It has been already shown that B C. BA. Sine B.

2

equal area of the triangle, ... B C. A C. Sine C.

2

area.

Hence B C. B A. Sine B=B C. A C. Sine C.

Multiply each side by B A, and we have B C. B A. Sine B-B C. A C. B A. Sine C. Divide this equation by B C, we have

B A3. Sine B=A C. B A. Sine C.

Multiply each side by Sine A.:

B A. Sine B. Sine A=A C. B A. Sine A. Sine C.

To find the perpendicular ordinates from the chord 6 of any arc of a railroad, in order to set off the curve correctly and speedily, without the help of an instrument, suppose it to be a 20° curve, the tangent 200. Find the radius, as formerly taught; multiply the radius by the natural co. sine of half the vertical angle, and you have the chord.

Multiply the radius by the natural sine of the same angle, and you have the distance from the centre to the middle of the chord, a constant number to be deducted. Now take any distance, suppose 10 feet, at which you choose to erect your

ordinates, and from the semi-chord subtract this number, square the remainder, and subtract it from the square of the radius; extract the square root, from which take the aforesaid constant number, and the remainder is the ordinate to be rightly applied, and so proceed till you arrive at

the middle of the chord; then the difference between the said constant number and the radius, is the versed sine or greatest ordinate, and now you are prepared to lay off the other side of your curve, and all this can be done in a few minutes in the field.

EXAMPLE. See last figure.

Ordinate. All this is plain from the figure, and when the radius and constant subtrahend are

found (which is only the work of a minute) all the others are nearly had at sight. This I consider quite superior to any other method now in practice. Otherwise thus: Let the radius, versed sine, chord, and constant

quantity D E, be found as

before, divide the semi-chord into any number of parts as e f g h i. From E C deduct one of the parts i C, leaves E i=m F, then D F (radius) squared-(m F)=(m D) the square root of which, minus the constant quantity, E D, gives the ordinate i F, and in like manner all the others are found, and thus the curve can be laid off in a few minutes in the most accurate manner, (by the 47th of the first of Euclid.)