a province of China, bounded on the east by Kiang-fu and Fokien; on the south, by the ocean; and on the west, by Tonquin. This province is diversified by valleys and mountains; and yields two crops of corn in a year. It abounds in gold, jewels, silk, pearls, tin, quicksilver, sugar, brass, iron, steel, salt-petre, ebony, and several sorts of odoriferous wood; besides fruits of all sorts proper to the climate. They have a prodigious number of ducks, whose eggs they hatch in ovens; and a tree, whose wood is remarkably hard and heavy, and thence called iron-wood. The mountains are covered with a fort of oaks which creep along the ground, and of which they make baskets, hurdles, mats, and ropes.

Although the climate of this province is warm, the air is pure, and the people are robust and healthy. They are very industrious; and it must be allowed that they possess in an eminent degree the talent of imitation: if they are only shown any of our European works, they execute others like them with the most surprising exactness. This province suffered much during the civil wars; but at present it is one of the most flourishing in the empire; and, as it is at a great distance from court, its government is one of the most important. This province is divided into ten districts, which contain 10 cities of the first class, and 84 of the second and third. Canton is the capital town.

Quantity, as explained by the great English Quantity lexicographer, is that property of any thing which may be increased or diminished. This interpretation of the word is certainly just, and for the purposes of common conversation it is sufficiently determinate; but the man of science may expect to find in a work like ours a definition of the thing signified. This, however, cannot be given him. A logical definition consists of the genus under which the thing defined is ranked, and the specific difference (see Logic, no. 22, &c.); but quantity is ranked under no genus. In that school where such definitions were most valued, it was considered as one of the ten categories, or general conceptions, under which all the objects of human apprehension were gathered, like soldiers in an army (see Category and Philosophy, no. 22.) On this account, even Aristotle himself, who delighted in definitions, and was not easily deterred from a favourite pursuit, could not consistently with his own rules attempt to define quantity. He characterizes it, however, in several parts of his works; and particularly in the 15th chapter of the 4th book of his metaphysics, where he gives the following account of the three first categories: ταύτα μεν ἐστιν, σὺ μεν ἐστιν, ὁμοίως δὲ ἐστιν ἐκ τῶν ἑαυτῶν. "Things are the same, of which the substance is one; similar, of which the quality is one; equal, of which the quantity is one. Again, he tells us*, that the chief characteristic of quantity is, that it may be denominated equal and unequal.

That any man can become wiser by reading such descriptions as these, none but an idolater of Aristotle will suppose. There is, indeed, no periphrasis by which we can explain what is meant by quantity to those who have not previously formed such a notion.—All that can be done by making the attempt is only to settle language, by stating exactly the cases in which we use this word in the greatest conformity to general custom; for there is a laxness or carelessness of expression in the language of most men, and our notions are frequently communicated by speech in a way by no means precise; so that it is often a great chance that the notions excited in the mind of the hearer are not exact counterparts of those in the mind of the speaker. The understandings of men differ in nothing more remarkably than in their power of abstraction, and of rapidly forming conceptions to general and simple as not to be clogged with distinguishing circumstances, which may be different in different minds while uttering and hearing the same words; and it is of great consequence to a man of scientific habits, either to cultivate, if possible, this talent, or to supersede its use, by studiously forming to himself notions of the most important universals in his own course of contemplation, by careful abstraction of every thing extraneous. His language by this means becomes doubly instructive by its extreme precision; and he will even judge with greater certainty of notions intended to be communicated by the more slovenly language of another person.

We cannot say that there is much ambiguity in the general use of the term quantity; but here, as in all other cases, a love of refinement, of novelty, and frequently of vanity, and the wish of appearing ingenious and original, have made men take advantage of even the smallest latitude with which the careless use of the word will furnish them, to amuse themselves and the public by giving the appearance of science to empty fountains.

