then, by ART. 7, the said triangles must be indubitably equal.

It is almost unnecessary to mention that we only give skeletons both of the construction and operation; just so much, in fact, as is necessary to exhibit the general principle, but not the details, which would be irrelevant to our purposes.

The practical utility of this theorem is as manifest as that contained in ARTS. 1, 2, and 3. It is applied in almost every mechanical process, and carpenters, masons, artificers of all ranks and descriptions, constantly employ it, though in nine cases out of every ten without being conscious that it is a geometrical principle which directs their operations.

LESSON II.

DEF. VI. "If one right line, standing upon another, make the adjacent angles equal to one another, each of these angles is called a right angle, and the right line which stands upon the other is called a perpendicular to it.”

TEACHER. The shortest way to know whether an angle,

such as DCB, be a right one, is to produce one side, as BC, through the vertex, and take CD, CB, CE, all equal. Then if the distances DB and DE be equal, the angle DCB is a right one. For the three sides DC, CB, DB, of the

D

C

B

We may observe, that in both these operations the sides of the equilateral triangle are not actually drawn as in the construction given in the Popular Geometry; but they are omitted as superfluous, because all we want is the vertex of the triangle, i. e., the point G in the first example, and the point н in the second.

It may be said that a shorter way in both examples would be to join E and F in the first, F and G in the second example, then to divide the joining line equally, and that the point of division: being joined with the vertex of the angle would divide it into two equal parts. This is perfectly true; but how is the joining line to be divided equally? Why, by means of this very problem. Where great accuracy is not required, the joining line may indeed be equally divided by trial; but, perhaps, after all, that will often be found more tedious than the scientifical method.

triangle DCB being respectively equal to the three sides DC, CE, DE, of the triangle DCE, the angle DCB opposite DB is equal, by ART. 6, to the angle DCE opposite DE. And these being adjacent angles, DCB must be a right one.

But if the lines CB, CD, be stiff and moveable, a yet shorter way would be to lay them on paper, and mark the angle DCB on it; then, to produce either marked line, and keeping the vertex c in its place, bring the side CB round to fall on the production CE. If now the other side coincided with the marked line CD, the angle DCB would be a right one.

G

It is in this way that a carpenter examines if his Square be true. He marks the outer edge of the blades CD and CE, producing either marked edge CE through the vertex to any convenient length CF. He then turns the instrument, so that the outer edge of either blade may lie along the line

C

E

PROB. V. "To divide a given finite right line into two equal parts."

The necessity of this problem is as obvious as that of the last but as it is more easily accomplished by trial, the geometrical construction does not seem to be so indispensable. Nevertheless, as we hinted above, perhaps the scientifical method will in many cases be found shorter than the experiment of opening and closing the legs of a compass, or any other mode of finding the exact middle point. An artizan wishes to divide a small block of ivory EADCBFG into two equal slips; that is, to cut through the middle length of the face ABEF. For this purpose he must divide the edge AB of the end ABCD, into two equal parts: how is he to do this?

A

With the points A and B respectively as centres, and the distance AB as radius, let him describe the two circular arches yzx, vzw, and draw right lines from their point of intersection z to a and B. Now let him take zm equal to zn, and with m and n respectively as centres describe

D

k

C

CF, and marks the outer edge CD' of the other blade (lying at the same side of EF as CD did). If the instrument be not true, that is, if DCE be not a right angle, suppose it less; then the blade CD in its first position will fall to the right of the true perpendicular line CG, and the angle DCG will be the error of the instrument. In its second position it will fall as much to the left of that line. Hence the marks CD and co' will exhibit an angle DCD' between them, which is twice the error of the instrument.

In the same manner, if the angle DCE of the square be greater than a right angle, the blade DC in its first position will fall to the left of the true perpendicular co, in its second to the right, and the angle DCD' will be twice the error of the instrument.

