EXAMPLES FOR PRACTICE.

Ex. 1. To find the side of a square, inscribed in a given semicircle, whose diameter is d. Ans. d√5

1

Ex. 2. To find the side of an equilateral triangle inscribed in a circle whose diameter is d; and that of another circumAns. d√3, and d√3 scribed about the same circle.

Ex. 3. To find the sides of a rectangle, the perimeter of which shall be equal to that of a square, whose side is a, and - a√2 its area half that of a square. Ans. a+a√2 and a — Ex. 4. Having given the perimeter (12) of a rhombus, and the sum (8) of its two diagonals, to find the diagonals.

Ans. √2+ √5 4+ √2 and 4-√2 Ex, 5. Required the area of a right angled triangle, whose hypothenuse is " and the base and perpendicular ** and * Ans. 1.029085 Ex. 6. Having given the two contiguous sides (a, b) of a parallelogram, and one of its diagonals (d), to find the other diAns. √(2a2+2b3 — ď3) agonal. Ex. 7. 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

Ex. 8. 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 Ex. 9. 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 Ans. 8.08004 and 5.87447 and perpendicular.

Ex. 10. 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 a circle.

Ans. ‡√ { †(a2+b")+2c2

}

Ex. 11. Two trees, standing on a horizontal plane, are 120 feet asunder; the height of the highest of which is 100 feet, and that of the shortest 80; where 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. 2021 feet from the bottom of the shortest

and 403 feet from the bottom of the other

Ex. 12. 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. Ans. (130x133) or 6.574572

Ex. 13. Supposing the town A to be 30 miles from B, в 25 miles from c, and c 20 miles from A; where must a house be erected that it shall be at an equal distance from each of them? Ans. 15.118578 miles from each, viz: in the centre of a circle whose circumference passes through each of the three towns.

Ex. 14. In a plane triangle, having given the perpendicular (p), and the radii (r R) of its inscribed and circumscribing circles, to determine the triangle.

DETERMINATION OF ALGEBRAICAL EXPRESSIONS FOR SURFACES AND SOLIDS.

7. We have seen in the Elements of Geometry, that surfaces depend upon the product of two dimensions, and solids upon the product of three dimensions; so that, if the several dimensions of one or two solids, or two surfaces, which we would compare, have to the several dimensions of the other, each the same ratio, the two surfaces will be to each other as the squares, and the two solids as the cubes, of the homologous dimensions; and more generally still, if any two quantities of the same nature are expressed each by the same number of factors, and if the several factors of the one have to the several factors of the other, each the same ratio, the two quantities will be to each other as their homologous factors, raised to a power whose exponent is equal to the number of factors. If, for example, the two quantities were a b c d, a' b' c' d', and we had

What is here said is true not only of simple quantities; the

same may be shown with respect to compound quantities. Let the quantities whose dimensions are proportional be

a b + c d, a' b' + c' d' ;

It follows, from what is here proved, that the surfaces of similar figures are as the squares of their homologous dimensions, and that the solidities of similar solids are as the cubes of their homologous dimensions; for, whatever these figures and these solids may be, the former may always be considered as composed of similar triangles, having their altitudes and bases proportional, (Prop. XXIII. B. IV. El, Geom.,) and the latter as composed of similar pyramids, having their three dimensions also proportional. (Prop. XXXII, Cor. 5, B. II, El. Sol. Geom.)

It will hence be perceived, that quantities may be readily compared, when they are expressed algebraically; and this may be done, whether the quantities be of the same or of a different species, as a cone and a sphere, a prism and a cylinder, provided only that they are of the same nature, that is, both solids, or both surfaces.

Let it be required to investigate the properties of a pyramid and and also of a frustum of a pyramid.

Let h= the altitude, s = the greater base and s' the smaller base, and h' the altitude of the vertical pyramid taken from the top of the frustum,

=

then we shall have s': s: h':h+h' or the altitude of the whole pyramid, and consequently,

and

(h + h') √s' = h' √ s = h √s' + h' √ s'

h' √s—h' √s' = h √s'

dividing by ✓s - √s'

Let now h+h' be represented by k, and we may have

then we shall have for the solidity of the whole pyramid (1)

sk

3

substituting for k its value, we have for the solidity of the

putting for h' its value found above we have,

that is, the solidity of the frustum is equal to the sum of the greater base, the smaller base, and a mean proportional between the two bases, multiplied by the altitude of the frustum; which agrees with the proposition in geometry.

And if the two bases are equal, viz: if s = s' then the solid becomes a prism, and the expression will become

h

3

1

(s + s + s) or h (3s) = hs - - - (4)

3

that is, the solidity of the prism is equal to its base multiplied by its altitude.

Let the lateral surface of the pyramid be used as an ele ment in its investigation.

To find the lateral surface of a frustum of a regular pyramid, having the two bases and slant height given, as well as the radius of the circle inscribed in the larger base.

Let the larger base be called s, the smaller s', the slant height h, and the radius of the circle inscribed in the larger base, r. The perimeter of the larger base will be

and the peri

meter of the smaller base may be found from the following ]

Or we may investigate the surface of the frustum in connection with the whole pyramid of which it is a part.

Thus

:

h√s' √s

h : the slant height of the

whole pyramid, which make = k, and the vertical pyramid cut from the frustum will be k― h.

Hence, we have

the whole pyramid.

(5)

And since the lateral surfaces of similar pyramids are pro

sk s'k

portional to their bases, we may make s: s:: : the sur

face of the vertical pyramid cut from the frustum.

Hence

sk

s'k

k

or (ss) the difference of the two bases, multiplied by the ratio of the slant height of the pyramid to the radius of the base, is cqual to the lateral surface of the frustum.

k

It may be observed that the ratio is constant whether ap

plied to the whole pyramid, to the pyramid cut off, or to the frustum; and is such as would be represented by the sine of the angle formed by its slant side with the plane of the base. Cor. Whence we have for the lateral surface of the frustum of a pyramid, this rule :

Multiply the difference of the bases by the sine of the angle which the slant side makes with the base, or by the ratio of the whole slant height of a perfect pyramid on the same base to the radius of the base, which will give the lateral surface.

8. If represent the ratio of the circumference of a circle to the diameter, a ratio which is known with sufficient accuracy for practical purposes (Prop. XIX. B. V. El. Geom.,) the circumference of any circle whose radius is r, will be 2rr - (1,) and its surface r3 (2.) Hence it is evident that the areas of circles increase as the squares of their radii, being always of the same value, the quantity depends on, and is proportional to

(3.)