'am

=p'a"xz, when it is found that p= - p is ascertained

to be equal to m.

m-1
2

2

a, because Dam.a"-1: but

then it is assumed, in the original series, that the second term is mar; an assumption which, though true, renders the demonstration less perfect.

From this demonstration, the writer proceeds to investigate the series for a*, which is expanded on similar principles; and he here makes the important and just observation, that the connexion between the theories of logarithms and curve lines is merely accidental and arbitrary.'

In fact, the arrangement of the truths of analytical science, such as history gives it, is very different from their logical and natural arrangement; and as, in the infancy of analysis, mathematicians were more solicitous to advance it, than to advance it by just and natural means, they frequently deviated into indirect and foreign demonstrations, and sometimes employed the geometrical method, with which they were well acquainted, to establish arithmetical and algebraical truths: the evil attending on this mode of procedure has been, that things in their nature totally independent have been thought to possess a real and necessary connexion, and, that the principles of a general method have been sought for, in some particular method, properly, that is according to the logical and natural order of ideas, to be comprehended under the general one. The properties of numbers established by means of the properties of extension, and the expansion of algebraic expressions by means of the properties of motion are curious facts in the history of science: a knowledge and examination of these facts would shew us the source of many confused notions, and conduct us to the solution of several paradoxes with which analytic science is supposed to be embarrassed.' P. 40.

From these, comparatively speaking, simple instances, we advance to the more complicated and embarrassed forms, through which our limits do not permit us to follow our author in so abstruse a subject.

The expressions, x, ax, 1.x are called functions of x, and are thus symbolically represented, by Fx, or x, orf x, or x, &c. and consequently F(x+Ax), or 4(x+Ax, or f(x+Ax), or, ↓(x+Ax), &c. mean either (x+4x)", or ax+ax, or 1(x+ax) ((x+Ax)= e'(x+4x)). Now the symbol D has been hitherto used only to denote that operation by which the coefficient of the second term in the expansion of (a+x)", or (x+Ax)≈ is formed. Let it now be made to denote the operation made on x”, ax, 1 .x, &c. by which the coefficient of the second term, in the expansions of (x+Qx)TM, ar+Ax, extax, L(x+Ax) (x+Ax=al(x+^x)) is formed, then under this signification, its former one will be comprehended, but it will no longer denote a similar operation, as when restricted to expressions such as "; P,

however, when applied to fx or x, &c. generally represents the second term of the series that arises from expanding fx or x, when for x, x+I is substituted: that second term can always be known, since all the expressions which ƒ (x+1), or (z+1) is made generally to represent have been previously expanded,

thus Dxmxm-1. P.42,

This leads us to the doctrine of fluxions; and the nature of vanishing fractions is treated with great propriety.

These vanishing fractions have caused many discussions amongst mathematicians; they have caused many false reasonings. It was not perceived that, to assign the value of (ra), there was an

xm-am

xn-an

absolute necessity of some definition, convention, or extension. The notion of an inherent signification, and of an essential value, belonging

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selves on the clearness of their apprehension, and the justness of their inferences.

The method of limits, or of prime and ultimate ratios, Landen's fx method, the method of finding the value of (fx, Fx 0) are Fx

related methods: they all demand the same arbitrary assumption, which has never been expressly made; they are all equally subject to an objection, which has never been satisfactorily removed.' P.71.

The contact of curves is then examined, and the mode of drawing tangents, and finding the radii of curvature, explained. On the latter subject, the following remark occurs, which will scarcely be thought satisfactory.

It is needless to give examples, since, in these cases, examples do not illustrate the theory: if given, they ought to be considered only as exercises by which the dexterity of the student is to be improved: they merely require the value of an analytical expression to be found, for which purpose rules have been previously laid down.'

P. 181.

This may be true: but experience informs us, that gene ral truths are fixed in our minds by examples; and that, if they be left to their own naked energy, their influence will be found, comparatively speaking, very small. The work before us is an instance in point: it will exercise the talents of the higher mathematicians, and will be neglected by all, even at Cambridge, who have not found a place among the first six on the first tripos.

Yet, though we apprehend the work doomed to have but few readers, we do not the less recommend it to those whose duty it is to give instruction on these subjects. They may bence discover the insufficiency of many modes of reasoning in which they have placed implicit confidence: their views will

be enlarged; and they may be profitably employed in bring ing Mr. Woodhouse's calculations down to the level of the humbler class of mathematicians. When they are disembarrassed from the novelty of their terms, and the terrors of their series when a few simple instances are added, by which a learner may catch the spirit of the general demonstrationwhen the whole is simplified and stated, so as to create a greater interest-the merit of this acute writer will be more generally acknowledged, and he will be deemed highly worthy of the patronage of the university by which the work has been printed.

ART. VII. - An Enquiry into the Necessity, Nature, and Evidences, of revealed Religion; by Thomas Robinson, A.M. &c. 8vo. 6s. Boards. Baldwin. 1803.

THE intent of this work is to show the necessity, nature, and evidences, of revelation, in a plain and popular manner, and to point out the incompetency of reason as a religious instructor. There are two ways of discovering the will of God--by reason and by revelation; and the state of the heathen nations, unenlightened by revelation, is brought before us, to show how inadequate a guide is the former in subjects of religion; while the confession of their philosophers teaches us that a divine revelation was even by themselves thought possible, probable, and even necessary. Such a revelation we have in the Bible, of which an account is given in the work before us, beginning with the five books of Moses, whose genuineness and authenticity are proved; while many very judicious observations are advanced on the three branches of the Mosaic law, the moral, ceremonial, and political. If the arguments be demonstrative on the genuineness and authenticity of these books, of which, notwithstanding some late cavils, we entertain no doubt, their divine authority is proved, as the author justly observes, by so strong and decisive a body of evidence, as cannot fail to remove every reasonable doubt, and satisfy every candid and unprejudiced inquirer.' Having brought those arguments before the reader, which are derived from the character, the miracles, and the prophecies, of Moses, which evidence his divine mission, the author next takes a summary view of the ensuing books, written by the prophets and other holy men, applying to them the same mode of proof which he has so successfully employed with respect to the Pentateuch; and con cluding his discussion on the Old Testament with an inquiry into its canon, and pointing out the alterations which have been made, and explanations given of certain passages by

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