RULE.

As any abfciffa

Is to the fquare of its ordinate,

So is any other abfciffa

To the fquare of its ordinate.

EXAMPLE I.

Let the abfciffa VC be 6, and its ordinate AC 5, required the ordinate DF, whofe abfciffa VF is 12.

6:25:12

12

6)300

50=DF*

and 50 = 7.071 Anf.

Ex. 2. The ordinates are 6 and 8, and the less abfciffa 9, required the greater.

Anf. 16.

Ex. 3. The ordinate is 18, and its abfciffa 27, the other ab fciffa is 48, required its correfponding ordinate.

Anf. 24.

PROBLEM XI.

To find the length of an arch of a parabolic curve, cut off by a double ordinate.

RULE.

To the fquare of the ordinate add of the square of the abfciffa, multiply this fum by 4, and the fquare root of the product will be the length of the curve required.

EXAMPLE

EXAMPLE I.

Let the abfciffa VF be 4, and its ordinate DF 12, required the length of the arch DAVBE.

Ex. 2. Required the length of the curve, when the abfciffa is 8, and the ordinate 16.

Anf. 36.951.

Ex. 3. Required the length of the curve, when the abfciflà

is 15, and ordinate 12.

Anf. 21.071.

Ff2

PROBLEM

PROBLEM XII.

To find the area of a parabola, the bafe and height being given.

RULE.

Multiply the base by the height, and the product will be the area required.

Note. Every parabola is equal to of the circumfcribing parallelogram.

EXAMPLE I.

Required the area of a parabola, whofe bafe is 16, and height 20.

16

20

320

2

3640

2131

Anf. 400.

Ex. 2. Required the area of a parabola, whofe bafe is 30, and height 20. Ex. 3. Required the area of a parabola, whofe bafe is 9, and height 14.

Anf. 84

Auf. 96.

Ex. 4. Required the area of a parabola, whose base is 12, and height 12. Ex. 5. Required the area of a parabola, whofe bafe and altitude are 15 and 22. Anf. 220. Ex. 6. Required the area, when the base and altitude are 3 and 4.

Anf. 8.

PROBLEM

PROBLEM XIII.

To find the area of the fruftum of a parabola.

RULE.

Divide the difference of the cubes of the two ends of the fruftum by the difference of their squares, multiply this quo tient by the altitude, and the product will be the area required.

EXAMPLE I.

In the parabolic fruftum DABE, the two parallel ends DE, AB, are 12 and 2c, and the altitude FC 6, required the area.

Ex. 2. The greater end of a fruftum is 20, the lefs 10, and

their distance 12, required the area.

Anf, 1863.

1

Ex. 3. The greater end of a fruftum is 30, the lefs 20, and their distance 15, required the area.

Anf. 380.

Ex. 4. The greater end of a fruftum is 9, the less 6, and their diftance 4, required the area. Anf. 113.

PROBLEM XIV.

To defcribe an hyperbola, the tranfverfe and conjugate diameters being given.

RULE.

Draw AB the tranfverfe diameter, and BC the conjugate at right angles to it; bifect AB in c, and and with the centre c, and radius cE, defcribe the circle EFDf, cutting AB produced in the points F, f, and these points will be the foci.

In AB produced take any convenient number of points x, x, &c. and from F and f as centres, and radii Bx, Ax, describe arches interfecting in the points m, m, &c. Join these points, and it will form the hyperbolic curve required.

Note. If through the points E and D ftraight lines be drawn from c, they will be the afymptotes of the hyperbola.

Any three of the four following particulars being given, to find a fourth, viz. the transverse, conjugate, ordinate, and its abfciffa.

PROBLEM XV.

The tranfverfe, conjugate, and abfciffa being given, to find the or

dinate.

RULE.