Mathematics is undoubtedly employed in discovering and stating many relations of quantity; and it is in this category alone that any thing is contemplated by the mathematician, whether in geometry, arithmetic, or algebra. Hence mathematics has been called the science of quantity. The simplicity of the object of the mathematician's contemplation, and the unparalleled distinctness with which he can perceive its modifications, have enabled him to erect a body of science, eminent not only for its certainty, but also for the great length to which he can carry his reasonings without danger of error; and the intimate connection which this science has with the arts of life, and the important services which it has performed, have procured it a most respectable place in the circle of the sciences. Ingenious men have availed themselves of this pre-eminence of mathematics, and have endeavoured to procure respect for their diligences on other subjects, by presenting them to the public as branches of mathematical science, and therefore susceptible of that accuracy and certainty which are its peculiar boast. Our moral affections, our sensations, our intellectual powers, are all susceptible of augmentation and diminution, are conceivable as greater and less when stated together, and are familiarly spoken of as admitting of degrees of comparison. We are perfectly well understood when we say that one pain, heat, grief, kindness, is greater than another; and as this is the distinguishing characteristic of quantity, and as quantity is the subject of mathematical discussion, we suppose that these subjects may be treated mathematically. Accordingly, a very celebrated and excellent Dr. Fran. Philosopher* has said, among many things of the same kind, that the greatness of a favour is in the direct compound ratio of the service performed and the dignity of the performer, and the inverse ratio of the merit and rank of the receiver; that the value of a character is in the compound ratio of the talents and virtue, introduced &c.; and he has delivered a number of formal propositions on the most interesting questions in morals, couched in this mathematical language, and even expressed by algebraic formulæ. But this is mere play, and conveys no instruction. We understand the words; they contain no absurdity; and in as far as they have a sense, we believe the propositions to be true. But they give no greater precision to our sentiments than the more usual expressions would do. If we attend closely to the meaning of any one of such propositions, we shall find that it only expresses some vague and indistinct notions of degrees of those emotions, sensations, or qualities, which would be just as well conceived by means of the expressions of ordinary language; and that it is only by a sort of analogy or resemblance that this mathematical language conveys any notions whatever of the subjects.

The object of contemplation to the mathematician is not whatever is susceptible of greater and less, but what is measurable; and mathematics is not the science of magnitude, in its most abstracted and general acceptation, but of magnitude which can be measured. It is, indeed, the science of measure, and whatever is measured in the way of mensuration is treated mathematically. Now, in the discourse of ordinary life and ordinary men, many things are called quantities which we cannot or do not measure. This is the case in the instances already given of the affections of the mind, pleasure, pain, beauty, wisdom, honour, &c. We do not say that they are incapable of measure; but we have not yet been able to measure them, nor do we think of measuring them when we speak rationally and usefully about them. We therefore do not consider them mathematically; nor can we introduce mathematical precision into our discussions of these subjects till we can, and actually do, measure them. Persons who are precise in their expression will even avoid such phrases on these subjects as suppose, or strictly express, such measurement. We should be much embarrassed how to answer the question, How much pain does the toothache give you just now; and how much is it earlier since yesterday? Yet the answer (if we had a measure) would be as easy as to the question, How many guineas did you win at cards? or how much land have you bought? Nay, though we say familiarly, "I know well how much such a misfortune would affect you," and are understood when we say it, it would be awkward language to say, "I know well the quantity of your grief." It is in vain, therefore, to expect mathematical precision in our discourse or conceptions of quantities in the most abstracted sense. Such precision is confined to quantity which may be and is measured (A). It is only

(A) To talk intelligibly of the quantity of a pain, we should have some standard by which to measure it; some known degree of it so well ascertained, that all men, when talking of it, should mean the same thing. And we should be able to compare other degrees of pain with this, so as to perceive distinctly, not only whether they exceed or fall short of it, but also how much, or in what proportion; whether by an half, or a fifth, or tenth. Reid. Quantity only trifling with the imagination when we employ mathematical language on subjects which have not this property.

It will therefore be of some service in science to discriminate quantities in this view; to point out what are susceptible of measure, and what are not.

What is measuring? It is one of these two things: it is either finding out some known magnitude of the thing measured, which we can demonstrate to be equal to it; or to find a known magnitude of it, which being taken so many times shall be equal to it. The geometer measures the contents of a parabolic space when he exhibits a parallelogram of known dimensions, and demonstrates that this parallelogram is equal to the parabolic space. In like manner, he measures the solid contents of an infinitely extended hyperbolic spindle, when he exhibits a cone of known dimensions, and demonstrates that three of these cones are equal to the spindle.

