investigation of their laws; still it must be kept distinctly in view, that they are alike the properties of space, whether occupied or unoccupied by matter; and that they belong to matter simply from its occupying space.

e. EXTENSION and DIRECTION become objects of science as admitting comparison and measurement, or, in other words, as possessing DIMENSION". The general term MAGNITUDE' is applied to whatever can be measured. Hence there are two kinds of magnitudes treated of in Geometry, those which may be referred to extension, and those which may be referred to direction; and we may use the term alike, whether treating of pure space, or of bodies occupying space.

f. In geometrical statement and reasoning, it is convenient to employ figures addressed to the eye, termed diagrams", and to designate the magnitudes of which we are speaking by letters or other marks placed upon these diagrams. But it must be distinctly understood, that, in the language employed, we are not speaking of the diagrams themselves, but universally of all such magnitudes, whether material or of pure space, as agree with the data; and that the diagrams are merely employed to represent these magnitudes to the eye, and to furnish a convenient language for speaking of them. The reasoning of Geometry is not inductive, from particulars to generals; but deductive or demonstrative, from generals to other generals; and it were as reasonable to suppose that the geographer, in his descriptions, is speaking of his maps, or the biographer, in his narrative, of the portrait prefixed to his volume, as that the geometer, in his reasonings, is treating of his diagrams.

§ 2. EXTENSION has three dimensions; LENGTH, BREADTH, and THICKNESS.

REMARKS. a. By one measurement, we obtain LENGTH; by a second, across the first, BREADTH (or WIDTH); by a third, through the other two (as up or down, if the other two

(s) Lat. dimensio, from dimetior, to measure. (t) L. magnitudo, from magnus, great, large. (u) Gr. Sáypaμμa, drawing, from Sxy piqu, to draw out, delineate.

are horizontal), THICKNESS (or HEIGHT). From the very nature of space, no fourth dimension is possible.

b. In measurement, the first dimension obtained or contemplated is termed length; the second, breadth; and the third, thickness. Where there are two or three dimensions, it is evident that the order of measurement, and consequently the application of the terms, may be varied.

§ 3. The MAGNITUDES OF EXTENSION are of three kinds, according to the number of their dimensions; SOLIDS', which have three dimensions ; SURFACES, which have two; and LINES', which have only one.

Surfaces are the mere while they have length Give to a surface any

REMARKS. a. All portions of space, and all bodies occupying space, have, of necessity, length, breadth, and thickness, and are consequently solids. outsides of solids; and consequently, and breadth, can have no thickness. thickness, and it is no longer the mere outside of a solid, but itself a solid. Lines are the mere limits or edges of surfaces; and consequently can have only length, without breadth or thickness. Give to a line breadth, and it is no longer the mere limit of a surface, but itself a surface.

b. The extremities of lines are termed POINTS". These have position, but can have no dimension. Give to a point length, and it is no longer the mere end of a line, but itself a line.

c. Thus solids are bounded by surfaces; surfaces, by lines; and lines, by points. Even a surface dividing a solid is only the common boundary of two solids; a line dividing a surface, of two surfaces; and a point dividing a line, of two lines. Surfaces, lines, and points exist as really as solids. But they exist only in solids, as properties of those solids. Still there is nothing to forbid our conceiving and speaking

(v) Lat. solidus. (w) French, from sur, upon, and face, face, that which is upon the face or outside; so, in Lat., superficies, from super and facies. (x) L. linea. (y) Fr. point, Old Fr. poinct, from Lat. punctum.

D

d. From the idea of extension arises that of distance; and by combining distance and direction, we obtain that of position. The ideas of extension, and of direction or position, produce that of form. Change of position constitutes motion; which, if not an essential idea in Geometry, is of great importance as an auxiliary. Thus, each of the magnitudes of extension may be regarded as produced by the motion of its limit; a line, by the motion of a point; a surface, by the motion of a line; and a solid, by the motion of a surface.

a.) Draw, between the points A and B, a line of which the direction shall be everywhere the same; that is, so that, if any number of points, as C, D, E, be taken in the line, the direction from A to C shall be the same with that from A to B; the direction from C to D, the same; and in like manner that from D to E, from E to B, from A to D, from C to E, &c.; or, reckoning from B, so that the direction from B to E, from E to D, from D to C, from C to A, from B to D, &c., shall be the same with that from B to A.

(z) Lat. distantia, from disto, to stand apart from. (a) L. positio, from pono, to place. (b) L. forma. (c) L. motio, from moveo, to more.

A line, of which the direction is everywhere the same, is called a STRAIGHT LINE. — - Is AC a straight line? Is AD a straight line? DB? CB?

Can a straight line have any bend or crook in it?

II.

D

b.) Draw, from A to B, the straight line ACB. Draw, below it, a second line AEFB, bent at E and F, but straight from A to E, from E to F, and from F to B. How many bends has the line? Of how many straight lines is it composed? Have these straight lines the same direction? How many times does the direction change?

E

F

B

A bent line composed of straight lines is called a BROKEN LINE. What kind of a line is AE? EF? AEF? FB? EFB? ACB? AEFB?

Draw a broken line with 3 bends; with 5; with 7. Of how many straight lines is each composed?

How many times does the direction change in a broken line composed of 4 straight lines? of 5 of 7? of 12 ?

c.) Draw from A to B, in the figure above, a line, as ADB, bent at every point. Is any part of the line straight? Does it change its direction at every point? - A line of which no part is straight is called a CURVED LINE, or a CURVE.

Draw, from A to B, another curve outside of ADB; and another still outside of this.

If these lines are all curves, do they bend at every point? Do they all bend alike? Have they the same, or different forms?

Can curves differ in form? Can broken lines? Can straight lines?

d.) In the surface AC of the block AF,* do you

see any bend? If it has
no bend, is the direction
of the surface everywhere
the same? A surface
of which the direction is
everywhere the same is
called a PLANE SURFACE,
or a PLANE.
AE?

Does AC appear to be a plane? DF?

Is every perfectly flat surface a plane ?

e.) Is the direction of the surface AC, above, the same with that of DF? Taken together, do they make a plane or a bent surface? In a perfectly round ball or marble, is any part plane?

A bent surface composed of planes is called a BROKEN SURFACE; one of which no part is plane, a CURVED SURFACE. What kind of a surface has a globe? the roof of a house?

Mention bodies which have plane surfaces; broken surfaces; curved surfaces.

Take a piece of paper, and hold it so as to represent a plane surface. Bend it so as to represent a curved surface. Bend it so as to represent a broken surface.

II. B.

§ 4. DIRECTION belongs to points, lines, and surfaces. The direction of lines is termed LINEAR'; of surfaces, SUPERFICIAL®.

*Surfaces and solids are often named by simply taking two letters at opposite points.

(d) Lat. lineāris. (e) L. superficialis. See Note w, page 37.