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they can never be equal to each other. Yet Newton, in his future reasonings, finding that the limits to which these quan. tities are approaching would answer his purpose, expresses his intentions by the quaint and incongruous term, ultimately equal.
It was natural that this innovation should excite, among the lovers of perspicuity, a great deal of disgust; and this was increased, when they found that it led them to what they conceived to be a still more perplexed subject-the doctrine of Auxions. The subtle notion of velocity was introduced into every question: quantity was divided into mi. nute parts; and some were rejected, others retained, appa- rently to those not acquainted with the new doctrines, at random. Hence the whole of this new science appeared, to a very amiable and intelligible writer, a complete tissue of sopbistry; and he attacked it with a vigour which the mathematicians in vain endeavoured to elude. They were, however, convinced that the conclusions obtained by their new science were accurate; and, little anxious about first principles, they left the good bishop Berkely in possession of his post, and pursued their investigations, though their defects,' as our author justly observes, ‘had in metaphysics and logic been clearly made out.' Some, however, since sir Isaac's time, have endeavoured to give that perspicuity to the doctrine of fiuxions, which would have been necessary to satisfy the rigid notions of the ancient geometricians: the # attempt does them credit; and their failure ought not to deter others; for 'mental satisfaction and improvement are more worthy objects, than simple rules and most compendious prucesses' in computation.
The intention, then, of our author is to conduct us, on the principles of true reasoning, through our analytic calculations; to separate what is strictly true, from mere articles of convention; and to show us where the mind can rest with satisfaction in the discoveries of the moderns. This task he has performed with much ingenuity ; but his technical language, together with the novelty and abstruseness of his discussions, will deter numbers from pursuing the chain of his reasonings; and they will rather acquiesce in the ipse dirit of Newton, than go through a laborious mental investigation, which may shake their faith in their former systems. We will point out a few instances in which the author detects the fal. lacy of modern reasonings.
The nature of positive and negative quantities is the first object that meets us in this work, arising from the explanation of the algebraic signs. The latter are allowed to have a place; but not because any proof can be given of the effects of addition, &c. upon them, but' for the sake of com.
modiousness in calculation ;' and all the rules relative to them are partly arbitrary, and suppose some previous convention. On this subject he is as much at variance with those who support as those who reject these quant ties: the former will not be pleased that he has destroyed all their proofs; the latter will think him inconsistent with himself, and retort upon him the passage we have just quoted, that mental satisfaction and improvement are more worthy objects, than simple rules and most compendious processes in computation. On this arduous subject, however, we will leave the author to speak for himself.
It has been already observed that, if negative quantities are made the object of demonstration, it must be in consequence of some arbitrary rule. The rule for transposition introduces negative quanti. ties, and leads to equations of no direct meaning; the rule for the multiplication of signs likewise introduces them, for although in all real questions, it can be proved that the rule leads to true results, and therefore, for commochousness is made general;' yet, in the reverse operation of extraction, the consequence of such a rule may be equations, considered separately from questions to which they belong, of no direct meaning ; thus if x=2?, * must be put + a, since according to the rule -ax-a gives a? as well as tax ta.
- The notions that have been sometimes formed of negative quantities are as faulty, as the methods by which rules for their multiplication have been proved; they have been considered as quantities less than o, or their nature has been attempted to be explained, by illus. tration drawn from mercantile transactions, or from the properties of geometrical figures; the first notion is manifestly an absurd one, the second merely illustrative and proves nothing; besides, it in some sort begs the question, for, that, a positive quantity representing a line drawn in one direction, a negative quantity must be used to denote a line drawn in an opposite direction, is by no means a self-evident truth, and is in fact a consequence of the definition, by which the application, of algebra to geometry is made, and of certain properties of negative symbols previously established by the rules of algebra.
• The method of proving the rule for the multiplication of signs, by multiplying (a- a) into I b contains a manifest fallacy, for it is essential to the proof, by which (a-a) Ibis made equal to ab F be, that, a-a should be a positive quantity: it cannot at once be put 50 and employed in reasoning as a quantity; or if first considered as a quantity and afterwards put o, the hypothesis is shifted, and the previous result must be abandoned.
