The only item in this account that requires a little explanation is the Balance at the bottom of the Cr. side. It is found in this way:-Having entered all the transactions as directed, we add up the Dr. column, and find £310 10s.; and on adding up the Cr. side we find £51 15s. 9d., which, taken from £310 10s., leaves £258 14s. 3d., the money remaining on hand.

Now, if we count the cash in our desk, drawer, or till, and find this exact sum, it proves almost conclusively that the account is correctly made out. The rule for Dr. and Cr. is, "All persons who receive are debtors; all persons we pay are creditors."

14. Open a cash account in due form and close it properly for a person who has on 6th Jan., 1882, on hand £258 14s. 3d. On the 7th he pays J. Good £31; on the 8th he buys furniture, £7 7s.; J. Jones, a tenant, pays him rent on the 10th, £25; 13th, he buys a suit of clothes, and pays for them £8 8s., being allowed 24 per cent. off. Find how much he has on hand on 18th, if his personal expenses to that date be £11 10s.

Most people who have money transactions, such as paying and receiving several times a day, find it convenient to keep their money in a bank, and instead of paying in gold, silver, or bank notes, simply write an order on their banker to

James H. Browne.

88

THER

HE method of expressing the magnitude of an angle in degrees, minutes, and seconds is sometimes called, from the degree being divided into 60 minutes, and the minutes into 60 seconds, the sexagesimal method; and, from its having been invariably adopted in English works and in practical calculations, the English method, to distinguish it from that proposed in France about a century ago in connection with the decimal system of weights and measures then adopted in that country.

However, as the adoption of the new French mode of reckoning the measure of an angle would require all the trigonometrical tables to be re-calculated in a form adapted to the new system, it failed to supersede that which has been familiarised by long-established usage.

Had it, however, at the first been introduced, calculations, at least into which minutes and seconds enter, would, in our system of decimal notation, be very much simpler.

In the proposed French method the right angle is divided into 100 equal parts called grades, the grade into 100 equal parts called minutes, and the minute into 100 equal parts called seconds. Hence this method is also called the Centesimal method.

Hence to convert DEGREES into GRADES, state:As 9 10 number of degrees: corresponding number of grades; that is, multiply the number of degrees by the fraction 10; or, which is the same, to the number of degrees add of that num} ber, and the result is the corresponding number of grades.

And to convert GRADES into DEGREES, state:

:

As 10 9 number of grades: corresponding number of degrees; that is, multiply the number of grades by the fraction; or, which is the same, from the number of grades subtract of that number, and the result is the corresponding number of degrees.

T

Since 1° of a right, and 18=1 of a right angle, .. 1° : 1 ::p'。: Too; i.e., 1° : 1 ::10:9 as before.

To convert sexagesimal or English minutes into centesimal or French minutes, and vice versa :—

Let N = number of English minutes in an angle; and n' = number of French minutes in the same angle. Then, since 90 × 60 = number of English minutes in a right angle; and 100 x 100 = number of French minutes in the same angle: .. 90 x 60 English minutes 100 × 100 French minutes; or, dividing each by 200, 27 English minutes 50 French minutes:

=

.. 27: 50: N': n'.

=

To convert English or sexagesimal seconds into French or centesimal seconds, and vice versa. Let

and

N"= number of English seconds in an angle,

Hence to convert English into French minutes state:

As 27 50 number of English minutes: corresponding number of French minutes.

That is, multiply the number of English minutes by the fraction, and the result is the corresponding number of French minutes.

To convert French into English minutes, state as 50:27:: number of French minutes : corresponding number of English minutes; that is, multiply the number of French minutes by the fraction, and the result is the corresponding number of English minutes.

Example 3.-How many French minutes in an angle of 54 English minutes; also in an angle of 2 degrees?

:

27:50 54: 100'18; from which it follows that 54 English minutes equal 1 grade.

and

... 171x 27

2° 51'=171' 50 171 x 50

==

27 3166-35 166

19 × 50 950 3 3

ANGLES OF RECTILINEAL FIGUres. In Euclid I. 32 it is proved that the three angles of every triangle are together equal to two right angles.

The angles of a triangle are therefore together equal to 180° or 2005, and if the three angles are equal, each is 60°, or 66% 66' 66"·6.

In corollaries to the same proposition it is shown that all the internal angles of any rectilineal figure, together with four right angles, are equal to twice as many right angies as the figure has sides; also, that all the external angles of any rectilineal figure formed by producing the sides in the same direction, are together equal to four right angles. If therefore n is the number of sides of a rectilineal figure, the sum of its internal angles =2(n-2) right angles=180(n-2) degrees or 200(n-2) grades. If the figure is equiangular, 2 (n−2) each of its internal angles = of a right 180 (n−2) 200 (n−2)

angle=

degrees=

n

4

=

n

grades,

and each of its external angles of a right angle

Express the measures of the following angles in the English and French systems:

1. An exterior angle of an equilateral triangle. 2. An internal angle of a regular pentagon. 3. An external angle of a regular hexagon. 4. The three angles of a triangle, the middle point of whose greatest side is equally distant from its angles.

5. All the angles of the triangle formed by producing two opposite sides of a pentagon till they meet.

6. The angle at the centre of a circle standing upon an arc, which is one-eighth of the circumference.

7. An internal angle of a regular octagon. 8. An interior angle of a regular decagon. 9. An interior angle of a regular duodecagon. 10. An internal angle of a regular nonagon. 11. The angle denoted by one point of the mariner's compass.

NOTE. Thirty-two points form the complete revolution.

12. The angles at the circumference of a circle standing on an arc, the chord of which is a side of the regular decagon inscribed in the circle.

13. The difference of two angles is 208 and their sum 62°. Express them in both measures. 14. The difference of the two acute angles of a right-angled triangle is 30%. Express them in both measures.

1. Divide 3,422 by 29, and explain the process. What would be the dividend if in a division sum 902,341 were the divisor, 978 the quotient, and 1,857 the remainder ?

2. If 1 ton 13 cwt. 1 qr. 9 lbs. 5 oz. cost £186 13s. 3 d., what will 17 cwt. 7 oz. cost.

3. An estate is divided into three portions of 250 acres, 62 acres 2 rds., and 19 acres 1 rd. 20 p.; these portions are let at £1 5s. 4d., £1 1s. 8d., and £3 per acre respectively. what uniform rent per acre might the whole estate be let so as to bring in the same rental?

At

4. Reduce to their lowest terms the fractions

8 and 1065 Find their difference, and also the quotient of the first divided by the second. Simplify

2 of -11 of 1÷54

5. Assuming that if 66420666 be divided by 7358, the quotient is 9027; write down the quotient of

(1) 6642-0666 divided by 7358000.
(2) of 066420666 divided by 007358.

Simplify

find to the nearest farthing, the value in English money of 3725.39 marks.

If a metre is 39-3708 inches, find how many metres there are in a mile, neglecting decimal parts of a metre.

7. Eleven cubic inches of iron weigh as much as seven cubic inches of lead, and the price per ton of lead is £15, of iron £4. The value of a certain block of lead is £36 17s. 11d. What would be the value of a block of iron of the same size?

8. I sell out £5,000 Four per cent. stock at 108, and with the proceeds buy Five per cents. at 120; what is the change in my income?

What must be the price of a six per cent. stock in order that the money invested in it may yield 4 per cent.?

9. Find the compound interest on £1,397 11s. 3d. for three years at 4 per cent.

Of what debt, due two years hence at 4 per cent. compound interest, is £1,397 11s. 3d. the present value? (Get your results right to the nearest penny).

.875 × 270 •125+125675

+ 333