In this process it will be found that he actually subdivides the quantity to be measured into parts of which it consists, and states these parts as actually making up the quantity, specifying each, and affixing its boundaries. He goes on with it, piece by piece, demonstrating the respective equalities as he goes along, till he has exhausted the figure, or considered all its parts. When he measures by means of a submultiple, as when he shows the surface of a sphere to be equal to four of its great circles, he stops, after having demonstrated the equality of one of these circles to one part of the surface; then he demonstrates that there are other three parts, each of which is precisely equal to the one he has minutely considered. In this part of the process he expressly affixes the whole surface into its distinct portions, of which he demonstrates the equality.

But there is another kind of geometrical measurement which proceeds on a very different principle. The geometer conceives a certain individual portion of his figure, whether line, angle, surface, or solid, as known in respect to its dimensions. He conceives this to be lifted from its place, and again laid down on the adjoining part of the figure, and that it is equal to the part which it now covers; and therefore that this part together with the first is double of the first: he lifts it again, and lays it down on the next adjoining part, and affirms that this, added to the two former, make up a quantity triple of the first. He goes on in this way, making similar inferences, till he can demonstrate that he has in this manner covered the whole figure by twenty applications, and that his moveable figure will cover no more; and he affirms that the figure is twenty times the part employed.

This mode is precisely similar to the manner of practical measurement in common life: we apply a foot-rule successively to two lines, and find that 30 applications exhaust the one, while it requires 35 to exhaust the other. We say, therefore, that the one line is 30 and the other 35 feet long; and that these two lines are to each other in the ratio of 30 to 35. Having measured two shorter lines by a similar application of a stick of an inch long 30 times to the one and 35 times to the other, we say that the ratio of the two first lines is the same with that of the two last. Euclid has taken this method of demonstrating the fourth proposition of the first book of his celebrated elements.

But all this process is a fiction of the mind, and it is the fiction of an impossibility. It is even inconceivable, that is, we cannot in imagination make this application Quantity of one figure to another; and we presume to say, that if the elements of geometry cannot be demonstrated in some other way, the science has not that title to pure, abstract, and infallible knowledge, which is usually allowed it. We cannot suppose one of the triangles lifted and laid on the other, without supposing it something different from a triangle in abstraction. The individuality of such a triangle consists solely in its being in the precise place where it is, and in occupying that portion of space. If we could distinctly conceive otherwise, we should perceive that, when we have lifted the triangle from its place, and applied it to the other, it is gone from its former place, and that there is no longer a triangle there. This is inconceivable, and space has always been acknowledged to be immovable. There is therefore some logical defect in Euclid's demonstration. We apprehend that he is labouring to demonstrate, or rather illustrate, a simple apprehension. This indeed is the utmost that can be done in any demonstration (see Metaphysics, n° 82.), but the mode by which he guides the mind to the apprehension of the truth of his fourth proposition is not consistent either with pure mathematics or with the laws of corporeal nature. The real process, as laid down by him, seems to be this. We suppose something different from the abstract triangle; some thing that, in conjunction with other properties, has the property of being triangular, with certain dimensions of two of its sides and the included angle. It has avowedly another property, not essential to, and not contained in, the abstract notion of a triangle, viz. mobility. We also suppose it permanent in shape and dimensions, or that although, during its motion, it does not occupy the same space, it continues, and all its parts, to occupy an equal space. In short, our conception is very mixed, and does not perceptibly differ from our conception of a triangular piece of matter, where the triangle is not the subject, but an adjunct, a quality. And when we suppose the application made, we are not in fact supposing two abstract triangles to coincide. This we cannot do with anything like distinctness; for our distinct conception now is, not that of two triangles coinciding, but of one triangle being now exactly occupied by that moveable thing which formerly occupied the other. In short, it is a vulgar measurement, restricted by suppositions which are inadmissible in all actual measurements in the present universe, in which no moveable material thing is known to be permanent, either in shape or magnitude.

