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because each symbol by itself expressing a number, each term, whether consisting of one or two or three factors, is a number also: and thus they may be added, and the whole expression is homogeneous and correct. Whereas, did such symbols represent concrete quantities, as, for example, geometrical lines, the first three terms would represent solid content, the next two superficial area, and the last lineal length: and thus they would be heterogeneous, and could not be added. Also from such a point of view, a term ay3, consisting of more than three dimensions, would be uninterpretable and impossible.

Hence we conclude, that

(1) The symbols, whose laws and combinations are considered in the Science of Number, express the number of times any thing is taken; and the science, disregarding the concrete thing, discusses the properties of the abstract number.

(2) As Infinitesimal Calculus is a branch of the Science of Number, the symbols which will be employed in the following work represent number only; and the properties of number will be considered only as they are represented by symbols.

This latter conclusion is important, as it restricts the subjectmatter to symbols, and our discussion to their laws and properties.

4.] The numbers or quantities which are employed in the following treatise are of two kinds, constants and variables.

Constant numbers are those which have the same determinate value throughout a given operation or problem: though in another operation, or considered in another relation, they may vary. Such are the symbols 2, 3, 4, and for the most parts those in algebra, which are represented by the early letters of the alphabet: constant numbers are specific in form and value.

A variable number is that which is capable of receiving values different from each other, and generally admits of any value, though it may by the conditions of a problem be restricted to values of a particular kind, or within certain limits; variable quantities are general in form, though they admit of specific values: they are generally represented by the later letters of the alphabet.

5.] That we may avoid misconception as to the following terms, which will be frequently employed, it is necessary to give detailed explanations of them; viz. Definite, Indefinite; Infinite, Finite, Infinitesimal.

Definite means determinate and assigned; thus constants are definite. Indefinite numbers are those whose values are not assigned; they are represented by general symbols, and may therefore, so far, have any value.

To form an accurate and due conception of the latter three terms, and of the means of symbolizing and estimating them, requires a knowledge of the whole Calculus: we must therefore have recourse to analogous illustration, in hope that the student may glean from it such notions, imperfect though they will be, as will enable him to understand the technical language of the science.

By finite we generally mean that which is within reach, or may be brought within reach, of our senses. Thus a ton, or an ounce, may be taken as the unit of weight, and any number of tons or ounces which the senses perceive would be considered a finite weight; and many animalculæ, which, on account of their minuteness, are beyond the power of unassisted vision, would nevertheless be considered finite, because they may be brought within it by means of the microscope, and may be measured. Or again, inasmuch as the senses are the media by which impressions of external objects are conveyed to the mind, and as the mind conceives them when so conveyed, we apply the term finite to those magnitudes, the relation of which to other magnitudes of the same kind the mind is capable of conceiving *. The powers therefore of our senses and mind place the limit to the finite; but those magnitudes which severally transcend these limits by reason of their being too great or too small, we call infinite and infinitesimal (or infinitely small). Thus, when our senses fail to perform their office of transmitting to the mind what it would think about, by reason of the object being too large or too small, or when for a similar reason the mind fails to be capable of considering the relation of such objects, and when the most delicate subsidiary instruments for assisting the senses are employed to bring within their reach what was beyond them, and yet in vain, then we are on the boundary of the infinite or infinitesimal: of the infinite if the object be too vast, of the infinitesimal if it be too minute. Physical Science affords instances of both these cases. The distances of those fixed stars of which the parallax has not

* Peacock's Algebra, vol. ii. p. 294.

been discovered must be so great, that 400 millions of miles are not appreciable in comparison of them; considering then these millions of miles to be a finite quantity, yet no sensible change is made in the distance of a star by the addition or subtraction of them. Nay, more than this; we can employ our millions of miles to greater advantage; we can make them the base of a triangle whose vertex is the star, and yet, great as is the delicacy of our astronomical instruments, the sides of the triangle are to all appearance parallel. This then is a case where we cannot compare two geometrical distances, on account of the immensity of one of them. Considering then the 400 millions of miles to be a finite quantity, the distance of the star is infinite. Again: if one grain weight of aloetic acid be added to five pounds of pure water, the whole will after a short time assume a fine crimson colour, which could not happen unless the grain of aloetic acid had been divided and equally diffused throughout the whole volume. Now it is possible to see a quantity of water as small as a thousandth part of a grain, and such a portion of the solution would contain a thirty-five millionth part of a grain of aloetic acid. We have therefore actually divided this substance into thirty-five millions of parts, and the most delicate microscope does not so far magnify the atoms of the acid that they should be separately visible in the water; yet there they are, and are so small as to be beyond the limit of our vision, even though it be increased many thousand times they are infinitesimal, though the sum of them is finite; and as they are so small, there must be an infinity of them. Hence also we have a new aspect of such quantities. In reference to a finite quantity, infinity and infinitesimal are reciprocal terms, each implying the other; the finite quantity may be the infinitesimal infinitely-quantupled, and the infinitesimal an element of the finite quantity, when it is resolved into an infinity of parts. Again: an infinite quantity may be so large, as not only to surpass the compass of our senses, but also to surpass quantities which are from their magnitude beyond them; that is, there may be infinite quantities beyond infinite quantities, and others again beyond these: and thus there may be quantities infinitely greater than infinities, and there may be orders of infinities. Astronomy supplies instances of such quantities, to an extent within the reach of sight, by means of the telescope, but beyond the range of the micrometer as a means

