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quantities b1, b2, bз, ... bn, by which process the signs are not

changed; then

a is > Lbi, < Gb1

ag is

Lbg, < Gbg

an is > Lbn, < ɑbn;

and therefore by addition

A1 + A2 + A3+...+a, is > L (b1 + b2+ b3 + +bn)

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...

is G (bi+b2+b3 + ... + bn),
>

is > L, <G, and therefore is equal

to some mean value of the fractions.

Q. E. D.

Secondly, let b1, b2, bз, ... b2 be negative; then, as before,

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let these inequalities be severally multiplied by the negative quantities b1, b2, bз, ...... bn, so that the signs of them are changed; then

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whence, bearing in mind that the sign of an incquality is changed when it is divided by a negative quantity,

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DIFFERENTIAL CALCULUS.

PART I.

ANALYTICAL INVESTIGATIONS.

CHAPTER I.

GENERAL PRINCIPLES, AND EXPLANATION OF TERMS. SECTION 1.-Introductory; on Number, its properties, affections, and science.

ARTICLE 1.] Infinitesimal Calculus is a branch of that science, the aggregate of the rules and operations of which French writers call "Le Calcul," but for which we have no more specific name than the Science of Number*. A metaphysical inquiry into the origin and nature of number would be out of place in a didactic treatise such as the present, and would also be superfluous; because it will be sufficient for the student to have that notion of it which an ordinary knowledge of arithmetic and algebra implies; but it is well to recall his attention to certain axiomatic properties of it, and to bring into greater prominence those from which the Infinitesimal Calculus is deduced.

Number is the ratio or relation which two quantities of the same kind bear to one another in respect of quantuplicity. By quantity I mean whatever is capable of measurement; whether it be geometrical space, or weight, or time, or heat, or light, or velocity, or any thing else; that, viz., of which we can predicate muchness in reply to the question "how much?" or number of times in answer to "how many times?" The above phænomena may be severally the substrata of mixed sciences, as they are called, but they can only be treated of in accord

* M. Comte writes, "Le Calcul a pour objet propre de résoudre toutes les questions de nombres.”—Philosophie Positive, vol. i. p. 143.

ance with the rules of the science of number, because they are capable of addition and division and measurement, at least in conception, if not in act; and it is only when a problem can be reduced to a question of pure number that it can be brought within the domain of the Calculus; and when this condition is satisfied, the science of number supplies, as it were, the skeleton or framework on which such mixed science is treated; and considers the subject-matter not in its concrete and physical and phænomenal, but in its abstract state, as measured, and in the measure, as the correct representative of it. It does not take cognizance of this or that weight or colour, but of the number of times such a weight contains another weight, and so and this is what I mean by the terms "in respect of quantuplicity" in the above definition of number; and thus it is that the science is so general, almost universal, because all the subject-matter of the Physical Sciences conforms to its requirements in respect of admitting of measurement.

on;

2.] There are two modes of measuring quantities, and thus of arriving at abstract numbers from the concrete magnitudes. Firstly, a certain amount of the given quantity is taken, which for the sake of convenience is called an unit of that particular quantity, and with it any other amount is compared, the principle of comparison being assigned by the particular science whose subject-matter the quantity is; and if the latter amount be divisible into two or three or more parts, severally equal to each other and to the unit, we say that the latter amount is twice or three times or more times the unit, and thus arrive at the abstract number; and the problem of determining other relations arising out of this one belongs to the science of number. The unit, it is to be observed, is arbitrary; thus, for instance, if a certain volume of matter of given density, say a cubic inch of distilled water, be considered the volume-unit, and its density be called the density-unit, then if two cubic inches of distilled water could be compressed into one inch, its density would be two. Hereby we arrive at abstract number by directly comparing any given concrete quantity with the unit of that quantity. Secondly, we may estimate quantity without formally introducing the unit; for suppose of two unequal quantities, one to be divisible into two equal parts, and the other into three parts, equal to each other and to each of the divided parts of the former quantity; then, although if one

