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the calcareous crusts of which were inflated with bubbles, so as to form a cancellated shell, in texture like pumice-stone. The most durable substances of animal bodies, such as the bones and teeth, are only partly vascular, since their calcareous materials are fixed by chemical precipitants, and remain under chemical laws. Injuries done to the horns of cattle, to the hoofs of animals, and to human nails, are never restored; these parts do not possess the power of self repair; and it is only by the mechanical wearing away, that such injuries are obliterated. Indeed the beneficent construction of animal nature is sufficiently manifested in the insensibility of all the exuvial coverings, and in the organic composition of many parts which are exposed to mechanical attrition, as the enamel of teeth, the horny beaks of birds, and the cuticular or horny coverings of feet. The same beneficence appears to be extended to many parts of the internal organic substances, by which painful sensations are obviated, whilst the substances themselves being less directly under the dominion of the vital superintendency become more permanent; such parts are the tendons, ligaments, cartilages, cellular tissue, the gelatine and lime of bones; even water is an essential constituent of the animal fluids, and affords the necessary. softness and flexibility to solids. But this subject, and its connexion with the vegetable composition and texture, extends far beyond the limits of a memoir; and I must therefore suspend my observations. (To be continued.)

ARTICLE IV.

Further Observations on Fluxions. By Alexander Christison, Esq F.R.S.E. Professor of Humanity in the University of Edinburgh.

MY DEAR SIR,

(To Dr. Thomson.)

Edinburgh, July 19, 1815.

AN experienced mathematician will find no difficulty in the reasoning, Annals of Philosophy, vol. v. pp. 330 and 331; a learner, however, will understand that reasoning better if he suppose the accent, which is put after the y at the top, to be put not at the top, but half way down the side of the y'in p. 330, line 40; and likewise wherever that letter so occurs afterwards with one or two accents, unless there be two letters in the numerator; and if he read i for i after the mark of equality in the last line but one of p. 330, and in the second line of p. 331.

You may insert the following observations.

It is evident from fig. 2, p. 328, that the ratio of the increments is never the ratio of the fluxions; for at F H, 5 minus one centillionth to 1 is too small, and 5 plus one centillionth to 1 is too

great. Newton's expression, therefore, "The ultimate ratio of the increments is the ratio of the fluxions," is incorrect, and seems to have misled the Bishop of Cloyne. If a man is not a soldier, he may be the last of the men in a train, but, in that train, he cannot be the last of the soldiers. Newton, therefore, must be understood rationally, not literally; the literal interpretation, indeed, is im possible. In Milton too, the literal interpretation of "The fairest of her daughters, Eve," is also impossible. Such incorrectness of expression is frequently found in Robins. I do not remember that Maclaurin has corrected it till article 505, in the second volume of Fluxions. Maseres has rectified it more directly in p. 21 of the preface to the fifth volume of the Logarithmic Writers. Euler has fallen into the same mistake in his Definition of the Differential Calculus, in p. 8 of the preface.

I am inclined to think that, in p. 468, Harvey's idea of developing generated quantities is better than mine of generating them. It was to avoid the idea of motion that, in the demonstration, which I think is new, I employed bisection like the ancients. I might have avoided the idea of motion in the solution too; for I might have solved as Lacroix does in the beginning of his Calcul in 8vo. As the fluxional calculus was derived from the celebrated problem of the tangents, I think that the easiest and shortest demonstration is to be obtained from the same source. I consider such a demonstration as an extension of Descartes' application of algebra to geometry. I think that no rigorous demonstration of the fluxional problem purely algebraical can be so short as that in pp. 330 and 331; it occupies no more than twelve entire lines, as it properly begins at line 33, p. 330, and ends at line 8, p. 331; for, in order to prove that the limits of a variable quantity are equal, I might have referred to Robins, vol. ii. p. 56, art. 120; or to Lacroix Calcul, vol. i. p. 18. D'Alembert observes, that all the differential calculus may be referred to the problem of the tangents.

