76. Definitions. When a straight line is divided internally and externally in the same ratio, it is said to be divided harmonically. Thus, AB is divided harmonically

of C from A and B is equal to the ratio of the distances of D from A and B.

Since this proportion may also be written in the form

the ratio of the distances of A from C and D is equal to the ratio of the distances of B from C and D; consequently the line CD is divided harmonically at A and B.

The four points A, B, C, D, thus related, are called harmonic points, and A and B are called conjugate points, as also C and D.

PROPOSITION XXVI.-PROBLEM.

77. To divide a given straight line harmonically in a given ratio. Let it be required to divide AB harmonically in the ratio of M to N.

Upon the indefinite line AX, lay off AE = M, and from E lay off EF and EG, each equal to N; join FB, GB; and draw EC parallel to FB, ED parallel to GB.

MH

therefore, by the definition (76), AB is divided harmonically at C and D, and in the given ratio.

78. Scholium. If the extreme points A and D are given, and it is required to insert their conjugate harmonic points B and C, the harmonic ratio being given = M: N, we take on AX, as before, AE =M and EF = EG = N, join ED, and draw GB parallel to ED, which determines B; then, join FB and draw EC parallel to FB, which determines C.

Also if, of four harmonic points A, B, C, D, any three are given, the fourth can be found.

PROPOSITION XXVII.-PROBLEM.

79. To find the locus of all the points whose distances from two given points are in a given ratio.

Let A and B be the given points, and let the given ratio be M : N. Suppose the problem solved, and

that P is a point of the required locus. Divide AB internally at Cand externally at D, in the ratio M: N, and join PA, PB, PC, PD. By the condition imposed upon P we must have

CA : CB = ᎠᎪ : ᎠᏴ ;

B

E

PA: PB =M: N: = therefore, PC bisects the angle APB, and PD bisects the exterior angle BPE (23). But the bisectors PC and PD are perpendicular to each other (I. 25); therefore, the point P is the vertex of a right angle whose sides pass through the fixed points C and D, and the locus of P is the circumference of a circle described upon CD as a diameter (II. 59, 97). Hence, we derive the following

Construction. Divide AB harmonically, at C and D, in the given ratio (77), and upon CD as a diameter describe a circumference. This circumference is the required locus.

124

PROPOSITION XXVIII.-PROBLEM.

80. On a given straight line, to construct a polygon similar to a given

and A'B'C' equal to BAC and ABC respectively; then, the triangle A'B'C' will be similar to ABC (25). In the same manner construct the triangle A'D'C' similar to ADC, A'E'D' similar to AED, and A'E'F' similar to AEF. Then, A'B'C'D'E'F' is the required polygon (38).

PROPOSITION XXIX.-PROBLEM.

81. To construct a polygon similar to a given polygon, the ratio of similitude of the two polygons being given.

any one of these lines, as OA, take ОA' a fourth proportional to M, N, and OA, that is, so that

M: N = OA: OA'.

In the angle AOB draw A'B' parallel to AB; then, in the angle BOC, B'C' parallel to BC, and so on. will be similar to ABCDE; for the two

The polygon A'B'C'D'E' polygons will be composed

the same number of triangles, additive or subtractive, similarly aced; and their ratio of similitude will evidently be the given

tio M: N. (40).

82. Scholium. The point O in the preceding construction is called e centre of similitude of the two polygons.

11*

BOOK IV.

COMPARISON AND MEASUREMENT OF THE SURFACES OF RECTILINEAR FIGURES.

1. DEFINITION. The area of a surface is its numerical measure, referred to some other surface as the unit; in other words, it is the ratio of the surface to the unit of surface (II. 43).

The most con

The unit of surface is called the superficial unit. venient superficial unit is the square whose side is the linear unit. 2. Definition. Equivalent figures are those whose areas are equal.

PROPOSITION I.-THEOREM.

3. Two rectangles having equal altitudes are to each other as their bases.

Let ABCD, AEFD, be two rectangles having equal altitudes, AB and AE their bases; then,

F

B

E

Suppose the bases to have a common measure which is contained, for example, 7 times in AB, and 4 times in AE; so that if AB is divided into 7 equal parts, AE will contain 4 of these parts; then, we have

If, now, at the several points of division of the bases, we erect perpendiculars to them, the rectangle ABCD will be divided into 7