1. Surface 4πr2× circumference x diameter = 3.1416 × square

of diameter.

=

2. Solidity=π = × circumference x (diameter)2 × 7854 × (diameter)3·5236 × (diameter)3.

ZONE OF SPHERE.

1. Surface =2πr (h1 − h2) = circumference of sphere × height of

2.

zone.

1

Solidity=h (3r12+3r,2+h2), where 1, 2 are radii of ends and h is the height of the zone.

=

SEGMENT OF SPHERE.

1. Surface 2πrh = circumference of sphere x height of segment. Solidity=h (3r12+ h2), where r1 is radius of end and h height of segment.

2.

SPHEROID (PROLATE OR OBLATE).

1. Solidity=5236 × square of revolving axis × fixed axis. 2. Solidity= volume of circumscribing cylinder.

PARABOLOID.

1. Solidity volume of circumscribing cylinder.

SIMILAR SOLIDS.

1. Volumes are as the cubes of corresponding edges.

Thirty days hath September,
April, June, and November,
February has twenty-eight alone,
And all the rest have thirty-one.
But leap-year coming once in four,
February then has one day more.

The civil or mean solar day (the standard unit for the measurement of time) is the mean time which elapses between two successive transits of the Sun across the meridian of any place.

The civil year consists of 365 mean solar days, but the actual time taken by the Sun to return to any point in the ecliptic is 365-242218 mean solar days.

Thus the civil year is about of a day less than the solar year.

To remedy this, Julius Caesar, as Pontifex Maximus, in B.C. 46 made every 4th year a leap-year or bissextile year of 366 days, February having a 29th (intercalary) day.

The Julian year is thus 365-25 days, while the solar year is 365-242218 days only-there is therefore an excess of 007782 days each year.

=

This difference becomes in 400 years 31128 days (400 × 007782), and therefore Pope Gregory in 1582 found the Vernal Equinox to be on March 11th instead of March 21st, which was the case in 325 at the Council of Nice (1257 x 007782 97819 or 10 days nearly). To correct the error, 10 days were left out, Oct. 15 following Oct. 4, 1582, and to prevent its recurrence, in every 400 years 3 of the leap-years are omitted, viz. those which complete a century not divisible by 4: thus 1200, 1600, 2000 are leap-years, but 1300, 1700, 1900 are not.

This Gregorian or New Style was adopted in England. on Sept. 2, 1752, when the error was 11 days.

THE CALENDAR.

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Russia is the only country of Europe not yet fully adopting the New Style.

The quarter days are March 25, Lady Day.

June 24,

Midsummer Day.

A lunar month is the time taken by the Moon to return to the same place in its orbit.

A calendar month is the time between the same dates in two consecutive months.

Special rule for a daily wage.

Given daily wage-to find yearly payment.

Wage in d. as £+ half wage in d. as £+5 (or 6) days' wages.

Special rule for a weekly wage.

Given weekly wage-to find yearly payment.

Twice wage in s. as £+ wage in s. as £+2 weeks' wages.

EXAMPLES.

1. Construct Inversion and Interchange Tables for 365, 366, 52, 312, 313.

2. Find annual wage-given daily wages as 1s. 7ąd., 1s. 10žd., 2s. 1d., 3s. 4d., 4s. 9d.

3. Find annual wage-given weekly wages as 15s. 8d., 178.9d., 25s. 6d., 32s., 35s., 52s., 63s.

Angular Measure.

The English or Sexagesimal Method.

60 seconds (") = 1 minute.

60 minutes () = 1 degree.

90 degrees (°) = 1 right angle.

30° = 1 sign (S.), 12 signs = 360° = 1 circle (C.).

The circumference of every circle is considered to be divided into 360 equal parts, each of which is called a degree as it subtends an angle of 1° at the centre of the circle.

The French or Centesimal Method.

100 seconds (") = 1 minute.
100 minutes () = 1 grade.

100 grades (*) = 1 right angle.

The advantage of this method is that 57° 33' 5" =57.3305=573305 right angles at once without any working other than moving the decimal point.

1 degree

=

=

of a right angle, 1 grade of a right angle,... 90° 100%,... 1° 10, or 18. Hence

=

=

(1) to convert degrees into grades—add §,

(2) to convert grades into degrees-subtract.

Longitude. A difference of 15° Longitude gives a difference of one hour in time.

(1) To convert angular measure into time.
Divide degrees by 15-quotient is hours.
Multiply remainder by 4-product is minutes.
Divide minutes by 15-quotient is minutes.
Multiply remainder by 4-product is seconds.
Divide seconds by 15-quotient is seconds.

DEGREES AND GRADES.

(2) To convert time into angular measure.
Multiply hours by 15-product is degrees.
Divide minutes by 4-quotient is degrees.
Multiply remainder by 15-product is minutes.
Divide seconds by 4-quotient is minutes.
Multiply remainder by 15-product is seconds.

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Temperature (Thermometers).

The chief scales are

(1) Centigrade. Freezing point 0°, boiling point 100°, no. of degrees between = 100.

(2) Fahrenheit. Freezing point 32°, boiling point 212°, no. of degrees between = 180.

(3) Reaumur. Freezing point 0°, boiling point 80°, no. of degrees between = 80.

The Centigrade or Celsius's scale is used in scientific operations and on the Continent.

Fahrenheit is commonly used in England, Reaumur in Russia.