PROBLEM VIII.

Required the area of a right angled triangle, whose hypothenuse is x, and the base and perpendicular and x. Ans. 1.029085

PROBLEM IX.

Having given the two contiguous sides (a, b) of a parallelogram, and one of its diagonals (d), to find the other diagonal. Ans. √(2a2 + 2b ̊ – d3)

PROBLEM X.

Having given the perpendicular (300) of a plane triangle, the sum of the two sides (1150), and the difference of the segments of the base (495), to find the base and the sides.

Ans. 945, 375, and 780

PROBLEM XI.

The lengths of three lines drawn from the three angles of a plane triangle to the middle of the opposite sides, being 18, 24, and 30, respectively; it is required to find the sides.

Ans. 20, 28.844, and 34.176

PROBLEM XII.

In a plane triangle, there is given the base (50), the area (796), and the difference of the sides (10), to find the sides and the perpendicular.

Ans. 36, 46, and 33.261

PROBLEM XIII.

Given the base (194) of a plane triangle, the line that bisects the vertical angle (66), and the diameter (200) of the circumscribing circle, to find the other two sides.

Ans. 81.36587 and 157.43865

PROBLEM XIV.

The lengths of two lines that bisect the acute angles of a right angled plane triangle, being 40 and 50 respectively, it is required to determine the three sides of the triangle.

Ans. 35.80737, 47.40728, and 59.41143

PROBLEM XV.

Given the altitude (4), the base (8), and the sum of the sides (12), of a plane triangle, to find the sides.

4

4

Ans. 6+√5 and 6-√5

PROBLEM XVI.

Having given the base of a plane triangle (15), its area (45), and the ratio of its other two sides as 2 to 3, it is required to determine the lengths of these sides.

Ans. 7.7915 and 11.6872

PROBLEM XVII.

Given the perpendicular (24), the line bisecting the base (40), and the line bisecting the vertical angle (25), to determine the triangle.

Given the hypothenuse (10) of a right angled triangle, and the difference of two lines drawn from its extremities to the centre of the inscribed circle (2), to determine the base and perpendicular.

Ans. 8.08004 and 5.87447

PROBLEM XIX.

Having given the lengths (a, b,) of two chords, cutting each other at right angles, in a circle, and the distance (c) of their point of intersection from the centre, to determine the diameter of the circle. Ans. {8(a+b) + 2c2 }

PROBLEM XX.

Two trees, standing on an horizontal plane, are 120 feet asunder; the height of the highest of which is 100 feet, and that of the shortest 80; whereabouts in the plane must a person place himself, so that his distance from the top of each tree, and the distance of the tops themselves, shall be all equal to each other?

Ans. 20/21 feet from the bottom of the shortest, and 40/3 feet from the bottom of the other

PROBLEM XXI.

Having given the sides of a trapezium, inscribed in a circle, equal to 6, 4, 5, and 3, respectively, to determine the diameter of the circle.

1

Ans. (130 x 153), or 7.051595

20

PROBLEM XXII.

Supposing the town A to be 30 miles from B, B 25 miles from e, and c 20 miles from A; whereabouts must a house be erected that shall be at an equal distance from each of them?

Ans. 15.118556 miles from each

PROBLEM XXIII.

In a plane triangle, having given the perpen

dicular (p), and the radii (r, R) of its inscribed and circumscribing circles, to determine the triangle. 2r (2pR-4rR-2)

Ans. The base

PROBLEM XXIV.

p-2r

Having given the base of a plane triangle equal to 2a, the perpendicular equal to a, and the sum of the cubes of its other two sides equal to three times the cube of the base; to determine the sides (g).

Ans. a(2+√6) and a(2 — ¦√6)

APPLICATION OF ALGEBRA TO THE DOCTRINE

OF CURVES.

THE use of Algebra, in the resolution of problems in common Geometry, appears to have been as ancient as the first introduction of that science into Europe; several propositions, relating to plane triangles, having been resolved, in this way, by

(g) This curious problem was first proposed in the Ladies' Diary, for the year 1794, in the following form:

In a palace of one of the Persian kings, there is said to have been a triangular figure such, that the sum of the cubes of two of its sides was equal to three times the cube of its base, or other side, which was 200 feet in length; and the area, or space, inclosed within its boundaries, contained just 10000 square feet from which data it is required to construct the triangle by common, or plane, geometry.

t;

This was accordingly done, in the Diary for the following year, where there was given the construction of the proposer, and two others, by different persons; for a farther account of which, as well as for several algebraical solutions of the problem, see the Scriptores Logarithmici of Maseres, Vol. 1v, p. 335, et seq.

Regiomontanus, in his Treatise of Trigonometry, written about the year 1464, which was some time prior in date to the Summa Arithmetica, and other works, of Lucas de Burgo, that have commonly been considered as the earliest modern productions on algebra now extant.

But the application of this science to the doctrine of curves, which began to be employed at a much later period, is due to the celebrated Descartes; who, in his Geometry, published in 1637, first laid down the method, now generally followed, of expressing the properties and relations of curves by means of algebraic equations; and thus gave an impulse to this branch of science, which, from the improvements of later writers, and the subsequent invention of fluxions, has become a subject of the greatest utility and importance.

Curves, or, as they are commonly called, geometrical lines, when treated of in this manner, are divided into classes, or orders, according to the dimensions of the equation that expresses the relation between their ordinates and abscissæ; or, which amounts to the same thing, according to the number of points by which they may be cut by a right line (h).

(h) The ancients admitted into geometry only right lines and the circle, and, in some cases, the conic sections; but the moderns, in consequence of the great extent to which this branch has since been carried, have agreed to introduce into it all kinds of lines, that can be expressed by equations. They have, also, laid it down as a general rule, that no line of a superior order is to be used in the construction of a problem when it can be done by one of an inferior order; and that the curve, which is the