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their Multiplication? Give the Note. What for their Division? Give
the Note. What is the rule in the first Case for their Reduction? In sec-
ond Case? In third? In fourth? In fifth? What are the rules in
Case first of Reduction of Currencies? In Case second? third? fourth?
fifth? What are the rules for changing Federal Money into the Currencies
of the several States? What for changing it to Canada and Nova Scotia
money? What to that of Great Britain? What does the Rule of Three
teach? Why so named? Why called Golden Rule? Give the rules and
notes for its operation. What is Practice? What the rule in Case first?
What says Case second, and what its rule? What are Tare and Tret?
Define all its terms. When is Case first used, and what its rule? When
Case second, and what the rule? Case third, and what the rule? Fourth,
and what the rule? What does the Double Rule of Three teach? How
many terms in its questions.? How distinguished? Give the rule and
notes. What is Conjoined Proportion? When is Case first used, and
what its rules? When Case second, and what its rule? What is Barter?
Its rule? What Loss and Gain? In what instruct merchants and traders?
How its questions solved? What its general law? What is Fellowship?
Its use? What Single Fellowship Its Rule? How proved? What
Double Fellowship? Its Rule? What is Interest? What the legal inter-
est? Define its terms. How many kinds? What Simple? Its Rule and
Note? Tell the Table of Aliquot Parts? What the Rule for Months, at


per cent.? For days at dito? What the Short Practical Rule for
pounds, &c. at ditto? How serve at 5 or 7 per cent.? What the Short
Rule for Federal Money, at 6 per cent.? How proceed at 5 or 7 per cent.?
What the first Rule to compute Interest on Notes, &c. having Endorse-
ments? How the Rule contracted? What the Rule in Massachusetts ?
What Compound Interest? What the Rule? What Commission and
Brokerage? What Insurance? What Discount? What Present Worth?
What the Rules? What the Rule when there are several sums to be paid,
&c.? What an Annuity? How in Arrears? What meant by Amount?
What by Present Worth? How is the Amount found at Simple Interest?
How the Present Worth? What Equation of Payments? What the
Rule? What known by Exchange? Tell the Table. What the Rules?
How many kinds of Vulgar Fractions? What a Proper one? Improp-
er? Single? Compound? Mixed? How turn a whole number to a
Fraction? What a Complex one? What does a fraction denote? What
its value equal to ? What meant by Common Measure? What by Com-
mon Multiple? Give the Rules of Problem first. Of Problem second.
What is Reduction of Vulgar Fractions? Repeat the last part of the rule
in Case first. What the Rule in Case second? What in Case third? In
Case fourth? Those in Case fifth? Tell the rule in Case sixth? What
the rule in Case eighth? What are the rule and notes in Addition of Vul-
gar Fractions?
What in Subtraction? What in Multiplication? What
in Division? How do you proceed in the Rule of Three in Vulgar Frac-
tions? How in the Rule of Three in Decimals? How in the Double Rule
of Three in Vulgar Fractions? Tell the Table of Ratios in Simple Inter-
est by Decimals. What is Ratie? How do you find the Interest? What
the Rule in Case second? In Case third? Case fourth? How find In-
terest for Days? How calculated on Cash Accounts where partial Pay
ments are made? How find Compound Interest by Decimals? How find
Amount of an Annuity at Compound Interest? How its Present Worth at
ditto? What is Involution? What the first Power? What the second
The third? The fourth? What Evolution or Extraction of Roots?
What the Root? Can the Root of any number be found? How approx-
imate towards it? What are Roots called? What is a Square? What
extracting the Square Root? What the Rules? What when a Vulgar
Fraction? What the Rules in its Application and Use? What a Cube?
What is, to extract the Cube Root? What the Rule? What the note,