of the equal parts were taken as the unit, one quantity would contain two and the other three units: yet we may omit the unit, and say that the ratio of one quantity to the other is that of the numbers two to three. Thus, if two lines admit of being resolved into 3 and 7 parts respectively, which are equal to each other, the ratio of the lines will be represented by the numerical ratio 37; and this mode of measuring quantities is independent of the amount of the chosen unit; for if the above lines had been divided into 6 and 14 equal parts respectively, or into 3n and 7n equal parts respectively, the unit would have been only one-half or one-nth part of what it was in the previous resolution; and yet the ratio of the lines, or of the numbers which represent them, viz. 6: 14, or 3n: 7n, would have been the same as before. And this mode of representing lines by numbers is equally applicable to areas, hours, weights, &c. all of which may be so related to each other as to admit of resolution into 3 and 7 equal parts respectively, and thus be represented by the ratio of the numbers 37. It is manifest that only quantities of the same kind can be measured in either of the above methods, and hence it is only from comparing homogeneous quantities that numbers can be formed. The principle and mode of comparison however must be assigned by the particular science whose subject-matter the quantity is; thereby are its concrete materials abstracted and brought within the range of the Science of Number, which has thus to deal with only abstract quantuplicities; and thus (which is a point of the utmost importance, and deserves the most careful attention) the symbols 2, 3, 4, .............. a, b, C,. x, y, x, do not represent

......

concrete quantities, such as 2 ounces, or a hours, or x feet, but abstract_numbers; and it is the properties of these that the Science of Number has to discuss *.

3.] The distinguishing characteristic of such numbers is, that they remain after any operation the same in kind which they were before. When two numbers are added or subtracted, the sum or difference is a number; when two numbers are multi

* The two modes of measuring quantities and of thereby forming numbers, correspond to the two aspects in which arithmetical fractions are viewed; one in which the denominator assigns the unit, and the numerator the number of times it is to be taken, and the other in which they are considered as the expressions of arithmetical ratios. See Peacock's Algebra, 2nd edition, vol. i. pp. 54, 155.

PRICE, VOL. I.

C

:

plied or divided, the resultant is the same in kind as each of the components: as they are numbers, so is it number. Thus 2x3= = 6, and 6 is an abstract number of the same kind as 2 and 3; that is, twice thrice is equivalent to six times; 8-4-2, that is, one-fourth part of eight times is twice. The same is also true of the operations of Involution and Evolution. These remarks are important, because if the symbols represent concrete quantities the results would be otherwise; thus in the analogous operation of geometrical multiplication, if two linear inches be "multiplied" by three linear inches, the result is six square inches and therefore in the process we have changed from linear to superficial quantity. And so, if the six superficial inches be multiplied by four linear inches, the result is 24 cubic inches, or inches of volume, and thus by the last process we have passed from superficial magnitude to solid content. By the operations then the kind has changed; and as all the dimensions which space admits of have been exhausted, it is impossible to multiply together more than three geometrical lines; thus two linear inches multiplied into themselves four times is an impossibility; or, in other words, a geometrical line cannot be raised to any power above the third. And the possibility of extracting roots is confined within narrower limits; thus the square root of a superficial area is a possible quantity, and so is the cube root of a solid content, being in each case a line. The case of geometrical multiplication is the most favourable one, for in other subject-matter, which is only uni-dimensional, we cannot at all multiply the concrete quantities, and can only divide them when they are homogeneous. It is absurd to speak of a pound multiplied into a pound, or of the product of an hour by an hour: such operations are impossible, and have no meaning; but we can divide two pounds by one pound, and thereby arrive at the abstract number 2, because the inverse process is possible, and may therefore be undone; that is, because we can multiply any concrete unit by an abstract number; but we can no more divide pounds weight by pounds sterling than we can multiply hours by degrees of heat. These remarks on the abstract character of Number are relevant to the present subject, inasmuch as they prove the correctness of an expression such as

ay2 + bxy + cx2 + dy + ex +f;

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