other extremity A draw any number of right lines, a C, A D, A E, &c. cutting the circle in the points R, O, м, &c. ; then, if c L be taken=A R, DO=A 0, E N=A M, &c., the curve passing through the points, A, L, O, N, &c. will be the cissoid.

1. Here the circle A O B O is called the generating circle; and A B is called the axis of the curves A L O N, &c. a lon, &c. which meet in a cusp at A, and, passing through the middle. points o, o, of the two semicircles, tend continually towards the directrix, e в E, which is their common asymptote.

2. If A o and A o are quadrants, the curve passes through o and o, or it bisects each semicircle.

3. Letting fall perpendiculars L P, R Q, from any corresponding points LR then is A P=B Q, and a L=C R.

4. AP PB:: P L2: A P3. So that, if the diameter A B of the circle = α, the absciss A P = x, the ordinate P L = y; then is. xax:: y2: x2, or x3-(a—-x) y', which is the equation to

the curve.

5. The right line e B E is an asymptote to the curve.

6. Arch A м of the circle=arch в R, and arch a m=B r. 7. The whole infinitely long cissoidal space, contained between the asymptote e в E and the curves N O L A, &c. A LON, &c. is equal to three times the area of the generating circle

A OBO.

SECTION III.-The Cycloid.

The cycloid, or trochoid, is an elegant mechanical curve first noticed by Descartes, and an account of it was published by Mersenne in 1615. It is, in fact, the curve described by a nail in the rim of a carriage-wheel while it makes one revolution on a flat horizontal plane.

F

E

1. Thus, if a circle E P F, keeping always in the same plane, be made to roll along the right line A B, until a fixed point P, in its circumference, which at first touched the line at A, touches it again after a complete revolution at в; the curve A P V P B described by the motion of the point P is called a cycloid.

2. The circle EPF is called the generating circle; and the right line A B, on which it revolves, is called the base of the cycloid.

Also, the right line, or diameter, c v, of the circle, which bisects the base A B at right angles, is called the axis of the cycloid; and the point v where it meets the curve, is the vertex of the cycloid.

3. If P be a point in the fixed diameter A F produced, and the circle A E F be made to roll along the line A в as before, so that the point A, which first touches it at one extremity, shall touch it again at B, the curve P V P, described by the point P, is called the curtate cycloid.

4. And, if the point P be any where in the unproduced diameter A P, and the circle A EF be made to roll along AB from A to B; the curve

PVP is, in that case,

E

called the inflected or prolate cycloid.

B

P

The following are the chief properties of the common cycloid.

1. The circular arc vE the line E G between the circle and cycloid, parallel to A B.

2. The semicircumf. v E c=the semibase C B.

3. The arc v G=2, the corresponding chord v E.

A

T

G

G'

D

E'

E

4. The semicycloidal arc v G B=2 diam. v c. 5. The tangent T G is parallel to the chord v E.

6. The radius of curvature at v=2 c v.

C

E

7. The area of the cycloid A V B C A is triple the circle CE V; and consequently that circle and the spaces v EC BG, VE' CAG', are equal to one another,

8. A body falls through any arc L K of a cycloid reverse, in the same time, whether that arc be great or small; that is, from any point L, to the lowest point K, which is the vertex reversed and that time is to the time of falling perpendicularly through the axis м K, as the semicircumference of a circle is to its diameter, or as 3.141593 to 2. And hence it follows that if a pendulum be made to vibrate in the arc L K N of a cycloid, all the vibrations will be performed in the same time.

M

L

K

9. The evolute of a cycloid is another equal cycloid, so that if two equal semicycloids o P, o q, be joined at o, so that oм be =м K the diameter of the generating circle, and the string of a pendulum hung up at o, having its length = 0 K or = the curve OP; then, by plying the string round the curve o P, to which it is equal, if the ball be let go, it will describe, and vibrate in the other cycloid P KQ; where o P=Q K, and o Q=P K.

