a floor six metres long and five and four-tenths metres wide? and what would be the cost of the carpet, at $2.50 a centiare?

Ans. M 36. $90.

6. In a farm consisting of four fields of the following dimensions, how many hectares? First field, length M 342, breadth M 273; second field, length M 634, breadth M 350; third field, length M 450, breadth M 329; fourth field, length M 730, breadth M C32.7. Ans. 2A 92.5187. 7. A pile of lumber was found to contain 150 boards M 4 long and M 4. wide, 225 boards M 6.2 long and 2M 52. wide, and 642 boards M 5.2 long and 2M 43 wide. How much was it worth, at $42. per are, face measure. Ans. $1008.38+.

8. How many bricks M 2.2 XM 1.1 would pave a side-walk M 842.6 long and M 2.2 wide? and what would be the whole cost at 60 cents per centiare. Ans. 76600 bricks. $1520.05+.

MEASURES OF VOLUMES, OR SOLIDS.

RELATIONS OF UNITS OF VOLUMES TO UNITS OF LENGTHS.

Millistere =

Centistere=

Decistere =

Stere =

Decastere =

Hectostere =

Kilostere =

Myriastere=

{

A cubic decimetre 1000 cubic centimetres.

10 cubic decimetres, or a volume, or solid, whose length is one metre, and breadth and thickness one decimetre.

10 centisteres = 100 cubic decimetres, or a volume whose length and breadth is one metre, and thickness one decimetre.

A cube metre = 10 decisteres = 100 centisteres = 1000 millisteres or cubic decimetres.

10 cubic metres, or a volume whose length is one decametre, and breadth and thickness one metre.

10 decasteres = 100 cubic metres, or a volume whose length and breath is one decametre, and thickness one metre.

A cubic decametre 1000 cubic metres.

10 kilosteres, or a volume whose length is one hectometre, and breadth and thickness each one decametre.

NUMERICAL EXPRESSION FOR VOLUME, OR SOLIDITY.

The solidity, or contents, of a volume is expressed numerically by giving the number of times it contains some given solid as the unit of volume.

The following illustrations will show how the various denominations of the table are used in numerical expressions of volume.

Millistere, or Cubic Decimetre.

10 millisteres, placed side by side, make a volume whose length is one metre, and breadth and thickness each one decimetre, thus,

Centistere = 10 Millisteres.

10 centistere, placed side by side, make a volume whose length and breadth is each one metre, and thickness one decimetre, thus,

Decistre 10 Centisteres 100 Millisteres.

10 decisteres, placed face to face, make a cube whose edge is one metre, thus,

Stere 10 Decisteres = 100 Centisteres = 1000 Millisteres.

From these illustrations, it is evident that the contents of a cubic metre may be expressed numerically, as S 1, 1S 10, 2S 100, S 1000.

The following figures illustrate the use of the same four denominations in expressing the contents of a cubic volume whose edge is one metre and one decimetre. The surface of one face of the volume contains one centiare, two milliares, and one decimilliare, thus,

Centiare.

Milliare.

Taking a slab of the face one decimetre thick, thus, and we have one decistere, two centisteres, and one millistere.

But the volume is eleven decimetres thick; therefore we have

Milliare

Decimilliare.

eleven such slabs, or eleven times one decistere, two centisteres, and one millistere.

11 millisteres = 1 centistere and 1 millistere = S 0.011 = 22 centisteres = 2 decisteres and 2 centisteres = S 0.22 11 decisteres = 1 stere and 1 decistere

M 1.1 X M 1.1 X M 1.1

= $1.1
= $1.331

From these illustrations, we derive the following rule for finding a numerical expression for a given volume of uniform length, breadth, and thickness:

RULE. Reduce the length, breadth, and thickness to the same denomination; find the product of the three dimensions, after reduction, and point off as many decimal places in this product as there are decimal places in the three dimensions.

The unit of the numerical expression thus found will be a millistere when the unit of length is a decimetre, a stere when the unit of length is a metre, a kilostere when the unit of length is a decametre.

