CENTRE OF, is a point of any body, or system of bodies, so selected, that we can estimate with propriety the situation and motion of the body or system, by the situation and motion of this point. It is very plain that, in all our attempts to accurate discussion of mechanical questions, especially in the present extended sense of the word mechanism, such a selection is necessary. Even in common conversation, we frequently find it necessary to ascertain the distance of objects with a certain precision, and we then perceive that we must make some such selection. We conceive the distance to be mentioned, neither with respect to the nearest, nor the remotest point of the object, but as a sort of average distance; and we conceive the point to be somewhere about the middle of the object. The more we reflect on this, we find it the more necessary to attend to many circumstances which we had overlooked. Were it the question, to decide in what precise part of a country parish the church should be placed, we find that the geometrical middle is not always the most proper. We must consider the population of the different quarters of the parish, and select a point such, that the distances of the inhabitants on each side, in every direction, shall be as equally balanced as possible.

In mechanical discussions, the point by whose position and distance we estimate the position and distance of the whole, must be so selected, that its position and distance, estimated in any direction whatever, shall be the average of the positions and distances of every particle of the assemblage, estimated in that direction.

This will be the case, if the point be so selected that, when a plane is made to pass through it in any direc- Position whatever, and perpendiculars are drawn to this plane from every particle in the body or system, the sum of all the perpendiculars on one side of this plane is equal to the sum of all the perpendiculars on the other side. If there be such a point in a body, the position and motion of this point is the average of the positions and motions of all the particles.

Plate XL. For if \( P \) (fig. 1.) be a point so situated, and if \( QR \) be a plane (perpendicular to the paper) at any distance from it, the distance \( PP' \) of the point from this plane is the average of the distances of all the particles from it. For let the plane \( APB \) be passed through \( P \), parallel to \( QR \). The distance \( CS \) of any particle \( C \) from the plane \( QR \) is equal to \( DS - DC \), or to \( PP' - DC \). And the distance \( GT \) of any particle \( G \), lying on the other side of \( APB \), is equal to \( HT + GH \), or to \( PP' + GH \).

Let \( n \) be the number of particles on that side of \( AB \) which is nearest to \( QR \), and let \( o \) be the number of those on the remote side of \( AB \), and let \( m \) be the number of particles in the whole body, and therefore equal to \( n + o \). It is evident that the sum of the distances of all the particles, such as \( C \), is \( n \times PP' \), after deducting all the distances, such as \( DC \). Also the sum of all the distances of the particles, such as \( G \), is \( o \times PP' \), together with the sum of all the distances, such as \( GH \). Therefore the sum of both sets is \( n \times PP' + o \times PP' + sum of GH - sum of DC \), or \( m \times PP' + sum of GH - sum of DC \). But the sum of \( GH \), wanting the sum of \( DC \), is nothing; by the supposed property of the point \( P \). Therefore \( m \times PP' \) is the sum of all the distances, and \( PP' \) is the \( m \)th part of this sum, or the average distance.

Now suppose that the body has changed both its place and its position with respect to the plane \( QR \), and that \( P \) (fig. 2.) is still the same point of the body, and \( PP' \) a plane parallel to \( QR \). Make \( PP' \) equal to \( PP' \) of fig. 1. It is plain that \( PP' \) is still the average distance, and that \( m \times PP' \) is the sum of all the present distances of the particles from \( QR \), and that \( m \times PP' \) is the sum of all the former distances. Therefore \( m \times PP' \) is the sum of all the changes of distance, or the whole quantity of motion estimated in the direction \( PP' \). \( PP' \) is the \( m \)th part of this sum, and is therefore the average motion in this direction. The point \( P \) has therefore been properly selected; and its position, and distance, and motion, in respect of any plane, is a proper representation of the situation and motion of the whole.

It follows from the preceding discussion, that if any particle \( C \) (fig. 1.) moves from \( C \) to \( N \), in the line \( CS \), the centre of the whole will be transferred from \( P \) to \( Q \), so that \( PQ \) is the \( m \)th part of \( CN \); for the sum of all the distances has been diminished by the quantity \( CN \), and therefore the average distance must be diminished by the \( m \)th part of \( CN \), or \( PQ = \frac{CN}{m} \).

But it may be doubted whether there is in every body a point, and but one point, such that if a plane passes through it, in any direction whatsoever, the sum of all the distances of the particles on one side of this plane is equal to the sum of all the distances on the other.

It is easy to show that such a point may be found, with respect to a plane parallel to \( QR \). For if the sum of all the distances \( DC \) exceed the sum of all the distances \( GH \), we have only to pass the plane \( AB \) a little nearer to \( QR \), but still parallel to it. This will diminish the sum of the lines \( DC \), and increase the sum of the lines \( GH \). We may do this till the sums are equal.

