PREFACE.

THE STATE SCHOOL ARITHMETIC aims m supplying a text-book suited at once to the requirements of the State School Programme of Instruction and to the several Examination Programmes prescribed by the Victorian Education Department for the classification of both teachers and pupil-teachers.

The exercises, which are both graduated and exhaustive, have been compiled chiefly from Teachers’ Examination Papers ; and as these papers have always been sought for and prized, as affording the best means of preparation available to pupil-teachers and candidates for scholastic honors, the adoption of this Arithmetic as a class-book offers the twofold advantage of exercising the scholar and preparing the young teacher for his own examinations.

Chapters I. to IV. are written respectively to suit the Third, Fourth, Fifth, and Sixth Classes.

Chapter V. is more especially adapted to more advanced scholars and teachers.

The success which has attended the sale of The State School Grammar leads me to look with confidence to the teachers of Victoria and the neighbouring colonies for a fair trial for this, the First Australian Arithmetic ; and I am vain enough to flatter myself that it needs but this trial to obtain for itself a place amongst the most popular of the many excellent treatiseson the subject already published.

As many new features will be found throughout the work, special attention is directed to the matter contained in the following pages:—Pages 32 to 42, 68, 69, 70, 124 to 129,

136 to 142, 148, 150,161, 170 to 175, 180, 181, 182, and 199 to 224.

From Head Teachers of State Schools I have received numerous letters complimenting me, in flattering terms, upon the suitability of my Arithmetic to the requirements of advanced classes, pupil-teachers, and candidates for examinations.

The first thousand copies sold in the short space of one month, thus showing an urgent demand for a new Arithmetic in our State Schools.

In revising my Arithmetic for the present edition (the fourth thousand), I have critically reviewed, and reconstructed, the definitions on pages 3 to 7 ; have carefully , revised the answers, and made some verbal alterations in the various rales, &c., through the book.

The now almost universal use of this Arithmetic as a pupil-teacher’s class-book, and the uniformly favourable opinion of my felk>w-teachers as to the eminent suitability of the S. S. A. to advanced scholars, lead me to think that the rapidly-increasing sale is indicative of a growing popularity ; and I hopefully look forward to its general adoption in public and private schools throughout the Australian colonies.

My best thanks are due to my brother, Mr. B. J. Burston, for his valuable assistance in this and in my new work,

“ Milton Parsed,” now at press.

JOHN J. BUBSTON,

State School No. 1267, Sandhurst.

Sandhurst, 31st January, 18SO.

Exercises on Practice    -    -    •    •

Ratio and Proportion .    -    .    •

Proportion (Direct and Inverse)    *    •    •

Exercise on Simple Proportion ... Compound Proportion    -    .    .

Exercise on Compound Proportion    -    *

Interest

Compound Interest ....

Exercises on Interest, from Teachers’ Examination Papers The Greatest Common Measure    -    -    -

The Least Common Multiple    ...

Notation and Numeration of Fractions    -    -

Vulgar Fractions -    -    -    -    -

Directions to be observed in Working Fractions -Decimal Fractions .....

Explanation of Recurring Decimals    -    -

Exercises on Vulgar Fractions    ...

Exercises on Decimal Fractions    -    -    -

Complex Fractions, from Teachers’ Examination Papers -Discount and Present Worth    -    -    -

Proportional Parts    ....

Exercise on Proportional Parts    -    -    -

The Reciprocal of Numbers ....

Profit and Loss    ....

Exercise on Profit and Loss ....

Arithmetical Equations    -    -    -

Average ......

Examples Worked out -    -    -    -

Miscellaneous Questions, from Teachers’ Examination Papers

ANSWERS .....

APPENDIX -

THE STATE SCHOOL ARITHMETIC.

Arithmetic, both as an abstract science and as applied to magnitudes of various kinds, treats of the value and properties of numbers ; the proper method of expressing them ; the various methods of making calculations; and the course of reasoning to be followed in order to arrive at the desired result.

Numbers are capable of increase or decrease, and all operations tend to affect numbers in one or other of these ways.

There are two methods of increase, Addition and Multiplication ; and two methods of decrease, Subtraction and Division.

These are in general use and are termed “ The Four Simple Rules.”

Most calculations are made by combinations of “ The Four Simple Rules.” It is frequently very difficult, however, to decide which of these processes is to be employed and, in some cases, a choice may be made, and the answer obtained in various ways. In fact, most questions requiring the employment of any two or more of the simple rules mav

be worked variously. Hence, no fixed rule can be given as the sine qud non to the solution of such questions.

The office of an Arithmetic is to describe the plan of working these “ simple rules,” to point out some of the simplest methods of applying them, and to give other rules for guidance in employing the most concise and convenient methods of solving ordinary practical questions.

DEFINITIONS.

1.    Arithmetic, the science of numbers.

2.    Axiom, a well-known or self-evident truth.

3.    Unit or Unity, a single one ; the number 1.

4.    Digits, the figures 1, 2, 3, 4, 5, 6, 7, 8, 9.

5.    Cypher, Zero, or Nought, the figure 0.

6.    Integer, a whole or entire number: as 7, 1760, 4840.

• 7. Fraction, a part or parts of unity ; as, £, f, |, §

(read one-half, two-thirds, three-fourths, three-fifths).

8.    Denominator, or Namer, the lower part of a fraction which gives a name to the parts ; thus, in §, 5 gives the name fifths.

9.    Numerator, or Numberer, the upper part of a fraction which tells how many parts ; thus, in 3 shows how many fifths.

10.    Vulgar Fraction, one whose numerator is placed above its denominator ; as, y’_r, V, y⁵.

11.    Decimal Fraction, one whose Denominator is a power of 10 and is expressed by a dot called a decimalpoint; thus, -5 -75 -625 -6.

12.    Repeating, Circulating, or Recurring Decimal, one having figures which constantly recur : as, -33333, &c. (expressed as -3); -0*125, or -0128. See page 179 et seq.

13.    Finite Decimal, one having no recurring figures.

14.    A Mixed Number consists of an integral number and a fraction : as, 5|, 30\.

15.    Reciprocal, that which any number must be multiplied by in order to produce unity. The quotient obtained by dividing unity by the number of which the reciprocal is required.

Note.—The reciprocal of a fraction is that fraction with its terms inverted ; thus, the reciprocal of _T^sr is y or 2^. The reciprocal of a whole number is the fraction formed by placing the figure 1 above it; thus, | is the reciprocal of 8, _Ty of 12, &c.

16.    Concrete Numbers refer to particular things : as, ten men,five days.

17.    Abstract Numbers do not refer to particular things.

18.    A Positive Number expresses an absolute amount.

19.    Negative Numbers are purely imaginary numbers, and are expressed by prefixing the sign, —, to denote a deficiency. This deficiency is supposed to represent as much less than nothing as the corresponding positive number represents more than nothing.

20.    Absolute or Intrinsic Value, the value of a di<rit

o

in itself; i.e., its value when standing alone.

21.    Relative or Local Value, the value of a digit from the place it occupies with respect to the units’ place.

In the number 1728, the local value of the figure 7 is seven hundred ; the intrinsic value is simply seven.

22.    Notation, expressing numbers in figures : as, 365.

23.    Numeration, the art of expressing numbers in words: as, Three ham We l and sixty-fi ve.

24.    Addends, numbers or quantities to be added together.

25.    Addition, the method of adding numbers together.

26.    Total, or Sum, the amount obtained by adding; the answer of addition ; also, an arithmetical problem.

27.    Subtraction, the method of subtracting one number from another, or of taking a part from the whole.

28.    Minuend, the number from which another number is to be taken. The upper line in subtraction.

29.    Subtrahend, the number which is to be subtracted.

30.    Difference or Excess, the remainder after part has been taken away ; the answer of subtraction ; the result of taking one number from another; or the amount by which one number exceeds another.

31.    Multiplication, a short method of adding the same number repeated any number of times.

32.    Multiplicand, the number to be multiplied.

33.    Multiplier, the number by which we multiply.

34.    Product, the result or answer of multiplication.

35.    Continued Product, the final result of successive multiplications. Earlier products are Partial Products.

36.    Division, the method of finding how often one number is contained in another; how often one number may be taken from another ; or of repeatedly subtracting the same number from another number.

37.    Dividend, the number which is to be divided.

38.    Divisor, the number by which we divide.

39.    Quotient, the result or answer of division. The quotient shows how often the dividend contains the divisor.

40.    Remainder, that which still remains in division after the integral part of the quotient has been obtained.

41a. Measure, an exact divisor, or one which leaves no remainder ; called also a factor or an aliquot part.

41 b. Aliquot Part, a measure, or a part of a concrete number represented by a simple fraction having 1 for its numerator : thus 10s., 6s. 8d., and 2s. 6d. are aliquot parts of a £, being    respectively.

42.    Common Measure or Common Factor, one which measures all the given numbers.

43.    Greatest Common Measure, the greatest number which measures the given numbers : thus, while 2, 3, 4, and 6 are all measures common to 24 and 36, 12 is the greatest common measure, or the Highest Common Factor.

44.    Factors, measures of a number; or numbers which taken any exact number of times will make the number : thus, 3 and 8, 4 and 6, and 2 and 12 are all factors of 24.

45.    Composite Number, a number having factors : as, 24.

46.    Prime Number, a number having no factors other than itself and unity : as, 5, 7, 11, 13, 17, 19, 23, 29, &c.

47.    Power, a product of a number multiplied by itself any number of times; or a product obtained by using the same number as a factor any number of times.

Note.—The number itself is called the first power or base. If the number enter as a factor twice, the product is called the square ; if thrice, the cube ; if four times, the fourth power, &c., &c. Thus 5 is the first power of 5 ; 5 times 5 or 25 is the square ; 5 x 5 x 5 or 125 is the cube, and 625 the fourth power.

48.    A Multiple of a number is the product of that number and any other number : thus, 9, 12, 15, and 18 are all multiples of 3.

49.    A Common Multiple contains each- of the given numbers an exact number of times. See page 163 et seq.

50.    The Least Common Multiple is the least multiple common to the given numbers; thus, 24 is the L. C. M. of 3, 6, 4, and 8. See page 163.

51.    Loot, the number which is multiplied by itself to produce any power ; thus, 4 is the square root of 16, or the cube root of 64 ; because it must be used as a factor twice to give 16, and thrice to give 64. See page 247 et seq.

52.    Ratio is the comparative magnitude of two abstract numbers, or of two quantities of the same kind; thus \ is the ratio of 3 to 6, or of 6d. to Is.

53.    Proportion, the equality of ratios. In proportion, the same result arises from two distinct comparisons, the ratios produced being identical. See page 119 et seq.