Without the aid of a diagram, the application to tangents, quadratures, cubatures, rectifications, and complanations, is much more difficult and tedious to a learner. This is evident from Lagrange.

Motion conceived may be rigorously mathematical; not so, motion executed. Now in fluxions it is motion conceived only that comes under consideration.

-

With regard to Newton's second lemma, as a square is simpler than an oblong, if we subtract the square of A a from that of Aa, there will remain 4 A a, of which the half is 2 A a; and then as the momentum is evidently greater than the decrement, and smaller than the increment, when the rate of change thus varies, we may prove by reduction to absurdity that the momentum of À A can be neither more nor less than 2 A a; for it may be demonstrated to differ less from 2 A a + a a, the increment, than by any assigned quantity how small soever: and, in 2, if the

momentum a be multiplied by i, an indeterminate quantity, and if 2 A ai – 2 ** We next,

a i

=

be substituted for A, we shall have by Maclaurin's process, Fluxions, art. 708, get the fluxion of an oblong, thence that of a cube, &c. Thus Newton's demonstration seems superior in brevity, and equal in rigour, to that of any of his contemporaries and successors at home or abroad; for it has evidently no dependance whatever on motion, or on infinitesimals, or on vanishing quantities, or even on limits.. It is wholly algebraical, but may, by a diagram, be rendered geometrical. I think the demonstration in Newton's second lemma one of the finest productions of his unequalled genius. The conception of motion, from which Maclaurin demonstrated so very tediously, belongs not to Newton's demonstration, but to his idea of the continuous generation of quantity. It seems to be through Maclaurin that some very eminent foreign mathematicians see and blame Newton.

Robins, from what Newton says himself, observes that Newton in his Mathematics uses the word momentum in two senses: first, for an infinitely small quantity, when he solves; and secondly, when he demonstrates, for an indeterminate quantity which is to be conceived to vanish: in the first sense, y = for example;

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here the quantities really employed are but it is evident that in the second lemma he uses the word momentum in a third sense for it is there neither a quantity which is to be conceived to vanish, nor is it ý or till it be multiplied by an indeter minate quantity i.

From Newton's second lemma we obtain the easiest demonstration of the binomial theorem for any exponent; because from the first fluxion we obtain the second, &c. Now these are the successive fluxional coefficients. We have therefore only to multiply them by the successive powers of i, and to divide the terms by 1, 1 x 2, 1 x 2 x 3, respectively. This would not be a legitimate demonstration, if the binomial theorem had been previously employed to find the fluxions. No one, I think, will say this is demonstrating the binomial theorem by employing the higher mathematics; for in my former paper I showed that much of fluxions belonged properly to the very elements of geometry and algebra. From fig. 3, p. 330, it is easy to demonstrate that any term 1 x i, for example, may be not greater only, but greater in any proportion than the sum of all the succeeding terms; for if nx be transferred, with the negative sign, to the other side, and if the equation be then divided by i, the thing is evident. Lagrange's demonstrations are not so easy: it is extremely tedious and teasing for a learner to proceed by his method to tangents, quadratures, &c.; a proof that his method of investigation and demonstration, how refined and convincing soever, is not short and

n.n

1.2

-2

easy, but circuitous and difficult. Thus the learner may think with regard to Lagrange's process; but the learned will admire its gene rality, vigour, consistency, and important applications. Why is not the Calculus of Variations, the noble discovery of Lagrange, admitted into our initiatory books? Much of it is quite elementary, and its nature is easily apprehended.

It appears to me also that much of the Méchanique Analytique is elementary, and may be taught early. Can any thing be easier and simpler than the two formulas, the one for statics, the other for dynamics? How delightful will the study of that comprehensive treatise, and of Laplace's masterly work the Méchanique Céleste, be, if the learner previously understand, as he easily may, the parallelopiped of forces, the three perpendicular axes of rotation, the three perpendicular co-ordinates, the three co-ordinated planes, the principle of virtual velocities, and be accustomed to introduce by substitution the sines and co-sines, &c.? Nothing will allure a learner more than to study the way in which Euler, yol. ii. of Introduction to the Analysis of Infinites, employs the sines and cosines in changing the position of the co-ordinates. May not the student also learn early, in that fine performance, the generation of curves from their equations, and the progressive induction of those equations without end?

