CHAPTER XII.
OF QUANTITY.
1. The definition of quantity.—2. The exposition of quantity, what it is.—3. How line, superficies, and solid, are exposed.—4. How time is exposed.—5. How number is exposed.—6. How velocity is exposed.—7. How weight is exposed.—8. How the proportion of magnitudes is exposed.—9. How the proportion of times and velocities is exposed.
[Sidenote: Definition of quantity.]
1. What and how manifold _dimension_ is, has been said in the 8th chapter, namely, that there are three dimensions, line or length, superficies, and solid; every one of which, if it be determined, that is, if the limits of it be made known, is commonly called _quantity_; for by _quantity_ all men understand that which is signified by that word, by which answer is made to the question, _How much is it?_ Whensoever, therefore, it is asked, for example, _How long is the journey?_ it is not answered indefinitely, _length_; nor, when it is asked, _How big is the field?_ is it answered indefinitely, _superficies_; nor, if a man ask, _How great is the bulk?_ indefinitely, _solid_: but it is answered determinately, the journey is a hundred miles; the field is a hundred acres; the bulk is a hundred cubical feet; or at least in some such manner, that the magnitude of the thing enquired after may by certain limits be comprehended in the mind. QUANTITY, therefore, cannot otherwise be defined, than to be _a dimension determined_, or _a dimension, whose limits are set out, either by their place, or by some comparison_.
[Sidenote: The exposition of quantity, what it is.]
2. And _quantity_ is determined two ways; one, by the sense, when some sensible object is set before it; as when a line, a superficies or solid, of a foot or cubit, marked out in some matter, is objected to the eyes; which way of determining, is called _exposition_, and the quantity so known is called _exposed quantity_; the other by memory, that is, by comparison with some exposed quantity. In the first manner, when it is asked of what quantity a thing is, it is answered, of such quantity as you see exposed. In the second manner, answer cannot be made but by comparison with some exposed quantity; for if it be asked, how long is the way? the answer is, so many thousand paces; that is, by comparing the way with a pace, or some other measure, determined and known by exposition; or the quantity of it is to some other quantity known by exposition, as the diameter of a square is to the side of the same, or by some other the like means. But it is to be understood, that the quantity exposed must be some standing or permanent thing, such as is marked out in consistent or durable matter; or at least something which is revocable to sense; for otherwise no comparison can be made by it. Seeing, therefore, by what has been said in the next preceding chapter, comparison of one magnitude with another is the same thing with proportion; it is manifest, that quantity determined in the second manner is nothing else but the proportion of a dimension not exposed to another which is exposed; that is, the comparison of the equality or inequality thereof with an exposed quantity.
[Sidenote: How line, superficies, and solids, are exposed.]
3. _Lines_, _superficies_, and _solids_, are exposed, first, by _motion_, in such manner as in the 8th chapter I have said they are generated; but so as that the marks of such motion be permanent; as when they are designed upon some matter, as a line upon paper; or graven in some durable matter. Secondly, by _apposition_; as when one line or length is applied to another line or length, one breadth to another breadth, and one thickness to another thickness; which is as much as to describe a line by points, a superficies by lines, and a solid by superficies; saving that by points in this place are to be understood very short lines; and, by superficies, very thin solids. Thirdly, lines and superficies may be exposed by _section_, namely, a line may be made by cutting an exposed superficies; and a superficies, by the cutting of an exposed solid.
[Sidenote: How time is exposed.]
4. _Time_ is exposed, not only by the exposition of a line, but also of some moveable thing, which is moved uniformly upon that line, or at least is supposed so to be moved. For, seeing time is an idea of motion, in which we consider former and latter, that is succession, it is not sufficient for the exposition of time that a line be described; but we must also have in our mind an imagination of some moveable thing passing over that line; and the motion of it must be uniform, that time may be divided and compounded as often as there shall be need. And, therefore, when philosophers, in their demonstrations, draw a line, and say, _Let that line be time_, it is to be understood as if they said, _Let the conception of uniform motion upon that line, be time_. For though the circles in dials be lines, yet they are not of themselves sufficient to note time by, except also there be, or be supposed to be, a motion of the shadow or the hand.