54.    Reduction, the process by which we reduce concrete numbers of one name to their equivalent in another name.

55.    Compound Quantities, quantities embracing things of different names ; thus, 2 cwt. 3 qrs. 14 lbs.

56.    Practice, a method of working compound multiplication by using aliquot parts and applying division.

57.    Interest, money paid for the loan of other money.

58.    Discount, a reduction allowed upon paying money; an allowance made on account of prepayment.

59.    Principal, the money lent, or invested in business.

60.    Amount, the amount to be repaid ; the sum comprising the principal together with its interest.

61.    Present Value or Present Worth, the‘principal less the discount; or the actual worth at the present time.

62.    Mean or Average, the medium of numbers; the result of taking one with another or with others. Of two numbers it is called the Mean.

Note.—The mean of two numbers is J their sum. The average of three numbers is £ of their sum ; of four, \ of their sum, &c.

63.    Proportional Parts, parts which bear a given relation to each other, and together make up the whole.

64.    Per Centage, parts of 100 ; thus 5 p. c. is _T(nyths.

65.    Stocks, Funds, Debentures, &c., shares in the public debt or in joint-stoclc companies.

66.    Surds, roots which cannot be exactly obtained, but are indicated by the signs f, ²\/> ³\/> &^c-; as, ^3, ³\/9.

Note.—1 is the only prime number whose roots can be found ; all roots of 1 are unity itself. Roots of all other prime numbers and of all negative numbers are surds. See pages 252, 253.

67.    The Complement of a number is its difference from the power of 10 next above it; thus, the complement of 639 is (1000 — 639) = 361. The complement of 78726 is (100000 — 78726) = 21274.

68.    Involution, the method of finding a power of a number.

69.    Evolution, the method of finding a root of a number.

70.    Unit of Measurement, the unit used as the standard of comparison in estimating the value of a concrete number ; thus, an inch is the “ unit of measurement ” in Long Measure.

71.    Logarithm, the exponent or index of the power to which another number must be raised in order to produce a given number. Thus, 10³ - 1000, where the index, 3, is the logarithm of 1000.

72.    Progression, numbers proceeding by equal differences, or by equal ratios, whether increasing or decreasing.

Note.—When numbers proceed hv equal differences, as 3, 6, 9, 12, they are said to be in Arithmetical Progression ; if they proceed by equal ratios, as 3, 9, 27, 81, they are said to be in Geometrical Progression.

ARITHMETICAL SYMBOLS.

+ (plus), means adcl, the sign of addition.

— (minus), means less, the sign of subtraction.

x (into), * means multiplied by.

-i- (by), means divided by.

^ (difference), means take the smaller from the greater, (therefore), means consequently.

V    (wherefore), means because.

; (is to), means compared with.

;: (so is), means is the same as.

As 2 : 8;: 3 : 12, reads, as 2 is to 8 so is 3 to 12 ; and means that 2 is the same part of 8 which 3 is of 12.

₌ (equality), denotes that the amounts on each side of it are equal in value.

V    (root), a corruption of r, the initial letter of radix, a root.

2 ^_} 3^    &c., the square root, cube root, fourth

root, &c.

10², 10³, 10⁴, (index), the small figures 2, 3, 4, denote the power,_the square, cube, and fourth power respectively.

()> \ }->[]> (brackets), enclose numbers to be acted upon as a whole; or denote that what is enclosed is to be treated collectively.

Note.—A straight line drawn over numbers has the same effect as enclosing them in brackets; thus, 6 — 8    5 = 6 — (8    5).

* The words “by ” and “ into” are more frequently used to mean « multiplied by ” and “ divided into ” than as the equivalent of the signs X and That is, such an expression as 6 X 8 is read as 6 by 8 ; and 12 -i- 3 as 3 into 12.

NOTATION AND NUMERATION.

All numbers are expressed by the nine significant figures, 1, 2, 3, 4, 5, 6, 7, 8, 9 (called digits), and the 11011-significant figure U (called a cypher, zero, or nought).

When used alone, or on the right of other figures, a digit is in the units place, and expresses the number which its name indicates ; thus, in the numbers, 9, 59, and 1009, the figure 9 denotes nine units, or single ones.

The units’ place is that on the right (except in decimals, when it is immediately 011 the left of the decimal point).

In the units’ place a figure has its ordinary or name value —called its nominal or intrinsic value.

In every other place the value of a figure depends upon its position, and this value varies in proportion to its distance from the units’ place.

The number 1, which, in the units’ place, simply means seven units, denotes seven tens (or seventy) when removed from the units place by a nought, and seven hundreds when removed two places by two noughts ; thus, 70, 700.

In like manner, the figure 5, which in 5000 indicates five thousands, signifies one-tenth of five thousands (or five hundred) when removed one place nearer to the units’ place, and one-tenth of five hundreds (orfifty) when removed two places; thus, 500, 50.

The value a figure thus receives from its position with regard to the units' place is called its local value.

The local value of a digit increases tenfold each remove to the left, and decreases tenfold each remove to the rifrht It is thus seen that the figure 0, which has no value in itself, affects the value of the significant figures by removing them from the units’ place.

A figure in the first place on the left of the units’ place will represent ten times as much as when in the units’ place; but only one-tenth as much when in the first place

on the riglbt of the units’ place.

Note.—Figures on the right of the units’ place are called

decimals.

Seeing that the value of digits increases or decreases at a tenfold rate with each remove of one place to the left or right, it is necessary to give names to the various places to the left or right of the units.

Let the cypher below occupy the units’ place, then the significant figures will occupy the places named :—

9876543210-123456789

CD ,3

3    a

a    o

⁰³ .2 S

All places between any particular place and the units’ place (together with the units’ place itself) must be filled up with significant figures or cyphers before any number of that particular place can be expressed : thus, for a digit to express millions, six other figures must follow it; while to express hundreds of millions, eight other figures must follow. In like manner any particular place in decimals is denoted by filling the places between it and unity with other digits or cyphers; but here the position of the units’ place is simply indicated by a dot, as -5, "25, -0125, -0002.

Children will do well to remember how many figures are required for each particular place : thus, for tens, 2 ; for hundreds, 3; for thousands, 4 ; for tens of thousands, 5 ; for hundreds of thousands, 6; for millions, 7, Ac.; for tenths, 1 (in addition to the decimal point); for hundredths, 2; for thousandths, 3; for tens of thousandths, 4; for hundreds of thousandths, 5, Ac., Ac.

Remark.—Decimal Notation would be less confusing if a plan of placing the decimal point over the units’ place, instead of after the units, were adopted. This would necessitate the filling of the units’ place with a cypher, when no whole number preceded the decimal, and thus make the number of figures necessary to express a millionth coincide with the number required for a million ; instead of, as at present, requiring one less.

Notation.—To Notate a Number is to Express it in Figures.

Numeration.—To Numerate a Number is to Express it in Words.

In order to enable us to notate and numerate more readily, ’ the figures which express large numbers are divided into groups, called periods.

There are two systems of notation, differing from each other in the number of figures composing a period :—

I.    The French or Italian Method, having three places in each period : units, tens, and hundreds.

II.    The English Method, having six places in each period : units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands.

The names of the periods in the English System are (1) Units’ Period, (2) Millions, (3) Billions, (4) Trillions, (5) Quadrillions, (6) Quintillions, (7) Sextillions, (8) Septillions, (9) Octillions, (10) Nonillions, (11) De-

CEMTILLIONS, &C., &C.

In the French System, a thousands’ period occurs between the units’ and millions’ period; while the billions follow directly upon the hundreds of millions,—there being no thousands of millions.

It will be observed that numbers of less than ten figures will read similarly in both systems; the first change occurring on the tenth figure.

Note 1.—The French or Italian System of Notation and Numeration has recently been adopted in England, in lieu of the English method above described. See Arithmetic by W. and R. Chambers.

Example: Numerate 785,010,011,012,900.

Old English Method.—Seven hundred and eighty-five billions, ten thousand and eleven millions, twelve thousand nine hundred. -

French or Continental Method. —Seven hundred and eighty-five trillions, ten billions, eleven millions, twelve thousand nine hundred.

Note 2.—It is customary to mark off the periods in English Notation by semicolons, and to subdivide them by commas.

In the French System the periods are marked off' by commas.

Note 3.—In speaking of a figure as before others, we mean on the

SUGGESTIONS UPON TEACHING TABLES. 13

SUGGESTIONS UPON TEACHING TABLES.

The following method of writing the Addition and Subtraction Table is here recommended, as being :—

1.    Analogous to Addition and Subtraction ; and hence

an introduction to them.

2.    A more effective plan than that usually adopted.

3.    A check upon the habit of learning by sequence.

4.    An excellent plan for ensuring ready replies to the pro

miscuous method of questioning known as “dodging."

How often do we hear a child who blunders at the first irregular question, affirm that he can say his tables “straight on !”

How many others hesitate to reply until they have mentally repeated from the beginning of the table !

The reason of this is to be found in the consecutive order employed in teaching the tables. The remedy would be to arrange them in an alternate order, as below :—

Example.—-Table of 8, Addition.

Note. —It is a good plan, in teaching the Subtraction Tables, to proceed thus S from 9 leaves 1, because 8 and 1 are 9 ; 8 from 11 leaves 3, because 8 and 3 are 11. Again, the Addition and Subtraction Tables may be taught together, thus : 8 and 1 are 9, 8 from 9 leaves 1 ; 8 and 3 are 11, 8 from 11 leaves 3.

The reciprocal connexion between the Addition and Subtraction Tables, and between the Multiplication and Division Tables, should be well impressed on the youthful mind.

If the ordinary method of arrangement be considered preferable, the Addition and Subtraction Tables might be combined by writing them side by side, and the alternate order still preserved. Thus : -

Table of 9, Addition and Subtraction.

9 and 1 are 10 ; then 9 from 10 leaves 1

&c.    &c.

MULTIPLICATION TABLE.

This may be taught in connexion with the Pence Table, with this advantage :—The two tables combined can be learnt in very little more time than each takes up singly; while the pernicious habit of learning the Pence Table by sequence is prevented. Who of us, if asked how many shillings there are in 87 pence, does not say to himself mentally, 84d. = 7s., then 87d. = 7s. 3d. i If the number of pence which are represented by the composite numbers of the Multiplication Table be well known, the whole Pence Table up to 150d. will be easily mastered. Another good feature in this plan is, that the scholar observes what any number of threepences, fourpences, sixpences, Ac., amount to.

Example.—Multiplication Table of 8 and 10.

In repeating the above, it will be simply necessary to say :_

8 ones are 8, 8d. are 8d. ; 8 threes are 24, 24d. are 2s. ; 8 fives are 40, 40d. are 3s. 4d., &c., &c. The repetition of the multiples 8, 24, 40, &c., impresses them upon the memory.

It is not thought necessary to give the Addition, Subtraction, and Multiplication Tables in full.

Another good combination of tables is that of the Multiplication and Division Tables, written thus :—

ADDITION AND SUBTRACTION.

