ample, if four equal_circles are to be inscribed, construct a square, A B CD, apart from the circle, on any line A B. Let O be the centre of this figure. Then, with an angular point A as centre, and half a side A B, as radius, describe a circle. Join O A, and produce it to meet the circle in M. Divide the radius of the given circle into parts proportional to the parts of O M; thus:-draw from O a line OR equal to this radius. Join R M, and draw AS parallel to MR. OS is the distance between the centres (C C', fig. 238).

Hence, if from the centre of the given circle we describe a circle with radius O S, and make angles at the centre equal to the angles at O in the figure subtended by the sides, the points of intersection of these lines with the circle last described will be the centres of the required circles.

To describe about the given circle a number of equal circles, we may proceed in exactly the same way, but if OA in the auxiliary figure cut the circle in m, we must produce the radius of the given circle until the part produced is to the radius as OA: 0m.

Applications of the preceding problems.

258. In machinery it is frequently necessary so to connect two parallel spindles or axles, that the rotatory motion of the one may be transmitted to the other. Two wheels or pulleys R and r are often fixed upon the spindles, and over them a cord or strap is stretched (fig. 249). Sometimes it is necessary, in order to maintain the necessary tension in the cord, to fix a roller g to one end of a bent arm, movable round an axis ; a weight applied at a causes the roller to press against the strap: with this arrangement the cord cannot slip. The motion of the one of the pulleys

may therefore be communicated to the other. It frequently happens that the forces applied are too great

R

Fig. 249.

for this method of communication; wheels are then arranged so that they may touch one another, either internally or externally, according to the direction in which the rotation is to be transmitted; and in order to insure that they will not slide one upon the other, they are both armed with teeth which fit one another, and are said to be in gear (fig. 250). The arrangement of the teeth is such that the motion is communicated with as

much regularity as if the two wheels were without teeth, and touched without sliding the one upon the other. The determination of the form of the teeth is a difficult question, which we shall not treat of here. It will suffice to say simply that when two wheels are in gear with one another, the teeth of the one are exactly of the same size as those of the other, and so are the spaces between the teeth, so that the number of teeth in the two wheels are in the same ratio to one another as the circumferences themselves. When one of the wheels is very small compared with the other, it is called a pinion. These contrivances are not only used for the transmission of motion, but are also employed to vary the speed of rotation. If a pinion has ten teeth and the wheel thirty-one, the pinion will make thirty-one revolutions while the wheel makes only ten.

The variation in the times of rotation is well illustrated in the works of a watch (fig. 251). The motion

originates with the spring A, and is transmitted through B, C, D, etc., to M.

259. Ovals and elliptical figures are formed of arcs touching one another.

If a circle be viewed obliquely, or its shadow be cast upon a plane which is not parallel to it, the figure presented is called an ellipse

Fig. 252.

(fig. 252). The general name of oval is given to curves which resemble an ellipse. The oval is formed of several arcs of a circle, the number of which is increased according to the degree with which we wish to approximate in form to the ellipse. This curve has, like the ellipse, a longer and a shorter axis, each of which is symmetrical. The meeting point of the two is the centre of the oval, and every line drawn through the centre is called a diameter. The semi-oval cut off by the greater axis has been called the basket-handle arch. It is employed in architecture in the drawing of elliptical vaults.

260. The basket-handle arch is generally formed of arcs of different radii which meet at their extremities, and which have given to the figure the name of the curve of many centres. Such curves are formed with 3, 5, 7, 9, and 11 centres. They are particularly applicable to the construction of arches in bridges. The basket-handle arch with three centres, is generally thus constructed: Let A A' be the span of the arch, and OB its height, corresponding respectively with the greater axis and half the smaller axis of the oval. Upon A A' as diameter, describe a semi-circumference Ab A', and divide it at the points m and m' into three equal parts; join Am, m b, bm' and m' A'; draw BM and BM' parallel respectively to bm and b m', and terminating in the chords, MA, M'A'; draw m O and m'O, and then M D and M'D' parallel to these lines. They

will intersect in BO produced, and cut A A' in C and C'. Since the four triangles which have their vertices at O are isosceles the four triangles, C MA, C'M'A,

BDM, BDM', which have their sides respectively parallel to the first four, are also isosceles triangles. Thus CM = CA, C'M' C'A', and DM BD=DM'. Hence the three arcs AM, A'M'and MBM' described respectively from the three centres, This method may

=

C, C, and D, meet in M and M. always be applied when the height O B is not less than three-fourths of the half of the span A A.

When the arch has a height less than three-fourths

Fig. 254.

of the semi-span, the following construction may be applied (fig. 254). Let A A' be the span, and O B the height; describe the circumferences, O B and O A. Suppose we wish to have an arc of nine centres, divide the semi-circumference in nine equal parts, and draw the radii OC, OD, O E, meeting the smaller circumference in the points c, d, e, etc. Draw through the