LIMITS.

Thus far the magnitudes considered have been of fixed greatness only. In Art. 227 we had occasion, however, to discuss certain approximate ratios tending towards a value they can never exactly attain, though they can approach it indefinitely near. In these we have examples of what are termed a variable and its limit, now about to be defined.

233. A constant is a magnitude or quantity whose greatness remains the same, neither increasing nor decreasing. 234. A variable is a magnitude or quantity whose greatness goes on increasing or decreasing.

A

Thus a chord AB, as long as it passes through the center o, remains constant in every position. But if it turn about an extremity 4, we have a decreasing variable chord AC, AC', etc., while in the angles BAC, etc., and in the arcs BC, BC', etc., we have increasing variables.

235. If a variable increases or decreases so as to approach indefinitely near to an equality with a certain constant, this constant is called the limit of the variable.

Thus in the figure above, the variable chord decreases towards zero as limit, the variable angle increases towards a right angle as limit, the variable arc increases towards a semicircumference as limit, and the distance of the variable chord from the center increases towards the radius as limit, none of which limits, evidently, can be attained as long as there is any chord.

Again, if a point P Q

move along from o

P P'

P" P" N

towards N, the distances OP and PN are variables; the one increasing towards ON, the other decreasing towards zero. If no condition were imposed upon the motion of P, it might

reach N, or even pass beyond it, and the distance ON would not be a limit of OP according to the definition. But if we impose the condition that at the end of the first second it reach P', half way to

N, at the end of the ọ next second reach P,"

P P'

"P" N

half way between P' and N, and so on, it is evident that P could never reach N, though it might come indefinitely near to it. For the fraction of the distance passed over in n seconds would be the sum of the series ++} + + a series that has 1 for its limit, a limit the series can never attain, no matter how many terms may be taken.

1

16

PROPOSITION I. THEOREM.

236. If two variables tending towards limits are always equal, these limits are also equal.

A

B

M

D P

N

Given: Two equal variables AM and BN, tending towards limits AP and BQ;

as

For if AP could be greater than BQ, some part of AP, AD, would be equal to BQ. stant difference DP could be, approach AP nearer than any constant difference (Hyp.), AM would become greater than AD or its equal BQ, while BN would always remain less than BQ (Hyp.). That is, BN would be both equal to and less than AM. In the same way it can be proved that BQ cannot be greater than AP. Hence BQ must be equal to AP. Q.E.D.

Now, however small the consince AM can increase so as to

PROPORTION. THEOREMS.

PROPOSITION II. THEOREM.

237. If four numbers are in proportion, the product of the extremes is equal to the product of the means.

238. COR. If the means are equal, that is, if

Q.E.D.

(237)

i.e., the product of the extremes is equal to the square of the mean.

239. DEFINITION. Here b is said to be a mean proportional between a and c, and c is a third proportional to a and b.

EXERCISE 295. If the bisector of an angle of a parallelogram passes through the opposite vertex, the figure must be equilateral.

296. Any parallelogram that can be circumscribed about a circle must be equilateral.

297. If each of three equal circles is tangent to the other two, their lines of centers form an equilateral triangle.

298. In triangle ABC, if D, E, the mid points of AB, AC, be joined, show that ▲ ABC: trapezoid DBCE = 4:3.

299. If A is an angle of an equilateral triangle and B is a right angle, show that ZA: ZB = 2: 3.

300. In the diagram for Art. 153, if / BAC: Z ACB = 7:8, and if LBAC: ZABC= 7: 3, show that ZAOC: Z ABC = 7:2.

301. Hence deduce the ratio of angle BOC to angle BAC and of angle AOB to angle ACB.

PROPOSITION III. THEOREM.

240. Conversely, if the product of two numbers is equal to the product of two others, either two may be made the extremes of a proportion, and the other two its means.

241. SCHOLIUM. If, instead of by bd, we divide by ab, ac,

EXERCISE 302. In the first diagram for Prop. XIII. (195), prove that if AC and BD be joined, angle BAC will be equal to angle ABD. 303. In the same diagram, show that if AD and BC be joined, they will intersect in MN.

304. In the third diagram for the same proposition, if a perpendicular be drawn through the mid point of EF, it will pass through M and N.

305. If two tangents to the same circle make equal angles with an intercept between them that passes through the center, they are equal.

306. If two secants to the same circle make equal angles with an intercept between them that passes through the center, they are equal.

PROPOSITION IV. THEOREM.

242. The products of the corresponding terms of two or more numerical proportions are in proportion.

In the same way the theorem can be proved for any number of proportions.

PROPOSITION V. THEOREM.

243. If four numbers are in proportion, like powers or like roots of those numbers are in proportion.

1 1

1

1

To Prove: a" : b" = c" : d", and a": b” = cn: dr.

.. taking like powers or like roots of both members,