To place numbers for Addition or Subtraction :—

Rule.—Place the numbers one under the other, so that the units shall be under units, tens under tens, hundreds under hundreds, thousands under thousands, dec., dec., in vertical columns, and then draw a straight line beneath them.

To add numbers

Rule.—Add up the figures in the units’ column; cut off the last figure and put it under tlbe units. Add up the number formed by the remaining figures with the digits in the next column ; cut off the last figure, as before, and carry the number expressed, by the others to the next column. Add each column in the same way ; putting down the ivhole result of the last column.

Examples :—    I.    II.

376495    1234567

729843    20796

26S594    379

726487    7005

2101419    1262747

In Example /., the sum of the first column is 19,—put down 9 and carry the one to the next column ; the 1 carried, with the figures of the second column, gives 31,—put down the 1 and carry the 3 to the hundreds’ column ; 3 added to the figures in this column gives 24,— put down the 4 and carry the 2 ; 2 added to the fourth column gives 31,—put down the 1 and carry the 3 ; 3 added to the fifth column gives 20,—put down the 0 and carry 2 ; 2 added to the sixth column gives 21,—put down the whole number, because there is no other column to add up.

Note.—It is usual to say, “ Put down the units and carry the tens.” Note.—In putting down the addends, it matters not in what order they are put: that is, which is put as the top line, which as the second line, third line, &c.

In Example II., the three last columns give sums each less than 10 ; and in this case there is no figure to carry.

Method of Proof :—

Rule.—Add the answer to the addends to obtain twice the

answer.

Note.—The following plan of reasoning Mull be found useful to prevent children from counting instead of adding, when learning addition :—

Suppose in adding a vertical column consisting of the figures 8, 7, 5, 9, 7 and 9, a boy hesitated at the last step of the addition to tell what 36 and 9 came to.

Place the number 9 under 6, 16, 26, 36, 46, 56, respectively, and then cause him to repeat—“Because 9 and 6 are 15, 19 and 6 are 25 ; 29 and 6 are 35 ; 39 and 6 are 45,” &c. He will then observe that any two numbers ending in 9 and 6 respectively M ill give a sum ending in 5.

The proof of the simple rules by casting out the nines is treated of at page 41. It will be sufficient, here, to point out how the teacher may construct addition sums, the answers of which he may see at a glance.

II.

2376594285 key-line 1235432683 j * 3421213104 [ 5343354212 ) 3251264107 ) 6213600291 [ 535135601 )

, 3795286086

In I., the figures of the first and second lines, added, will give a line of 9’s, thus, 999999999 ; the figures of the third and fourth lines also give a line of 9’s; as also do the figures of the fifth and sixth lines. N o vv, three such lines of nines will be but 3 less than 3000000000, and this added to the bottom line will make 37952S6089 all but 3' Hence, the answer may be at once written by placing 3 before the bottom line (called the key-line) and subtracting 3 from the units’ figure of that line. Hence, A ns. =3795286086.

In II., the top is taken as the key-line ; while the second, third, and fourth lines give ten nines ; and the fifth, sixth, and seventh give another line of ten nines. Now, two lines of ten nines make but 2 short of 20000000000. Hence, if we take 2 from the key-line, and place 2 before it, we obtain the answer at once, =22376594283, Ans.

Note—Any line may be taken as the key-line; while the sum may be constructed with any number of lines.

SUBTRACTION.

Subtraction is the method of finding the difference between two numbers by taking one from the other.

It is also used to take part of any number from the whole.

We take one number from another when we want to find (1) by how much the one exceeds the other; (2) by how much the less is in defect of the other.

If the former be the object of Subtraction, the difference is called the Excess ; if the latter, it is called the Defect.

When Subtraction is performed with the object of taking away part of a number from the whole, the difference is called the Remainder. The term Remainder, however, is generally applied to the number which remains over after division has been performed.

Difference is, therefore, a general term for the Excess, Defect, or Remainder.

The number from which another number has to be taken is called the Minuend ; the number to be taken away is called the Subtrahend.

Note—Minuend, from Minuo, I diminish or lessen; and Subtrahend, from Subtraho, I subtract. As observed by Cornwall and Fitch, in their “ Science of Arithmetic,” words ending in and or end from Latin participles in andus and endue, signify an action which has to be done to a thing ;—'“ Thus the Minuend is that which has to be diminished ; Subtrahend, that which has to be subtracted ; Multiplicand, that which has to be multiplied; Dividend, that which has to he divided.”

Illustration.—If 12 balls be placed on the top wire of a ball-frame, or Arithmetican, and then 7 be taken away, there will be 5 left. The whole 12 is the Minuend ; the 7 taken away the Subtrahend ; and the 5 left the Remainder.

Again, if 12 balls be placed on the top wire and 9 on the next, we see that there are 3 more on the top wire than on the second wire ; or that the number 12 exceeds the number 9 by 3 ; so that 3 is the excess. We also observe that 3 more require to be placed on the bottom wire to make up 12, so that 3 is the defect of 9 as compared with 12.

To find the difference between two numbers by considering how much one exceeds or is less than the other, is very easy in the case of small numbers; but with large numbers the case is different.

I. We observe that any number of tens, or hundreds, or thousands, taken from another number of tens, hundreds, or thousands, will leave the same number of tens, hundreds, or thousands, as the cor responding number of units as Subtrahend from the corresponding number of units as Minuend will leave units.

For example, because 3 from 8 leaves 5 ; 3 tens from 8 tens leave 5 tens ; 3 hundreds from 8 hundreds leave 5 hundreds ; 3 thousands from 8 thousands leave 5 thousands, &c., &c.

On account of this correspondence between the remainders in the case of units, tens, hundreds, See., we say, when dealing with the hundreds or thousands, &c., 3 from 8 leaves 5, instead of 3 hundreds from 8 hundreds leave 5 hundreds ; 3 thousands from 8 thousands leave 5 thousands, &c., &c., taking care, however, to place the diffeience under the hundreds, or thousands, or whatever we may be dealing with.

11. Again, v e observe that the aijfevence between two numbers is not altered by adding the same number to each.

Thus, if one boy have 9 marbles and another boy 5 marbles, the difference, 4, will not be altered by my giving 10 marbles to each j for they will then have 19 and 15 respectively, or the one will still have 4 more than the other.

From I. and II. we obtain the ordinary method of working.

Rule. — Place the number to be subtracted under the other, units under units, tens under tens, dec. Take each figure from the one above. If the upper figure be the less, add 10 to it, and then subtract. Add 1 to the next figure of the bottom line before subtracting it from the figure above it.

The rule more fully expressed is :—

Rule.—After placing the numbers as directed (page 16), commence with the units, and take the figures in the Subtrahend from those directly above them in the Minuend. If this cannot be done on account of the lower figure being greater than the upper, add 10 to the upper figure, subtract the lower, and put down the remainder. To prevent the difference being altered hy the addition of 10 to the Minuend, add 1 to the next figure in the Subtrahend before subtracting.

Should nothing be left when taking the lower figure from the upper, put down a nought ; unless it is seen that no significant figure will come before it, in which case put nothing at all down.

Note.—The local value of the 1 added to the figure of the subtrahend is equal in value to the 10 added to the minuend, since its position is one place farther to the left. Hence, in adding 10 to the upper figure, and carrying 1 to the next figure in the minuend, we are simply adding an equal amount to two unequal numbers (viz., to the subtrahend and minuend) ; and it has been shown that an equal amount added to two unequal numbers does not alter tlieir difference.

In working Subtraction by the above method, the terms “ borrow” and “ carry ” are generally used to remind us where to add 1 ten to the bottom line after having added 10 to the top line. These are excellent mnemonical aids ; but since they have been condemned on the grounds that they are inappropriate and apt to mislead as to the real nature of the process employed, they are not used here.

That the terms “ borrow ” and “ carry ” are inappropriate will be seen by applying them to the subtraction of 18 from 24. Here we say S from 4 cannot be taken, “borrow 10.'’ Whence are we to borrow? If the 10 were borrowed from the subtrahend 18 there would be no ten left; yet we say “return the, 10 we borrowed, 1 ■and 1 are two tens” Now, if the ten were borrowed and returned, there would be but one ten. Hence, we should rather say, add 10 to the minuend (of course meaning 10 units, 10 tens, 10 hundreds, &c., according to the local value of the figure with which we are dealing), and then, after subtracting, say, add 1 ten to the subtrahend, so that the difference shall be unaltered.

Examples.—From 207368, take 193295 ; and take 990998 from

1001001.

14073 difference.    10003 difference.

I.    Take the number 5 from the number 8, leaves 3 ; 9 from 6 we cannot (take), add 10 to the 6, 9 from 16 leaves 7 ; add 1 to the 2, 3 from 3 leaves nothing, put down 0 (nought) ; 3 from 7 leaves 4 ; 9 from 0 we cannot, add 10, 9 from 10 leaves 1 ; add 1, 2 from 2 leaves 0, and since a nought is of no value on the left of a number, leave it out.

II.    8 from 1 we cannot, add 10, 8 from 11 leaves 3 ; add 1, 1 and 9 are 10, and 10 from 0 we cannot, add 10, then 10 from 10 leaves 0 ; add 1, 1 and 9 are 10, 10 from 0 we cannot, add 10, 10 from 10 leaves 0 ; add 1, 1 from 1 leaves 0 ; 9 from 0 we cannot, add 10, 9 frpm 10 leaves 1 ; add 1, 1 and 9 are 10, and 10 from 10 leaves 0, do not put it down, as it is seen that the next figure will be a nought also.

Remember that when we say “ add 10,” it is to the minuend ; while “ add 1 ” refers to the subtrahend.

Note.—The subtrahend can in no case be greater than the minuend ; for it is impossible to take a greater number from a less.

Second Method :—

Rule II.—Proceed as before to subtract the under digit from the upper. If it cannot be taken, remove 1 from the next significant figure of the Minuend, and change this into its corresponding value in the position which the figure we are dealing with occupies, and put a stroke through the figure from which 1 has been removed to show that it now denotes 1 less.

Example I.—Take 4128237, from 9243672.

9243672 minuend.

4128237 subtrahend.

5115435 difference.

Process. —7 from 2 we cannot, remove 1 from the 7, the 1 removed = 10 units, 10 and 2 are 12, and 7 from 12 leaves 5. Next, 3 from 6 leaves 3 (not 3 from 7, as 1 was removed from the 7). Then 2 from 6 leaves 4 ; 8 thousand from 3 thousand we cannot, remove 1 ten thousand from the 4 tens of thousands, 10 and 3 are 13 thousands and 8 thousands from 13 thousands leaves five thousands. Next, 2 from 3 (one having been removed from the 4), leaves 1. 1 from 2 leaves 1 ; and 4 from 9 leaves 5.

When the next figures of the Minuend are noughts, go to the next significant figure, as before, substitute 9 for every nought and add 10 to the figure being dealt with.

Example II.—Find the difference between 30002 and 12345.

999

30002 minuend.

12345 subtrahend.

