for the payment of an amount not exceeding forty shillings, but for no greater amount. The intrinsic value of a shilling coin varies with the price of silver in terms of gold, but it is always considerably less than the twentieth part of a pound. It is about the one twenty-fifth part. Hence the need for the limitation mentioned. Silver is more suitable than gold for coins of less value than the half-sovereign, because gold coins truly representative of the values would be inconveniently small. As the silver coins are not full equivalents of what they pass for, they are called token-coins.

ART. 7.--Bronze Coins. The bronze coins (introduced in 1860) comprise the penny, halfpenny, and farthing. They are made of a bronze alloy, the composition of which is 95 parts by weight of copper to 4 of tin and 1 of zinc. A pound avoirdupois of the bronze is coined into 48 penny pieces, or 80 halfpenny pieces, or 160 farthings. Thus 3 pennies, or five halfpennies, or 10 farthings weigh one ounce avoirdupois. These coins are also made so as to serve as measures of length. The diameter of the penny is one tenth of a foot, that of the halfpenny is an inch, and that of the farthing four fifths of an inch.

Bronze coins are legal tender for the payment of an amount not exceeding one shilling, but for no greater amount. Their intrinsic value is very much less than their nominal value, so that they are tokens rather than coins. The intrinsic value of the penny is about one fourth of a penny.

There is no least current weight for either silver or bronze The light ones are selected when passing through the Bank of England, and the loss involved in re-coining is borne by the Government.

ART. 6.-Paper Currency. A bank note is a promise on the part of the bank to pay on demand the sum specified on the note. Notes for £5, £10, £20, £50, £100, £500, and £1000 are issued by the Banks of England, Scotland, and Ireland, and notes for £1

by the Banks of Scotland and Ireland. Bank of England notes are a legal tender for any sum, except at the Bank and in Scotland and Ireland. The issue of notes by a bank is restricted by law to a fixed amount depending upon the securities held by the bank; any further issue of notes must be represented by an equivalent amount of specie (that is, coined or uncoined gold or silver) in the head-office of the bank. Of this specie one fifth may be silver.

ART. 9.-Equivalence. The definitions of the units farthing, penny, and shilling are usually written in the form

These are not equations, but equivalences. In an equation the unit is the same for both sides, and it is not necessary that it should be expressed explicitly; in an equivalence the units are different and require to be explicitly expressed.

An equivalence remains true when both sides are multiplied by the same number, or both are divided by the same number. From this follows a rule for elimination, when one unit is given in terms of a second, and that second in terms of a third. Take, for example,

12 pence = 1 shilling,

20 shillings = 1 pound.

ART. 10.-Chain Rule. The rule may be stated as follows :Arrange the equivalences so that the consequent unit of the uppermost shall be the antecedent unit of the second, and the consequent unit of the second the antecedent unit of the third, and so on for any number of equivalences; then the unit on the left-hand which does not appear on the right-hand side multiplied by all the multipliers on its own side is equivalent to the unit on the right-hand side which does not appear on the left-hand side, multiplied by all the multipliers on its own side. This rule is called the "Chain Rule," and its application is very extensive in physical arithmetic.

ART. 11.-Reciprocal. Each equivalence, such as

has a reciprocal form. That of the equivalence mentioned is

or

20

pound 1 shilling,

=

05 pound = 1 shilling.

The number indicated by, namely 05, is called the reciprocal of the number 20. The value of the reciprocal rate is the reciprocal of the value of the primary rate.

20

It has become customary among scientific writers, after the example of Stokes, to write the fraction in the form 1/20, especially when the fraction occurs not in an equation but in the text. The slant bar is more convenient than the horizontal bar in several respects.

ART. 12.-Conversion. The simplest kind of reduction is called conversion; it consists in changing from a single unit to another single unit. Example: convert £123 into shillings.

The expression is wanted entirely in terms of the pound, not as £22 16s.

The Chain Rule applies to calculations of this kind; when there is only one odd unit we get a quantity and not an equivalence as the result of the reasoning. In a simple application of the Chain Rule it is not necessary to repeat the unit the second time; it may be understood. But it is important to arrange each unit in strict alternate order.

ART. 13.--Reduction. In reduction we are given a quantity expressed in a plurality of units, and are required to express it in terms of one unit; or we are given the quantity expressed in terms of one unit, and are required to express it in terms of a plurality. Example: Reduce £97 16s. 5d. to farthings.

Here we have multiplied the pounds by 20 × 12 × 4, the shillings by 12 x 4, and the pence by 4. These three operations are combined, so far as they permit, to expedite the calculation.

Again, to express the same quantity in terms of the pound,

Here we have divided the 3 by 4 × 12 × 20, the 5 by 12 × 20, the 16 by 20; and the operations have been combined so far as possible for the purpose of accelerating the computation.

An example of the second kind of question is: Express 84783 farthings in terms of pounds, shillings, pence, and farthings.

This is equivalent to dividing 84783 by 20 × 12 × 4 to get the number of pounds, dividing the remainder by 12 × 4 to get the number of odd shillings, and dividing the remainder of the remainder by 4 to get the number of odd pence.

The laborious processes of common reduction arise entirely from the fact that a quantity is expressed, not in terms of one unit and its decimal derivatives, but in terms of a series of units related by numbers which have no system in themselves, and are not identical with or multiples or sub-multiples of 10, the base of the notation of Arithmetic.

ART. 14-Decimalization of Money. It is very important to have a ready rule for transforming any sum expressed in terms of l. s. d. into one expressed in terms of l. alone. This is done by taking first the shillings, and second the pence and farthings reduced to farthings.

First. Since 2s. =l., the number of the shillings when even divided by 2 will give the number of tenths; when odd, the number of tenths and in addition five hundredths.

Second. 1 farthing = pound

=(1000+ D80-1ooo) pound

= (1 + 224) TO100 pound.

=

Hence the number of pence and farthings reduced to farthings