But if the angle DCE be exactly equal to a right angle, the blade CD will lie along the true perpendicular CG in its first position, and in its second likewise. So that there will

D

G

D G D

F

E

E

the circular arches prq, srt, with the common distance mn as radius. The right line zr, joining these two points of intersection, will, if continued, divide AB equally at i.

For, the above is exactly the construction given in PROB. V, GEOMETRY, for the division of the right line AB into two equal parts. First the equilateral triangle AzB is constructed, and then its vertical angle at z is divided into two equal parts by zr, or zi.

By the same method it is evident that each of the slips zikCBFG, zikDAE, may be divided into two equal slips, thus subdividing the original block into four equal slips. And so on; it is clear that we may subdivide by this means the original block into 8, 16, 32, &c. equal slips.

It is true that in many instances this nicety of subdivision would be superfluous: and instead of employing the above geometrical problem, it would be sufficient to take the length of AB with a string, which being doubled would give the half of AB; or to measure AB on a scale, and take half the number of inches; or to adopt any other such experimental process But there are

C

be no angle between both, and therefore no error in the instrument.

By the help of a string we may easily tell whether a straight beam, pole, post, or pillar, be perfectly upright on the surface where it stands; for we have only to measure equal distances with the string at opposite sides of the foot. Then let the string be stretched from the extremities of these several distances to the same height on the shaft from the ground; and if the part stretched be of the same length at all the opposite sides, the shaft is perfectly upright; but if this part be shorter at one side than the opposite, the shaft leans to that side. Any thing with which we can measure will, it is evident, answer for the above purpose. ART. 8." All right angles are equal."

This principle is of perpetual recurrence in practice. The side-boards of a drawer are placed at right angles to the end-boards, in order that the corners thus formed may exactly fit into the corners of the chest, which are also right angles. The angles at a and a of the brackets (Art. 7) are made right ones, and thence the shelf will rest evenly

numberless cases in which these rude methods would be altogether inapplicable; the scientifical one must then be followed. PROB. VI. "To draw a perpendicular to a given right line, from a given point without it.'

This problem and the next are of such frequent occurrence in practice, that an instrument has been made for the purpose of mechanically accomplishing them. The Carpenters' Square is an instrument composed of two straight and evenly edged blades AB, CD, set at a right angle BAN; so that when one, as CD, is moved along any right line xy, and the edge of the other, AB, brought over a given point, v, this edge being traced, will mark out a right line, vs, perpendicular to

XY.

V

L

Such an instrument, if accurate, almost supersedes the use of the geometrical construction; but as it is frequently very inaccurate, and moreover is sometimes not at hand, we must often recur to the scientifical method.

upon them, and be perfectly horizontal,—which would not be the case if these angles were unequal. Carpenters' Squares, though made by different hands, and by different means, have all their corners exactly equal (that is, if the instruments are all true), because they are all right angles, &c. &c.

ART. 9. "When a right line meeting another right line makes angles with it, these angles are together equal to two right angles."

This theorem is of extensive use in Astronomy, as will be found in the study of that science.

We are frequently able to measure the angle at one side of a line which stands on a second, without having the power to measure the adjacent angle; but the former being had, we need only subduct it from two right angles in order to have the latter, by this ART. Thus, if BAC were the corner of a field

B

or enclosure which we could not enter; in order to get the size of this angle, we have only to measure the angle BAD (contained by one side BA of the enclosure, and the pro

If, for example, a surveyor, after having laid down the plan ABCD of a field, wishes to know how broad it is from any point in the side BC, as E, to the opposite side AD, he proceeds thus:

With E as a centre, and the distance from E to any point н on the other side of AD as radius, he describes the circular portion FHG cutting AD in F

and G. With these points respectively as centre, and the interval FG as radius, he describes the circular portions nom, hog, and draws the straight line Eo from their intersection o to the given point E. This will be perpendicular to AD at I; and EI being measured will show the breadth required.

For, EF and EG are equal, as radii of the same circle; and EI divides the angle FEG equally, by PROB. IV. Consequently, EI