duction AD of the other); this being subducted from two right angles, the remainder is the angle BAC required.

ART. 10. "When two right lines meet another at the same point, but at different sides, and make angles with it which are together equal to two right angles, those right lines are in one continued right line.

D

AA'

BB'

A A

Practical examples of this theorem are common. Thus, if the corners B, B', of two accurate carpenters' squares be applied vertex to vertex, and also two of their blades BA, and B'A', the other blades BD, B'D', will form one straight edge DD'. And the same would be true if the blades BA, B'A', were bent from the square, as in fig. 2,—their corners, however, still making up together two right angles. Also, if a mason build with bricks, he will have much less difficulty in keeping the straight line with his

D

2

B B

D'

being common to the triangles FEI, GEI, the adjacent angles at r must be equal, and therefore EI is perpendicular to AD*. This problem may be executed by means of a string only, as follows: ABCD is a garden-bed, straight across which the gardener wishes to make a foot-path, the borders setting out from E and F respectively. He has only his line to accomplish this with; so proceeds thus:

Fixing a spud at E, he stretches his line from it to L at the other side

L

H

K

of AD, and keeping it stretched, traces with the end of it at L the

* In the above process we divide the angle FEG equally, instead of the interval FG, as ordered by the geometrical construction; because it saves the necessity of constructing an equilateral triangle on Fe, and then dividing its vertical angle into two equal parts. We, in fact, save a step; and in not doing so, the geometrical construction of this Problem, from the age of Euclid to the present day, has been supererogatory. Instead of telling us to divide the interval, it should tell us to divide the angle, equally. It is thus that practice serves to correct theory, or at least to better it.

coins and his layers, than if he build with stones, unless they be as accurately squared as the other material. For the corners of the bricks being all right angles, each pair when neatly put together

will make up two right angles, and therefore form a right line AE, CD, or EF. And he would find the same ease in keeping his level if the bricks or stones were all of the same lozenge shape, as in the upper layers of this figure,because the two corners which come together would exactly make up two right angles. But thus shaped, the other edges EG, GI, though they would form a right line EI, would not form a perpendicular line of coins, but a slanting one, as

GK OF FL.

It is from the squared form of stones now used that the pavement acquires its regularity, appearing to run in straight lines across the street. The curb-stones of the foot-path are chiselled into the same right-angled shape, in order that when the ends are put together they may have their outermost sides in one continued right line bordering the carriage-way.

In a diamond-paned window, if properly glazed, the cross bars are generally right lines, on the same principle: each pair of adjacent corners make up together two right angles. Also, the outer sides of the triangular panes are made to form one straight line, and thus to fit the straight sides of the window-frame, because the neighbouring

angles abf and

circular arch GLH, cutting AD in G and H. Then, fixing his line at G, and stretching it to н, he traces the arch poq with the moveable end. He does the same at H, tracing the arch mon. Finally, he stretches his line from o, the intersection of these latter arches, to the given point E, and this will be one borderline, perpendicular to AD, by PROB. VI. By the same means he finds the other border-line; and so marks out the foot-path required.

cbd, together with angle abc of the interposed lozenge-pane, make up together exactly two right angles.v

ART. 11. "If two right lines intersect one another, the vertically opposite angles are equal."

J2

A very familiar application of this principle may be seen in a clock with a double minute-hand. Whenever the pointed hand has past the centre line (supposed to be drawn from 12 to 6), the opposite end will also have passed this line exactly as much to the other side of it. Thus, if the former point to 1, or the fiveminute division beyond 12, the latter will point to 7, or the five-minute division beyond 6. In other words, the angle between the central line

and the pointed index will be exactly equal to that between the central line and the opposite index.

The above truth is a consequence of the nature of straight lines: whatever inclination they had to each other before intersection, the same inclination will they preserve after their intersection by reason of their straightness.

