the angle E, and the angle C = the angle F; also, the like sides AB, DE, and AC, DF, being those opposite the equal angles; then will the rectangle of AB, DF be equal to the rectangle of AC, DE.

In BA, produced, take AG equal to DF; and through the three points B, C, G, conceive a circle BCGH to be described, meeting CA, produced, at H, and join GH.

Then the angle G is equal to the angle C on the same arc BH, and the angle H equal to the angle B on the same arc CG (th. 39); also, the opposite angles at A are equal (th. 7): therefore the triangle AGH is equiangular to the triangle ACB, and consequently to the triangle DFE also. But the two like sides AG, DF are also equal, by construction; consequently, the two triangles AGH, DFE are identical (th. 2), and have the two sides AG, AH equal to the two DF, DE, each to each.

But the rectangle GA. AB is equal to the rectangle HA. AC (th. 42); consequently, the rectangle DF. AB is equal to the rectangle DĚ. AC. Q. E. D.

EXERCISES.

1. Find the length of an arc of 20° 45′ to a radius of 10.

2. Through two given points, to draw a circumference of given radius.

3. Divide an arc into 2, 4, 8, 16.... equal parts.

4. Prove that every other chord is less than the diameter.

5. That parallel tangents include semicircumferences between their points of contact.

6. Draw a tangent to a given circle parallel to a given line.

7. To describe a circle of given radius tangent to a given line at a given point.

8. To describe a circle of given radius touching the two sides of a given angle.

9. To describe a circumference which shall be embraced between two parallels, and pass through a given point.

10. To place a chord of given length and direction in a given circle.

11. Prove that the chords of equal arcs are equal, and the converse.

12. To find in one side of a triangle the center of a circle which shall tonch the other two sides.

13. To find the radius of a circle when a chord and perpendicular from the centre to the chord are given.

14. With given radii to describe two circumferences which shall intersect in a given point, and have their centers in a given line.

15. With given radii to describe two circles which shall touch each other either externally or internally.

16. Three circles with equal given radii touching each other externally.

17. The same with unequal radii.

18. Through a given point on a circumference, and another given point without, to describe a circle touching the given circumference. 19. The same when, instead of the point upon the circumference, the radius of the required circle is given.

20. To describe a circle of given radius touching two given circles. 21. To construct a right-angled triangle with the hypothenuse and one of the perpendicular sides given.

22. In a given circle to inscribe a right angle, one side of which is given.

23. In a given circle to construct an inscribed triangle of given altitude and vertical angle.

24. Also, a quadrangle, when one side and two angles not adjacent this side are given. (See Exercise 31, below.)

25. To find the center of a circle in which two given lines meeting in a point shall be a tangent and chord.

26. In a given circle to inscribe a triangle equiangular to a given triangle.

27. Show how to circumscribe a square about a given circle, and how to inscribe a circle in a given square.

28. That a straight line touching a circle can have with it but one point of contact.

29. To inscribe in an equilateral triangle three equal circles touching each other, and the sides of the triangle.

30. Prove that an eccentric angle is measured by half the sum of the opposite arcs subtending it, if the vertex be within the circle; and by half the difference of the arcs if it be without.

31. That the opposite angles of an inscribed quadrilateral are supplements.

32. That if one of the sides of an inscribed quadrilateral be produced out, the outward angle will be equal to the inward opposite angle. 33. That the sums of the opposite sides are equal.

34. That a regular polygon may be circumscribed with a circle. 35. That a circle may be inscribed in any regular polygon.

36. If one circle touch another externally or internally, any straight line drawn through the point of contact will cut off similar segments.* 37. Prove that only one tangent can be drawn to a circle at a given point on the circumference.

38. That of two chords the greater is nearer the center of the circle.

Numerical Problems.

1. In a triangle suppose two of the sides to be 8.76 and 5.26, and the perpendicular from the vertex in which they meet 4.38; required the third side.

Suppose the two segments of the required side to be represented by z and

or,

y.

x=√ (8.76)2 — (4.38)3, y=√ (5.26)o — (4.38)2,

≈=√(8.76+4.38) (8.76—4.38), y=√(5.26 +4.38) (5.26 −4.38). log. 13.14

1.1185954

y=2.9126 ..x+y=10.4990

for the value of the third side if the perpendicular falls within the triangle; and

x~y=4.67

for the value if the perpendicular falls without.

