the distance Fc: but CD and AB being also parallel to each other, the whole distance a b will be equal to the whole distance c d. From these equal quantities therefore, taking away the equal quantities a E and c F, the remainders must be equal; that is, Eb must be equal to F d; but two right lines which are so situated as to be, in different points of their extent, equally distant the one from the other, are parallel; therefore the line EF, which was given parallel to CD, is also parallel to AB, which was given parallel to CD.

From what has been said, it is evident that two right lines cannot inclose a space, or form a plane figure; for, to inclose a space, the lines must meet in some point, as in B, (fig. 3,) but they never can meet any more, as the longer they are made, the greater must be the distance between their extremities.

Suppose now these extremities to be joined by another right line, as in fig. 6, Pl. 1, where AC and AB meeting in the point A, have their extremities B and C joined by the right line BC: here we have a figure containing three angles and three sides, and from the former circumstance it obtains the name of a Triangle. When the three sides are all of the same length, as in this fig. 6, the triangle is said to be equilateral, that is, equal-sided. When only two of the sides are equal, as in fig. 7, where AB and AC are of equal length, but the third side AC is shorter, the triangle is said to be isosceles or equal-legged. Lastly, if the three sides be all of different lengths, as in fig. 8, the triangle is said to be scalene, that is, leaning, as a ladder against a wall.

If there be a right angle in a triangle, it is called rightangled, as fig. 9; and the side opposite to the right angle is called the hypothenuse, as BC, while the side AB is also called the perpendicular, and the side AC is called the base; this last term is also applied to the lower side of all other triangles; as AC in fig. 6, 7, and 8. If a line be let fall perpendicularly on the base of a triangle from the opposite angle, this line will give the altitude of the

triangle:

triangle; whether it fall within the triangle, as BD in fig. 6, or without it, as BD on the base CA produced in fig. 8; or coincide with one of the sides, as BA in fig. 9.

If a figure has four sides all equal, and the four angles all right; that is, if each side be perpendicular to the two adjoining sides, such a figure is called a square, as fig. 10, where the four sides AC, CD, DB, and BA are all equal to one another, and the four angles at A, C, D, and B are all right, and consequently equal to one another. Again, if all the sides be equal, but the angles not right, the figure is termed a rhombus, or distorted square; as fig. 11, where the four sides AC, CD, DB and BA are equal, but none of the angles are right.

Four-sided figures, having only the two opposite sides equal and parallel to each other, are in general termed parallelograms: but when such a figure contains a right angle (the other three being necessarily right also) it is called a rectangle: see fig. 12, where the opposite sides, AB and CD are equal, and CA and DB are also equal; and the four angles at A, C, D, and B, are all equal and right. Again, if the opposites sides of a four-sided figure be respectively equal, but none of the angles right, although respectively equal, such a figure is called a rhomboides, or distorted parallelogram. See fig. 13, where AB and CD are equal, and AC and BD are equal, and the opposite angles at A and D, and at C and B are equal, but none of them right angles.

When the four sides are all unequal, and none of them parallel to its opposite side, the figure is called a trapezium, or quadrilateral, such as in fig. 14; but if two opposite sides be parallel, although all the sides be unequal, the figure is termed a trapezoid, as in fig. 15.

A line joining opposite angles in a figure, is termed a diagonal; as the dotted lines AD and CB in fig. 10, AD in fig. 11, 12, 13, 14, and 15 and a line let fall perpen

dicularly from an opposite angle on the base of any parallelogram, gives the altitude of the figure: as the dotted line CE in fig. 11, 13, and 15. When the parallelogram is right angled, the line of altitude coincides with the perpendicular side, as CA or DB in the square fig. 10, and CA in fig. 12.

Figures composed of more than four sides are distinguished by various names, descriptive of their number of sides thus, one of five sides is called a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of nine sides, a nonagon; one of ten sides, a decagon; one of twelve sides, a dodecagon; and so on; but in general they are termed polygons, a name merely indicating that they are figures of many sides.