This is an undeniable consequence of the principle of universal gravitation, and the compressibility of every kind of tangible matter with which we are acquainted. Remove the brass rule but one inch from its place; its gravitation to the earth and to the rest of the universe is immediately changed, and its dimensions change of consequence. A change of temperature will produce a similar effect; and this is attended to and considered in all nice mensurations. We do the best we can to assure ourselves that our rule always occupies a sensibly equal space; and we must be contented with chances of error which we can neither perceive nor remove.

We might (were this a proper place) take notice of some other logical defects in the reasoning of this celebrated proposition: but they are beside our present purpose of explaining the different modes of mathemati- We think that the only circumstance in which all modes of mensuration agree, or the only notion that is found in them all, is, that the quantity is conceived as consisting of parts, distinguishable from each other, and separated by assignable boundaries; so that they are at once conceived separately and jointly. We venture to assert that no quantity is directly measured which we cannot conceive in this way, and that such quantities only are the immediate objects of mathematical contemplation, and should be distinguished by a generic name. Let them be called Mathematical Quantities.

Extension, Duration, Number, and Proportion, have this characteristic, and they are the only quantities which have it. Any person will be convinced of the first assertion by attending to his own thoughts when contemplating these notions. He will find that he conceives every one of them as made up of its own parts, which are distinguishable from each other, and have assignable boundaries, and that it is only in consequence of involving this conception that they can be added to or subtracted from each other; that they can be multiplied, divided, and conceived in any proportion to each other.

He may perhaps find considerable difficulty in acquiring perfectly distinct notions of the mensurability, and the accuracy of the modes of mensuration. He will find that the way in which he measures duration is very similar to that in which he measures space or extension. He does not know, or does not attend to, anything which hinders the brafs foot-rule in his hand from continuing to occupy equal spaces during his use of it, in measuring the distance of two bodies. In like manner he selects an event which nature or art can repeat continually, and in which the circumstances which contribute to its accomplishment are invariably the same, or their variations and their effects are insensible. He concludes that it will always occupy an equal portion of time for its accomplishment, or always last an equal time. Then, observing that, during the event whose duration he wishes to measure, this standard event is accomplished 29 times, and that it is repeated 3654 times during the accomplishment of another event, he affirms that the durations of these are in the ratio of 29:3654. It is thus (and with the same logical defect as in the measuring a line by a brafs rod) that the astronomer measures the celestial revolutions by means of the rotation of the earth round its axis, or by the vibrations of a pendulum.

We are indebted for most of the preceding observations to Dr Reid, the celebrated author of the Inquiry into the Human Mind on the Principles of Common Sense, and of the Essays on the Intellectual and Active Powers of Man. He has published a dissertation on this subject in the 45th volume of the Philosophical Transactions, p. 489, which we recommend to our philosophical readers as a performance eminent for precision and acuteness. If we presume to differ from him in any trivial circumstance, it is with that deference and respect which is due to his talents and his worth.

Dr Reid justly observes, that as nothing has proportion which has not either extension, duration, or number, the characters of mathematical quantity may be restricted to these three. He calls them proper quantities, and all others he calls improper. We believe that, in the utmost precision of the English language, this denomination is very opposite, and that the word quantity, derived from quantum, always supposes measurement: But the word is frequently used in cases where its original is not kept in view, and we use other words as synonymous with it, when all mensuration, whether possible or not, is out of our thoughts. According to practice, therefore, the jus et norma loquendi, there seems to be no impropriety in giving this name, in our language at least, to whatever can be conceived as great or little. There is no impropriety in saying that the pain occasioned by the stone is greater than that of the toothache; and when we search for the category to which the affection may be referred, we cannot find any other than quantity. We may be allowed therefore to say, with almost all our scientific countrymen, that every thing is conceivable in respect of quantity which we can think or speak of as greater and less; and that this notion is the characteristic of quantity as a genus, while measurableleness is the characteristic of mathematical quantity as a species.

But do we not measure many quantities, and consider them mathematically, which have not this characteristic of being made up of their own distinguishable parts? What else is the employment of the mechanician, when speaking of velocities, forces, attractions, repulsions, magnetic influence, chemical affinity, &c., &c.? Are not these mathematical sciences? And if the precision and certainty of mathematics arise from the nature of their specific object, are not all the claims of the mechanician and physical astronomer ill-founded pretensions? These questions require and deserve a serious answer.

