three of the sides of the pentagon and one of

its angles?

392. Let there be two rectangles of different magnitudes, but similar in form; it is required to determine the size of another similar one that shall equal their sum.

393. Given two triangles dissimilar and unequal, can you make a triangle equal to their sum?

394. Can you make a triangle equal to the difference of two triangles?

395. Can you make a rectangle equal to the difference of two rectangles?

396. Can you place three circles of equal radii to touch each other?

397. Place a regular octagon in a square, so that four sides of the octagon may touch the four sides of the square.

398. There is one class of triangles that will divide into two triangles that are both equal and similar; there is another class that will bear dividing into two triangles that are similar,

but not equal; and a third class that may be divided into two that are equal, but not similar. Give an example of each.

399. Can you divide a trapezium into two equal parts by a line drawn from a point in one of the sides?

400. Can you place three circles, whose diameters are 3, 4, and 5, to touch one another?

401. Make of strong cardboard a box open at one end, and large enough to receive a pack of cards, and make a lid that shall slide on that end and go over it three-quarters of an inch.

402. Can you make a square that shall contain three-quarters of another square?

403. Can you place a square in an equilateral triangle?

404. Can you place a square in an isosceles triangle?

405. Can you place a square in a quadrant ? 406. Can you place a square in a semicircle? 407. Can you place a square in any triangle?

408. Can you place a square in a pentagon?

409. Determine the form of that rectangle which will bear halving by a line drawn parallel to its shortest side, without altering its form.

410. Show that there is a polygon, the interior of which may, by four lines, be divided into nine figures; one being a square, four reciprocal rectangles, and the remaining four reciprocal triangles.

411. Geometricians have asserted that, when in a circle one chord halves another chord, the rectangle contained by the segments of the halving chord is equal to the square of one half of the chord which is halved; and that, when one chord in a circle halves another chord at right angles, one half of the halved chord is a mean proportional between the segments of the halving chord. Determine, as nearly as you can, by a scale, whether it is true.

One-half of the sum of any two numbers or any two lines is called the arithmetic mean to those numbers, or to those lines.

412. Show by a figure the arithmetical mean to 3 and 12.

The arithmetic mean has the same distance from the less extreme that the greater extreme has from it.

The square root of the product of two numbers is called the geometric mean to those numbers.

413. Show by a figure the geometric mean to 3 and 12.

The geometric mean has the same ratio to one extreme that the other extreme has to it, thus 3: 66:12. This is why it also takes the name of mean proportional.

414. Find the arithmetic mean and the geometric mean to 4 and 9. Say which mean is the greater.

415. Determine by geometry and prove by calculation the side of a square that shall just contain an acre.

416. Extract by geometry the square root of 5, and prove by arithmetic.

The angle which the chord of a segment

makes with the tangent of the segment is called the angle of the segment.

417. Can you determine the angle of a segment of 90° ?

418. Can you determine which two lines drawn from the extremities of the chord of a segment so as to meet together in the arc of the segment will make the greatest angle?

419. Can you determine the angle in a quadrantal segment?

420. Can you ascertain the relation existing betwixt the angle of a segment and the angle in a segment?

421. Can you give an instance where the angle in the segment and the angle of the segment are equal?

A line that begins outside a circle, and on being produced enters it, and traverses it until stopped by the other side of it, is called a secant to a circle.

422. Make a few circles and fit a secant to each.