17657 difference.

Pi •ocess.—5 from 2 we cannot, go to the next significant figure in the minuend, viz., 3 ; cross it out and change the 1 taken from it into 9 thousands, 9 hundreds, 9 tens and 10 units. Then 10 and 2 are 12, 5 from 12 leaves 7. Then 4 from 9 leaves 5 ; 3 from 9 leaves 6 ; 2 from 9 leaves 7 ; and 1 from 2 leaves 1.

It is necessary,' in working by this method, to remember that any digit from which 1 has been removed now represents 1 less than its original intrinsic value.

Third Method :—

In working Subtraction by the first, or ordinary method, a plan is sometimes adopted by which subtraction from any number greater than 10 is avoided.

Rule III.—When the lower figure cannot he taken from the upper, take it f rom 10, add the result to the upper figure, and put down the sum. Carry 1 to the next figure in the Subtrahend, and proceed as usual.

Example.—By how much does 30165940 exceed 18139400 ?

30165940 minuend.

18139400 subtrahend.

12026540 excess.

Process.—In dealing with the thousands, say 9 from 5 we cannot, take 9 from 10 leaves 1 ; then, 1 and 5 are 6, put it down. Add 1 to the 3 in the subtrahend, and continue the subtraction.

Note 1.—This method is very suitable for young children, as the difficulty which they experience in subtracting from large numbers is entirely avoided.

Note 2 —In practice, the second method presents more difficulties than either the first or third ■. but the principles upon which it is worked are more easy of comprehension. The third method, which is hut a modification of the first, is the easiest both to work by and to teach.

Method of Proof :—

The Excess + the Subtrahend — the Minuend.

For, the excess of the greater number is also the defect of the less, and if the defect be added to the less number the result will equal the greater. Hence—

Rule I.—Add together the two bottom lines to obtain the top'line.

Again, The Minuend — the Difference = the Subtrahend.

For if the excess be taken from the larger number (or Minuend) that number will no longer exceed the smaller, but will be equal to it. Hence—

Rule II.—Take the answer from the Minuend to obtain the Subtrahend.

MULTIPLICATION.

To place Numbers for Multiplication :—

Rule.—Place the Multiplier under the Multiplicand with the last significant figure of the former beneath the units of the latter. If the Multiplier end in cyphers, place them to the right of the figure under the units.

Example:—Multiply 307695by 2307 and 870900 respectively.

307695 Multiplicand.    307695 Multiplicand.

2307 Multiplier.    870900 Multiplier.

Note.—In speaking of a figure as first or last in a number, first means farthest on the left, and last, farthest on the right.

RULE FOR MULTIPLICATION.

Rule.—Multiply the whole of the figures in the Multiplicand, beginning at the units, by the last significant figure of the Multiplier. Multiply the Multiplicand by the next significant figure, and then by the next, and so on ; taking care to place the first obtained figure of each partial product directly under the figure you are Multiplying by.

When the product of any two figures exceeds 10, put down the units and carry the tens, as in addition.

Wherever a cypher occurs in the multiplier, put a nought directly under it, and proceed to the next significant figure.

The partial products are placed beneath each other and finally added up, as shown in the following example :—

Example I.—Find the product of 150672 and 504200.

Note.—When the product of two abstract numbers is required, either numbers may be used as the multiplier.

150672 Multiplicand.    504200

504200 Multiplier.    150672

I. - IL --

30134400 \    1008400

602688    > Partial Products.    3529400

7533600    )    3025200

--- 25210000

75968822400 Product.    504200

75968822400

Example II.—Find the continued product of 8070, 30500, and 370.

Here we first find the product of any two of the given numbers, and multiply this product by the third number.

8070

30500

4035000

242100 246135000

370

17229450000

738405000 91069950000 Continued Product.

Multiplication by Factors :—

When the multiplier is a composite number, the product is the same as the continued product of the multiplicand and the factors of the multiplier. Hence, to multiply by factors :—

Rule.—Multiply the Multiplicand by the First

Factor; multiply the Product thus obtained by the next Factor, and so on, using a complete set of Factors.

Example III.—Multiply 76809504 by 1728.

The Factors of 1728 are 12, 12, and 12

76809504

12

921714048

12

11060568576

12

132726822912 Answer.

Multiplication by Prime Numbers is often effected:—

I.    By multiplying by the factors of the next lower composite number and adding to the result the product of the multiplicand and the difference between the composite number selected and the proper prime multiplier.

II.    By multiplying by the factors of the next higher composite number, and subtracting from the result the product of the multiplicand and the difference between the composite Dumber selected and the proper prime multiplier..

Examples.—Multiply 3928 by 89.

1st Plan.    2nd Plan.

3928 X 89    3928 x 89

8    9

31424 = 8 times 3928    35352 = 9 times top line

11    10

345664 =    88 times 3928    353520 =    90 times top line

add 3928 = 1 time 3928    subtract 3928 = 1 time top line

In such a sum as the preceding the second plan is the better, since the multiplication by 10 can he effected at once by adding a cypher to 9 times the top line : that is, we might have multiplied by 90 in one line, and then have subtracted the top line, thus—

3928 X 89 90

353520    90 times 3928

3928 = 1 time 3928

319592 = 89 times 3928.

To multiply by 10, 100, 1000, 10000, &c., at sight:—

Rule. —Simply add as many cyphers to the multiplicand as there are in the multiplier :

Thus, 87 X 1000 = 87000.

To multiply by 25 :—

Rule.—Add two noughts and divide by 4 :

Thus, 64 x 25 = 6400 4- 4 = 1600.

To multiply by 99 :—

Rule.—Add two noughts and subtract the multiplicand :

Thus, 789 X 99 = 78900 - 789 = 78111.

To multiply by 125 :—

Rule.—Add three noughts and divide by 8 :

Thus, 848 X 125 = 848000 4- 8 = 106000.

To divide by 10, 100, 1000, &c., at sight:—

Rule.—Cut off as many figures as there are cyphers in the divisor: Thus, 370 -4- 10 = 37; 3700 -P 100 = 87; 365000 -p 1000 = 365.

To divide by 50, 500, or 5000, &c. :—

Rule.—Multiply by 2, and divide by 100, 1000, 10000, <tc.

Thus, 72900 -P 50 = 145800 -4- 100 — 1458.

To divide by 25, 75, 125, 175, 225, or 275 : —

Rule.—Multiply by 4, and divide by 100, 300, 500, 700, 900, or 1100. Thus, 5625 -4- 225 = 5625 x 4- 900 = 25.

To divide by 15, 35, 45, or 55, respectively :—

Rule.—Multiply by 2, cut off the last figure of the product, and divide the result by 3, 7, 9, or 11.

Thus, 360 -4- 15 = 72/0 -4- 3 = 24.

SHORT DIVISION.

Division by any number less than 13 is called Short Division.

Short Division is worked as under :—

Divisor. 9) 763848 Dividend.

84872 Quotient.

Here, 9 is not contained in 7 ; so we see how often it is contained in 76. It is contained 8 times and 4 over. Put down the 8 and place the 4 before the next figure 3; then 9 into 43 = 4, and 7 over. Place the 7 before the next figure 8 ; then 9 into 78 = 8, and 6 over. Place the 6 before the 4 ; then 9 into 64 = 7, and 1 over. Place the 1 before the 8 ; then 9 into 18 = 2.

To divide by any composite number greater than 12 :—

Rule.—Divide the dividend by one factor, and then divide result by the next factor, and so on, until each factor has been used.

Example I.—Divide 132726822912 by 1728.

Here the factors of the divisor are 12, 12, and 12.

12) 132726822912 Dividend.

12) * 11060568576 Partial Quotient.

12)    * '921714048 Partial Quotient.

76809504 Quotient, or Answer.

Compare the working of the above sum with the example of multiplication by factors on page 26, Example 111.

Example II.—How many times can 378 be taken from 9094680 %

Note.—Division is a short method of finding how often one number can be taken from another.

The factors of 378 are 9, 7, and 6.

9 ) 9094680 7 ) 1010520 6 ) 144360

24060

When remainders occur at the several steps of the operation in dividing by factors, the true remainder is obtained thus:—

Rule.—Multiply each remainder by the factors preceding that which produced the remainder, and add together the several results.

Example.—Divide 31255 into (1) 240, and (2), 5760 equal parts.

Here, 12 and 20 are convenient factors of 240; and 2), 3, 8, and 12 are the most convenient factors of 5760.

II.

20)31255

3)1562 and 15 remainder.

8)520 and 2 remainder.

12)65 and 0 remainder.

5 and 5 remainder.

I. True Remainder (4 X 12) + 7 = 55.

II. True Remainder (5 X 8 X 3 X 20) + (2 x 20) + 15 = 2455.

To the scholar who can work Reduction, the reason for the above will be made clear by considering the first example as one of reduction of pence to pounds; and the second example as one of the reduction of grs. to lbs. Troy. The remainders 4 and 7 will then represent 4 shillings and 7 pence, or 55 pence ; while the remainders 5, 0, 2, 15, in example II., will represent 5 ozs., 0 dwts., 2 scr. 15 grs., or 2455 grains.

To find the True Remainder when two factors only are used :_

Rule. —Multiply the last remainder hy the first divisor, and add in the first remainder.

LONG DIVISION.

To Place Numbers for Division :—

Place the numbers thus :

Divisor.) Dividend. (Quotient.

To divide numbers :—

Rule.—Mark off from the left of the Dividend as many figures as will express a number not less than the Divisor. See how often the Divisor is contained in the number cut of, and put the result in the Quotient. Multiply the Divisor by the figure placed in the Quotient, and subtract the product thus obtained from the part of the Quotient cut off. Bring down the next figure of the Dividend, and divide the number thus formed in the same way, putting the result in the Quotient,— multiplying and subtracting, as before. Continue in this way until all the figures of the Dividend have been brought down.

Important Note.—Whenever the divisor is not contained in the number formed by bringing down the next figure of the dividend, put a nought in the quotient, bring down the next figure, and continue the division. Should the divisor still not be contained, put another nought in the quotient, bring down the next figure, &c. Remember, there will be one more figure in the quotient or answer than there are figures following the part at first marked off.

Example I.—Divide 349592 by 89.

Divisor Dividend Quotient.

89 ) 349/592 ( 3928 267

825

801

249

178 712

712

In this example there is no remainder.

Example II.—How often is 8070 contained in 246135600 ?

Divisor. Dividend. Quotient.

8070) 24613/5600 (30500 24210

40356

40350

• • ■ ^-600 Remainder.

When the product of the divisor and first figure is subtracted and the figure 5 brought down from the dividend, the number formed does not contain the divisor ; hence a nought is placed in the quotient and the next figure, 6, is brought down. The divisor is now contained 5 times, leaving 6 as a remainder, which becomes 60 when the next figure is brought down. 8,070 is not contained in 60, hence we put a nought in the quotient and bring down the next figure of the dividend. The divisor is still not contained, so we place another nought in the quotient ; and, as there is no other figure to bring down, the sum is finished. The answer is 30500, and the remainder 600.

