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

27° Prob. To find out of a given circle the point of meeting of two tangents, drawn through the extremities of a chord which contains a given point.

Solution. The geometric locus which resolves this problem is the polar line corresponding to the given point. (See Appendix II.)

The solutions above given are easily deduced from well-known theorems of Geometry. A great number of problems, both simple and indeterminate, could be pointed out, the solutions of which would reduce themselves, in a similar manner, to systems of right lines and circumferences of circles. Let it be observed, moreover, that from the solutions of n problems of this kind, in each of which the unknown point is subjected to a single condition, we can deduce immediately the solutions of "(n+1) simple and determinate problems, in each of which the unknown point is subjected to two conditions. For, to obtain a simple and determinate problem, it is sufficient to combine two conditions corresponding to two simple but indeterminate problems, or even two conditions alike and corresponding to a single indeterminate problem. But the number of combinations of n quantities, two and two, is (see Alg., art. 203),

2

n(n-1);

2

and, adding to this number that of the quantities themselves, the result is,

n(n−1) _n(n+1).
'+n="

2

2

This result increases very rapidly with n. Thus, if n=27,

n(n+1)

2

378; that is, the solution of the 27 indeterminate problems enunciated above furnishes already the means of resolving 378 simple and determinate problems.

In order that the principles just brought to view may be the better apprehended, they will now be applied to the solution of some determinate problems.

Suppose, first, that it is required to draw a tangent to a circle through a point without. The question may be reduced to seeking the unknown point of contact of the tangent with the circle. The two conditions which this point must satisfy are, 10. That it shall be upon the circumference of the given circle. 20. That lines drawn from this point to the given point and the center of the circle should make a right angle with each other. Then the question to be resolved will be a determinate problem, resulting from the combinations of the indeterminate problems, 2 and 10.

The combined solutions of 2 and 10 furnish, in fact, the solutions heretofore given (Prob. 20, p. 83).

Suppose, secondly, that it is required to circumscribe a circle about a given triangle. The question can be reduced to seeking for the cen

E

ter of the circle. But the two conditions which this center must satisfy will be those of being not only at equal distances from the first and second vertex of the given triangle, but also at equal distances from the first and third. Then the question to resolve will be a determinate problem resulting from the combination of two indeterminate problems identical with each other and with problem 6. In fact, the solution of problem 6, twice repeated, will furnish two geometric loci, reduced to two right lines, which cut each other in a single point, and thus will be obtained the known solution of the problem proposed.

Suppose, next, that the question is how to draw a circle tangent to the three sides of a given triangle.

The question can be reduced to finding the center of the given circle. But the two conditions which this center must satisfy will be not only to be at equal distances from the first and second side of the given triangle, but also at equal distances from the first and third sides. Then the question to be resolved will be a determinate problem, resulting from the combination of two indeterminate problems identical with one another, and with problem 8. In fact, the solution of problem 8 twice will furnish two geometric loci, which, reduced each to the system of two right lines, will cut each other in four points, and thus four solutions will be obtained of the proposed problem.

Suppose, finally, that the question is to inscribe between a chord of a circle and its circumference a line equal and parallel to a given line. The question can be reduced to seeking either one of the two points which will form the extremities of this line, and, consequently, to a determinate problem resulting from the combination of two indeterminate problems, to wit, problems 1 and 17, or problems 2 and 16. In fact, by the aid of this combination, the question proposed is resolved without difficulty. And one of the extremities of the line sought will be found determined either by the meeting of the circumference of the given circle with a new line, or by the meeting of the given chord with a new circumference.

It is seen here how the solution obtained may be modified when the order comes to be inverted in which the unknown points are determined.

The construction of the geometric locus which corresponds to a simple and indeterminate problem may itself require the resolution of one or more determinate problems. It should be observed, upon this subject, that in the case where the problem is resolvable by the rule and compass, the geometric locus should reduce to a system of right lines and circles. Then, since each line or each circumference finds itself completely determined when there are known two or three points of it, the construction of the geometric locus, corresponding to a simple and indeterminate problem, can always be deduced from the construction of a certain number of points suitable to verify the condition which ought to be fulfilled, in virtue of the enunciation of the problem, by the unknown point.

Thus, for example, to resolve problem 6; that is to say, to find a point which shall be situated at equal distances from two given points, and, consequently, to construct the geometric locus which shall contain every point suitable to fulfill this condition. We begin by seeking such a point, for example, one the distance of which from the

given points is sufficiently great. But the solution of this last problem deduces itself immediately from problem 3, which, twice repeated, will furnish at once two points that fulfill the proposed condition; consequently, two points which suffice to determine the geometric locus required.