It is most certain that we consider the notions which velocity, are expressed by these terms velocity, force, density, and force, denoting the like, as susceptible of measure, and we consider them measured mathematically.

Some of these terms are nothing but names for relations of measurable quantity, and only require a little reflection to show themselves such. Velocity is one of these. It is only a name expressing a relation between the space described by a moving body and the time which elapses during its description. Certain moderate rates of motion are familiar to us. What greatly exceeds this, such as the flight of a bird when compared with our walking, excites our attention, and this excess gets a name. A motion not so rapid as we are familiar with, or as we wish, also gets a name because in this the excess or defect may interest us. We wish for the flight of the hawk; we elude the tardy pace of our messenger; but it is scientific curiosity which first considers this relation as a separate object of contemplation, and the philosopher must have a name for it. He has not formed a new one, but makes use of a word of common language, whose natural meaning is the combination of a great space with a short time. Having once appropriated it, in his scientific vocabulary, to this very general use, it loses with him its true signification. Tardity would have done just as well, though its true meaning is diametrically opposite; and there is no greater impropriety in saying the tardity of a cannon bullet than in saying the velocity of the hour-hand of a watch. Quantity. Velocity is a quality or affection of motion, the notion of which includes the notions of space and duration (two mathematical quantities), and no other. It does not therefore express a mathematical quantity itself, but a relation, a combination of two mathematical quantities of different kinds; and as it is measurable in the quantities so combined, its measure must be a unit of its own kind, that is, an unit of space as combined with an unit of time.

Density is another word of the same kind, expressing a combination of space with number. Dense arbores means trees standing at a small distance from each other; and the word is used in the same sense when we say that quicksilver is denser than water. The expression always suggests to the reflecting mind the notions of particles and their distances. We are indeed so habituated to complicated views of things, that we can see remote connections with astonishing rapidity; and a very few circumstances are sufficient for leading forward the mind in a train of investigation. Common discourse is a most wonderful instance of this. It is in this way that we say, that we found by weighing them that inflammable air had not the fifth part of the density of common air. Supposing all matter to consist of equal atoms equally heavy, and knowing that the weight of a bladder of air is the sum of the weights of all the atoms, and also knowing that the vicinity of the atoms is in a certain proportion of the number contained in a given bulk, we affirm that common air is more than five times denser than inflammable air; but this rapid decision is entirely the effect of habit, which makes us familiar with certain groups of conceptions, and we instantaneously distinguish them from others, and thus think and discourse rationally. The Latin language employs the word frequens to express both the combination of space and number, and that of time and number.

There are perhaps a few more words which express combinations of mathematical quantities of different kinds; and the corresponding ideas or notions are therefore proper and immediate subjects of mathematical discussion: But there are many words which are expressive of things, or at least of notions, to which this way of considering them will not apply. All those affections or qualities of external bodies, by which they are conceived to act on each other, are of this kind: Impulsive force, weight, centripetal and centrifugal force, magnetic, electrical, chemical attractions and repulsions; in short, all that we consider as the immediate causes of natural phenomena. These we familiarly measure, and consider mathematically.

What was said on this subject in the article Physics will give us clear conceptions of this process of the mind. These forces or causes are not immediate objects of contemplation, and are known only by and in the phenomena which we consider as their effects. The phenomenon is not only the indication of the agency of any cause, and the characteristic of its kind, but the measure of its degree. The necessary circumstances in this train of human thought are, 1st, The notion of the force as something susceptible of augmentation and diminution. 2d, The notion of an inseparable connection of the force with the effect produced, and of every degree of the one with a corresponding degree of the other. From these is formed the notion that the phenomenon or effect is the proper measure of the force or cause. Quantity. All this is strictly logical.

But when we are considering these subjects mathematically, the immediate objects of our contemplation are not the forces which we are thus treating. It is not their relations which we perceive, and which we combine with such complication of circumstances and certainty of inference as are unknown in all other sciences: by no means; they are the phenomena only, which are subjects of purely mathematical discussion. They are motions, which involve only the notions of space and time; and when we have finished an accurate mathematical investigation, and make our affirmation concerning the forces, we are certain of its truth, because we suppose the forces to have the proportions and relations, and no other, which we observe in the phenomena. Thus, after having demonstrated, by the geometrical comparison of the lines and angles and surfaces of an ellipse, that the momentary deflection of the moon from the tangent of her orbit is the 360th part of the simultaneous deflection of a stone from the tangent of its parabolic path, Newton affirms, that the force by which a particle of the moon is retained in her orbit is the 360th part of the weight of a particle of the stone; and having farther shown, from fact and observation, that these momentary deflections are inversely as the squares of the distances from the centre of the earth, he affirms, that all this is produced by a force which varies its intensity in this manner.

