a.) Take this block of wood (represented above),

and tell me if it has any length. Has it any breadth, or width?

is it?

How long is it?

How broad, or wide,

Has it any thickness, or height? How thick, or high, is it?

LENGTH, BREADTH, and THICKNESS (or height) are called DIMENSIONS. How many dimensions has the

block? What are they?

Can you find a block of wood which has not length, breadth, and thickness? Can you find a piece of any other matter which has not?

How many dimensions, then, has every block, or piece of matter?

b.) Has this room any length? any width? any height? How many dimensions then has it?

What is this room? ANSWER. A portion of space, inclosed within the sides, ceiling, and floor.

Has the space in that box any length? any breadth?

any height (or depth, which is the same dimension, measured down)?

How many dimensions has the space in a drawer ? in a cellar ? Can you find or think of any portion of space which has not these three dimensions?

c.) That which has LENGTH, BREADTH, and THICKNESS is called a SOLID. - In Geometry, whatever has the three dimensions of length, breadth (or width), and thickness (or height), is called a solid, whether it be hard or soft, whether matter or mere space.

Is this block a solid? this book? Can you find any piece of matter which is not a solid ? According to the definition, is this room a solid? the space in a box or drawer? every portion of space?

d.) How many FACES or SURFACES has this block? Which is the upper surface? which, the under? Which are the largest surfaces? which, the smallest ?

Take the upper surface (which is marked above by the letters A, B, C, D at the corners), and tell me if it has any length. How long is it?

Has it

any

breadth? How broad is it?

Has it any thickness? Is the surface all upon the outside? Does it then go down into the wood at all? Can it then have any thickness?

How many dimensions then has it? Which dimension does it want?

How long is the surface marked by the letters C, D, E, F at the corners ? how broad? how thick ?

How long is the surface marked ADEG? how broad? how thick?

Can any surface, as it is nothing but outside, have any thickness?

How many dimensions then has a surface? What are they? Which does it want?

If you define a SOLID, as above (c), that which has, &c.; how would you define a SURFACE? ANS. That which has without (in answering, fill up the blanks with the proper words).

and

e.) How many EDGES has the upper surface of the block (represented above by ABCD), or, in other words, how many LINES are there round it? Point them out, and likewise name them by the letters at the corners, as the line AB, the line BC, &c.

Has the edge, or boundary-line AB, any length? How long is it?

Has it any breadth, or width? Does it all lie at the very outside of the surface ? Can it then come into the surface at all? Can it then have any breadth ?

As it is merely the boundary-line of a surface, can it have any thickness?

How many dimensions then has it? which? Which dimensions does it want?

Has the line AD any length? any breadth? any thickness?

How long is the line BC? how broad? how thick? Can any boundary-line of a surface have any breadth or thickness? — The only lines spoken of in Geometry are boundary-lines of surfaces, since a division-line is only a boundary-line between two surfaces.

What dimension then has a line? What dimensions does it want?

If you define a SOLID, that which has, &c. (c); and a SURFACE, that which has, &c. (d); how would you define a LINE? ANS. That which has

or

,

without

f.) How many ENDS, or END-POINTS, has the line AB? These may be called, from the letters near them, the point A, and the point B.

As the point A is merely the end of a line, can it have any breadth ? any thickness?

Has it any length? As it is only the end of the line, does it extend into the line at all?

any length?

Has it then any dimension?

Can it then have

What property then has it? ANs. Nothing but position, that is, a place at the end of the line.

Has the point B any length, breadth, or thickness? Can the end-point of any line have any dimension? -The only points spoken of in Geometry are the end-points of lines, since a division-point is only the common end-point of two lines.

What is then the only property which a point has? What are the three dimensions which it wants?

How then would you define a POINT? ANS. That which has but neither

I. B.

nor

§ 1. GEOMETRY" is the science" of EXTENSION and DIRECTION".

(a) From the Greek Tswμsтpia, compounded of yй, land, and μergiw, to measure. The science was so called from its early application to the measuring of land, especially among the Egyptians, to whom it was peculiarly important as a means of ascertaining their boundaries after the inundations of the Nile. See Herodotus, ii. 109. (b) From the Latin scientia, knowl edge. (c) Lat. extensio, from extendo, to stretch out. (d) L. directio, from

REMARKS.

a. Geometry, as a science, first defines the objects of which it treats; and states or implies certain selfevident truths in respect to them. These truths are termed AXIOMS', and are the primary laws of the reasoning which it employs. From these definitions and axioms, it then, by a method of strict proof, or demonstration, deduces THEOREMS", that is, general truths or laws established by proof. It lastly shows the application of its truths to the solution or performance of PROBLEMS'.

b. The DEFINITIONS, AXIOMS, and THEOREMS constitute the theoretical part of the science; and the PROBLEMS, the practical. The theorems and problems, from the usual method of presenting them, have been classed together under the general head of PROPOSITIONS; the theorem proposing something to be proved, and the problem, something to be performed. A LEMMA' is a theorem introductory to another theorem, or to a problem. A COROLLARY is an inference from that which precedes. A SCHOLIUM" is an accompanying

remark.

c. In every proposition, it is essential to distinguish accurately between that which is given, and that which is required. Things given are called DATAo; and the data of a proposition are termed its HYPOTHESISP. A supposition not included in the data is termed an ASSUMPTION".

d. EXTENSION and DIRECTION are simple ideas, and are implied in every conception of space, or of bodies occupying space. They are first suggested to the mind by material objects, and these objects afterwards assist the mind in the

dirigo, to direct. (e) Lat. definio, to limit. (f) Gr. žiμ, from žiów, to deem worthy, suppose, take for granted. (g) L. demonstratio, from demonstro, to point out, prove. (h) Gr. Gwenu, from 6swpiw, to view, contemplate. (i) Gr. рóbλnua, from рobáλλ, to throw or lay before. (k) L. propositio, from propōno, to place before, to propose. (1) Gr. añμμa, premise, from λaubávw, to take. (m) L. corollarium, something given over and above, from corolla, a wreath, a common present or mark of honor. (n) Gr. oxorov, comment, from oxox, leisure, in a special sense, leisure devoted to learning, school. (0) L., from do, to give. (p) Gr. væóbeσis, foundation, from voτiêu, to place beneath, lay down. (q) L. assumptio, from assümo, to assume. (r) L. spatium.