Section 2. The Theory of Limits.

DEFINITIONS. A quantity is called a variable if, in the course of the same investigation, it may take indefinitely many values; on the other hand, a quantity is called a constant if, in the course of the same investigation, it keeps the same value.

E.g. if a line AB is bisected at M1, and M1B at M2, and M2B at Ms, and so on, and if x represents the line from A to any of the points M1, M2, then x is a variable, but AB is a constant.

It is customary, as in algebra, to represent variables by the last letters of the alphabet, and constants by the first letters.

If a variable x approaches nearer and nearer a constant a, so that the difference between x and a can become and remain smaller than any quantity that may be assigned, then a is called the limit of x.

E.g. in the above figure, AB is the limit of x.

That "x approaches as its limit a" is indicated by the symbol x = a. COROLLARY. If xa, then a x is a variable whose limit is zero;

that is, a x ≤ 0.

Theorem of Limits. If, while approaching their respective limits, two variables have a constant ratio, their limits have that same ratio.

B

Given

X and X', two variables, such that as they increase they approach their respective limits AB, or L, and AC, or L', and have a constant ratio r.

To prove
Proof.

that L L'r, or that X: X' = L : L'

If the ratio X: X' is not equal to the ratio L: L', then

(1) it must equal the ratio of L to something less than L', or (2) it must equal the ratio of L to something greater than L'.

It will be shown that both of these suppositions are absurd. I. To show that (1) is absurd.

3. Then

... X=rX'

Lr (L'

DC).

L— X=r (L' — DC — X').

4. But L'X' may be as small as we please,

Def. ratio

Ax. 3

5. .. a positive quantity L-X would equal a negative quantity r(L'— DC — X').

For LX, so that LX is positive,

and if L'X' < DC, then r (L' — DC — X') is negative.

II. To show that (2) is absurd.

1. Suppose X: X' L:L'+ CD'.

2. Then

=

L-X=r(L'+ CD'- X'), as in 3, above. 3. But because X' always <L', ... L'— X'— something, and.. r (L'+ CD'X') always > r. CD'.

4. But LX0, because L is the limit of X.

5. ... an indefinitely small quantity, L-X, would equal a quantity greater than r CD'.

COROLLARIES.

1. If, while approaching their respective limits, two variables are always equal, their limits are equal. For their ratio is always 1.

2. If, while approaching their respective limits, two variables have a constant ratio, and one of them is always greater than the other, the limit of the first is greater than the limit of the second.

Section 3.

A Pencil of Lines Cut by Parallels.

DEFINITIONS. Through a point any number of lines can be passed. Such lines are said to form a pencil of lines. The point through which a pencil of lines passes is called the vertex of the pencil.

C
B

A pencil of three lines.

A pencil of four parallels.

The annexed pencil of three lines is named “V-ABC”.

To conform to the Principle of Continuity, the word pencil is also applied to parallel lines, the vertex being spoken of as "at infinity ".

Theorem 1. The segments of a transversal of a pencil of parallels are proportional to the corresponding segments of any other transversal of the same pencil. Given the pencil of parallels P,

cutting from two transversals T and T' the segments A, B, and C, D, respectively.

m

NOTE. The preceding proof assumes that A and B are commensurable. The following proof is valid if A and B are incommensurable.

ence, by increasing n, .. to assume any difference leads to an absurdity.

DEFINITION. Two lines are said to be divided proportionally when the segments of the one have the same ratio as the corresponding segments of the other.

COROLLARIES. 1. A line parallel to one side of a triangle

divides the other two sides proportionally.

For in the annexed figure, if BCO is the triangle, the lines OB, OC are cut by parallels. Hence BB1 B10 CC1 : C10, and so for B2, C2, and for B3, C3.

2. In the annexed figure, AB : A1B1 = OA: OA1.

Proved by drawing from A1 a line parallel to OB; it then follows from th. 1. Proof and general statement of the corollary left for the student.

=

C, BA,

Αν

Α

B C

D

A2

B2

C2

3. In the annexed figure, AB BC A,B, By cor. 2, AB: A1B1 this corollary is true.

=

=

BO: B10 BC: B1C1. Hence, by fund. prop. III, Give the general statement of the corollary.

4. In the annexed figure, OA:OA1=OB:OB1=OC:OC1=· Give the general statement and proof.

EXERCISES. 391. What is the limit of 1/x as x increases indefinitely? of 1/(1+x) as x0? as x=1?

=

392. In ABC, P is any point in AB, and Q is such a point in CA that CQ PB; if PQ and BC, produced if necessary, meet at X, prove that CA AB PX : QX. (From P draw a line || AC.)

=

393. In the annexed figure of a "Diagonal Scale," AB is 1 centimeter. Show how, by means of the scale and a pair of dividers, to lay off 1 millimeter, 0.5 millimeter, 0.3 millimeter, etc. On what proposition or corollary does this measurement of fractions of a millimeter depend?

394. Show that the diagonals of a trapezoid cut each other in the same ratio.

10

9

7

6

3

B

395. ABCD is a parallelogram; from A a line is drawn cutting BD in E, BC in F, and DC produced in G. Prove that AE is a mean proportional between EF and EG.

396. ABC is a triangle, and through D, any point in c, DE is drawn Il a to meet b in E; through C, CF is drawn || EB to meet c produced in Prove that AB is a mean proportional between AD and AF.

F.