In like manner we can do this with respect to a plane \( LM \) (also perpendicular to the paper), perpendicular to the plane \( AB \). The point wanted is somewhere in the plane \( AB \), and somewhere in the plane \( LM \). Therefore it is somewhere in the line in which these two planes intersect each other. This line passes through the point \( P \) of the paper where the two lines \( AB \) and \( LM \) cut each other. These two lines represent planes, but are, in fact, only the intersection of those planes with the plane of the paper. Part of the body must be conceived as being above the paper, and part of it behind or below the paper. The plane of the paper therefore divides the body into two parts. It may be so situated, therefore, that the sum of all the distances from it to the particles lying above it shall be equal to the sum of all the distances of those which are below it. Therefore the situation of the point \( P \) is now determined, namely, at the common intersection of three planes perpendicular to each other. It is evident that this point alone can have the condition required in respect of these three planes.

But it still remains to be determined whether the same condition will hold true for the point thus found, in respect to any other plane passing through it; that is, whether the sum of all the perpendiculars on one side of this fourth plane is equal to the sum of all the perpendiculars on the other side. Therefore

Let \( AGHB \) (fig. 3.), \( AXYB \), and \( CDFE \), be three planes intersecting each other perpendicularly in the point \( C \); and let \( CIKL \) be any other plane, intersecting the first in the line \( CI \), and the second in the line \( CL \). Let \( P \) be any particle of matter in the body or system. Draw \( PM, PO, PR \), perpendicular to the first three planes respectively, and let \( PR \), when produced, meet the oblique plane in \( V \); draw \( MN, ON \), perpendicular to \( CB \). They will meet in one point \( N \). Then \( PMNO \) is a rectangular parallelogram. Also draw \( MQ \) perpendicular to \( CB \), and therefore parallel to \( AB \), and meeting \( CI \) in \( S \). Draw \( SV \); also draw \( ST \) perpendicular to \( VP \). It is evident that \( SV \) is parallel to \( CL \), and that \( STRQ \) and \( STPM \) are rectangles.

All the perpendiculars, such as \( PR \), on one side of the plane \( CDFE \), being equal to all those on the other side, they may be considered as compensating each other; the one being considered as positive or additive quantities, the other are negative or subtractive. There is no difference between their sums, and the sum of both sets may be called \( o \) or nothing. The same must be affirmed of all the perpendiculars \( PM \), and of all the perpendiculars \( PO \).

Every line, such as \( RT \), or its equal \( QS \), is in a certain invariable ratio to its corresponding \( QC \), or its equal \( PO \). Therefore the positive lines \( RT \) are compensated by the negative, and the sum total is nothing.

Every line, such as \( TV \), is in a certain invariable ratio to its corresponding \( ST \), or its equal \( PM \), and therefore their sum total is nothing.

Therefore the sum of all the lines \( PV \) is nothing; but each is in an invariable ratio to a corresponding perpendicular from \( P \) on the oblique plane \( CIKL \). Therefore the sum of all the positive perpendiculars on this plane is equal to the sum of all the negative perpendiculars; and the proposition is demonstrated, viz., that in every body, or system of bodies, there is a point such, that if a plane be passed through it in any direction whatever, the sum of all the perpendiculars on one side of the plane is equal to the sum of all the perpendiculars on the other side.

The point P, thus selected, may, with great propriety, be called the centre of position of the body or system.

If A and B (fig. 4.) be the centres of position of two bodies, whose quantities of matter (or numbers of equal particles) are \(a\) and \(b\), the centre C lies in the straight line joining A and B, and AC : CB = \(b : a\), or its distance from the centres of each are inversely as their quantities of matter. For let \(C'\) be any plane passing through C. Draw \(A'\), \(B'\), perpendicular to this plane. Then we have \(a \times A' = b \times B'\), and \(A' : B' = b : a\), and, by similarity of triangles, CA : CB = \(b : a\).

If a third body D, whose quantity of matter is \(d\), be added, the common centre of position E of the three bodies is in the straight line DC, joining the centre D of the third body with the centre C of the other two, and DE : EC = \(a + b : d\). For, passing the plane \(E'\) through E, and drawing the perpendiculars \(D'\), \(C'\), the sum of the perpendiculars from D is \(d \times D'\); and the sum of the perpendiculars from A and B is \(a + b \times C'\), and we have \(d \times D' = a + b \times C'\); and therefore DE : EC = \(a + b : d\).

In like manner, if a fourth body be added, the common centre is in the line joining the fourth with the centre of the other three, and its distance from this centre and from the fourth is inversely as the quantities of matter; and so on for any number of bodies.

If all the particles of any system be moving uniformly, in straight lines, in any directions, and with any velocities whatever, the centre of the system is either moving uniformly in a straight line, or is at rest.