2. Given in a triangle the base 88; one of the sides 128.49; and the perpendicular upon the base from the vertex opposite 96.45, to find the third side. Ans. 96.50.

3. Two chords cut each other in a circle; the segments of the one are 13 and 25; the segments of the other are in the ratio of 4 to 7; required the length of the latter chord. Ans. 37.47.

4. To find the absolute length of an arc of 45° 20′ in a circle whose radius is 5.4, supposing the ratio of the circumference to the diameter of a circle to be 3.1416. Ans. 4.2726.

5. The side of a square being given 0.25, to find the side of an equilateral triangle equal to the square. Ans. 0.37994.

6. Given the area of a circle 33.1830, to find its radius.

Ans. 3.25.

7. Find the chord of the sum of two arcs, the chords of the arcs

being given 10 and 12, and the radius 16.

8. Find the chord of half an arc, the chord of the whole arc being 12 and 16.

* Similar segments are those which correspond to similar arcs.

C

OF RATIOS AND PROPORTIONS.

DEFINITIONS.

DEF. 75. RATIO is the proportion or relation which one magnitude bears to another magnitude of the same kind with respect to quantity.

Note. The measure or quantity of a ratio is conceived by considering what part or parts the leading quantity, called the Antecedent, is of the other, called the Consequent ;* or what part or parts the number expressing the quantity of the former is of the number denoting, in like manner, the latter. So the ratio of a quantity, expressed by the number 2 to a like quantity expressed by the number 6, is denoted by 2 divided by 6, or or; the number 2 being 3 times contained in 6, or the third part of it. In like manner, the ratio of the quantity, 3 to 6, is measured by or; the ratio of 4 to 6 is or; that of 6 to 4 is or, &c. The ratio of two lines is the ratio of the number of times which each contains the common measure of the two lines. When the terms of a ratio are equal, it is called a ratio of Equality. When unequal, a ratio of Inequality.

76. Proportion is an equality of ratios. Thus,

77. Three quantities are said to be proportional when the ratio of the first to the second is equal to the ratio of the second to the third. As of the three quantities, A2, B=4, C=8, where ==, both the same ratio.

78. Four quantities are said to be proportional when the ratio of the first to the second is the same as the ratio of the third to the fourth. As of the four, A (4), B (2), C (10), D (5), where 2, both the same ratio.

=

=

* The antecedent and consequent are called the terms of a ratio.

Note. To denote that four quantities, A, B, C, D, are proportional, they are usually stated or placed thus, A:B::C:D; and read thus, A is to B as C is to D. The two dots: must be understood as representing the sign of division; the four dots :: the sign of equality. The same proportion or equality of ratios or A: B-C: D. When

may be written thus,

A

B

=

C

D'

three quantities are proportional, the middle one is repeated, and they are written thus, A: B:: B: C.

79. Of three proportional quantities, the middle one is said to be a Mean Proportional between the other two; and the last a Third Proportional to the first and second.

80. Of four proportional quantities, the last is said to be a Fourth Proportional to the other three, taken in order.

81. Quantities are said to be Continually Proportional, or in Continued Proportion, when the ratio is the same between every two adjacent terms, viz., when the first is to the second as the second to the third, as the third to the fourth, as the fourth to the fifth, and so on, all in the same common ratio.

As in the quantities 1, 2, 4, 8, 16, &c., where the common ratio is equal to 2.

82. Of any number of quantities, A, B, C, D, the ratio of the first A, to the last D, is said to be compounded of the ratios of the first to the second, of the second to the third, and so on to the last.

83. Inverse ratio is, where the antecedent is made the consequent, and the consequent the antecedent. Thus, if 1:2::3:6; then inversely, or by inversion, 2:1::6:3.

84. Alternate proportion is where antecedent is compared with antecedent, and consequent with consequent. As, if 1:2::3:6; then, by alternation or permutation, it will be 1:3::2:6.

85. Compound ratio is, where the sum of the antecedent and consequent is compared either with the consequent or with the antecedent. Thus, if 1:2::