of measuring distance; assuming as the unit of length the mean radius of the earth's orbit, which is about 95 millions of miles, we can compare with it the mean radii of the orbits of the other planets, and thus determine the relative sizes of their orbits; but when we extend our observations to other bodies in the celestial space, we find stars situated at such a distance from the sun, that, taking the star to be the vertex of an isosceles triangle, and the base to be a line through the sun's centre and of 190 millions of miles in length, the vertical angle of the triangle is less than 1", and in the case of Capella is computed to be 0.046; were the vertical angle 1", it can easily be shewn that the distance would be 20 billions of miles; and as it is determinable, we may say that it is comparable with finite quantities, though on the verge of the infinite. "In such numbers the imagination is lost; the mode we have of conceiving such intervals at all is by the time which it will take light to traverse them. Light, we know, travels at the rate of 192 thousand miles per second; it will occupy therefore three years and eighty-three days to traverse the distance in question. Now as this is an inferior limit, which it is already ascertained that even the brightest and therefore (in the absence of all other indications) the nearest stars exceed, what are we to allow for those innumerable stars of the smaller magnitudes which the telescope discloses to us? What for the dimensions of the galaxy in whose remoter regions the united lustre of myriads of stars is perceptible in powerful telescopes as a feeble nebulous gleam *?" Here then we have not only finite but also infinite distances, and spaces infinitely greater than these infinite distances; that is, we have successive orders of infinities, and in an ascending scale from finite distances, of which our senses are cognizant, to those infinite spaces which surpass our powers of

measurement.

So again may any one of the small particles of the aloetic acid which has been dissolved in the water be conceived to be analysed into other parts infinite in number, each one of which will therefore be infinitesimally small in comparison of its original particle; that is, one small particle may be conceived to be distributed through a finite volume, and thus to be resolved into other particles infinitely less than itself: and thus we may

* Sir John Herschel's Outlines of Astronomy, Art. 800 and following.

arrive at orders of infinitesimals, each one being infinitesimally less than that of which it is an element. It is also to be observed, that in the resolution of a finite quantity into infinitesimals of successive orders, different orders of infinities arise which severally correspond to the orders of infinitesimals, to which they are so related that the product of the infinity and infinitesimal of the same order is equal to the original finite quantity.

Here perhaps it may be asked, when does a quantity pass from the finite to the infinite and to the infinitesimal? How many finite quantities must be added to make an infinity, and into how many parts must a finite quantity be resolved so that each should be infinitesimal? An answer to such questions may be beyond our power; and it may be a matter of words only: but it is also beside the object and requirements of the Calculus. We have nothing to do with concrete quantities; the instances above cited are for the sake of illustration only: to give the reader a rough notion of the principles; such as may serve their purpose until other and more accurate ones take their place; our subject-matter is number, and number as represented by symbols and the form of the symbols, and the subsidiary symbols which will be derived from them, as will be shewn in the sequel, enable us to overcome the apparent difficulty. For we shall create our numbers, and our subsidiary numbers, subject to certain laws: and therefore, so long as they are applied within the conditions prescribed by these laws, all results correctly inferred will also be correct.

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6.] To resume then the course of the exposition from the end of the 4th Article: variable number may change value in two ways, either continuously or discontinuously.

A quantity or number varies discontinuously when it passes abruptly from one value to another, as by the addition of a finite quantity. Thus the passage from 1 to 2, and from 2 to 3, and so on, is made discontinuously, viz. by the abrupt addition of the number 1; similarly, if x is finite, we pass from 2x to 4x, and from 4x to 8x, by the successive addition of 2x and 4x; and the changes are made "per saltus."

But continuous increase is when number grows, that is, passes from one value to another only by going through all the intermediate numbers, whereby the successive increments or augments which the numbers receive are infinitesimal; thus, if we

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