For, let \(m\) be the number of particles in the system. Suppose any particle to move uniformly in any direction. It is evident from the reasoning in a former paragraph, that the motion of the common centre is the \(m\)th part of this motion, and is in the same direction. The same must be said of every particle. Therefore the motion of the centre is the motion which is compounded of the \(m\)th part of the motion of each particle. And because each of these was supposed to be uniform and rectilinear, the motion compounded of them all is also uniform and rectilinear; or it may happen that they will so compensate each other that there will be no diagonal, and the common centre will remain at rest.

Cor. 1. If the centres of any number of bodies move uniformly in straight lines, whatever may have been the motions of each particle of each body, by rotation or otherwise, the motion of the common centre will be uniform and rectilinear.

Cor. 2. The quantity of motion of such a system is the sum of the quantities of motion of each body, reduced to the direction of the centre's motion. And it is had by multiplying the quantity of matter in the system by the velocity of the centre.

The velocity of the centre is had by reducing the motion of each particle to the direction of the centre's motion, and then dividing the sum of those reduced motions by the quantity of matter in the system.

By the selection of this point, we render the investigation of the motions and actions of bodies incomparably more simple and easy, freeing our discussions from numberless intricate complications of motion, which would frequently make our progress almost impossible.

Position, in arithmetic, called also False Position, or Supposition, or Rule of False, is a rule so called, because it consists in calculating by false numbers supposed or taken at random, according to the process described in any question or problem proposed, as if they were the true numbers, and then from the results, compared with that given in the question, the true numbers are found.

Thus, take or assume any number at pleasure for the number sought, and proceed with it as if it were the true number, that is, perform the same operations with it as, in the question, are described to be performed with the number required; then if the result of those operations be the same with that mentioned or given in the question, the supposed number is the same as the true one that was required; but if it be not, make this proportion, viz., as your result is to that in the question, so is your supposed false number to the true one required.

Example. What number is that, to which if we add \(1\), \(1\), and \(1\) of itself, the sum will be 240?

Suppose 99

\[ \begin{align*} 49.5 &= \frac{1}{3} \\ 33. &= \frac{1}{3} \\ 24.75 &= \frac{1}{3} \\ 16.5 &= \frac{1}{3} \\ \end{align*} \]

Then, as 222.75 : 240 :: 99 : 106.6 = Answer.

\[ \begin{align*} 53.3 &= \frac{1}{3} \\ 35.5 &= \frac{1}{3} \\ 26.6 &= \frac{1}{3} \\ 17.7 &= \frac{1}{3} \\ \end{align*} \]

240 = proof.

This is single position.

Sometimes it is necessary to make two different suppositions or assumptions, when the same operations must be performed with each as in the single rule. If neither of the supposed numbers solve the question, find the differences between the results and the given number; multiply each of these differences into the other's position; and if the errors in both suppositions be of the same kind, i.e., if both suppositions be either less or greater than the given number, divide the differences of the products by the differences of the errors. If the errors be not of the same kind, i.e., if the one be greater and the other less than the given number, divide the sum of the products by the sum of the errors. The quotient, in either case, will be the answer.

Example. Three partners, A, B, and C, bought a sugar-work which cost them £1,000; of which A paid a certain sum unknown; B paid as much as A, and £50 over; C paid as much as them both, and £50 over: What sum did each pay?

(1.) Suppose A paid £500

\[ \begin{align*} B &= 550 \\ C &= 1075 \\ \end{align*} \]

2125

2000

125 = error of excess.

(2.) Sup- This is called double position.

POTTERY is an art of very considerable importance; and in addition to what has been said on it in the Encyclopedia, the following reflections, by that eminent chemist Vauquelin, will probably be acceptable to many of our readers.

Four things (says he) may occasion difference in the qualities of earthen-ware: 1st. The nature or composition of the matter; 2d. The mode of preparation; 3d. The dimensions given to the vessels; 4th. The baking to which they are subjected. By composition of the matter, the author understands the nature and proportions of the elements of which it is formed. These elements, in the greater part of earthen-ware, either valuable or common, are silex, argil, lime, and sometimes a little oxyd of iron. Hence it is evident that it is not so much by the diversity of the elements that good earthen-ware differs from bad, as by the proportion in which they are united. Silex or quartz makes always two-thirds at least of earthen-ware; argil or pure clay, from a fifth to a third; lime, from 5 to 20 parts in the hundred; and iron, from 0 to 12 or 15 parts in the hundred. Silex gives hardness, insufficiency, and unalterability; argil makes the paste pliable, and renders it fit to be kneaded, moulded, and turned at pleasure. It possesses at the same time the property of being partially fused by the heat which unites its parts with those of the silex; but it must not be too abundant, as it would render the earthen-ware too fusible and too brittle to be used over the fire.

