PROBLEM XIII.

TO DRAW A SQUARE, GIVEN IN MAGNITUDE, WITHIN A LARGER SQUARE GIVEN IN POSITION AND MAGNITUDE; THE SIDES OF THE TWO SQUARES BEING PARALLEL.

LET A B, Fig. 34., be the sight-magnitude of the

side of the smaller square, and a c that of the side of the larger square.

Draw the larger square. Let D E F G be the square so drawn.

Join E G and D F.

On either D E or D G set off, in perspective ratio, Dи equal to one half of B C. Through н draw H K to

the vanishing-point of D E, cutting D F in I and E G in

K. Through I and K draw I м, K L, to vanishing-point of DG, cutting D F in L and E G in M. Join L M.

Then I K L M is the smaller square, inscribed as

required.*

COROLLARY.

If, instead of one square within another, it be required to draw one circle within another, the dimen

Fig. 36.

sions of both being

given, enclose each

circle in a square.

Draw the squares

first, and then the circles within, as in Fig. 36.

If either of the sides of the greater square is parallel to the

PROBLEM XIV.

TO DRAW A TRUNCATED CIRCULAR CONE, GIVEN IN POSITION AND MAGNITUDE, THE TRUNCATIONS BEING

IN HORIZONTAL PLANES, AND THE AXIS OF THE CONE

VERTICAL.

LET A B C D, Fig. 37., be the portion of the cone required.

As it is given in magnitude, its diameters must be given at the base and summit, A B and C D; and its vertical height, c E.*

*Or if the length of its side, A C, is given instead, take a e, Fig. 87., equal to half the excess of A B over σ D; from the point e raise the perpendicular c e. With centre a, and distance A o, describe a

circle cutting ce in c.

of cone required, or o E

Then c e is the vertical height of the portion

And as it is given in position, the centre of its

base must be given.

Draw in position about this centre,* the square

In the square of its top, e f g h, inscribe concentrically a circle whose diameter shall equal c D. (Coroll. Prob. XIII.)

Join the extremities of the circles by the right lines k l, nm.

Then k l n m is the portion of cone required.

The direction of the side of the square will of course be regu lated by convenience.

COROLLARY I.

If similar polygons be inscribed in similar positions in the circles k n and m (Coroll. Prob. XII.), and the corresponding angles of the polygons joined by right lines, the resulting figure will be a portion of a polygonal pyramid. (The dotted lines in Fig. 38., connecting the extremities of two diameters and one diagonal in the respective circles, occupy the position of the three nearest angles of a regular octagonal pyramid, having its angles set on the diagonals and diameters of the square a d, enclosing its base.)

If the cone or polygonal pyramid is not truncated, its apex will be the centre of the upper square, as in Fig. 26.

COROLLARY II.

If equal circles, or equal and similar polygons, be inscribed in the upper and lower squares in Fig. 38., the resulting figure will be a vertical cylinder, or a