[Sidenote: How number is exposed.]
5. _Number_ is exposed, either by the exposition of points, or of the names of number, _one_, _two_, _three_, _&c._; and those points must not be contiguous, so as that they cannot be distinguished by notes, but they must be so placed that they may be _discerned_ one from another; for, from this it is, that number is called _discreet quantity_, whereas all quantity, which is designed by motion, is called _continual quantity_. But that number may be exposed by the names of number, it is necessary that they be recited by heart and in order, as one, two, three, &c.; for by saying one, one, one, and so forward, we know not what number we are at beyond two or three; which also appear to us in this manner, not as number, but as figure.
[Sidenote: How velocity is exposed.]
6. For the exposition of _velocity_, which, by the definition thereof, is a motion which, in a certain time, passeth over a certain space, it is requisite, not only that time be exposed, but that there be also exposed that space which is transmitted by the body, whose velocity we would determine; and that a body be understood to be moved in that space also; so that there must be exposed two lines, upon one of which uniform motion must be understood to be made, that the time may be determined; and, upon the other, the velocity is to be computed. As if we would expose the velocity of the we would expose the velocity of the body A, we draw two lines A B and C D, and place a body in C also; which done, we say the velocity of the body A is so great, that it passeth over the line A B in the same time in which the body C passeth over the line C D with uniform motion.
A B ------ C D ------
[Sidenote: How weight is exposed.]
7. _Weight_ is exposed by any heavy body, of what matter soever, so it be always alike heavy.
[Sidenote: How the proportion of magnitudes is exposed.]
8. The _proportion_ of two magnitudes is then exposed, when the magnitudes themselves are exposed, namely, the proportion of equality, when the magnitudes are equal; and of inequality, when they are unequal. For seeing, by the 5th article of the preceding chapter, the proportion of two unequal magnitudes consists in their difference, compared with either of them; and when two unequal magnitudes are exposed, their difference is also exposed; it follows, that when magnitudes, which have proportion to one another, are exposed, their proportion also is exposed with them; and, in like manner, the proportion of equals, which consists in this, that there is no difference of magnitude betwixt them, is exposed at the same time when the equal magnitudes themselves are exposed. For example, if the exposed lines A B and C D be equal, the proportion of equality is exposed in them; and if the exposed lines, E F and E G be unequal, the proportion which E F has to E G, and that which E G has to E F are also exposed in them; for not only the lines themselves, but also their difference, G F, is exposed. The proportion of unequals is quantity; for the difference, G F, in which it consists, is quantity. But the proportion of equality is not quantity; because, between equals, there is no difference; nor is one equality greater than another, as one inequality is greater than another inequality.
A B ------- C D ------- E F F -------
[Sidenote: How the proportion of times and velocities is exposed.]
The proportion of two _times_, or of two _uniform velocities_, is then exposed, when two lines are exposed by which two bodies are understood to be moved uniformly; and therefore the same two lines serve to exhibit both their own proportion, and that of the times and velocities, according as they are considered to be exposed for the magnitudes themselves, or for the times or velocities. For let the two lines A and B be exposed; their proportion therefore (by the last foregoing article) is exposed; and if they be considered as drawn with equal and uniform velocity, then, seeing their times are greater, or equal, or less, according as the same spaces are transmitted in greater, or equal, or less time, the lines A and B will exhibit the equality or inequality, that is, the proportion of the times. To conclude, if the same lines, A and B, be considered as drawn in the same time, then, seeing their velocities are greater, or equal, or less, according as they pass over in the same time longer, or equal, or shorter lines, the same lines, A and B, will exhibit the equality, or inequality, that is, the proportion of their velocities.
A --- B
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