EXAMPLES FOR PRACTICE.

1. How many steres in a wall twenty-four metres long, eight and five-tenth metres high, and fifty-two centimetres thick? And what would be the cost of building it, at $4.25 a stere?

Ans. S 106.08. Cost, $450.84.

2. What would be the cost of a pile of wood fifteen and seventenths metres long, three metres high, and seven and fifty-two hundredths metres wide, at $1.50 a stere? Ans. $531.29.

3. What would be the cost of excavating a cellar eighteen and three-tenths metres long, ten and seventy-three hundredths metres wide, and three and four-tenths metres deep, at 15 cents per stere? Ans. $100.14+.

4. How deep must a box be, the surface of whose base is thirty-two milliares to contain seven and thirty-six hundredths steres? Ans. M 23.

5. How many steres in five sticks of timber of the following dimensions: First, M 5.2 by M 7.3, and M 13 long; second, ‚M 43. by 2M 65, and M 17.5 long; third,,M 5.3 by M 3.7, and M 15.42 long; fourth, M 39 by M 56, and M 14 long; fifth,,M 4.52 by M 3.78, and M 15 long. Ans. S 18.470352. What must be the height of a load of wood, M 3.2 long and M 1.1 wide, to contain S 4.0128. Ans. M 1.14.

6.

MEASUREMENT OF ANGLES.

In the ordinary or sexagesimal system, a right-angle, which is used as the measure of all plane angles, is divided into 90 equal parts, called degrees; a degree is divided into 60 equal parts, called minutes; and a minute into 60 equal parts, called seconds.

In the centesimal or French system, a right-angle is divided into 100 equal parts, called grades; a grade into 100 equal parts, called minutes; and a minute into 100 equal parts, called seconds.

The former is called the sexagesimal system, on account of the occurrence of the number sixty in forming the subdivisions of a degree; and the latter centesimal, on account of the occurrence of the number one hundred.

Grades, minutes, and seconds are usually written thus: 35o 42' 24"; read, thirty-five grades, forty-two minutes, twenty-four seconds. Since the scale is centesimal, minutes may be expressed as hundredths, and seconds as ten-thousandths; hence any number of grades, minutes, and seconds may be expressed decimally thus: 738 4569; read, seventy-three grades, forty-five minutes, sixty-nine seconds.

In a right-angle, there are 100 grades, or 90 degrees; hence, for every 10 grades there are 9 degrees. Dividing the 10 grades into 9 equal parts or degrees, each part will contain 1 grades; therefore a degree s equal to 13 grades. Hence, in any number of grades there are as many degrees as 13 is contained times in the given number of grades; and, conversely, in any number of degrees there are 1 times as many grades as there are degrees. Hence the following rules:

TO CHANGE THE CENTESIMAL MEASURE TO THE SEXAGESIMAL.

RULE. Express the minutes and seconds as a decimal of a grade; divide by 13: the quotient will express the number of degrees and decimals of a degree in the given number of grades, minutes, and seconds.

EXAMPLES.

Change the following quantities from the centesimal measure to the sexagesimal.

1. 258 34 42".

2. 57 93.

3. 83 13' 87".

4. 36 98' 15".

5. 14 15' 60".

6. 90 90 90".

7. 18 50' 25".

Ans. 22° 48′ 35.208".

Ans. 31' 16.932".

Ans. 74° 49′ 29.388′′.

Ans. 33° 17' .06".

Ans. 12° 44′ 25.44′′.

Ans. 81° 49′ 5.16".

Ans. 16° 39′ 8.1".

TO CHANGE THE SEXAGESIMAL MEASURE TO THE CENTESIMAL.

RULE. Reduce the minutes and seconds to a decimal of a degree; multiply the degrees and decimal of a degree by 14: the product is the number of grades, minutes, and seconds in the given number of degrees, minutes, and seconds.

EXAMPLES.

Change the following quantities from the sexagesimal measure to the centesimal.

1. 36° 18′ 27′′. 27". 2. 56′ 54′′.

Ans. 40 34 163".

Ans. 18 5 377".