260. DEF. A tangent from an external point to a circle is the part of the tangent between the external point and the point of contact.

PROPOSITION XII. THEOREM.

261. The tangents to a circle drawn from an external point are equal, and make equal angles with the line joining the point to the centre.

B

A

Let AB and AC be tangents from A to the circle whose centre is 0, and let AO be the line joining A to the centre O.

(a tangent to a circle is to the radius drawn to the point of contact). The rt. A OAB and OAC are equal.

§ 151

For OA is common, and the radii OB and OC are equal.

§ 217

.. AB

=

AC, and BAO = ▲ CAO.

§ 128

Q. E. D.

262. DEF. The line joining the centres of two circles is called the line of centres.

263. DEF. A tangent to two circles is called a common external tangent if it does not cut the line of centres, and a common internal tangent if it cuts the line of centres.

264. If two circles intersect each other, the line of centres is perpendicular to their common chord at its middle point.

Let C and C be the centres of the two circles, AB the common chord, and CC' the line of centres.

To prove that CC' is 1 to AB at its middle point.

.. C and C" are two points, each equidistant from A and B.

.. CC is the perpendicular bisector of AB.

§ 161

Q. E. D.

Ex. 92. Describe the relative position of two circles if the line of centres :

(1) is greater than the sum of the radii;

(2) is equal to the sum of the radii;

(3) is less than the sum but greater than the difference of the radii ;

(4) is equal to the difference of the radii;

(5) is less than the difference of the radii. Illustrate each case by a figure.

Ex. 93. The straight line drawn from the middle point of a chord to the middle point of its subtended arc is perpendicular to the chord.

Ex. 94. The line which passes through the middle points of two parallel chords passes through the centre of the circle.

PROPOSITION XIV. THEOREM.

265. If two circles are tangent to each other, the line of centres passes through the point of contact.

C

Let the two circles, whose centres are C and C', be tangent to the straight line AB at 0, and CC' the line of centres.

To prove that O is in the straight line CC'.

Proof. AL to AB, drawn through the point O, passes through the centres C and C',

§ 255

(a to a tangent at the point of contact passes through the centre

of the circle).

.. the line CC, having two points in common with this must coincide with it.

$ 47

.. O is in the straight line CC'.

Q. E. D.

Ex. 95. Describe the relative position of two circles if they may have: (1) two common external and two common internal tangents; (2) two common external tangents and one common internal tangent; (3) two common external tangents and no common internal tangent; (4) one common external and no common internal tangent;

(5) no common tangent.

Illustrate each case by a figure.

Ex. 96. The line drawn from the centre of a circle to the point of intersection of two tangents is the perpendicular bisector of the chord joining the points of contact.

MEASUREMENT.

266. To measure a quantity of any kind is to find the number of times it contains a known quantity of the same kind, called the unit of measure.

The number which shows the number of times a quantity contains the unit of measure is called the numerical measure of that quantity.

267. No quantity is great or small except by comparison with another quantity of the same kind. This comparison is made by finding the numerical measures of the two quantities in terms of a common unit, and then dividing one of the measures by the other.

In other words the ratio

The quotient is called their ratio. of two quantities of the same kind is the ratio of their numerical measures expressed in terms of a common unit.

The ratio of a to b is written ab, or

a

b

268. Two quantities that can be expressed in integers in terms of a common unit are said to be commensurable, and the exact value of their ratio can be found. The common unit is called their common measure, and each quantity is called a multiple of this common measure.

Thus, a common measure of 21 feet and 33 feet is of a foot, which is contained 15 times in 2 feet, and 22 times in 33 feet. Hence, 21 feet and 33 feet are multiples of of a foot, since 21 feet may be obtained by taking of a foot 15 times, and 33 feet by taking of a foot 22 times. The ratio of 2 feet to 3 feet is expressed by the fraction 15.

269. Two quantities of the same kind that cannot both be expressed in integers in terms of a common unit, are said to be incommensurable, and the exact value of their ratio cannot be found. But by taking the unit sufficiently small, an approximate value can be found that shall differ from the true value of the ratio by less than any assigned value, however small.

Thus, suppose the ratio,

Now √21.41421356

but less than 1.414214.

α

=√2.

a value greater than 1.414213,

If, then, a millionth part of b is taken as the unit of measure,

α

the value of lies between 1.414213 and 1.414214, and therefore b

differs from either of these values by less than 0.000001.

By carrying the decimal further, an approximate value may be found that will differ from the true value of the ratio by less than a billionth, a trillionth, or any other assigned value. , then the error in taking

m

In general, if > but b n

either of these values for

m + 1

n

a

b

between these two fractions.

1

nitely, can be decreased indefinitely, and a value of the ratio

n

can be found within any required degree of accuracy.

270. The ratio of two incommensurable quantities is called an incommensurable ratio; and is a fixed value which its successive approximate values constantly approach.

THE THEORY OF LIMITS.

271. When a quantity is regarded as having a fixed value throughout the same discussion, it is called a constant; but when it is regarded, under the conditions imposed upon it, as having different successive values, it is called a variable.

If a variable, by having different successive values, can be made to differ from a given constant by less than any assigned value, however small, but cannot be made absolutely equal to the constant, that constant is called the limit of the variable, and the variable is said to approach the constant as its limit.