The remainder is frequently placed after the quotient, with the divisor beneath it; thus—Quotient 30500 -g^^.

Method of Proof :—

To prove Division :—Rule.—Multiply together the Divisor and Quotient, and add the Remainder to obtain the Dividend.

8070 x 30500 = 246135000 (see example of multiplication), and 246135000+ 600 = 246135600 = the Dividend.

The Dividend is thus the product of the Divisor and Quotient, + the Remainder. Hence, the Divisor may always be found by subtracting the Remainder from the Dividend and dividing by the Quotient.    .

Thus (246135600-600) + 30500 = 8070, the Divisor.

HINTS UPON CANCELLING BY INSPECTION.

1.    All even numbers are exactly divisible by 2.

2.    3 will measure every number the sum of whose digits is a multiple of 3.

3.    4 will measure any number whose two last figures denote a number divisible by 4.

4.    5 will measure every number ending in 5 or 0.

5.    6 will measure every even number the sum of whose digits is a multiple of 3.

6.    8 will measure every number whose three last figures denote a number divisible by 8.

7.    9 will measure every number whose digits give a sum divisible by 9.

8.    10 will measure every number ending in 0.

9.    11 will measure any number whose alternate figures added give the same sum as the other figures ; as, 785268, 374 ; or, whose alternate figures give a sum which is 11, 22, or some multiple of 11, in excess or defect of the sum of the other figures, as 8170943.

10.    12 will measure every number whose last two figures denote a number divisible by 4, and whose digits added give a multiple of 3 as a sum.

11.    15 will measure every number answering to the requirements of (2) and (4) above, and 18 will measure every even number whose digits give a sum divisible by 9.

12.    20 will measure every number answering to the requirements of (3) and (4).

13.    25 will measure every number ending in 25, 50, 75, or 00.

14.    Numbers divisible by 14 are also divisible by 2 and 7; numbers divisible by 16 are divisible by 4 and 4, or by 2 and 8, &c.; numbers divisible by 18 are divisible by 9 and 2, or by 3 and 6, &c. ; and, generally, numbers divisible by any composite number are divisible by the factors of that number

15.    Odd numbers multiplied by odd give an odd product.

16.    Even numbers multiplied by odd or even give an even product. The product of two or more odd numbers is odd ; all other products are even.

17.    Even numbers divided by even give an even remainder, if any;

HINTS UPON CANCELLING BY INSPECTION. 33

odd by even always give an odd remainder. In all other cases the remainders may be either odd or even.

18.    Numbers divisible by 7 can be told by trial only.

19.    Every prime number greater than 7 ends in 1, 3, 7, or 9 ; for all even numbers, and numbers ending in 5, are composite.

20.    Every prime number greater than 3 is either 1 more or 1 less than a multiple of 6 : thus, 13 is 1 more than twice 6, 17 is 1 less than three times 6.

21.    Every measure of two numbers is also a measure of their sum, or difference.

22.    Since 13, 17, and 19 are prime numbers, and their products, together with those of the figure 7, can be told by trial only, it is necessary to learn the multiplication tables of these numbers in order to facilitate cancelling.

23.    Any number consisting of two figures repeated in the same order an even number of times is divisible by 101 ; as, 5454.

For 5454 := 4 times 101 + 50 times 101, or 54 times 101.

24.    Any number consisting of three figures repeated an even number of times is divisible by 1001 ; as, 159159, 240240240240, 29029.

For 159159 = 9 times 1001 + 50 times 1001 + 100 times 1001 ; or, 159 times 1001.

Hence, (by 14), such a number will also be divisible by the factors of 1001 ; that is, by 7, 11, and 13.

25.    Any number consisting of four figures repeated twice, four times, &c., is divisible by 10001 ; or, by its factors, 73 and 137.

Note. (23), (24), and (25) are useful in cancelling a circulating decimal; while the following will enable us to measure a pure repeater of one figure

26 Any number made up of the same digits three times repeated —as 222, 555, 444666,—is divisible by 111; and, consequently, by 3 and 37, the factors of 111.

Cancellation.—Cancelling is a method of shortening the labour of multiplication and division when both operations have successively to be performed.

The process depends upon the principle that to multiply by a certain part of one number, and to divide by the same part of a second number, produces the same result as multiplying and dividing by the numbers themselves.

PRINCIPLES IN CONNECTION WITH THE FOUR SIMPLE RULES.

Axiom I.—Of any two numbers—

(a)    The sum + the difference = twice the larger.

(b)    The sum — the difference — twice the smaller.

Reason.—The difference of two numbers is the excess of the larger over the smaller, or the deficiency of the smaller as compared with the larger.

Hence, The smaller -I- the difference gives the larger.

And, The larger - the difference leaves the smaller.

Now, since the sum is the larger and the smaller, by adding the defect of the smaller to the sum we obtain twice the larger; and, by taking away the excess of the greater from the sum, we get twice the smaller.

For, adding the difference to the sum is equivalent to adding the defect to the smaller and then adding the result to the larger, thus making twice the larger.

Again, taking the difference from the sum is equivalent to taking the excess from the larger and then adding the result to the smaller, thus making twice the smaller.

Having the sum and difference of two numbers, we thus obtain the following rule for finding the numbers :—

1.    To find the larger. Rule.—Add together the sum and difference

and divide by 2.

2.    To find the smaller. Rule.—From the sum take the difference

and divide by 2.

Or, as expressed by Algebraists,

\ sum + £ difference = the larger number.

| sum - | difference = the smaller number.

Problem 1.—The sum of two numbers is 20659 and their difference is 6077 ; find the numbers.

S;lution.—The largerof (20659 + 6077) = ^ of 26736 = 1336S.

The smaller = y of (20659 - 6077) = j of 14582 = 7291.

Problem 2.—The sum of the ages of a father and son is 37, and the father was 23 when the son was born ; find their ages.

Solution.—Here the difference between the ages is 23 years.

The father’s age = \ of (37 + 23) = 30 years.

The son’s age = £ of (37 - 23) = 7 years.

Axiom II.—The sum of numbers is the same in whatever order they are added:

Thus, 100 + 15 + 75 + 2 gives the same sum as 75 + 100 + 2 + 15.

Axiom III.—When addition and subtraction have severally to be 'performed, the residt is the same in whatever order the operations are performed:

Thus, 100 — 6 + 12 — 80 + 34 gives the same result as 100 + 34 + 12 — 80 — 6 = 146 — 80 — 6 = 66 — 6 = 60.

Axiom IV.— When several numbers have to be successively taken away, the result will be the same, if the sum of the numbers to be taken be subtracted :

Thus, 146 - 80 - 6 = 146 - (80 + 6) = 146 - 86 = 60.

Note.—When addition and subtraction occur promiscuously, it is frequently convenient to subtract the sum of the numbers preceded by the sign — from the sum of the other numbers.

The algebraic rule is—add together separately the positive and negative numbers, find the difference of these sums, and prefix the sign of the greater sum:

Thus, 8 + 9- 19 -8 + 10 - 5 = 27 - 32    - 5 ans.

Note.—Negative numbers belong more properly to algebra; but, since questions with negative answers have been proposed at recent examinations, an explanation of a negative number may not be inappropriate here.

Negative numbers, then, are always preceded by a minus sign ( —), and represent a deficiency. It is easy to understand that 12-7 signifies the number twelve together with a deficiency of seven, which gives the positive number five as the result ; but that 7-12 leaves a negative nifinber — 5, as the deficiency, is not so easily realized.

A negative number standing alone (thus, - 2) is purely an imaginary number, expressing (if the analogy may be pardoned), as much less than nothing, as its corresponding positive number expresses more than nothing. 20-35 = - 15, for since taking 20 from 20 would leave nothing, we can imagine that taking 15 more away would leave 15 less than nothing. In the same way, the property of a man who has £1000, and owes £1250, may be represented by - £250, since he is £250 worse off than if he simply had nothing, and were clear of debt. Negative numbers are subject to the same laws of increase and decrease as ordinary or positive numbers ; and are frequently useful to assist in solving difficult problems.

Negative numbers X negative numbers give a positive answer : thus, - 6 X - 8 = + 48.

Negative numbers X positive numbers give a negative answer; thus, - 6 X 8 zr - 48.

Negative numbers 4- negative numbers give a positive answer : thus, - 6 4- - 2 — 3.

Negative numbers 4- positive numbers give a negative answer : thus, - 6 4 2 ~ - 3.

Positive numbers 4- negative numbers give a negative answer : thus, 6 4- - 2 — - 3.

The product and quotient of two numbers being given, to find the numbers.

I.    To find the number which was divided by the other :—

Rule.—Multiply the product by the quotient, and extract

the square root of the result.

II.    To find the number used as divisor :—

Rule.—Divide the product by the quotient, and extract the square root of the result.

Axiom V.—To multiply or divide by the factors of a number is to multiply or divide by the number itself :

Thus, 240 divided by 3 and 4, or by 2 and 6, gives the same as 240 4- 12.

Axiom VI.—The product of two numbers divided by either gives the other :

Thus, if S7 is the product of 29 and another number, that other number must be 87 divided by 29, or 3.

Axiom "VII.—Every dividend is the product of the divisor and quotient :

Hence, the dividend divided by the quotient will give the divisor.

Note.—If there be a remainder, subtract it from the dividend before dividing.

Axiom VIII.—The product of the sum and difference of two numbers equals the difference of their squares:

Thus, (8 + 3) x (8 - 3) = 8² - 3² = 55.

Axiom IX.—The square of the sum of two numbers equals the sum of their squares plus twice their product :

Thus, (8 + 3)² = 8² + 3² + (8 x 3) x 2, or 64 + 9 + 4S = 121.

Axiom X.—Concrete numbers cannot be multiplied together :

Thus, we cannot say £5 times £8 ; but simply five times £8, or eight times £5. Such a sum as £19 19s. 9|d. x £19 19s. 9|d. is therefore without meaning. The multiplier must always be an abstract number.

Axiom XI.—Concrete numbers can be added or subtracted <mly when they refer to the same kind of thing :

Thus, if we consider horses and cows under the common head, animals, we can say that 5 horses and 8 cows make 13 animals ; but we cannot say that 5 horses and 8 cows make 13 horses or 13 cows.

Axiom XII.—The value of a number is not altered by removing one part and adding it to another part:

4

Hence, in Subtraction, when we cannot take the under number from the upper, we remove one of the figures in the place next on the left and add its equivalent to the number we wish to subtract from :

Thus, in taking 57 from 92, we say 7 from 2 we cannot, remove one ten from the 9 tens (leaving 8 tens), 10 and 2 are 12 and 7 from 12 leaves 5. We then take the 5 tens from the 8 tens, and thus obtain 3 tens. Then 3 tens and 5 = 35.

