+--------+--------+--------+----------+----------------+---------+
| Ounce. | Pound. | Stone. | Quarter. | Hundredweight. | Ton. |
+--------+--------+--------+----------+----------------+---------+
| 1 | 1/16 | 1/224 | 1/448 | 1/1792 | 1/33840 |
+--------+--------+--------+----------+----------------+---------+
| 16 | 1 | 1/14 | 1/28 | 1/112 | 1/2240 |
+--------+--------+--------+----------+----------------+---------+
| 224 | 14 | 1 | 1/2 | 1/8 | 1/160 |
+--------+--------+--------+----------+----------------+---------+
| 448 | 28 | 2 | 1 | 1/4 | 1/80 |
+--------+--------+--------+----------+----------------+---------+
| 1792 | 112 | 8 | 4 | 1 | 1/20 |
+--------+--------+--------+----------+----------------+---------+
| 33840 | 2240 | 160 | 80 | 20 | 1 |
+--------+--------+--------+----------+----------------+---------+
(Also 7000 grains = 1 lb. avoirdupois.)
120. _Change of System._--It is sometimes necessary, when a quantity is
expressed in one system, to express it in another, The following are
the ratios of some of the units; each unit is expressed approximately as
a decimal of the other, and their ratio is shown as a continued product
(S 116), a few of the corresponding convergents to the continued
fraction (S 117) being added in brackets. It must be remembered that the
number expressing any quantity in terms of a unit is _inversely
proportional_ to the magnitude of the unit, i.e. the number of new units
is to be found by multiplying the number of old units by the ratio of
the old unit to the new unit.
Yard 9144 10000 11 884 8225 /11 32 8 4 235\
----- = ----- = ----- = -- . --- . ---- ... ( --,-- = -.-,--- )
Metre 10000 10935 12 385 8224 \32 35 7 5 257/.
Inch 25400 10000 5 66 1651 /5 33 127\
---------- = ----- = ----- = - . -- . ---- ... ( -, --, --- )
Centimetre 10000 3937 2 65 1650 \2 13 50/.
Mile 16093 10000 8 185 2369 /8 37 103\
--------- = ----- = ----- = - . --- . ---- ... ( -, --, --- )
Kilometre 10000 6214 5 184 2368 \5 23 64/.
Square Yard 8361 10000 5 306 15250 /5 51 250\
------------ = ----- = ----- = - . --- . ----- ... ( -, --, --- )
Square Metre 10000 11960 6 305 15249 \6 61 299/.
Acre 4047 10000 2 85 5320 /2 17 380\
------- = ----- = ----- = - . -- . ---- ... ( -, --, --- )
Hectare 10000 24711 5 84 5321 \5 42 939/.
Quart 11365 10000 8 175 8976 /8 25 408\
----- = ----- = ----- = - . --- . ---- ...( -, --, --- )
Litre 10000 8799 7 176 8975 \7 22 359/.
Pound 4536 10000 1 10 484 29391 /1 5 44 303\
---------- = ----- = ----- = - . -- . --- . ----- .. ( -, --, --, --- )
Kilogramme 10000 22046 2 11 485 29392 \2 11 97 668/.
(ii.) _Special Applications._
121. _Commercial Arithmetic._--This term covers practically all dealings
with money which involve the application of the principle of proportion.
A simple class of cases is that which deals with equivalence of sums of
money in different currencies; these cases really come under S 120. In
other cases we are concerned with a proportion stated as a _numerical
percentage_, or as a _money percentage_ (i.e. a sum of money per L100),
or as a _rate_ in the L or the shilling. The following are some
examples. Percentage: _Brokerage, commission, discount, dividend,
interest, investment, profit and loss._ Rate in the L: _Discount,
dividend, rates, taxes._ Rate in the shilling: _Discount._
Text-books on arithmetic usually contain explanations of the chief
commercial transactions in which arithmetical calculations arise; it
will be sufficient in the present article to deal with interest and
discount, and to give some notes on percentages and rates in the L.
_Insurance_ and _Annuities_ are matters of general importance, which are
dealt with elsewhere under their own headings.
122. _Percentages and Rates in the L._--In dealing with percentages and
rates it is important to notice whether the sum which is expressed as a
percentage of a rate on another sum is a part of or an addition to that
sum, or whether they are independent of one another. Income tax, for
instance, is calculated on income, and is in the nature of a deduction
from the income; but local rates are calculated in proportion to certain
other payments, actual or potential, and could without absurdity exceed
20s. in the L.
It is also important to note that if the increase or decrease of an
amount A by a certain percentage produces B, it will require a different
percentage to decrease or increase B to A. Thus, if B is 20% less than
A, A is 25% greater than B.
123. _Interest_ is usually calculated yearly or half-yearly, at a
certain rate per cent. on the principal. In legal documents the rate is
sometimes expressed as a certain sum of money "per centum per annum";
here "centum" must be taken to mean "L100."