· The method of proving the rule, by reasoning on the signification of the word subtraction, is not more satisfactory; for, to subtract a negative quantity, does not necessarily mean to add a positive orie; if the phrases are equivalent, they must be made so by definition; there is a wide difference, between what is agreeable to the analogy of language, and what is admissible in strict demonstration.' P.7.
The author makes a just distinction on the meaning of the sign =, which is confounded by most mathematicians.
Thus, when we say a +x=b, we mean that a + x is the sum of two numbers which is denoted by b: but, when =l+x+32+x, ,.,,. &c. then = is only the mark of operation, and shows that being expanded, produces 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 nature of the equation, Thus -- = 1+x+xat.x3.....
+2. In this case,
i 1-X 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, D?, D", D4, are notes of operations períormed on other quantities. Thus Dam represents mam-, D denoting the operation on the term am, 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 the m, the index of a. DD an signifies m Dam->, the first denoting the preceding operation on am; and now, for the term m dana, the same process is to be followed, when it will be seen that DD am represent m. m-1.am-2. "Let therefore, (a +r)"=2" + mam-ix++2+qx} + &c. (1),
or, =a" +0 a".*-+ px? +973 + &c. (2). Increase x by Z, then (a +*+z)"=((a+z; +*)"= (a +(x+e))" and to expand ((a+z)++)", in series (2) for a put atz, and in 8, 9,1, for a, put a +%, then since p, q, s, &c. are of the form xa', N, a numeral coefficient, when a is increased by Z, the expansion of (a+z)'will be a' tsiz+.za +, &c. or a + Dar mts.z. and consequently, P,9,1, &c. will become of the form ptop.x+ &c. 9+Dqz+ &c.
hente, ((a+x) +*)" = (a +z)* +Da+z)".*+(P+0p.z+ &c.) *2 + &c. or a" + Da"Z+px? +4*3+ &c. Ot Da-x+0°4" „z+DP.*?%
+ &c. + &c. • To expand (@+(*+z))", in series (2) for *, *?, &c. put (57%), (x+z)?, &c.
then (a +(x+2))"-" +D a" (x+x) + P(x2+2xz+ &c.). +q (+3+3x2++ &c.)
i Now, since the same operation is made on a +*+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 2p*and d’a" .22, 2p must=Doa”, and p= Doc"
D.D." Dan y comparing 39.*z and Dp.**2, 39=Dp=
in like manner 4 r=Dg and r=
220.127.116.11 and if the coefficients of the terms affected with x*-*, *" in the ex. pansion of (a +x)" are N, -, NENO
DN(n-1) then D N (-1)= n N (n) and Nw= =.*),
I n - D?!4 _m(m-1)(n-2) ... (m-m+2). and since Ninj
1 1 sta m utlu _m(m— 1) (m— 2) (m-3) ... (m-n+1)
1.2.3 + &c.
economi a hence (a + x)" =a" + Dan .x+
, d’a”., D3a".x) or, expanding the coefficients by the operations that D, D*, &c. Do, denote (@+x)" = a" +man-4.*+ *012
' T"" " T 1.2 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)=a’+D ara+pra, how he may carry on this method with certainty to the higher powers. We may observe also, that, in comparing 2 p.x,
to be equal to m." an, because da" = m.2*l: but then it is assumed, in the original series, that the second term is m am-1x; an assumption which, though true, renders the demonstration less perfect.
From this demonstration, the writer proceeds to investigate the series for ax, 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 con. duct us to the solution of several paradoxes with which analytic sçi.. ence 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, xw, ax, l.x are called functions of x, and are thus symbolically represented, by Fx, or ® *, or f x, or * *, &c. and consequently Fix+ar), or Q(x+ax, or f(x + ar), or, 4(x+ax), &c. mean either (r+ ax)", or ax+4x, or 1(*+^x) ((*+^x)= 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 ex. pansion of (a +*)m, or (x+Ax)is formed. Let it now be made to denote the operation made on *", ax, 1.4, &c. by which the coefficient of the second term, in the expansions of (x + Ax), artar, er+ax, L(*-- Ax) (*-*Ax-al(x+4x)) 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 **; ,