Now, since the area or space of a rectangle, is expressed by the product of the base and height (cor. 2, th. 81, Geom.), and since a triangle is equal to half a rectangle of equal base and height (cor. 1, th. 26), it follows that,

But the three last triangles make up, or are equal to, the whole former or great triangle;

that is, x2√3 = ax + bx + cx: hence, dividing by x, gives

x √3 = a+b+c, and dividing by √3, gives

x=

a+b+c. half the side of the triangle sought.
√3

,

Also, since the whole perpendicular CH is = x√3, it is therefore = a + b+c. That is, the whole perpendicular CH, is just equal to the sum of all the three smaller perpendiculars DE+ DF + DG taken together, wherever the point D is situated.

PROBLEM VI.

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

PROBLEM VII.

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

PROBLEM VIII,

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

PROBLEM IX.

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.

PROBLEM X.

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.

PROBLEM XI.

In a triangle, having given the two sides about the vertical angle, with the line bisecting that angle, and terminating in the base; to find the base.

PROBLEM XII.

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 XIII.

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

PROBLEM XIV.

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

PROBLEM XV.

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

PROBLEM XVI.

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

PROBLEM XVII.

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 hypothenuse; to determine the triangle, that is, its sides.

PROBLEM XVIII.

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

PROBLEM XIX.

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

PROBLEM XX.

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

PROBLEM XXI.

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

PROBLEM XXII.

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

PROBLEM XXIII.

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

PROBLEM XXIV.

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 centre of that circle.

PROBLEM XXV.

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

PROBLEM XXVI.

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

PROBLEMS ON MAXIMA AND MINIMA.

TO BE SOLVED GEOMETRICALLY.

1. Divide a right line into two parts so that their rectangle shall be a maximum.

2. Find a point in a given straight line, from which if two straight lines be drawn to two given points on the same side of the given line, and in the same plane with it, their sum shall be a maximum.

3. Let ABC be a right-angled triangle of which AB is the hypothenuse. Draw through the angular point, C, a right line such, that the sum of two perpendiculars let fall upon it from A and B, respectively, shall be a minimum.

4. Through a given point within a circle, which is not the centre, to draw the least chord.

5. Through either of the points of intersection of two given circles that cut each other, to draw the greatest of all straight lines, passing through that point, and terminated both ways by the two circumferences.

6. Two semicircles whose radii are in a known ratio, lie on contrary sides of the same right line, the circumference of one terminating in the centre of the other. Draw the greatest right line perpendicular to the common diametral line, and terminated both ways by the two curves.

7. Through a given point in a given circle, out of the centre, draw a chord which shall cut off the least segment.

8. To find a point in the circumference of a given circle, at which any given straight line drawn from the centre, but less than the radius of the circle, shall subtend the greatest angle.

9. Given the base and the ratio of the sides, to determine the triangle when its area is a maximum,·

CONIC SECTIONS.

THERE are three curves, whose properties are extensively applied in Mathematical investigations, which, from a circumstance hereafter to be explained, are called the Conic Sections. These are,

1. THE PARABOLA.

2. THE ELLIPSE.

3. THE HYPERBOLA.

Before entering upon the discussion of their properties, it may be useful to enumerate the more useful theorems of proportion which have been proved in the treatises on Algebra and Geometry, or which are immediately deducible from those already established. For convenience in reference, they may be

PARABOLA.

DEFINITIONS.

1. A PARABOLA is a plane curve, such, that if from any point in the curve two straight lines be drawn; one to a given fixed point, the other perpendicular to a straight line given in position: these two straight lines will always be equal to one another.

2. The given fixed point is called the focus of the parabola.

3. The straight line given in position, is called the directrix of the parabola.

4. A straight line drawn perpendicular to

the directrix, and cutting the curve, is called a diameter; and the point in which it cuts the curve is called the vertex of the diameter.

5. The diameter which passes through the focus is called the axis, and the point in which it cuts the curve is called the principal vertex.

Thus: draw N, P, W1, Ng P2 W2, N3 Pa W3, KASX, through the points P1, P2, P3, S, perpendicular to the directrix; each of these lines is a diameter; P1, P2, P3, A, are the vertices of these diameters; ASX is the axis of the parabola, A the principal vertex.

6. A straight line which meets the curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point.

N

PL

Ni

Wi

N2

P

-W2

KA

S

X

Na

W3

P3

7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter.

8. The ordinate which passes through the focus, is called the parameter of that diameter.