proposition, and rule in its Application and Use? What the Rules to ex-
tract Roots generally? What the Note? When are Numbers in Arith-
metical Progression? What form increasing? What decreasing? How
named? What TERMS given? What found? What the whole number
called? What Problem first and rule? Second and rule? Third and
rule? When are numbers in Geometrical Progression? What called
ratio? What Problem first and rule? What Problem second, Case first
and rules? What Case second, rules and note? What does Alligation
teach? How distinguished? What Alligation Medial? What the rule?
What Alternate? What the rules? What Position? How many kinds?
What taught by Single Position? What the rule? What by Double
Position? What the rules and note? What Permutation? What Com-
- bination? What Problem first and rule? Problem second and rule?
Problem third and rule? How are Grindstones sold? How are their
Contents found? What is Superficial Measure? How made up? What
is measured by it? What Case first and rule? Case second and rule?
What the Note? What a Triangle? How its Surface measured? How
the Superficies of Joists and Planks found? How measure irregular Sur-
faces? What a Circle? How find circumference if diameter be given?
How diameter if circumference be given? How find Area? How, if cir-
cumference alone be given? How find diameter by the area? How cir-
cumference by the area? What is a Sector? How measured? By Rule
second? What a Segment of a Circle? How find its area? How measure
a regular Polygon? How describe an ellipse or oval? How find its area?
What is a Sphere or Globe? How find its area? How are solids measur-
ed? What is a Cube? How measured? What Case second and rule?
What noted? What Case third and rule? What a Cylinder? How
measured? What Case fifth and rule? Case sixth and rule? Case
seventh and rule? What is a Cone or Pyramid; How its solidity found?
What the Note? What a Frustum of a Cone? How find its solidity, if a
square pyramid? How, if a triangular pyramid? How, if a circular
pyramid, or cone? What is a Globe? How find its solid content? What
a Frustum of a sphere How find its solid content? What is Guaging?
What its rule and notes? How use the Sliding rule? How guage round
tubs? How a square vessel? What the Note? How find a ship's
Tonnage? What the Note? What Section fifth and rule? Section sixth
and rule? How find the solidity of Wood and Bark? How find the Cords
in a pile of either? What the principal rules in Assessing Taxes? What
the general Rules in common Book-keeping.


The sign of equality and is pronounced, equal to; The sign of Addition, and is pronounced, added to; -The sign of Subtraction, and is pronounced, subtracted by.


12+7=19, twelve added to seven will be equal to


23-8=15, twenty-three subtracted by eight, equal fif


>The sign of Multiplication, and is pronounced, multiplied into ;

The sign of Division, and is pronounced, divided by.


8x7-56, eight multiplied into seven equal fifty-six. 36-49, thirty-six divided by four, equal nine. Division is also implied by the signs 3)6(2 and 9=2, six divided by three, equal two.

The sign of Proportion, and is pronounced, is ta, so is, to.


6:9: 8:12, as 6 is to 9 so is 8 to 12.


✔or signifies the Square Root: thus 81 is read, the

square root of 81; or 81 is read 81 in the square root.

or denotes the Cube Root, &c. 3 squared, or to be multiplied, by itself.

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3 means that 3 is

48 shows that 48 must

19+3x9-198 means that 19 added to 3, and the sum multiplied by 9, equal 198.



3 shows that 12 less the product of 2 multiplied by 3, and divided by 2, equal 3.



ARITHMETIC is the art and science of numbers, and has for its operation four fundamental rules, viz. Addition, Subtraction, Multiplication, and Division. To understand these, it is necessary to have a perfect knowledge of our method of Numeration or Notation.


TEACHES to express numbers by words or characters. When performed by means of characters or figures, ten are employed. Nine of these are of intrinsic value and are called digits, or significant figures, being written and named thus:

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The tenth figure, namely, 0, is called nought or cipher, and denotes a want of value wherever it is found.

Besides the simple value of the digits, as noted above, they have each a local one, which depends on the following principle.

In a combination of figures, reckoning from right to left, the figure in the first place represents its simple value; that in the second place ten times its simple value; that in the third place an hundred times its simple value; and so on; each figure acquiring anew a tenfold value for every higher place it occupies. Hence our system of arithmetic is called decimal.

The names of places are denominated according to their order. The first is the place of units; the second of tens; the third of hundreds; the fourth of thousands; the fifth of ten thousands; the sixth of hundred thousands; the seventh of millions; and so on. Thus in the number 8888888; 8 in the first place signifies only eight; 8 in the second place eight tens or eighty; 8 in the third place eight hundred; 8 in the fourth place eight thousand;

Read as follows:

8 in the fifth place eighty thousand; 8 in the sixth place eight hundred thousand; 8 in the seventh place eight millions. The whole number is read thus, eight millions, eight hundred and eighty-eight thousand, eight hundred and eighty-eight.

Though a cipher has no value of itself, yet it occupies a place; and when set on the right hand of other figures it increases their value in the same tenfold proportion : Thus in the number 8080; the ciphers in the first and third places denote, that, though no simple unit or hundreds are reckoned, yet the place of units and that of hundreds are to be kept up to assist in reckoning the tens and thousands. The above number (8080) is read eight thousand and eighty, which, without the two ciphers, would be read eighty-eight.

Large numbers are divided into periods and half periods, each half period consisting of three figures. The name of the first period is units; of the second millions; of the third billions; of the fourth trillions; and also, the first part of any period is so many units of it; and the latter part so many thousands of it.*


Trillions. Billions. Millions.

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