Hitherto it has not been proved by experience that lime is necessary in the composition of pottery; and if traces of it are constantly found in that substance, it is because it is always mixed with the other earths, from which the washings and other manipulations have not been able to separate it. When this earth, however, does not exceed five or six parts in a hundred, it appears that it is not hurtful to the quality of the pottery; but if more abundant, it renders it too fusible.

The oxyd of iron, besides the inconvenience of communicating a red or brown colour, according to the degree of baking, to the vessels in which it forms a part, has the property of rendering them fusible, and even in a greater degree than lime.

As some kinds of pottery are destined to melt very penetrating substances, such as salts, metallic oxides, glaas, &c., they require a fine kind of paste, which is obtained only by reducing the earths employed to very minute particles. Others destined for melting metals, and substances not very penetrating, and which must be able to support, without breaking, a sudden transition from great heat to great cold, require for their fabrication a mixture of calcined argil with raw argil. By these means you obtain pottery, the coarse paste of which resembles breccia, or small-grained pudding-stone, and which can endure sudden changes of temperature.

The baking of pottery is also an object of great importance. The heat must be capable of expelling humidity, and agglutinating the parts which enter into the composition of the paste, but not strong enough to produce fusion; which, if too far advanced, gives to pottery a homogeneous mass that renders it brittle. The same effect takes place in regard to the fine pottery, because the very minute division given to the earth reduces them nearly to the same state as if this matter had been fused. This is the reason why porcelain strongly baked is more or less brittle, and cannot easily endure alternations of temperature. Hence coarse porcelain, in the composition of which a certain quantity of calcined argil is employed, porcelain retorts, crucibles, tubes, and common pottery, the paste of which is coarse, are much less brittle than dishes and saucers formed of the same substance, ground with more labour.

The general and respective dimensions of the different parts of vessels of earthen-ware have also considerable influence on their capability to stand the fire.

In some cases the glazing or covering, especially when too thick, and of a nature different from the body of the pottery, also renders them liable to break. Thus, in making some kinds of pottery, it is always essential, 1st. To follow the best proportion in the principles; 2d. To give to the particles of the paste, by grinding, a minuteness suited to the purpose for which it is intended, and to all the parts the same dimensions as far as possible; 3d. To carry the baking to the highest degree that the matter can bear without being fused; 4th. To apply the glazing in thin layers, the fineness of which ought to approach as near as possible to that of the matter, in order that it may be more intimately united.

C. Vauquelin, being persuaded that the quality of good pottery depends chiefly on using proper proportions of the earthy matters, thought it might be of importance, to those engaged in this branch of manufacture, to make known the analysis of different natural clays employed for this purpose, and of pottery produced by some of them, in order that, when a new earth is discovered, it may be known by a simple analysis whether it will be proper for the same object, and to what kind of pottery already known it bears the greatest resemblance.

| Silex | Argil | Lime | Oxyd of iron | Water | |------|-------|------|--------------|-------| | 69 | 43'5 | 3'5 | 8 | 1 | | 61 | 33'2 | 3'5 | 1 | 18 | | 64'2 | 28 | 6 | 0'5 | 6'2 |

Raw Raw kaolin 100 parts.—Silex 74, argil 165; lime 2, water 7. A hundred parts of this earth gave eight of alum, after being treated with the sulphuric acid.

Washed kaolin 100 parts.—Silex 55, argil 27, lime 2, iron 05, water 14. This kaolin, treated with the sulphuric acid, gave about 45 or 50 per cent. of alum.

Petunzé.—Silex 74, argil 145, lime 35, lofs 6. A hundred parts of this substance, treated with the sulphuric acid, gave seven or eight parts of alum. But this quantity does not equal the lofs sustained.

Porcelain of retorts.—Silex 64, argil 28-8, lime 455, iron 050, lofs 277. Treated with the sulphuric acid, this porcelain gave no alum.

There is a kind of earthen vessels, called Alcarrezas, used in Spain for cooling the water intended to be drunk. These vessels consist of 60 parts of calcareous earth, mixed with alumina and a little oxyd of iron, and 36½ of siliceous earth, also mixed with alumina and the same oxyd. The quantity of iron may be eliminated at almost one hundredth part of the whole. This earth is first kneaded into a tough paste, being for that purpose previously diluted with water; formed into a cake of about six inches in thickness, and left in that state till it begins to crack. It is then kneaded with the feet, the workman gradually adding to it a quantity of sea-salt, in the proportion of seven pounds to a hundred and fifty; after which it is applied to the lath, and baked in any kind of furnace used by potters. The alcarrezas, however, are only about half as much baked as the better kinds of common earthen ware; and being exceedingly porous, water oozes through them on all sides. Hence the air, which comes in contact with it by making it evaporate, carries off the caloric contained in the water in the vessel, which is thus rendered remarkably cool.