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. To find the side of a square, inscribed in a given semicircle, whose diameter is d. - 1 - Ans. #ovs


Having given the hypothenuse (13) of a right angled triangle, and the difference between the other two sides

(7), to find these sides (f). Ans. 5 and 12


To find the side of an equilateral triangle, inscribed in a circle, whose diameter is d, and that of another circumscribed about the same circle. Ans. }dx^3, and d.V3

§ PROBLEM IV, To find the side of a regular pentagon, inscrib in a circle, whose diameter is d. Ans. 4dv/(10–2 v3)


(e) Newton, in his Universal.Arithmetic, English edition, 1728, has resolved this problem in a variety of different ways, in order to show, that some methods of proceeding, in cases of this kind, frequently lead to more elegant solutions than others; and that a ready knowledge of these can only be obtained by practice.

(f) Such of these questions as are proposed in numbers, should first be resolved generally, by means of the usual symbols, and then reduced to the answers above given, by substituting the numeral values of the letters in the results thus obtained.


To find the sides of a rectangle, the perimeter of which

shall be equal to that of a square, whose side is a, and its area half that of a square.

Ans. a+}av/2 and a-40 v2


Having given the side (10) of an equilateral triangle, to find the radii of its inscribed and circumscribing circles. Ans. 2.8868 and 5.7736


Having given the perimeter (12) of a rhombus, and the sum (8) of its two diagonals, to find the diagonals. Ans. 4+V2 and 4–V2


Required the area of a right angled triangle, whose hypothenuse is a 3", and the base and perpendicular wo and aco. Ans. 1.029085


Having given the two contiguous sides (a, b) of a parallelogram, and one of its diagonals (d), to find the other

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aving 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


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 .


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


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, 47,407.28, and 59.41 i43


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

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Having given the base of a plane triangle (15), i (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 xv.11.

Given the perpendiculaf (24), the line bisecting the

o: (40), and the line bisecting the vertical angle (25) to det ine the triangle. ermine the triangle Ans. The base * v/7

From which the other two sides may be readily found.

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