Axiom XIII.—The difference between tivo numbers is not altered by adding an equal amount to each:

Thus, the difference between 57 + 10 and 92 + 10 is the same as the difference between 57 and 92. Hence, Subtraction is often worked thus, 7 from 2 we cannot, add ten to the upper number (7 from 12 leaves 5) ; then, to prevent the difference from being altered, add 1 ten to the bottom line, and say 1 ten and 5 tens are 6 tens, 6 from 9 leaves 3.

Axiom XIV.—The product of two numbers plus the square of half their difference = the square of their average :

Thus, 12 x 8 + 2² = 10² = 100.

Hence, the product and difference of two numbers being given, to find the numbers :—

Rule.—To the product, add the square of half the difference and extract the square root for their half sum.    Then, the

half-sum + the half-difference gives the larger number, and the half-sum - the half-difference gives the smaller number.

This rule, algebraically expressed, is :—

Rule.—To four times the product, add the square of the difference and extract the square root for the sum. Then, having the sum and difference, find the numbers by Axiom /.

To find two numbers, whose sum and product are given:—

Rule.—From the square of half the sum (which is then-average) take the product, and extract the square root of the remainder to obtain the half-difference. To this half-difference add the half-sum for the larger, and take the half-difference from the half-sum for the smaller.

To prove Axiom XIV., take any numbers in succession, as, for instance, 4, 5, 6, 7, 8, 9, 10, 11, 12 ; square one of them (say 8), then the product of any two equidistant from it will be less than this square by the square of half the difference of these numbers. Thus, 7 x 9, 6 times 10, 5 times 11, and 4 times 12 will be 1, 4, 9, and 16 respectively less than 8² or 64.

Axiom XV.—In every multiple of 9, the digits when added make an exact number of nines. For illustration see Multiplication Table of 9.

Axiom XVI.—Every number, minus the sum of its digits, leaves a midtiple of nine.

Thus, 100 - 1 = 99, and 524 - 11 = 513 = 57 times 9 ; because, taking away the figure in the units’ place leaves 0 or no times 9, and taking away as many ones as there are tens will leave so many nines, and taking away as many units as there are hundreds will leave so many 99’s, and taking away as many units as there are thousands will leave so many 999’s, &c., &c.

Upon Axioms XV. and XVI. the rules for proving all the simple rules by casting out the nines depend. See page 41.

Axiom XVII.—Of any two numbers, the products of each part of one into all the parts of the other give a sum equal to the product of the numbers themselves.

If 10    7 + 3, and 12 = 6 + 4 + 2, then, 7x6+7x4+7x

2 + 3 X 6 + 3X4 + 3X 2 = 42 + 28 + 14 + 18 + 12 + 6 = 120 ; or 10 times 12.

Again, 16 X 15    (10 + 6) X (10 + 5) = 100 + 50 + 60 + 30

= 100 + 30 + 60 + 50 = 240.

We thus obtain the following rule to multiply one binominal by another.

Rule.—Add together the products of the first terms, the last terms, the means, and the extremes :

Thus (10 + 7) x (12 - 3) zz 120 - 21 + 84 - 30 zz 204 - 51 = 153 ; or, 17 times 9.

Note.—A binominal is a bracket enclosing two numbers or terms. The means are the inner of four numbers, the extremes the outer.

Corollary of Axiom XIY.—Of any two numbers with equal sums, the two which are nearest equal have the greatest ‘product:

Thus, 8 times 8 zi 64; while 7 times 9, 6 times 10, 5 times 11, &c., give products each less than 64, and less than the preceding product.

Corollary of Axiom XIV.—The square of any number = the product obtained by doubling the unit and multiplying the number thus formed by the tens and adding the square of the unit:

Thus, 63 x 63 zz 66 x 60 + 9 zz 3969.

Corollary of Axiom VIII.—The square of any number exceeds the square of the number next below it by the sum of the two numbers.

Thus, 13" zz 12| + (13 and 12), or 144 -f- 25 zz 169, which is the square of 13.    99'- zz 100² - 199 zz 10000 - 199 zz 9801.

Corollary of Axiom VIII.—Of any two numbers whose difference is one, their sum = the difference of their squares.

For, since the sum X the difference zz the difference of their squares (Axiom 8), their sum X unity (or the unaltered sum) is the difference of their squares. Of any two fractions whose sum is unity, their difference zz the difference of their squares ; for the sum (or 1) X the difference zz difference of their squares, that is one time (the sum being 1) the difference zz the difference of their squares.

Note.—Multiplying or dividing by unity does not alter a number.

Axiom XVIII.—The sum of the dividend and divisor contains the divisor once more than the dividend itself does.

Axiom XIX.—The dffere'nce of the dividend and divisor contains the divisor once less than the dividend itself does.

Note.—Hence the rules, on page 55, for finding two numbers whose sum and quotient, or difference and quotient, are given.

PROVING BY CASTING OUT THE NINES. 41

RULES FOR PROVING BY CASTING OUT THE NINES.

Addition.—Rule.—Cast the nines out of all the digits in the Adderids (in any order whateverJ, and the remainder will be the same as that produced by casting the nines out of the Sum or answer.

Subtraction.—Rule.—Cast the nines out of the Minuend and Subtrahend; subtract the latter remainder from the former. This should give the same as casting the nines out of the Difference or answer.

Note.—If the remainder from the minuend cannot be taken from the remainder of the subtrahend, add 9 to the latter and then subtract.

Multiplication.—Rule.—Cast the nines out of the Multiplicand and Multiplier ; multiply the remainders together, a,nd cast out the nines. This should give the same as cays ting the nines out of the Product.

Division.—Rule.—Cast the nines out of the Divisor and Quotient; multiply the remainders together, and add in the excess of nines in the Remainder of the sum. This should give the same as casting the nines out of the Dividend.

Examples:—

1. Addition. 79890 Leaving out the 9’s and 0’s we have 7 and 36542    8 are 15 = 9 and 6 over ; 6 and 3 are 9 ;

90970    6 and 5 are 11 =9 and 2 over ; 2 and 4

-    and 2 and 7 are 15 = 9 and 6 over.

207402

Casting the nines out of the sum, we get, 2 + 7 + 4 + 2 = 15 = 9 and 6 over.

2.    Subtraction. 79890 Casting out nines leaves 6 over.

36542    ,,    ,,    ,,    2 over.

43348    ,,    ,,    ,,    4 over    = 6-2.

3.    Multiplication. 36542 Casting out nines leaves 2.

90970    „    ,,    ,,    7.

2357940

328878    7 x 2 = 14 = 9 and 5 over.

32887S0

3324225740 Casting out nines leaves 5 over.

4.    Division.    36542) 3324225748 (90970

2 over. 4 over. 7 over.

Remainder 8.

Then 7x2 = 14 + 8 = 22 = 2 nines and 4 over, and casting the nines out of the dividend leaves 4 over.

Note.—When an error of 9 or 0, or of misplacement of any digit or partial product, &c., has been made, casting out the nines will not detect the error.

PROOF BY CASTING OUT THE ELEVENS.

To cast out the elevens from a number : —

Rule.—First cast out the elevens from the sum of the digits in the odd places, and then from the sum of the digits in the even places. Subtract the smaller remainder from the greater.

To prove by casting out the elevens :—

Rule.—Cast out the elevens, and proceed as in proof after casting out the nines.

Note.—The final results should either coincide or else give 11 as a sum.

The above rules depend upon the following principle :

Every odd power of 10 (as 10, 1000, 100000, &c.), is 1 less than a vnidtiple of 11, and every even power of 10 (as 100, 10000, 1000000, ¿cc.), is 1 more than a multiple of 11.

Thus, (10 + 1), (1000 + 1), (100 - 1), (10000 - 1), are all divisible by 11.

Exercise I.—NOTATION, NUMERATION, AND THE SIMPLE RULES.

1.    In the English method of notation, state what are the values of the figures which occur in the places named below :—the 1st, 3rd, 7th, 8th, and 13th.

2.    Write in the English method one billion sixty-seven millions five hundred and nine.

3.    Write down ten millions one hundred, and one million ten thousand; and find the product of their sum and difference.

4.    Find the product of the sum and difference of seventy millions six hundred and eighty thousand and forty, and nine millions nine hundred and nine thousand and fifty-four. Express the result in words.

5.    Find the sum, difference, and product of the following numbers. Sixty millions thirty-six thousand, eight hundred and five millions four hundred thousand and nine. Numerate the product according to the English method.

6.    If three thousand nine hundred and eighty-one millions seventy-three thousand seven hundred and twenty be diminished by seven thousand nine hundred and forty-eight, and the remainder again diminished by the same number, and so on ; find how many times the operation must be performed before there is no longer any remainder.

7.    Find the product of ten millions one thousand and ten, and one million eleven thousand, and express the result in words.

8.    In a division sum the divisor is seventy thousand nine hundred and eight, the quotient five hundred and eight thousand nine hundred and seventy, and the remainder six thousand and twenty. Find the dividend, and express it in words.

9. The dividend of a sum being four billions three hundred and thirty-nine thousand three hundred and forty-six millions two hundred and forty-two thousand two hundred and forty; and the quotient five hundred and seventy thousand eight hundred and ninety-six. Find the divisor.

10 (a). Distinguish between the intrinsic and local value of a figure.

(b) . Set down at full length the local value of each figure in 9072'06.

(c) . What is meant by the power of a number ? Give an instance, and explain it.

11.    Find the quotient of six hundred and thirty millions seventy-eight thousand and four by eighty-six thousand five hundred and forty-nine; and prove the correctness of

the operation.

12.    Find the quotient of five hundred millions six hundred and eighty thousand and forty-nine, by seventy-eight thousand five hundred and ninety-six. Prove the correctness of the operation.

13.    The product of two numbers increased by four thousand seven hundred and eighty-four (which is just one half of one of the numbers) is six thousand seven hundred and four millions two hundred and six thousand seven hundred and four. Find the numbers.

14.    Find the product of ten billions one thousand and ten and one million eleven thousand, and express the result in words.

15.    Find tlie sum, difference, and product of the following numbers :—Sixty millions thirty-six thousand eight hundred and seven, and five millions four hundred thousand and nine. Numerate the product according to the English method.

16.    Divide two hundred and thirty-eight billions four hundred and eighteen thousand six hundred and nine millions two hundred and twenty thousand nine hundred and forty-six by four millions three hundred and seven. Numerate the quotient according to the English method.

17.    Multiply twelve millions seven thousand and fifteen by one hundred and seven, and state in words the difference between the product and one thousand millions.

18.    Divide ten thousand one hundred and five millions twenty thousand and seven by fifty thousand and thirteen, and numerate the quotient and remainder.

19.    Division bears the same reference to Subtraction as Multiplication does to Addition. Explain this statement.

20.    What number multiplied by six hundred and seventy-nine will give the same result as five thousand six hundred and seven multiplied by four hundred and eighty-five 1

21.    Divide 47139 by 35, making use of factors. Explain how the full remainder is arrived at, and give reasons for the process.