Now all this investigation proceeds on the two suppositions mentioned above, and the measures of the forces are in fact the measures of the phenomena. The whole of physical astronomy, and indeed the whole of mechanical philosophy, might be taught and understood, without ever introducing the word force, or the notion which it is supposed to express: for our mathematical reasonings are really about the phenomena, which are subjects purely mathematical.

The precision, therefore, that we presume to affirm to attend these investigations, arises entirely from the measurable nature of the quantities which are the real objects of our contemplation, and the suitability and propriety of the measures which we adopt in our comparisons.

Since then, the phenomena are the immediate subjects of our discussion, and the operating powers are only inferences from the phenomena considered as effects, the quantity ascribed to them must also be an inference from the quantity of the effect, or of some circumstance in the effect. The measure, therefore, of the cause, or natural power or force, cannot be one of its own parts; for the whole and the part are equally unperceived by us. Our measure, therefore, must be a measure of some interesting part, or of the only interesting part of the phenomenon. It is therefore in a manner arbitrary, and depends chiefly on the interest we take in the phenomenon. It must, however, be settled with precision, so that all men in using it may mean the same thing. It must be settled, therefore, by the description of that part or circumstance of the phenomenon which is characteristic of the natural power. This description is the definition of the measure.

Thus Newton assumes as his measure of the centripetal force, the momentary deviation from uniform rectilinear motion. Others, and sometimes Newton himself, Quantity itself assumes the momentary change of velocity, which again is measured by twice this deviation. These measures, being thus selected, are always proper in a mathematical sense; and if strictly adhered to, can never lead us into any paradoxism. They may, however, be physically wrong: there may not be that indissoluble connection between the phenomenon and the supposed cause. But this is no mathematical error, nor does it invalidate any of our mathematical inferences; it only makes them useless for explaining the phenomenon by the principle which we adopted; but it prepares a modification of the phenomenon for some more fortunate application of physical principles.

All that can be defined in the definitions or descriptions of these measures is, that they may not deviate from the ordinary use of the terms, because this would always create confusion, and occasion mistakes. Dr Reid has given an example of an impropriety of this kind, which has been the subject of much debate among the writers on natural philosophy. We mean the measure of the force inherent in a body in motion. Descartes, and all the writers of his time, assumed the velocity produced in a body as the measure of the force which produces it; and observing that a body, in consequence of its being in motion, produces changes in the state or motion of other bodies, and that these changes are in the proportion of the velocity of the changing body, they asserted that there is in a moving body a vis insita, an inherent force, and that this is proportional to its velocity; saying that its force is twice or thrice as great, when it moves twice or thrice as fast at one time as at another. But Leibnitz observed, that a body which moves twice as fast, rises four times as high, against the uniform action of gravity; that it penetrates four times as deep into a piece of uniform clay; that it bends four times as many springs, or a spring four times as strong, to the same degree; and produces a great many effects which are four times greater than those produced by a body which has half the initial velocity. If the velocity be triple, quadruple, &c. the effects are nine times, sixteen times, &c. greater; and, in short, are proportional, not to the velocity, but to its square. This observation had been made before by Dr Hooke, who has enumerated a prodigious variety of important cases in which this proportion of effect is observed. Leibnitz, therefore, affirmed, that the force inherent in a moving body is proportional to the square of the velocity.