_Simple interest_ arises where unpaid interest accumulates as a debt not
itself bearing interest; but, if this debt bears interest, the total,
i.e. interest and interest on interest, is called _compound interest._
If 100r is the rate per cent. per annum, the simple interest on LA for n
years is LnrA, and the compound interest (supposing interest payable
yearly) is L[(1 + r)^n - 1]A. If n is large, the compound interest is
most easily calculated by means of logarithms.
124. _Discount_ is of various kinds. Tradesmen allow discount for ready
money, this being usually at so much in the shilling or L. Discount may
be allowed twice in succession off quoted prices; in such cases the
second discount is off the reduced price, and therefore it is not
correct to add the two rates of discount together. Thus a discount of
20%, followed by a further discount of 25%, gives a total discount of
40%, not 45%, off the original amount. When an amount will fall due at
some future date, the _present value_ of the debt is found by deducting
discount at some rate per cent. for the intervening period, in the same
way as interest to be added is calculated. This discount, of course, is
not equal to the interest which the present value would produce at that
rate of interest, but is rather greater, so that the present value as
calculated in this way is less than the theoretical present value.
125. Applications to _Physics_ are numerous, but are usually only of
special interest. A case of general interest is the measurement of
_temperature._ The graduation of a thermometer is determined by the
freezing-point and the boiling-point of water, the interval between
these being divided into a certain number of degrees, representing equal
increases of temperature. On the Fahrenheit scale the points are
respectively 32 deg. and 212 deg.; on the Centigrade scale they are 0
deg. and 100 deg.; and on the Reaumur they are 0 deg. and 80 deg. From
these data a temperature as measured on one scale can be expressed on
either of the other two scales.
126. _Averages_ occur in statistics, economics, &c. An average is found
by adding together several measurements of the same kind and dividing by
the number of measurements. In calculating an average it should be
observed that the addition of any numerical quantity (positive or
negative) to each of the measurements produces the addition of the same
quantity to the average, so that the calculation may often be simplified
by taking some particular measurement as a new zero from which to
measure.
AUTHORITIES.--For the history of the subject, see W.W.R. Ball,
_Short History of Mathematics_ (1901), and F. Cajori, _History of
Elementary Mathematics_ (1896); or more detailed information in M.
Cantor, _Vorlesungen uber Geschichte der Mathematik_ (1894-1901). L.
C. Conant, _The Number-Concept_ (1896), gives a very full account of
systems of numeration. For the latter, and for systems of notation,
reference may also be made to Peacock's article "Arithmetic" in the
_Encyclopaedia Metropolitana_, which contains a detailed account of
the Greek system. F. Galton, _Inquiries into Human Faculty_ (1883),
contains the first account of number-forms; for further examples and
references see D.E. Phillips, "Genesis of Number-Forms," _American
Journal of Psychology_, vol. viii. (1897). There are very few works
dealing adequately but simply with the principles of arithmetic.
Homersham Cox, _Principles of Arithmetic_ (1885), is brief and lucid,
but is out of print. _The Psychology of Number_, by J.A. McLellan and
J. Dewey (1895), contains valuable suggestions (some of which have
been utilized in the present article), but it deals only with number
as the measure of quantity, and requires to be read critically. This
work contains references to Grube's system, which has been much
discussed in America: for a brief explanation, see L. Seeley, _The
Grube Method of Teaching Arithmetic_ (1890). On the teaching of
arithmetic, and of elementary mathematics generally, see J.W.A.
Young, _The Teaching of Mathematics in the Elementary and the
Secondary School_ (1907); D.E. Smith, _The Teaching of Elementary
Mathematics_ (1900), also contains an interesting general sketch; W.
P. Turnbull, _The Teaching of Arithmetic_ (1903), is more elaborate.
E.M. Langley, _A Treatise on Computation_ (1895), has notes on
approximate and abbreviated calculation. Text-books on arithmetic in
general and on particular applications are numerous, and any list
would soon be out of date. Recent English works have been influenced
by the brief _Report on the Teaching of Elementary Mathematics_,
issued by the Mathematical Association (1905); but this is critical
rather than constructive. The Association has also issued a _Report on
the Teaching of Mathematics in Preparatory Schools_ (1907). In the
United States of America the _Report of the Committee of Ten_ on
secondary school studies (1893) and the _Report of the Committee of
Fifteen_ on elementary education (1893-1894), both issued by the
United States Bureau of Education, have attracted a good deal of
attention. Sir O. Lodge, _Easy Mathematics, chiefly Arithmetic_
(1905), treats the subject broadly in its practical aspects. The
student who is interested in elementary teaching should consult the
annual bibliographies in the _Pedagogical Seminary_; an article by D.
E. Phillips in vol. v. (October 1897) contains references to works
dealing with the psychological aspect of number. For an account of
German methods, see W. King, _Report on Teaching of Arithmetic and
Mathematics in the Higher Schools of Germany_ (1903). (W. F. Sh.)