P.-It is bounded by twenty equilateral and equiangular triangles; there are five plane angles about each of its solid angles: it has thirty edges, and fifteen solid angles.

M.-See whether it is so. Which of the remaining two solids is bounded by quadrilateral faces? What is their number? It is therefore called hexahedron (from the Greek, six, and ëồpa, a seat), or cube (from the Greek Kubos, a cube).-How many plane angles are there about each of its solid angles ? Calculate the number of its edges and solid angles.

P.-The hexahedron, or cube, is a solid bounded by six quadrilateral faces; three plane angles are about each of its solid angles: it has twelve edges and eight solid angles.

M.-Compare the sides and the angles of the faces of the hexahedron.

P.-The sides are all equal to one another: the angles are likewise equal.

M. Represent such a face on your slate. The angles of this figure are called right angles. Draw a right angle on your slate.-How many right angles make with two lines?

can you

P-Either one right angle, or two, or four.

M.-Compare these two angles with each other.
P.-They are equal.

M.-Are the two angles which one line makes with

another line always equal?

P.-No.

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M. When one line makes two angles with another line, these are said to be adjacent (from the Latin ad, near, and jacens, lying) angles. When these adjacent angles are equal, each of them is called a right angle. If these adjacent angles are not equal, what can be said of them?

P.-One is greater than a right angle, the other is less.

M.—An angle which is greater than a right angle is called obtuse (from the Latin obtusus, blunted) angle; the angle which is less than a right angle is called acute (from the Latin acutus, pointed) angle.-Describe on your slates right angles, obtuse, and acute angles. Describe as many different quadrilateral figures as you are able, and first let their difference consist in their sides.

P.-The four sides may be equal; the opposite sides may be equal; the adjacent sides may be equal; three sides may be equal, and the fourth unequal; all four may be unequal.

M.-Can you describe two quadrilateral figures

having equal sides, and yet being different?

P.-Yes: in the one, all the angles

may be equal; in the other, the sides

ᄆᄆ

may be equal, but its angles not all equal,-only two pairs of opposite angles are equal.

M.-A quadrilateral figure whose sides are equal,

and angles right angles, is called a square. A quadrilateral figure having equal sides, but whose angles are not equal, is called a rhomb (from the Latin rhombus). Can you describe two quadrilateral figures having their opposite sides in each equal, and yet be different?

P.-Yes: the angles in the one may

all be equal,—that is, right angles; in the other they are not all equal,—only those which are opposite each other.

M. What do you observe respecting the distance of those opposite and equal lines?

P.-Their distance is everywhere the same.

M.-Draw two such lines upon your slates, and two others whose distance is not everywhere the same. P.

M.-What happens if the last two lines be lengthened?

P. They will cross each other.

M.—And what happens if the first two lines be lengthened?

P.-They will not cross each other.

M.-Such lines are called parallel (from the Greek Tapa, beside, and aλλnλv, each other) lines, and hence these figures are called parallelograms (from the Greek παράλληλος, and γραμμα, a figure); the former a rectangular parallelogram, or simply rectangle, and the latter merely parallelogram. The other quadrilateral figures of which three sides only are equal, or

of which all four are unequal, are called trapeziums (from the Greek тparéžiov, a small table).

SUBSTANCE OF THE LESSON.

1. The icosahedron is a solid bounded by twenty equilateral and equiangular triangles: there are five plane angles about each of its solid angles.

2. The hexahedron, or cube, is a solid bounded by six squares: there are three plane angles about each of its solid angles.

3. When one line standing on another line makes the adjacent angles equal to one another, each of them is called a right angle.

4. An angle which is greater than a right angle is called an obtuse angle.

5.-An angle which is less than a right angle is called an acute angle.

6.—A quadrilateral figure which has equal sides, and its angles right angles, is called a square.

7. A quadrilateral figure which has equal sides and two pairs of equal opposite angles is called a rhomb.

8. A quadrilateral figure which has two pairs of equal opposite and parallel sides, and its angles right angles, is called a rectangle.

9.—A quadrilateral figure which has two pairs of equal opposite and parallel sides, but its angles not right angles, is called a parallelogram.

10. All other quadrilateral figures are called trapeziums.

LESSON VI.

M.-Examine the last of these five solids.-What are the faces which bound this solid, and what is their number? It is therefore called dodecahedron (from the Greek dudeкa, twelve, and “dpa, a seat).—How many plane angles are there about each of its solid angles? Calculate the number of its edges and solid angles.

P.-The dodecahedron is a solid bounded by twelve pentagons; there are three plane angles about each of its solid angles: it has thirty edges, and twenty solid angles.

M.-Examine the sides and angles of the pentagons. P.-They are equal.

M.-Are the sides and angles of every pentagon equal?

P.-No; for pentagons may be described of which the sides are unequal, as also the angles.

M.-How will you distinguish the former from the latter?

P. The former is an equilateral and equiangular pentágon, a regular pentagon; the latter an irregular pentagon.

THE RHOMBOIDAL DODECAHEDRON.

M.-What are the faces of this solid, and what is their number?

P.-The faces are rhombs, and their number is twelve.