These things being premised, the student will be prepared to understand the following series of propositions, containing some fundamental theorems illustrative of the most important properties of geometrical figures,

PROPOSITION I. When one right line meets another right line, the angles formed at the point of meeting are either two right angles, or they are, when taken together, equal to two right angles.

(Pl. 1, fig. 4.) Suppose the right line FC to meet the right line AB in the point C, the two angles, FCA and FCB, are either two right angles, or, when taken together, equal to two right angles. If the line FC stood perpendicularly upon AB at the point C, then the angles on each side, as has been already said, would be both right; but in this case, as FC does not stand perpendicularly on AB, let some other line, as DC, be drawn perpendicular to AB, at the same point C; and the angles DCA and DCB will be both right; but the two angles DCA and DCB are equal to the three angles ACF, FCD and DCB, therefore the sum of these three angles is equal to two right angles;

but

but the angles FCD and DCB are equal to the angle FCB, therefore the two angles FCA and FCB are equal to two right angles.

Corollary. However many angles may be formed, on the same side of a line, at any given point, their sum will still be equal to the sum of two right angles. In the same figure the angles ACF, FCD, DCE, and ECB, all formed at the point C, on the same side of the line AB, are, taken together, equal to two right angles; for ACD is a right angle, and it is equal to the two angles ACF, and FCD, and the other right angle DCB, is equal to the remaining two angles DCE and ECB.

2d Corollary. From this proposition it follows, that all the angles that can be formed at one point, on both sides of a right line, are equal to four right angles.

PROPOSITION II. If two right lines intersect each other, the vertical or opposite angles are equal to one another. Let the right lines AB and CD (Pl. 1, fig. 16,) cut one another in the point E, then the vertical or opposite angles will be respectively equal to each other: that is, the angle AEC will be equal to the angle BED, and CEB will be equal to AED. For, as the line CE meets AB in the point E, it follows, from the 1st Proposition, that the angles AEC and CEB are equal to two right angles: again, as the line BE meets CD in the point E, the angles CEB and BED are also equal to two right angles; therefore the sum of AEC and CEB is equal to the sum of the same CEB and BED. Now if from equal quantities we take away equal quantities, or a quantity included in, and thereby common to both quantities, the remainder must be equal: take away, therefore, from AEC + CEB, and from CEB + BED, the common angle CEB, and the remaining angle in the one case AEC will be equal to the remaining angle in the other case BED. But these angles are opposite and vertical, agreeably to the proposition; and in the same manner it

may

may be shown that the other opposite angles, CEB and AED, are equal to one another.

PROP. III, fig. 17. Two triangles are equal to one another, when the one has an angle and the two sides containing it, equal to an angle, and the two sides containing it in the other, each to each. If the two triangles ABC and DEF, be of such a nature that the angle ABC in the one is equal to the angle DEF in the other; the side AB in the one equal to the side DE of the other; and the side BC in the one, to the side EF in the other; then these two triangles will be equal the one to the other. For supposing the triangle ABC to be moveable, let it be placed upon the triangle DEF, so that the angular point B shall coincide with the angular point E; the line AB with the line DE, and the line BC with the line EF; then will the point A coincide with the point D, and the point C with the point F, and consequently the line or base AC with the base DF. But it was already stated as an axiom, that magnitudes, whether lines, surfaces, or solids, which, when applied to one another, perfectly coincide in all their parts, are equal to one another; the two triangles, therefore, given in the proposition must be equal to each other; and the remaining angles of the one will be equal to the remaining angles of the other, each to each; that is BAC to EDF, and BCA to EFD, and the remaining side AC to the remaining side DF.

1st Corollary. From this proposition it follows that, if two triangles have two sides of the one respectively equal to two sides of the other, but the angle formed by these two sides in the one greater than the corresponding angle in the other, the base of the first will be greater than the base of the second; and if the contained angle in the first triangle be less than the contained angle in the second, the base of the first will be less than the base of the second triangle.

2d Gorollary. Any two sides of a triangle are together

greater