But, as EC and AB are two chords that intersect each other in a

circle, we have,

Therefore,

cw=xy xy+c2=ab

But, as CD bisects the vertical angle, we have,

(th. 17, b. 3)

(2)

Now, as x and y are determined, the base is determined.

N. B. Observe that equation (2) is theorem 20, book 3.

PROBLEM 4.

To determine a triangle, from the base, the line bisecting the vertical angle, and the diameter of the circumscribing circle.

Describe the circle on the given diameter, AB, and divide it in two parts, in the point D, so that ADXDB shall be equal to the square of one half the given base.

Through D draw EDG at right angles to AB, and EG will be the given base of the triangle.

Put AD=n, DB=m, AB=d, DG=b.

Then, n+m=d, and nm=b2; and these two equations will determine n and m; and therefore, n and m we shall consider as known. Now, suppose EHG to be the required A, and join HIB and HA. The two As, AHB, DBI, are equiangular, and therefore, we have,

AB: HB:: IB: DB.

But HI is a given line, that we will represent by c; and if we put IB=w, we shall have HB=c+w; then the above proportion becomes, dc+ww: m

Now, w can be determined by a quadratic equation; and therefore, IB is a known line.

In the right angled ▲ DBI, the hypotenuse IB, and base DB, are known; therefore, DI is known (th. 36, b. 1); and if DI is known, EI and IG are known.

Lastly, let EH-x, HG=y, and put EI=p, and IG=q.

And, from equations (1) and (2) we can determine x and y, the sides of the A; and thus the determination has been attained, carefully and easily, step by step.

PROBLEM 5.

Three equal circles touch each other externally, and thus inclose one acre of ground; what is the diameter in rods of each of these circles?

Draw three equal circles to touch each other externally, and join the three centers, thus forming a triangle. The lines joining the centers will pass through the points of contact (th. 7, b. 3).

Let R represent the radius of these equal circles; then it is obvious that each side of this A is equal to 2R. The triangle is therefore

equilateral, and it incloses the given area, and three equal sectors. As each sector is a third of two right angles, the three sectors are, together, equal to a semicircle; but the area of a semicircle, whose яR2 2

radius is R, is expressed by (th. 3, b. 5, and th. 1, b. 5); and the

яR2
2

area of the whole triangle must be +160; but the area of the A is also equal to R multiplied by the perpendicular altitude, which

In a right angled triangle, having given the base and the sum of the

perpendicular and hypotenuse, to find these two sides.

PROBLEM 7.

Given, the base and altitude of a triangle, to divide it into three equal parts, by lines parallel to the base.

PROBLEM 8.

In any equilateral A, given the length of the three perpendiculars drawn from any point within, to the three sides, to determine the sides.

In a right angled triangle, having given the base (3), and the difference between the hypotenuse and perpendicular (1), to find both these two sides.

PROBLEM 10.

In a right angled triangle, haviny given the hypotenuse (5), and the difference between the base and perpendicular (1), to determine both these two sides.

Having given, the area or measure of the space of a rectangle inscribed in a given triangle, to determine the sides of the rectangle.

In a triangle, having given the ratio of the two sides, together with both the segments of the base, made by a perpendicular from the vertical angle, to determine the sides of the triangle.

In a triangle, having given the base, the sum of the other two sides, and the length of a line drawn from the vertical angle to the middle of the base, to find the sides of the triangle.

To determine a right angled triangle; having given the lengths of two lines drawn from the acute angles to the middle of the opposite sides.

PROBLEM 15.

To determine a right angled triangle; having given the perimeter, and the radius of its inscribed circle.

PROBLEM 16.

To determine a triangle; having given the base, the perpendicular, and the ratio of the two sides.

PROBLEM 17.

To determine a right angled triangle; having given the hypotenuse, and the side of the inscribed square.

To determine the radii of three equal circles, inscribed in a given circle, to touch each other, and also the circumference of the given circle.

PROBLEM 19.

In a right angled triangle, having given the perimeter, or sum of all the sides, and the perpendicular let fall from the right angle on the hypotenuse, to determine the triangle; that is, its sides.

To determine a right angled triangle; having given the hypotenuse and the difference of two lines, drawn from the two acute angles to the center of the inscribed circle.

PROBLEM 21.

To determine a triangle; having given the base, the perpendicular, and the difference of the two other sides.

PROBLEM 22.

To determine a triangle; having given the base, the perpendicular, and the rectangle, or product of the two sides.

To determine a triangle; having given the lengths of three lines drawn from the three angles to the middle of the opposite sides.

PROBLEM 24.

In a triangle, having given all the three sides, to find the radius of the inscribed circle.

To determine a right angled triangle; having given the side of the inscribed square, and the radius of the inscribed circle.

PROBLEM 26.

To determine a triangle, and the radius of the inscribed circle; having given the lengths of three lines drawn from the three angles to the center of that circle.

To determine a right angled triangle; having given the hypotenuse, and the radius of the inscribed circle.

THE 14th definition of book 1, defines a plane. It is a superfices, having length and breadth, but no thickness.

The surface of still water, the side of a sheet of paper, may give a person some idea of a plane.

A curved surface is not a plane; although we sometimes say, "the plane of the earth's surface."

1. If any two points be taken in a plane, and a straight line join the points, every point in that line is in the plane.

2. If any point in such a line should be either above or below the surface, such a surface would not be a plane.

3. A straight line is perpendicular to a plane, when it makes right angles with every straight line which it meets in that plane. 4. Two planes are perpendicular to each other when any straight line drawn in one of the planes, perpendicular to their common section, is perpendicular to the other plane.

5. If two planes cut each other, and from any point in the line of their common section, two straight lines be drawn, at right angles to that line, one in the one plane, and the other in the other plane, the angle contained by these two lines is the angle made by the planes.

6. A straight line is parallel to a plane when it does not meet the plane, though produced ever so far.

7. Planes are parallel to each other when they do not meet, though produced to any extent.

8. A solid angle is one which is formed by the meeting, in one point, of more than two plane angles, which are not in the same plane with each other.