GO, FO, at the middle of the chords, will meet in its. center O (schol. to th. 34), and the distances OG, OF, &c., of these chords are equal (th. 35).

PROBLEM XXXVI.

On a given line to construct a rectilinear figure similar to a given rectilinear figure.

Let abcde be the given rectilinear figure, and AB the side of the proposed similar figure that is similarly posited with ab.

E

D

B

a

Place AB in the prolongation of ab, or parallel to it. A Draw AC, AD, AE, &c. parallel to ac, ad, ae, respectively. Draw BC parallel to bc, meeting AC in C; CD parallel to cd, and meeting AD in D; DE parallel to de, and meeting AE in E; and so on till the figure is completed. Then ABCDE will be similar to abcde, from the nature of parallel lines and similar figures (th. 68).

Otherwise, divide the given figure up into triangles as in the diagram; then upon ab make the triangle abc equiangular with the triangle ABC, and upon ac, acd, equiangular with ACD, and so on till the figure is completed. (See the demonstr. of th. 68.)

1

GENERAL NOTE UPON THE METHOD OF SOLUTION OF PROBLEMS.

In every problem of Plane Geometry, it is necessary to trace upon a plane, in accordance with given conditions, one or more right lines or curves, one or more angles, one or more points.

The problem can be solved by the aid of the rule and compass, if the entire system of lines to be traced, and the lines of construction, are reduced to a system of right lines and circumferences of circles.

A line is determined when two of its points are known; a circumference when its center and a point of it, or when three of its points; and an angle when the two sides, or the vertex and another point in each of the sides. The tracing, then, of a system of lines, circles, angles, and points, and, consequently, the solution of a problem of Geometry, when this problem is resolvable by the aid of the rule and compass, can be reduced to the determination of a certain number of unknown points.

We may call that a simple problem which is reduced to the determination of a single unknown point, and that a compound problem which requires the determination of several points. For a compound problem, the nature of the solution may vary not only with the number and nature of the points proposed to be determined, but also with the order of their determination; and it is easy to perceive from hence how the same problem of Geometry admits of different solutions more or less elegant. But as the different unknown points must be determined one after another, it is clear that, to resolve a compound problem, it is only necessary to resolve successively a number of simple problems.

It remains to consider how a simple problem is to be solved.

In every simple problem the unknown point is generally determined by two conditions. By virtue of one of these conditions alone the unknown point is not completely determined; it will only be subjected to the necessity of coinciding with one of the points situated upon a certain right line or curve corresponding to this condition. But if we have regard to the conditions united, the unknown point must be situated, at the same time, upon the two lines corresponding to the two conditions, and can, therefore, only be at one of the points common to these two lines. Then, if the two lines do not meet, the proposed geometrical problem is impossible, or incapable of solution. It will admit of a single solution, if the two lines meet in a single point; it will admit of several distinct solutions, if the two lines intersect in several points. Thus a simple determinate problem may be considered as resulting from the combination of two other simple problems, but indeterminate, each of which consists in finding a point which fulfills a single condition, or, rather, the geometric locus of all the points (infinite in number) which fulfill the given condition. If this condition is reduced to that of the unknown point being found upon a certain line, the geometric locus sought will evidently be this line itself. It may be added, that very often the geometric locus corresponding to a given condition will comprehend the system of a number of right lines or curves. Thus, for instance, if the unknown

point is required to be at a given distance from a given right line, the geometric locus sought will be the system of two parallel lines drawn at the given distance from this line.

Let it be observed, moreover, that a simple problem, determinate or indeterminate, will be resolvable by the rule and dividers, if each of the geometric loci which serve to resolve it is reduced to a system of right lines or circumferences.

To illustrate the above, the solutions of some simple and indeterminate problems will now be indicated.

10 Prob. To find a point which shall be situated upon a given line. Solution. The geometric locus which resolves this problem is the line itself.

2° Prob. To find a point which shall be situate upon the circumference of a given circle.

Solution. The geometric locus which resolves this problem is the circumference of the given circle itself.

3° Prob. To find a point which shall be at a given distance from a given point.

Solution. The geometric locus which resolves this problem is the circumference of a circle described with the given point as a center, and with a radius equal to the given distance.

4° Prob. To find a point which shall be situated at a given distance from a given line.

Solution. The geometric locus which resolves this problem is the system of two lines drawn parallel to the given line, and separated from it by the given distance.

5° Prob. To find a point which shall be at a given distance from the circumference of a given circle.

Solution. The geometric locus which resolves this problem is the system of two circumferences of circles which are concentric with the given circle, and have radii equal to its radius, increased or diminished by the given distance.

6° Prob. To find a point which shall be situated at equal distances from two given points.

Solution. The geometric locus which resolves this problem is the perpendicular erected at the middle of the line which joins the two given points.

7° Prob. To find a point which shall be situated at equal distances from two given parallel lines.

