PART I.

ARITHMETIC.

I. In order to avoid using a new symbol and word for every different number, all nations, who have attained any proficiency in counting at all, have adopted a radix, or turning point, after which no new symbols are used; and farther numbers are expressed by the combination of old symbols, and by the positions which they relatively occupy. This radix has in all civilized nations been the number ten; not that that is the most convenient number, (twelve would be the most convenient), but because the system was adopted while they were still in a savage state, and did not reason on its convenience or inconvenience for elaborate computation. They were led to employ the number ten by reason of the ten fingers of the hand, and no doubt at first counted with small round stones or any similar substances. The words digit and calculation refer us back to a rude origin of Arithmetic, or rather, of counting. A clear and interesting explanation of numeration and notation will be found in the early part of De Morgan's Arithmetic.

2. Numeration and Notation.

Begin at the unit figure and separate the figures into groups of six each. Then the figures in the second group will give the number of millions, in the third group the number of billions, in the fourth group the number of trillions, and so on.

Thus, take the number

12345678905698.

Group it thus:

12,345678,905698.

It will be read as follows:

12 billions, 345 thousand 678 millions, 905 thousand, 698.

N.B. The rule given above is not universally adopted. Some writers, (all continental nations we believe), consider a billion to be a thousand millions, a trillion, a thousand billions, and so on. In this way, the above number would be read, 12 trillions, 345 billions, 678 millions, 905 thousand, 698.

3. Division.

The definition of division sometimes given in Arithmetical treatises, is, that it is "the process of finding how often one number is contained in another." This scarcely seems to be sufficient, for it does not include all classes of Arithmetical cases. For abstract numbers it is true, but not necessarily for concrete numbers. In order to take in all cases, and to retain strictly Arithmetical language and ideas, it would seem that two definitions are necessary, as thus.

By the division of one concrete quantity by another of the same kind is meant the process of finding how

often the second is contained in the first, or how many parcels, each containing the second number of things, there will be in the first number of things: e. g. 12 apples ÷ 3 apples. Ans. 4.

By the division of a concrete quantity by an abstract number, is meant the process. by which we find, if we separate the first into as many equal parcels as are represented by the second, how many units each parcel will contain: e.g. £2. 165. ÷ Ans. 18s. 8d.

We shall return to these definitions in Note B, on the Division of Fractions.

The following tabular results are very important, and are far too often forgotten.

In Avoirdupois weight;

112 lbs. make I cwt.

The lb. Av. contains 7000 grs. Tr.

I

30 sq. yds. make 1 sq. rod, pole, or perch.

4

4840 sq. yds. make 1 acre.

In Cubic measure ;

1728 cubic in. make 1 cubic ft.

I

The following weights and measures, though not required for ordinary Arithmetic, should be observed.

In Length measure;

I

12 lines make 1 inch.

9 inches make i span.

18 inches make I cubit.

69 miles make 1 geographical degree.

9,

I

69 miles is the length of part of the circumference

9

of the earth at the equator.

In Cloth measure;

360

In Surveying ;

Gunter's chain is 22 yds. long, and contains

100 links.

Hence an acre contains 10 square chains.

A cub. foot of distilled water weighs 1000 oz.

A lb. of gold contains 24 carats.

From this fact we express the fineness of gold by saying that it is so many carats fine; i. e. so many parts out of 24.

Thus, if a mass of gold were 15 carats fine,

15

24

of the mass

would be "alloy," a

9

would be gold, and the rest, viz.

" 24

substance depending on the nature of the mass.

Standard gold is 22 carats fine, the remaining two parts

3

being a mixture of silver and copper. Nearly 46 sovereigns

are coined out of a lb. Tr. of standard gold. gold is 18 carats fine.

4

Jewellers'

Standard silver contains 37 parts out of 40 pure silver, the remaining three parts being copper. A lb. Tr. of standard silver yields 66 shillings.

24 pence are coined out of a lb. Av. of copper.

5. Square and Cubic Measure.

A common or linear yard is length, and measures length.

A sq. yard is surface, and measures surface, or area. It is a square surface whose length and breadth are both a yard. The length round a sq. yard is 4 yards.

A cubic yard is a solid, and measures solidity, or cubic contents. It is a solid whose length, breadth, and height are each a yard.