It is evident that a body, moving with the same velocity, has the same inherent force, whether this be employed to move another body, to bend springs, to rise in opposition to gravity, or to penetrate a mass of soft matter. Therefore these measures, which are so widely different, while each is agreeable to a numerous class of facts, are not measures of this something inherent in the moving body which we call its force, but are the measures of its exertions when modified according to the circumstances of the case; or, to speak still more cautiously and securely, they are the measures of certain classes of phenomena consequent on the action of a moving body. It is in vain, therefore, to attempt to support either of them by a demonstration. The measure itself is nothing but a definition. The Cartesian calls that a double force which produces a double velocity in the body on which it acts. The Leibnitzian calls that a quadruple force which makes a quadruple penetration. The reasonings of both in the demonstration of a proposition in dynamics may be the same, as also the result, though expressed in different numbers.

But the two measures are far from being equally proper: for the Leibnitzian measure obliges us to do continual violence to the common use of words. When two bodies moving in opposite directions meet, strike each other, and stop, all men will say that their forces are equal, because they have the best test of equality which we can devise. Or when two bodies in motion strike the parts of a machine, such as the opposite arms of a lever, and are thus brought completely to rest, we and all men will pronounce their mutual energies by the intervention of the machine to be equal. Now, in all these cases, it is well known that a perfect equality is found in the products of the quantities of matter and velocity. Thus a ball of two pounds, moving with the velocity of four feet in a second, will stop a ball of eight pounds moving with the velocity of one foot per second. But the followers of Leibnitz say, that the force of the first ball is four times that of the second.

All parties are agreed in calling gravity a uniform or invariable accelerating force; and the definition which they give of such a force is, that it always produces the same acceleration, that is, equal accelerations in equal times, and therefore produces augmentations of velocity proportionate to the times in which they are produced. The only effect ascribed to this force, and consequently the only thing which indicates, characterizes, and measures it, is the augmentation of velocity. What is this velocity, considered not merely as a mathematical term, but as a phenomenon, as an event, a production by the operation of a natural cause? It cannot be conceived any other way than as a determination to move on forever at a certain rate, if nothing shall change it. We cannot conceive this very clearly. We feel ourselves forced to animate, as it were, the body, and give it not only a will and intention to move in this manner, but a real exertion of some faculty in consequence of this determination of mind. We are conscious of such a train of operations in ourselves; and the last step of this train is the exertion or energy of some natural faculty, which we, in the utmost propriety of language, call force. By such analogical conception, we suppose a something, an energy, inherent in the moving body; and its only office is the production and continuation of this motion, as in our own case. Scientific curiosity was among our latest wants, and language was formed long before its appearance: as we formed analogical conceptions, we contented ourselves with the words already familiar to us, and to this something we gave the name Force, which expressed that energy in ourselves which bears some resemblance (in office at least) to the determination of a body to move on at a certain rate. This sort of allegory pervades the whole of our conceptions of natural operations, and we can hardly think or speak of any operation without a language, which supposes the animation of matter. And, in the present case, there are so many points of resemblance between the effects of our exertions and the operations of nature, that the language is most expressive, and has the strongest appearance of propriety. By exerting our force, we not only move and keep in motion, but we move other bodies. Just as a ball not only moves, but... Quantity, but puts other bodies in motion, or penetrates them, &c.—This is the origin of that conception which forcibly obtrudes itself into our thoughts, that there is inherent in a moving body a force by which it produces changes in other bodies. No such thing appears in the same body if it be not in motion. We therefore conclude, that it is the production of the moving force, whatever that has been. If so, it must be conceived as proportional to its producing cause. Now this force, thus produced or exerted in the moving body, is only another way of conceiving that determination which we call velocity, when it is conceived as a natural event. We can form no other notion of it. The vis infusa, the determination to move at a certain rate, and the velocity, are one and the same thing, considered in different relations.

Therefore the vis infusa corpori moventi, the determination to move at a certain rate, and the velocity, should have one and the same measure, or any one of them may be taken for the measure of the other. The velocity being an object of perception, is therefore a proper measure of the inherent force; and the propriety is more evident by the perfect agreement of this use of the words with common language. For we conceive and express the action of gravity as uniform, when we think and say that its effects are proportional to the times of its action. Now all agree, that the velocity produced by gravity is proportional to the time of its action; And thus the measure of force, in reference to its producing cause, perfectly agrees with its measure, independent of this consideration.

But this agreement is totally lost in the Leibnitzian doctrine; for the body which has fallen four times as far, and has sustained the action of gravity twice as long, is said to have four times the force.

