therefore is calling the objects we choose by the letters of the alphabet-the first A, the second B, and so on, just as in conversation we talk of "Mr. A." or " Mr. B." when we are not speaking of any particular person but merely wish to put a case. By this means not only is the repetition of the same word avoided, but the objects themselves are distinguished from each other: a fact of no slight importance as the number becomes greater. We may observe further that there is no need to preserve the strict alphabetical order; and very often we shall deviate from it; for example, the first object is sometimes called P, and the second G, or it may be H, or any other letter.

CHAPTER V.

INCLINATION.

A.

1.

B

2.

We have now our two objects, A and B, and the line between them (1,) which is called AB. But there can be no reason why there should be but two objects on the paper, why not a third, C, and a line AC (2.) Then we have the two lines AB and AC, and something which we have not had before. For if we were

at C and wished to measure the distance to A, the line CA would represent it; but the distance from C to B would be along the line CB (3,) and not from C to A and then from A to B. There is therefore some connection

A.

between the two lines CA and CB, A which must now be examined. First

3.

then let us give the new idea a name, which shall be "an Angle.” Secondly, we must inquire what an angle represents.

Suppose, now, that there be a post EF erected in the ground, near a brick wall GH; the post and the wall will apparently have nothing to do with each other (4.)

If however the post should lean a little towards the wall, its head will be nearer than its foot (5.)

And if the foot of the post were not fixed in the ground at all, then the post not being able to stand up of itself would lean for support against the wall (6.)

F

6.

And in this case the post would be in exactly the same position with regard to the wall that CA was with regard to AB, that is EF would make an angle with GH. Hence an angle is the "inclination or leaning of two lines to each other, and therefore does not depend upon the size of the lines themselves. We shall return again to the subject of angles; for the present this definition must be sufficient. We may however observe that there is a curious use of the same word in morals of a similar nature. For the exact measure of a man's inclination towards any object is just how much he will go out of his way to attain it.

CHAPTER

V I.

FIGURES AND SURFACES.

HITHERTO we have had but three objects on paper; now let us suppose a fourth D, and lines drawn from one to the other AB, AC, CD, DB.

B

Here, then, again we meet with an idea unknown hitherto, for the lines connecting A, B, C, and D all meet each other. Let us call it a "Surface." For example, suppose that at each of the four corners of a table there were placed some object, say a mustardpot, an inkstand, a candlestick, and a book, and lines drawn from one to the other; by these lines the outline of a Figure" would be formed, and the "Surface" of the table would be enclosed. To examine what this surface is more closely, it must be remembered that the line AB represents the distance between the objects A and B

66

1.

D

And let AC and BD vary, then the surface grows larger, (5) or smaller (6) as C and D C recede (5.) or approach (6.)

D

B

Hence the size of the surface depends upon two things; one the distance of B from A, the other of C from A. Now the distance of B c from A is already called the length; to distinguish the other direction therefore, let us

a

call it by another name, viz. Breadth, then we may say that “ Surface represents Length and Breadth," just as we know it does in the case of the table of which we spoke above.

Two kinds of " 'Magnitudes," that is of things which can be measured, have now been mentioned, those namely the size of which depends only upon their length, and those of which the size depends upon two things-length and breadth. Is there any kind which depends upon more than two? There is one kind more, namely that, the size of which depends upon three "dimensions" as they are called, that is upon three things any one of

More

which may change its size without altering that of either of the two others. This third kind of magnitude is called a Solid," or better a "Volume," and the third dimension "Thickness." dimensions than these cannot be conceived, though why they cannot is beyond our present work to inquire, even if it be possible to discover. Thus far therefore we have obtained the following definitions::

OUR next step will be to find out, if possible, some relation between these three kinds of magnitude, the solid, the surface, and the line.

Suppose now that we have a pack of cards; this will be a solid, inasmuch as it has both length breadth and thickness; but the top card of it will be a surface, as is shown by the very fact of its being called the top card. If then the top card be taken away, the one immediately beneath it will become the top card in its turn and if this be done fifty-one times, each card will become the surface until the whole pack is reduced to one card, and the solid will itself have become a surface. But further, this last card may be pared and pared away without any limit to the fineness to which it may be reduced, and neither its length nor breadth need be altered at all. But, meanwhile, however small it may become, some thickness must be left unless we take away both length and breadth also. Hence we conclude that the existence of length and