I wish Lagrange had been more precise in the titles of his two books, Theory of Analytical Functions, Calculus of Functions; for, as his Theory does not include geometrical analysis, it relates to algebraical functions only, and not to them all; for it does not relate to common functions of known and unknown, of constant and variable, quantities; it therefore relates to derived functions only; and not even to them all; for let any one consider Arbogast's Derivations, and he will see that it does not relate to derived functions where the operations, not the quantities, are derived from each other; it is, consequently, the theory of fluxional or differential functions direct and inverse.

j dy
*' d x'

dy

Here let me remark, that the views of perhaps all the writers on the important subject of fluxions relate more or less directly to the doctrine of ratios, fx = 14, according to Lagrange's own statement; for, in every fraction, is not the numerator the antecedent, and the denominator the consequent, of a ratio?

d x'

The observations of Lacroix and other eminent mathematicians may remove the difficulties which learners always find, in consequence of the differential and the integral notation, as the differences of the absciss and of the ordinate are not employed, nor the integer of a fraction, nor the sum of quantities; the notation, however, is extremely convenient, and will not puzzle a learner, if its defect be supplied by a very careful explanation.

Even variation is not a very happy word, for variation may be either starting or continuous. Fluxion is the happiest word that I know, as it marks a continuous, not a starting, change: and since

variations as a calculus succeed fluxions in the order both of nature
and of invention, the proper appellation, perhaps, would have been
subfluxions, with a suitable notation. It would be improper, how-
ever, to propose any change.

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With regard to the fluxional notation, seems as convenient as

d x"

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dy
while the latter d is preserved for algebraic operations; and ƒ
seems as convenient as s for marking the fluent. In a philosophical
point of view, there is no comparison.

I sometimes hear mathematicians say, We ought to adopt the
foreign notation. Would not such adoption be to attempt, as far
as it is in our power, to efface the knowledge of one of Newton's
greatest discoveries? Would it not be also unpatriotic? Inde-
pendently of a natural patriotism, and of the respect due to Newton,
would a change rather unphilosophical be a change for the better?
1st x A
To some it may seem a digression, that the formula
3d x A is derivable by a boy from the simplest operation in the

4th x B

2d x B

=

Rule of Three; that in the eighth of a line it contains Euclid's
fifth definition of eight lines in his fifth book; that it comprehends
all proportional quantities, whether commensurable or incommen-
surable; and that Euclid, it is probable, thus deduced the defi-
nition.

The mistake of a very able mathematician, Carnot, in his Méta-
physique du Calcul Infinitésimal, where he endeavours to show that
the differential equations are imperfect, seems to arise from his not
distinguishing sufficiently the differences or increments from the
fluxions or differentials.

From all that has been said we may conclude, that no demonstration ought to depend on motion, if motion can be avoided, but that motion is either mathematical or mechanical: that no demonstration of the fluxional problem can be rigorous and satisfactory that depends on infinitesimals and on vanishing quantities that though, in compliance with custom, I said in p. 331, line 24,

vanishing quantity," yet it is not strictly a vanishing quantity, but a quantity which, by the continued bisection of the increment of the abscissa, may become less than any assigned quantity how small soever; that in my former paper I might without fig. 1 or 2 have stated and demonstrated by fig. 3 the doctrine of fluxion in the form of a theorem; or in the form of a problem thus, prop. problem, to find the fluxion of any function of a variable quantity: or thus, prop. problem, to find the rate, &c. To find the rate of change in a quantity and its function. This procedure would have been more scientific and elegant, not more intelligible, than that which I employed: that Newton's lemma consists of two parts; first, of the conception of the generation of quantity by motion; and, secondly, of the demonstration which relates neither to mo

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