figure by determining all the angular points (fig. 359). The plan of drawing on squares (§ 327) may be ap

Fig. 359.

plied to the construction of a figure similar to another (figs. 360, 361), but the reduction is effected more easily

with the pantograph (§ 190). O is the fixed point of the instrument.

Two regular polygons of the same name are similar. 339. For if we join to the centre the angular points

B

of each of them,

the other, for the angles are respectively equal; therefore the two polygons are similar, and we have

Fig. 363.

AB A'B'.

AO A'O

Hence we may construct a regular polygon similar to a given one, and having a given side, by taking the centre of the given polygon centre of similitude (fig.

as a

363.)

To lay down plans with the portable table.

340. It has been before stated (§ 177), that a plan of a meadow is a figure similar to the boundary of the meadow drawn upon paper, and methods have been given for producing the plan when the dimensions of the land to be represented are known.

By using the portable table about to be described, we obtain directly upon the paper a figure similar to that formed by the different lines which join the points marked upon the piece of ground.

The complete apparatus consists of a plane table for drawing, mounted upon a stand, so that it may be placed in a horizontal position (fig. 364), and of a cross bar (fig. 365) of copper, having two "sights" at its extremities; that is to say, two copper plates parallel to one another, and perforated by narrow slits, along which are stretched fine black lines. The bar forms a ruler constructed so that a line drawn along its edge would pass through the prolongation of the threads in the "sights."

Let us use the instrument to draw the plan required (fig. 366.) Select in the meadow two points M and N, from which all the extremities of the lines to be laid down may be seen. Measure the distance M N, and represent it upon the table by mn. MN must

be of such a lengthe nd placed in such a position

that the plan may not pass

C

the edges of the paper. Fix a needle at m, and

E

Fig. 366.

place the table so that the point m, may be over the point M, and the line m n may have as nearly as possible the direction M N. Fix the feet of the stand firmly in the soil, and see that the table is quite horizontal, and that when the

cross-bar is placed in the direction mn, on looking through the slit, a staff placed at N may be seen. Fix the table in this position. Taking care not to touch the table, turn the cross-bar round the needle fixed at m, until the line of sight passes through the staff fixed at A; draw along the edge of the ruler the line, ma ; the angle amn will be equal to AMN. Direct in succession the cross-staff to all the points marked B, C, D, etc., and in each case draw mb, m c, m d, etc. From time to time make certain that the table has not been moved; that is to say, that mn has exactly the direction M N.

Now, take up the instrument, and fix it at the other station N. Repeat the same operations at this point, and draw the lines na, nb, nc, etc., the angle anm will be equal to the angle A N M, etc. Compare now the figure thus obtained with the polygon to be laid down. The triangles ANM, anm, are similar to each other because they are equiangular, and it is the same with all the others, the ratio of similarity being that of M N to m n. It follows that the ratios of MA, MB, M C, to ma, mb, m c, are equal, and the polygon abcd is similar to the polygon A B CD.

The perimeters of similar polygons have the same ratio as their homologous sides.

341. For since the ratio of each side to the side homologous to it is the same, the ratio of the sum of sides of the first polygon to the sum of the sides of the second is equal to the ratio of any two homologous sides. If, for example, each side of one is three times the homologous side of the other, the perimeter of the first is three times the perimeter of the second. The surfaces of similar polygons have the same ratio as the squares of the homologous sides.

342. We have already seen (§ 281) that the ratio of the surfaces of two similar triangles is equal to the ratio of the squares of the homologous sides. But, two similar polygons may be decomposed into the same number of triangles similar each to each, and having the same ratio of similarity; therefore, as the ratio of the surfaces of two homologous triangles is equal to the ratio of the squares of their homologous sides, the same ratio must exist between the sum of the triangles which form the first polygon and the sum of those which form the second.

Circumferences have the same ratio to one another as their radii. Circles have the same ratio to one another as the squares of their radii.

343. Suppose that two circumferences are each divided into the same number of equal parts, sufficiently numerous to be very small compared with the radius, two similar polygons will be formed. The greater the number of sides in the polygons, the less will be the difference between their perimeters and the circumferences, and the less the difference between their surfaces and the circles. However great the number, the ratio of the perimeters of the two polygons remains always equal to the ratio of their radii,