EXAMPLE 2.

The diameter of a circle is 24; required the diameter of a circle 4 times less.

Answer, 12. 24 × 24 = 5764144 of which is 12 the answer. NOTE 1.-To find the diameter and circumference of a circle the one from the other.

Multiply the diameter by 3.1416. the product will be the circumference:

Therefore,

Divide the circumference by 3.1416 and the quotient will be the diameter.

NOTE 2.-When the diameter is given to find the area of a circle. RULES.

RULE 1.-Multiply the square of the diameter by .785398.

Or,

RULE 2.-Multiply the square of the semidiameter by

RULE 3.-Multiply half the circumference by half the diameter.

Or,

RULE 4.-Multiply the square of the diameter by .7854, the product in either case will be the area.

NOTE 3.-To find the area of an Ellipsis or Oval.

RULE.

Multiply the transverse diameter by the conjugate and their product by .7854, the last product will be the area of the Ellipsis.*

LESSON 17

The conjugate and transverse diameters of an Ellipsis given to find the diameter of a circle equal in area.

RULE.

Multiply the two diameters of the ellipsis together, and the square root of their product will be the diameter of a circle equal in area, to the ellipsis.

* Conjugate diameter is the longest line passing through the centre of

an oval.

Transverse diameter is the shorter line passing through the centre of an an oval at right angles with the conjugate,

EXAMPLE.

The area of an ellipsis is 1244 inches, the conjugate diameter is 44 and the transverse is 36; of what length must be the diameter of a circle equal in area? 39.8 inches nearly. OPERATION.

Work by Note 2, and Rule 4, under Lesson 16.

Here note, if the diameter of the circle had been carried further in the decimal parts, say .79 instead of .8, the proof would have come out more exactly.

QUESTIONS FOR EXERCISE.

LESSON 18.

If a pipe of an inch diameter, will fill a cistern in 4 hours, what must be the diameter of another pipe which will fill the same cistern in 2 hours.

Answer, 4" 2" 10" or .35355 + of an inch.

RULE.

As the time given,

is to the square of the given diameter; So is the time required,

to the square of the required diameter.

OPERATION.

As 4 hours are to 12 of .25, so are 2 hours to of the diameter required. Inverse Proportion.

If a pipe of an inch diameter will fill a cistern in 8 hours, in what time will a pipe whose diameter is 14 inch fill the same cistern? Ans. 53 min. and 20 sec.

When the answer consists of time, the second term will be in time; and the first and third terms in squares of given diameters.

If 4 pipes of 3 inches diameter each, will be 4 hours in discharging a cistern of water, what time will 3 pipes of 4 inches diameter, require to discharge another cistern, which contains 5 times that quantity? Ans. 15 hours. 3 inches diam.

LESSON 21.

In the midst of a meadow well cover'd with grass,
I will take just two acres to tether my horse;
How long must the cord be, when he's feeding around,
To command him within these two acres of ground?

Answer, 252.313+ links: or, 166 ft. 6 in. 4 sec
See the rules under lesson 15.

Common multiplier, 1,273241
Links in 2 acres,

200000

✔ 25,46,48,20,00,00(504.626+

25

1004) 4648
4016

10086) 63220
60516

100922) 270400
201844

1009246). 6855600

6055476

,800124

Diameter, 504.626+

Semidiameter, 252.313 or length of the cord.

1 link = 7.92 inches. .313 of a link =24 in!