10. The cycloid is the curve of swiftest descent: or a heavy body will fall from one given point to another, by the way of the arc of a cycloid passing through those two points, in a less time, than by any other route. Hence this curve is at once interesting to men of science and to practical mechanics.

SECTION IV.-The Quadratrix.

The quadratrix is a species of curve by means of which the quadrature of the circle and other curves is determined mechanically. For the quadrature of the circle, curves of this class were invented by Dinostrates and Tschirnhausen, and for that of the hyperbola by Mr. Perks. We shall simply describe in in this place the quadratrix of Tschirnhausen; and that in order to show its use in the division of an arc or angle.

N

A

E

22

M

P

P

m

To construct this quadratrix, divide the quadrantal arc a B into any number of equal parts, A N, N n, nn', nв; and the radius A c into the same number of equal parts a P, P P, p p', p' c. Draw radii c N, c n, &c. to the points of division upon the arc; and let lines P м, p m, &c. drawn perpendicularly to a c from the several points of division upon it, meet the radii in м, m, m', &c. respectively. The curve A мm m' D that passes through the points of intersection M, m, &c. is the quadratrix of Tschirnhausen.

B

m'

D

The figure AC Dm'm м A thus constructed may be cut out from a thin plate of brass, horn, or pasteboard, and employed in the division of a circular arc.

D

Thus, suppose the arc I L or the angle I K L is to be divided into five equal parts. Apply the side A B of the quadratrix upon IK, the point в corresponding with the angle K. Draw a line along the curve a s, cutting K L in F. Remove the instrument, and from Flet fall the perpendicular F E upon I K. Divide E I into five equal parts by prob. 5, Practical Geometry, and through the points of division draw c м, D N, &c. parallel to E F. Then from their intersections, M, N, O, P, draw the lines K M, K N, K O, K P, and they will divide the angle I K L into five equal parts, as required.

C

M

F

E

K

Note 1.-If, instead of dividing the arc into equal parts, it were proposed to divide it into a certain number of parts having given ratios to each other; it would only be necessary to divide E I into parts having the given ratio, and proceed in other respects as above.

Note 2.-If the arc or angle to be divided exceed 90 degrees, bisect it; divide that bisected arc or angle into the proposed number of parts, and take two of them for one of the required divisions of the whole arc.

SECTION V.-The Catenary, and its application.

The catenary is a mechanical curve, being that which is 6sumed by a chain or cord of uniform substance and texture, when it is hung upon two points or pins of suspension (whether those points be in a horizontal plane or not), and left to adjust itself in equilibrio in a vertical plane.

This curve is of great interest to practical men on account of its connexion with bridges of suspension, or chain bridges. Its consideration cannot, therefore, with propriety be omitted, although it involves mechanical propositions which will be announced subsequently.

A

D

B

d

B

Let A, B, be the points of suspension of such a cord, a ac bв the cord itself when hanging at rest in a vertical position. Then the two equal and symmetrical portions A a c, cb в, both exposed to the force of gravity upon every particle, balance each other precisely at c. And, if one half, as c b в, were taken away, the other half, A a c, would immediately adjust itself in the vertical position under the point A were it not prevented. Suppose it to be prevented by a force acting horizontally at c, and equal to the weight of a portion of the cord or chain equal in length to cм; then is c м the measure of the tension at the vertex of the curve; it is also regarded as the parameter of the catenary. Whether the portion A a c hang from A, or a shorter portion, as a c, hang from a, the tension at c is evidently the same for in the latter case the resistance of the pin at a, accomplishes the same as the tension of the line at a when the whole A a c hangs from A.*

M

Let the line c м which measures the tension at the vertex be =p, let c d=x, a d=d b=y, c a = c b=z, c D = h, A B = d, cа А=c bв=l.

Then

* This may easily be determined experimentally, by letting the cord hang very freely over a pulley at c, and lengthening or shortening the portion there suspended, until it keeps à a c in its due position; then is the portion so hanging beyond the pulley equal in length to c м.