Thus, again, to resolve problem 15; that is to say, to find a point the distances of which, from two points given, shall furnish squares the sum of which shall be equal to a given square, and, consequently, to construct the geometric locus of every point suitable to fulfill this condition. We can commence by seeking such a point; for example, that which shall be situated at equal distances from the two given points, and, consequently, separated from each of them by a distance equal to half the diagonal of the given square. But the solution of this last problem deduces itself immediately from the solution of problem 3, and, twice repeated, will furnish at once, also, two points which will fulfill the proposed condition. Moreover, these two points are precisely the extremities of a diameter of the circle, the circumference of which represents the geometric locus required.

The above note is from a recent article by the celebrated Cauchy. Although designed by him as an introduction to a new method of resolving determinate geometric problems by means of the Indeterminate Analysis (for an exhibition of which, see Comptes Rendus de L'Acadamie des Sciences, No. 17, 21 Avril, 1843, p. 867, and No. 19, 15 Mai, 1843, p. 1039), yet it is calculated to afford important aid to the solution of problems by the processes of ordinary geometry.

M. Cauchy acknowledges that it is but the development of some principles, the memory of which he has preserved, which were contained in the course of lectures given by Dinet, at the Lycée Napoleon, some forty years ago.

MISCELLANEOUS EXERCISES IN PLANE GEOMETRY.

1. Prove that all regular polygons of the same number of sides are similar figures.

2. That if a line join the middle points of two sides of a triangle, it will be parallel to the third side and equal to its half.

3. To describe a circle about a given square.

4. To divide a right angle into three equal parts.

5. To circumscribe about a given circle a triangle one side of which is given.

6. Find the length of the circumference of a circle in seconds of a degree.

7. Find the length of the radius in seconds of a degree.

8. Find the length of 1" to radius 1.

9. Prove that a straight line can meet a circumference in but two points.

10. From a given point without a circle to draw a secant such that the part within the circle shall be equal to a given line.

11. To draw to a circle a tangent of given length, and terminating at a given line, which cuts the circumference.

12. An inscribed polygon being given, to circumscribe another similar.

13. Prove that of two convex lines, broken or curved, terminating at the same points, the enveloped is less than the enveloping line.

14. Prove that the difference between the sum of the two perpendicular sides of a right-angled triangle and the hypothepuse is equal to the diameter of the inscribed circle.

15. To trisect a given finite straight line.

16. Prove that if, from the extremities of the diameter of a semicircle, perpendiculars be let fall on any line cutting the semicircle, the parts intercepted between those perpendiculars and the circumference are equal.

17. If, on each side of any point in a circle, any number of equal arcs be taken, and the extremities of each pair joined, the sum of the chords so drawn will be equal to the last chord produced to meet a line drawn from the given point through the extremity of the first arc.

18. That if two circles touch each other, and also touch a straight line, the part of the line between the points of contact is a mean proportional between the diameters of the circles.

19. From two given points in the circumference of a given circle to draw two lines to a point in the same circumference, which shall

cut a line given in position, so that the part of it intercepted by them may be equal to a given line.

20. Prove that if, from any point within an equilateral triangle, perpendiculars be drawn to the sides, they are together equal to a perpendicular drawn from any of the angles to the opposite side.

21. That if the three sides of a triangle be bisected, the perpendiculars drawn to the sides, at the three points of bisection, will meet in the same point.

22. If from the three vertices of a triangle lines be drawn to the points of bisection of the opposite sides, these lines intersect each other in the same point.

23. The three straight lines which bisect the three angles of a triangle meet in the same point.

24. If from the angles of a triangle perpendiculars be drawn to the opposite sides, they will intersect in the same point.

25. If any two chords be drawn in a circle, to intersect at right angles, the sum of the squares of the four segments is equal to the square of the diameter of the circle.

26. In a given triangle to inscribe a rectangle whose sides shall have a given ratio.

27. Prove that the two sides of a triangle are together greater than the double of the straight line which joins the vertex and the bisection of the base.

28. That if, in the sides of a square, at equal distances from the four angles, four other points be taken, one in each side, the figure contained by the straight lines which join them shall also be a square.

29. That if the sides of an equilateral and equiangular pentagon be produced to meet, the angles formed by these lines are together equal to two right angles.

30. That if the sides of an equilateral and equiangular hexagon be produced to meet, the angles formed by these lines are together equal to four right angles.

31. If squares be described on the three sides of a right-angled triangle, and the extremities of the adjacent sides of any two squares be joined, the triangles so formed are equal, though not identical, to the given triangle, and to one another.

32. If the squares be described on the hypothenuse and sides of a right-angled triangle, and the extremities of the sides of the former square, and those of the adjacent sides of the others, be joined, the sum of the squares of the lines joining them will be equal to five times the square of the hypothenuse.

33. To bisect a triangle by a line drawn parallel to one of its sides.