22.    Divide 16798 by 48, making use of factors.

23.    Define the terms notation, addend, quotient. Find how often seven thousand six hundred and nine can be subtracted from thirty-eight millions six hundred and forty thousand and five, and state what the remainder will be.

24.    The sum of two numbers is 7410967470, and the greater exceeds the less by 7290785910. Find their product and numerate the result, stating what system of notation you adopt.

25.    Explain the process in working a sum in Long Division, and show that Division is but a shorter method of performing Subtraction.

26.    Find the product of the sum and difference of 746969164 and 38068794 and numerate the result after the English method.

27.    Define the terms Numeration, Minuend, Product. Find how often eight thousand nine hundred and seven must be added before it exceeds seventy millions nine thousand six hundred and four.

28.    Find the number which, when multiplied by 8405, will give 293839455590. Express the answer in words.

29.    Find and write down in words the number which, when divided by six hundred and eighty-five, will give for a quotient seventy-three thousand four hundred and thirty-six, with a remainder equal to three hundred and forty-seven.

30.    Find and write down, in words, the number which, when multiplied by seven hundred and sixty-five, will give such a product as, when increased by seven hundred and forty-two, will amount to sixty millions fifty-four thousand and seven.

31.    The product of two numbers is fifty-eight billions nine hundred and eighty-two thousand eight hundred and ninety-nine millions three hundred and ninety-five thousand and sixty, and the less of them is six millions eighty thousand and ninety-seven : find the greater.

32.    The greater of two numbers is five hundred and forty millions three hundred and sixty-four thousand eight hundred, and the less is one-sixth of the greater. Numerate their product according to the English method.

33.    What number divided by seven hundred and nine thousand five hundred and thirty is greater by five hundred and three thousand nine hundred and ninety-eight than nine hundred and sixty-three thousand and seven 1 Numerate it.

34.    How many times must eight thousand seven hundred and nine be added to itself to give the sum of eight hundred and thirty-six billions seven hundred and sixty thousand seven hundred and twenty millions ?

Note.—8709 is contained in 836760720000000 exactly 96080000000 times. See Note to Sum 38.

35.    How many times can the difference between one million, and three hundred, be subtracted from seven thousand and forty millions nine thousand and eighteen 1 What number must be added to this latter number that the above difference may be subtracted just once more¹?

36.    The divisor is six hundred and nine millions four hundred and twenty-six thousand and seven, and the quotient is three hundred and six thousand and ninety-seven, and the remainder seven millions eight hundred and seventy-nine thousand six hundred and four. Write the dividend in words according to the English method.

37.    The quotient of a sum is seven millions nine thousand eight hundred and thirty and the divisor exceeds the ninth part thereof by 1240. What is the dividend 1

38.    How many times must eight thousand seven hundred and nine be added to itself to give the sum of eight hundred and thirty-six millions seven hundred and sixty thousand seven hundred and twenty 1

Note.—8709 is contained in 836,760,720 exactly 96080 times ; but, since the question is how many times must 8709 be added to itself, the correct answer is 96079, or one less than the number of times which it is contained. If 8 be added to itself once, the sum will be 16, or twice 8 ; if twice, the sum will be 24, or thrice 8; if three times, the sum will be 32, or four times 8, &c., &c.

In the answer given to sum (3) of I. of Ex. IX., Barnard Smith’s Arithmetic (page 31), the principle that a number added to itself 398 times will give 399 times that number has been overlooked. If the words “ to itself ” were omitted, the answer given would be correct.

39.    Write down in words the number which will give four thousand and forty as a remainder when eight thousand and fifteen has been subtracted from it seventy-five thousand and twenty-two times.

40.    Write down in words the number which will give three thousand one hundred and eighty-nine as remainder after forty thousand nine hundred and eighty have been subtracted from it five thousand and seven times.

41.    The sum of two numbers is six millions four hundred and eight thousand four hundred and thirty-two, and the greater is fifteen times the less. Find their product.

42.    The sum of two numbers is 2986900, and the greater is to the less as 7 : 3. Find their product.

43.    The sum of two numbers is 61589979, and their difference 61588441. Find how often the less is contained in the greater.

44.    Find a number which, on being multiplied by 769078 will give a product 5230337971620.

45.    Multiply 6784096 by 8726481, using only four multipliers.

Note.—This is done by first multiplying by the 8 in the millions’ place ; next, this product by 9 for the 72 ; this again by 9 for the 648 ; and, lastly, the top line by 1. Of course, the partial products must be carried to the right instead of to the left, and the proper number of cyphers appended before they are added.

46.    By what number must eight thousand and one be multiplied that the product may exceed fifty-six millions seventy-eight thousand and eight by one thousand and one 1

47.    What number multiplied by four thousand seven hundred and nine will give the same result as two millions three hundred and seven thousand four hundred and ten multiplied by ninety-four thousand and seventy ?

48.    How many times must seven hundred and eight thousand nine hundred and sixty-five be used as an addend before it exceeds 800,000,000 1

49.    What number diminished by three hundred and ninety millions twenty-three thousand seven hundred will be equal to the product of fifty millions nine hundred thousand six hundred and eighty, and seven millions eight hundred and nine thousand and four ? Express the answer in words.

50.    Write down one million and ten, and ten millions one thousand ; and prove that the product of their sum and difference is ecpial to the difference between their squares

51.    The divisor being 6809507, the quotient 786090, and the remainder 40967, find the dividend and enumerate it, according to the English method.

52.    What number, increased by four hundred and eight millions two thousand and sixteen will be equal to the product of (a) sixty-nine millions eighty thousand and seventy, and (b) eight millions seven hundred thousand nine hundred and five 1 Express your answer in words.

Exekcise II.—EXERCISE UPON THE SIMPLE

RULES.

1.    Add together 1300, 2750, 97, 6394 and 27942.

2.    Find the sum of 30709, 5625, 73297, 1200 and 605.

3.    Find the difference between 1700960 and 799990.

4.    By how much does 305060 exceed 290006 1

5.    How much less is 3706209 than 9190909 1

6.    What number must be added to 37609 to make 100000 1

7.    What is the defect of 9100905 compared with 100000001

8.    What number taken from a million will leave 2990091

9.    What is the amount of 500, 2001, 976, and 131

10.    Which is the greater—10000 diminished by 9654, or 3901

11.    When 50976 is taken from 59076, what is left 1

12.    From ten millions five hundred, take seven millions and ninety.

13.    Take 806398 from one million ten thousand and ten.

14.    What remains when fifty thousand nine hundred is taken from 568701

15.    To 50607, add 5689, and take 55689 from the sum.

16.    From 82000 take seventy thousand nine hundred and nine.

17.    How many years from the beginning of 1847 to the end of 19001

18.    Take eleven hundred and eleven from one thousand two hundred.

19.    What is the difference between thirteen hundred and one thousand two hundred 1

20.    By how much does seventeen thousand seventeen hundred exceed 18650 1

21.    What number added to 70286 will make 90495 1

22.    What number taken from 30706 will leave 260901

23.    From what number must 30706 be taken to leave 260901

24.    If 3026, 509 and 7000, successively taken from a certain number left 6095, what was the original number 1

25.    Add together the sum and difference of 20609 and 80964.

26.    From the sum of 8090 and 9090, take the difference.

27.    Add together ¿£1248, ¿£295, ¿£864, ¿£2900, and ¿£87.

28.    How many more days in a year of 365 days than in seven months of 31 days each ?

29.    The months of the year contain 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 days respectively. Find the number of days in twelve months.

30.    Find the value of 8600 + 32609 — 726 + 3727 — 9000.

31.    Find the difference between the sum of the positive and negative numbers contained in the following expression:—

36 - 40 + 509 - 67 + 837 + 497 - 364 - 246.

32.    To what number must 12976 be added to give 59688 ?

33.    By how much is the difference between 80096 and 70984 less than their suin'?

34.    What is the remainder of 706072 ^ 907062?

35.    Find the value of 3968 - 495 + 826 - 963 + 120096.

36.    Multiply 6806009 by 489700,and numerate theanswer.

37.    Find the product of seven thousand six hundred and eighty-nine and ninety-seven.

38.    Multiply six thousand seven hundred and eighty-nine by sixty-nine.

39.    Add together eight thousand seven hundred and ninety-six, seven thousand six hundred and eighty-nine, nine hundred and fifty-seven, and five thousand eight hundred and seventy-four.

40.    From five hundred and sixty thousand and forty-nine lake fifty-nine thousand nine hundred and sixty-four. Numerate the difference.

41.    Express in figures five millions eighty thousand and seven.

42.    From three hundred and seventy thousand and thirty-seven take fifty-nine thousand nine hundred and forty-three.

43.    Write down in words—7009030, and express in figures four millions eight hundred thousand and five.

44.    Express in figures nine millions seventy thousand and six.

45.    Divide eight hundred thousand one hundred and sixty-seven by seventy-nine.

46.    What is the least number which, taken from 764982, leaves a remainder divisible by 96 ; and what is the least number which, added to 70609, gives a sum divisible by 87?

47.    Find the product of 12345670 and 245497 by means of three lines of multiplication, and numerate the answer.

48.    Divide 357357 by 21021, using the factors 1001, 3 and 7.

49.    The difference between two numbers is 936 and the larger is 1267. Find the other.

50.    The difference between two numbers is 936 and the smaller is 1267. Find the other.

51.    The sum of two numbers is 2648 and one of them exceeds the other by 170. What are the numbers 1

52.    Find a number such that, if it be added to 768953, the sum will be 986742.

53.    The 96th part of a number is 2798. What is the number ?

54.    The product of two numbers is 15580656, and one of them is 13104. What is the other number¹?

55.    The sum of two numbers is 987653, and their difference is 4781. Find the numbers.

56.    The quotient of a division sum is 2908, the divisor is 546, and the remainder 297. Find the dividend.

57.    The quotient arising from the division of 868000 by a certain number is 879, and the remainder 427. Find.the divisor.

58.    The product of two numbers is 1587768, and their difference is 2362. Find the numbers.

59.    The product of two numbers is 10137600, and their sum is 7520. What are the numbers 1

60.    Three numbers are such that the third is the same number of times the second that the second is of the first. The second number is 48, and the difference between this and the third is 96. Find the three numbers.

61.    Three numbers are such that the third exceeds the second by as much as the second exceeds the first—namely, by 9. If the sum of the numbers be 78, what are they¹?

62.    The quotient of a division sum equals five times the divisor, and the divisor is five times the remainder. If the three together amount to 217, find the dividend.

63.    Multiply 987654321 by 576729, using three multipliers only. Numerate the answer.

64.    What number exceeds the sum of its half and 399

by 777 ?    _

65.    What number is less than the sum of its half and 4840 by 2200 1

66.    The continued product of 17, 7, and another number is 2261. Find the other number.

67.    Divide the sum of six hundred millions five hundred thousand and twelve and five hundred and ninety-nine millions nine thousand and nine by their difference.