The quaintness and continual paradox of expression which this measure of inherent force leads us into, would have quickly exploded it, had it not been that its chief abettors were leagued in a keen and acrimonious warfare with the British mathematicians who supported the claim of Sir Isaac Newton to the invention of fluxions. They rejoiced to find in the elegant writings of Huyghens a physical principle of great extent, such as this is, which could be set in comparison with some of the wonderful discoveries in Newton's Principia. The fact, that in the mutual actions of bodies on each other the products of the masses and the squares of the velocities remain always the same (which they call the conservatio virium vivarum), is of almost universal extent; and the knowledge of it enabled them to give ready and elegant solutions of the most abstruse and intricate problem, by which they acquired a great and deserved celebrity. Dr Robert Hooke, whose observation hardly anything escaped, was the first (long before Huyghens) who remarked *, that in all cases of the gradual production and extinction of motion, the sensible phenomenon is proportional to the square of the produced or extinguished velocity.

John Bernoulli brought all these facts together, and systematized them according to the principle advanced by Huyghens in his treatise on the centre of oscillation. He and Daniel Bernoulli gave most beautiful specimens of the prodigious use of this principle for the solution of difficult physical problems in their dissertations on the motion and impulse of fluids, and on the communication of motion. It was however very early objected to them (we think by Marquis Poleni), that in the collision of bodies perfectly hard there was no such conservatio virium vivarum; and that, in this case, the forces must be acknowledged to be proportional to the velocities. The objections were unanswerable.—But John Bernoulli evaded their force, by affirming that there were and could be no bodies perfectly hard. This was the origin of another celebrated doctrine, on which Leibnitz greatly pleased himself, the Law of Continuity, viz. that nothing is observed to change abruptly, or per saltum. But no one will pretend to say that a perfectly hard body is an inconceivable thing; on the contrary, all will allow that softness and compressibility are adjunct ideas, and not in the least necessary to the conception of a particle of matter, nay totally incompatible with our notion of an ultimate atom.

Sir Isaac Newton never could be provoked to engage in this dispute. He always considered it as a wilful abuse of words, and unworthy of his attention. He guarded against all possibility of cavil, by giving the most precise and perspicuous definitions of those measures of forces, and all other quantities which he had occasion to consider, and by carefully adhering to them. And in one proposition of about 20 lines, viz. Great superiority of Newton.

In the 39th of the 1st book of the Principia, he explained every phenomenon adduced in support of the Leibnitzian doctrine, showing them to be immediate consequences of the action of a force measured by the velocity which it produces or extinguishes. There it appears that the heights to which bodies will rise in opposition to the uniform action of gravity are as the squares of the initial velocities: So are the depths to which they will penetrate uniformly resisting matter: So is the number of equal springs which they will bend to the same degree, &c. &c. &c. We have had frequent occasion to mention this proposition as the most extensively useful of all Newton's discoveries. It is this which gives the immediate application of mechanical principles to the explanation of natural phenomena. It is incessantly employed in every problem by the very persons who hold by the other measure of forces, although such conduct is virtually giving up that measure. They all adopt, in every investigation, the two theorems $f = v^2$ and $f' = v^2$, both of which suppose an accelerating force $f'$ proportional to the velocity $v$ which it produces by its uniform action during the time $t$, and the theorem $f' = v^2$ is the 39th 1. Principle, and is the conservatio virium vivarum.

This famous dispute (the only one in the circle of mathematical science) has led us somewhat aside. But we have little more to remark with respect to measurable quantity. We cannot say what varieties of quantity are susceptible of strict measure, or that it is impossible to give accurate measures of everything susceptible of augmentation and diminution. We affirm, however, with confidence, that pain, pleasure, joy, &c., are not made up of their own parts, which can be contemplated separately: but they may chance to be associated by nature with something that is measurable; and we may one day be able to assign their degrees with as much precision as we now ascertain the degrees of warmth by the expansion of the fluid in the thermometer. There is one sense in which they may all Quantity be measured, viz. numerically, as Newton measures density, vis matrix, &c. We can conceive the pain of each of a dozen men to be the same. Then it is evident that the pain of eight of these men is to that of the remaining four as two to one; but from such measurement we do not foresee any benefit likely to arise.