It is on the above principle that a pair of scales adjusts itself. The heavier side, suppose d', pulls down the corresponding arm CE below the horizontal line AB, and by this means (the whole length DE being straight and rigid), it raises the other arm CD just as much above the horizontal line: that is to say, both arms, CE, CD, play through exactly the same angular space, or, in other words, the vertical angles BCE, DCA, are equal. Additional weight being thrown into the opposite scale d, the

A

D

B

E

corresponding arm, cn, descends, and the other, CE, rises,

both through the same angle, till at length, the weights in both dishes being equalized, the whole length, DE, of the arms becomes horizontal.

The action of a common pump (which is only a kind of balance, where the pressure of the workman on the arm overpowers the weight of water in the bucket, or on the piston,) similarly illustrates this Article. The inner part, dc, of the arm, at first below the horizontal line ab, when the bucket or piston is down, revolves gradually round the pivot, d, as the outward part, de, of the arm is depressed. The latter now gets below the line ab, and the other plays through an equal angular space above it.

LESSON III.

DEF. IX. "Two right lines are said to be equally distant from one another when any two points whatsoever in the one not the greater, and any two equally remote points in the other, being taken, the right lines which join each opposite pair of points towards the same hand are equal to each other."

It is a pity that the doctrine of parallel lines should be in the slightest measure imperfect, for it is the most beautiful as well as the most useful in all Geometry. We shall find that after passing the threshold, in which this definition. lies as a stumbling-block, the structure itself opens widely and nobly, rearing its head to an eminence which will at once astonish and delight us. You have of course observed how frequently parallel lines occur in the works of Nature and of Art, especially in the latter: the trunks of trees in a forest, the columns of a temple, the pillars of a basaltic range, the shafts of a cloistered aisle, the banks of a stream, the opposite edges of a table, nay, the form of a crystal, the shape of a die,—all present to the mind a specimen, more or less rude and inaccurate, of parallelism. It is this relation which the eye is so constituted as to think most beautiful, or at least most regular. If any one chose to build his house broader at the bottom than the top, or to have his windows in the shape of a triangle, he would be deemed a very odd sort of person, and of extremely bad taste, if not of deranged intellect. How curious, how any

thing but pleasing, do we think the façade of the Egyptian Hall, with its leaning pilasters and narrowing windows! Even in those works where the lines are necessarily curved, mutually inclined, or totally irregular, there will be almost always discovered a tendency in some part or other towards parallelism. Take the human subject, "the most replenished sweet work of nature," and to the beauty of which the "waving line" is considered indispensable,-even here you will find a sort of parallelism between the line of the eyebrows and that of the lips; the temporal bones, or sides of the forehead are generally parallelous; and the visible parts of the teeth approach nearly to the form of parallel oblong figures, being considered handsome exactly in proportion as they approach this shape. The arms and legs have, in an upright unexerted position, their central lines or axes nearly parallel where there is no deformity of person. Again, examine a landscape: if there be rocks, you will often find their sides taking a square or parallel form; if there be trees, their stems will usually be parallel to each other; if there be houses, they will have parallel sides or gables. The roads, hedges, palings, fields, garden-walks, &c. &c. all assume this quality in some degree. Nay, the very ocean itself, unruly, wild, and turbulent as it may be, carries its billows in parallel ridges, and rolls its waters in parallel breakers on the shore. The poet says, no less appropriately than finely,

66 Regular as rolling water;"

its regularity being the effect of its parallel motion. The stars likewise seem to perform their aerial courses in parallel lines: : geologists have shown that the earth is formed of layers, or strata, different in matter and thickness, whose surfaces are parallel, or nearly so: the cloudy belts, or sashes, as they are called, which astronomers behold encompassing several of the planets, as Jupiter, Saturn, &c., lie parallel to each other; rivers scoop out their channels in winding parallels wherever the nature of the soil permits them; a flock of birds in steady flight, or a herd of animals running, keep between themselves a parallel course; rain pours, hail pelts, and the forked lightning points its blades in lines of a nearly parallel direction.