68.    Divide 3795462 by 198, using factors.

69.    Find the sum of all prime numbers less than 100.

70.    Find the sum of all composite numbers from 40 to 100 inclusive.

71.    The less of two numbers is 567 and their difference 453. Fmd their product, and numerate it.

72.    Find the difference between 780 multiplied by itself and 67900. Numerate it.

73.    The quotent of two numbers is 71 and the larger number is 7029. Fmd the other number.

7 4. Find a number which is seventeen times as great as seventy-five times nine score.

75.    What number multiplied by 57 and divided by 79 will equal 56943 1

76.    What number will equal 108737 when 507 has been added to it and the sum multiplied by 97 1

77.    What number will make 1000 when it has been divided by 9, the quotient thus obtained multiplied by 7 and 237 added to the product 1

78.    What number is it from which, if 32 be taken, 900 be added to the remainder, and the sum thus obtained be multiplied by itself will amount to a million 1

79.    By what number must we multiply one thousand in order that, after dividing the product by 1080, we shall obtain 6001

80.    What number must be multiplied by 20 in order that, by continued division by 4, 7, 8 and 10, the quotient will be 11

81.    What number must be divided by 15, and the quotient thus obtained multiplied by 7, in order to give 1611

82.    If 74 be added to a certain number and 90 taken from the sum, the result will be 100. What is the number!

Note —In working the last nine sums, it will be well to remember that, to undo what has been done, the opposite operation to that which is mentioned must be performed ; thus, if the number 240 multiplied by 17, and the product divided by 12 equals 340, then 340 x 12 —17 will equal 240.

Addition and Subtraction, Multiplication and Division, Involution and Evolution are operations the converse of each other.

83.    Find and numerate the continued product of 90070 X 5090 x 3060800. Before performing the operation required, say how many figures the answer will contain.

84.    Without performing the operation of multiplication, say how many cyphers the product of 375 multiplied by 76000 will contain.

85.    Increase 270960 ninety thousand and fifty-two times. Numerate your answer.

86.    Square 1728 ; that is, nudtiply 1728 by itself.

87.    Cube 365; that is, multiply 365 times 365 by 365.

88.    Find the least number that will contain 13, 17, 19, 23, and 29, each an exact number of times. This is done by finding the continued product of the numbers.

89.    What number taken from the square of 165 will leave 175 times 155 1

90.    What number is that to which, if 5000 be added and 4926 be taken from the sum, will equal 864741 See note to sum 82.

91.    Divide the fourth part of 31416 by 66 both by short and by long division. In the latter case prove the correctness of the answer by adding the three subtrahends as they occur one above the other. This should give the dividend. By what number of one digit is the answer divisible 1

92.    Divide 277274 by 79 and prove the correctness of the answer by adding the remainder to the various subtrahends as they appear above one another.

93.    Divide the sum of 1760, 5280, 4840, and 1728, by the difference between 365 and 313. Prove by casting out the nines.

94.    The product of two numbers is 78 times the larger;

what is the larger number, if the eighth part of the product equals 8780851    _    ^

95.    Find the value of the expression: 6372+-18 + 37 x

34    2 — 639 -r- 71 + 3065 - 3729 + 70007 x 13 x 11 -

350. Numerate the answer.

96.    What does (1224 — 2560 -h 4 + 336 -f 84 x 7 — 512 + 126 r 6 x 9 — 100) x 189 amount to?

97.    The product of two numbers is 40320000 and their difference is 1240. Find the numbers.

98.    The product of two numbers is 1728 and the quotient of the larger divided by the smaller is 6|. Find the numbers.

99.    A father was 35 years of age when his eldest son was born, and the sum of their present ages is 51. Find their ages.

100.    Find the sum, difference, product, and quotient of 450723000 and 635. Numerate the answers according to both the English and the Continental method.

101.    Multiply 12345679 by 9, 18, 27, 36, 45, 54, 63, 72, and 81 respectively, and find the sum of the products.

102.    Multiply 12345679 by 171 and by 153, and find and numerate the differerice of the products.

103.    The minuend is 9876543210 and the difference is 8758832033. Find the subtrahend, and numerate it.

104.    The product of two numbers is 605377, and the quotient obtained by dividing the larger by the smaller is

97. Find the numbers.

105.    The difference of two numbers contains the smaller once less than the larger number contains it. Hence : To find two numbers whose difference and quotient are given.

Rule.—Divide the Difference by one less than the Quotient to obtain the smaller number, and Divide the Product of the Quotient and Difference by one less than the Quotient to find the larger number. From this find the numbers whose difference is 76, and quotient 5.

106.    The sum of two numbers contains the smaller once more than the larger number contains it. Hence : To find two numbers whose sum and quotient are given.

Rule.—Divide the Sum by one more than the Quotient to obtain the smaller number ; and the Product of the Quotient and Sum by one more than the Quotient to find the larger number. By these rules find the numbers whose sum is 776, and quotient 7.

CHAPTER II.

TABLES.

Money Table.

1    Farthing = One Quarter of a Penny, written |d.

2    Farthings = One Half-Penny, written |d.

3    Farthings = Three Quarters of a Penny, written |d.

4    Farthings = 1 Penny, written Id.

12 Pence = 1 Shilling, written Is.

20 Shillings = 1 Pound or Sovereign, written XI.

ADDITIONAL.

Pounds, shillings, and pence, together with the other coins of the realm, are called Sterling Money.

The coins not mentioned above bear names significant of the number of pence they contain: they are, the threepence, fourpence, and sixpence.

An Alloy is a mixture of gold or silver with an inferior metal.

Standard Gold is an alloy of 22 parts of pure gold and 2 parts of copper.

A Carat means a twenty-fourth part; so that if 22 parts out of 24 be pure gold, the alloy is said to be 22 carats fine.

Standard gold is 22 carats fine.

Jewellers’ gold is 18 carats fine.

A Sovereign weighs 123-hyl grains, while 5,607 sovereigns contain 110 lbs. Troy of pure gold and 10 lbs of copper.

£, s., d., and q., signifying pounds, shillings, pence, and farthings, are the initial letters of Libra, Solidus, Denarius, and Quadrans, Latin words, names of Roman coins.

Obsolete coins—A Noble = 6s. 8d. ; an Angel = 10s. ; a    Mark =    13s.    4d. ; a Guinea =    21s. ;    a    Carolus    =    23s.    ;

a    Jacobus    =    25s. ; a Moidore =    27s. j    a    Groat    =    4d.;    a

Tester = 6d. ; a Half-Guinea = 10s. 6d.

Decimal coinage—

10 Mils = 1    Cent.

10 Cents = 1    Florin

10 Florins = 1    £

Note.—A Cent is the one hundredth part of a £, or 2|d. A Mil is rather less than a farthing.

Note.—Bronze coins are not legal payment for more than a shilling; nor is silver a legal tender for more than forty shillings.

Lineal Measure.

The unit of measurement is a straight line an inch long; thus, ___    .

ADDITIONAL.

7 Yards = 1    Irish Perch    22    Yards    =    1    Chain

3 Inches = 1    Palm    4    Inches    =    1    Hand

9 Inches = 1    Span    18    Inches    =    1    Cubit

Links and Chains are used in measuring land, and the Hand in measuring the height of a horse.

Note.—Lineal Measure, or Long Measure, is used to find the length, breadth, depth, or height of anything.

Cloth Measure.

1 Nail (na.)

1 Quarter (qr.)

1 Yard (yd.)

1 English Ell (En. ell.)

ADDITIONAL.

3 Quarters = 1 Flemish Ell; 6 Quarters = 1 French Ell.

N_0TE-—This Measure was used in measuring cloth ; but the nail is now very seldom employed, being supplanted by its equivalent, one-sixteenth of a yard.

Square Measure, or Surface Measure.

144 Square Inches = 1 Square Foot (sq. ft.)

9 Square Feet = 1 Square Yard (sq. yd.)

30-| Square Yards = 1 Square Pole or Perch (sq.po. or sq. per.) 40 Perches    =    1 Pood (ro. or rd.)

4 Poods    =    1 Acre (ac.)

640 Acres    =    1 Square Mile (sq. ml.)

ADDITIONAL.

49 Sq. Yds. =1 Sq. Perch Irish 484 Sq. Yds.    = 1 Sq. Chain

62-7264 Sq. In.    = 1 Sq. Link

10,000 Sq. Links = 1 Sq. Chain 100,000 Sq. Links = 1 Acre 10 Sq. Chains = 1 Acre 272^ Sq. Feet = 1 Square Foci (The Sq. Fod is used in Brickwork.)

4,840 Sq. Yards = 1 Acre 121 Irish Acres = 196 English Acres

The Unit of Measurement is a Square Inch ; that is, a square measuring am inch each way ; thus,—

A Square Inch.

The Feet, Yards, and Perches of Square Measure are obtained from Lineal Measure by squaring the numbers representing (a) the No. of inches in a foot, (b) the No. of feet in a yard, and (c) the No. of yards in a perch.

Note—To square a number is to multiply it by itself.

12 In. = 1    Ft. ; then, 12 times 12 (or 144) Sq.    In.    =    1    Sq.    Ft.

3 Ft. =1    Yd. ; then, 3 times 3 (or 9) Sq. Ft.    =    1    Sq.    Yd.

Yds. = 1    Per.; then, 5|times5.) (or30£) Sq. Yds.    =    1    Sq.    Per.

Note—Square Measure is used for measuring    the    area of    any

surface, whether land; the walls, floor, or ceiling of a room; the surface of a plank, or of a roll of paper; or anything where the extent of surface is required. The area of any surface is obtained by multiplying together its two dimensions ; whether length and breadth, length and height, or length and depth. No account is taken of the third dimension, which is generally the thickness of the thing measured. Links and chains are usually employed in measuring land. They afford a much readier means of calculation than yards, perches, roods, &c.; for acres may be at once expressed in links or chains, and links or chains as acres, by merely removing the decimal point. Furthermore, the work of Addition, Subtraction, Multiplication, and Division, requires the application of the simple rules only.

Examples:—

35-760 acres = 357’60 chains = 3576000 square links.

3265000 square links = 326'5000 square chains = 32'65 acres.

528 acres = 5280 square chains = 52800000 square links.

From these examples it will be seen that acres are expressed as square chains by removing the decimal point one place to the right ; and as square links by removing it five places. Again, to express square links as acres we have merely to remove the decimal point five places to the left; or, in case there is no decimal point, to cut off the last five figures of the number expressing the links.

This matter will be treated of more fully under Decimal Fractions.

Cubic Measure.

1728 Cubic Inches = 1 Cubic Foot (c. ft.) 27 Cubic Feet = 1 Cubic Yard (c. yd.)

ADDITIONAL.