of the four segments AC, CB, AD, DB, must be opposite in sign from the other three; and so, if attention be given to signs, we should write either

AC: CB: = AD : — DB,

or

AC: CBAD: BD.

Choosing the latter form, we may write

a formula of great value in the theory of Harmonic Division.

(1)

286. Another important relation among harmonic points on a line can be deduced in a similar way.

Let A, C, B, D be a har

monic range, A and B being A

harmonically separated by

M

C B

D

C and D, and let M be the mid-point of the line-segment CD.

Proof. Since the points A, C, B, D are harmonic, A, B and C, D being the pairs of conjugates,

AC AD

=

CB BD

Moreover, giving attention to the signs of segments,

Whence, multiplying by the common denominator and collecting terms,

AM. BMCM · CM = CM2.

EXERCISES

1. Show that two points of a straight line which are harmonically separated by two others are both on the same side of the mid-point of the line-segment determined by the two others.

2. The base of any triangle is cut harmonically by the bisectors of the internal and external vertical angles.

3. The hypotenuse of a right triangle is cut harmonically by two lines through the vertex of the right angle which make equal angles with one of the sides.

4. A straight line meets two intersecting circles at P, Q, R, S, and their common chord at 0; prove that OP, OQ, OR, OS, taken in a proper order, are proportional.

MISCELLANEOUS EXERCISES

1. If two triangles have one angle of the one equal to one angle of the other, and a second angle of the one supplementary to a second angle of the other, then the sides about the third angles are proportional.

2. AE bisects the vertical angle of the triangle ABC and meets the base in E; show that if circles are described about the triangles ABE and ACE, their diameters are to each other in the same ratio as the segments of the base.

3. Two circles touch internally at O; AB a chord of the larger circle touches the smaller at C which is cut by the lines OA, OB at P and Q ; show that OP: OQ = AC: CB.

4. If two triangles have their sides parallel in pairs, the straight lines joining their vertices meet in a point, or are parallel.

5. If any two similar polygons have three pairs of corresponding sides parallel, the straight lines joining the corresponding vertices meet in a point or are parallel.

6. If A, B, C, D are any four points on a circle and E, F, G, H are the mid-points of the arcs AB, BC, CD, DA, respectively, prove that the straight lines EG and FH are at right angles.

7. The sum of the perpendiculars drawn from any point within an equilateral triangle on the three sides is invariable.

8. Prove that the straight lines which trisect one angle of a triangle do not trisect the opposite side.

9. That part of any tangent to a circle which is intercepted between tangents at the extremities of a diameter is divided at the point of contact into segments such that the radius of the circle is a mean proportional between them.

10. If two chords AB and AC, drawn from a point A on a circle ABC, are produced to meet the tangent at the other extremity of the diameter through A, in the points D and E respectively, show that the triangle AED is similar to the triangle ABC.

11. On a circle of which AB is a diameter take any point P. Draw PC and PD on opposite sides of AP and equally inclined to it, meeting AB at C and D. Prove AC: BC: = AD: BD.

1. DEFINITIONS.

SUMMARY OF CHAPTER III

(1) To Measure to find out by experiment how many times a given magnitude will contain a chosen unit.

(2) Multiple of a Given Magnitude

§ 211.

a magnitude which will contain that magnitude an integral number of times. § 211. (3) Measure of a Given Magnitude—a magnitude which is contained in that magnitude an integral number of times. § 211. (4) Commensurable Magnitudes· such as can be measured with a common unit. § 213.

(5) Incommensurable Magnitudes — such as cannot be measured by any common unit. § 213.

216.

(6) Ratio of Two Quantities—their relative magnitude, i.e. how
many times one is as great as the other.
(7) Ratio of Two Commensurable Magnitudes
numerical measures by a common unit. § 220.

(8) Ratio of Two Incommensurable Magnitudes ·

the ratio of their

the limit which

the ratio of their approximate measures by a common unit approaches, as this unit is indefinitely diminished. § 229. (9) Limit of a Variable Quantity · a fixed quantity to which the variable approaches nearer than for any assignable difference, though it cannot be made absolutely identical with it. § 227. (10) Proportion · - a statement of the equality of two ratios. § 232. (11) Continued proportion, mean proportional, third proportional. See § 236.

(12) Mutually Equiangular Polygons· those having their angles equal, each to each, and in the same order. § 247. (13) Similar Polygons — two polygons which have the same number of sides, are mutually equiangular, and have their pairs of corresponding sides proportional. § 248.

(14) Radical Axis of Two Intersecting Circles—the line of their common chord. § 263.

(15) Coaxial System of Circles the circles through two fixed points. § 263.

(16) Division in Extreme and Mean Ratio — division of a line-segment into two parts, such that one of them is a mean proportional between the whole segment and the other part. § 274.

(17) Harmonic Division — division of a line-segment internally and externally in the same ratio. § 283.

2. POSTULATES.

(1) If P and Q are any two equal magnitudes, and R is a third magnitude of the same kind, then the ratio of P to R is equal to the ratio of to R, i.e. if P= Q, then P:R Q :R; and conversely, if P and Q are such that P:R= Q: R, then P= Q. (Postulate 6.) § 218.

=

(2) If P and Q are two unequal magnitudes, and R is a third magni-
tude of the same kind, then the ratio of P to R is greater or
less than the ratio of Q to R, according as P is greater or less
than Q.
(Postulate 7.) § 218.

3. PROBLEMS.

(1) To divide a given line-segment into any required number of equal parts. § 269.

(2) To divide a given line-segment into parts proportional to other given line-segments. § 270.

(3) To divide a given line-segment internally or externally in a given ratio. § 271.

(4) To find a fourth proportional to three given line-segments. § 272.

(5) To find a mean proportional between two given line-segments. $ 273.

(6) To divide a given line-segment in extreme and mean ratio. § 275. (7) Upon a given line-segment to construct a polygon similar to a given polygon, and such that the given line-segment shall be homologous to a given side.

§ 276.

(8) To find the locus of a point whose distances from two fixed points are in a constant ratio different from unity. § 282.

4. THEOREM ON LIMITS.

If there are two variable quantities dependent on the same quantity in such a way that they remain always equal while each approaches a limit, then their limits are equal. § 230.

5. THEOREMS ON PROPORTION.

(1) If four numbers are in proportion, the product of the extremes equals the product of the means. § 233.

(2) If a: b = c: d, then by inversion bad: c, and by alternation

a:cb: d. § 234.

(3) If a : b = c:d, then by composition a + b: b

=c+d:d, by division ab:bc-d:d, and by composition and division a+ba-b=c+d:c - d. § 237.