Solution. The geometric locus which resolves this problem is a third line parallel to the two others, and which divides their mutual distance into two equal parts.

8° Prob. To find a point which shall be at equal distances from two lines which intersect.

Solution. The geometric locus which resolves this problem is the system of two new lines which bisect the angles comprehended between the given lines.

9° Prob. To find a point situated at equal distances from the circumferences of two given concentric circles.

Solution. The geometric locus which resolves this problem is a third circumference concentric to the other two, and which divides their mutual distance into equal parts.

10° Prob. To find a point from which lines drawn to the extrem

ities of a line given in length and position, form, with each other, a right angle.

Solution. The geometric locus which resolves this problem is the circumference of a circle which has the given line for a diameter. 11° Prob. To find a point from which lines drawn to the extremities of a given line form, with each other, an obtuse or acute angle. Solution. The geometric locus which resolves this problem is the system of two segments of a circle described on the given line as a chord, and capable of containing the given angle.

120 Prob. To find a point the distances of which, from two given points, shall have a given ratio.

Solution. The geometric locus which resolves this problem is the circumference of a circle, one diameter of which has, for extremities, the two points which fulfill the prescribed condition upon the line drawn through the two given points.

13° Prob. To find a point the distances of which, from two given lines, shall be in a given ratio.

Solution. The geometric locus which resolves this problem is the system of two new lines which divide the angles comprehended between the given lines into parts, the trigonometric sines of which have the given ratio.*

14° Prob. To find a point the distances of which, from two given points, are the sides of squares, the difference of which is equal to a given square.

Solution. The geometric locus which resolves this problem is the perpendicular erected upon the line which joins the two given points at the point of this line which fulfills the given condition.

15° Prob. To find a point the distances of which, from two given points, are sides of squares, the sum of which is equal to a given

square.

Solution. The geometric locus which resolves this problem is the circumference of a circle, one diameter of which has, for extremities, the two points which fulfill the prescribed condition, upon the line joining the two given points.

160 Prob. To find a point such that the oblique line drawn from this point to a given line, under a given angle, shall have a given length.

Solution. The geometric locus which resolves this problem is a system of two lines drawn parallel to the given line through the extremities of a secant line, which, having its middle point upon the given line, cuts it at the given angle, and has a length double the given length.

17 Prob. To find a point such that the secant, drawn from this point to the circumference of a given circle and parallel to a given line, shall be of given length.

Solution. The geometric locus which resolves this problem is the system of two new circumferences, the radii of which are equal to that of the given circumference, and the centers of which are the extremities of a line which, having its middle point at the center of the given circle, is parallel to the given line, and of a length equal to double the given length.

* This solution, of course, requires a knowledge of the first principles of Trigonometry.

18° Prob. A point and a line being given, to find a second point which shall be the middle of a secant drawn from the given point to the given line.

Solution. The geometric locus which resolves this problem is a new line drawn parallel to the given line, and which divides into equal parts the distance of the given point from this line.

19° Prob. A point and the circumference of a circle being given, to find a second point which shall be the middle of a secant drawn from this point to the circumference.

Solution. The geometric locus which resolves this problem is a new circumference of a circle which has for its radius the half of the radius of the given circumference, and for its center the middle of the distance of the given point, from the center of the given circle.

20° Prob. To find a point the distance of which, from a given point, has its middle upon a given line.

Solution. The geometric locus which resolves this problem is a new line drawn parallel to the given line, at a distance equal to that which separates this line from the given point.

21° Prob. To find a point the distance of which, from a given point, has its middle upon the circumference of a given circle.

Solution. The geometric locus which resolves this problem is a new circumference which has for its radius the double of the radius of the given circumference, and for its center the extremity of a line, the half of which is the distance of the given point from the center of the given circle.

220 Prob. Two points being given, symmetrically placed on opposite sides of a given axis, to find a third point such that the line drawn from this third point to the first shall meet the given axis at equal distances from the second and third points.

Solution. The geometric locus which resolves this problem is a line drawn parallel to the given axis, at a distance equal to that which separates it from the given point.

23° Prob. A circle being given and a chord, to find a point such that the line drawn from this point to one of the extremities of a chord shall meet the circumference of the circle at equal distances from this point and from the other extremity.

Solution. The geometric locus which resolves this problem is the system of two new circumferences of circles which have for a common chord the given chord, and for centers the extremities of the diameter perpendicular to this chord in the given circle.

240 Prob. Two lines perpendicular to each other being given, to find a point which shall be at the middle of a secant of given length comprehended between these two lines.

Solution. The geometric locus which resolves this problem is a circumference of a circle which has for its center the point common to the two lines, and for its radius half the given length.

250 Prob. To find in a given circle a point which shall be the middle of a chord of given length.

Solution. The geometric locus which resolves this problem is a circumference which has for its center the center of the given circle, and for its radius the distance from this center to any one of the chords drawn so as to be of the given length.

26° Prob. To find out of a given circle a point which must be the extremity of a tangent of given length,