Origin of measures of length, of surface, of capacity and

of solidity,

Origin of weights, and the balance,

Denominations of coin,

Division of time,

Calculations in simple and compound numbers compared,

THE FRENCH MONEY, WEIGHTS AND MEASUres,

British do.

New York & Connecticut do.

Promiscuous Examples,

RULE OF THREE by Analysis,

FRACTIONS,-general principles,

Multiples a least com. mult.,

66

To reduce them to their lowest terms,

144

Prime numbers-measures-to determine the measures

of a number,

145

To find common measures-a greatest com. meas.

146

To find a com. denominator,

148

149

151

153

154

156

PRACTICE,

157

158

To find a least com. den.,

Addition,

Multiplication,-a Fraction by a whole number,

A whole number by a Fraction,

To multiply a Fraction by a Fraction,

Compound Fractions,

FELLOWSHIP,-simple and compound, by analysis and ratio, 159–61

To reduce whole numbers to Fractions of higher denominations, 167

fractions to do.

fractions to do. of lower denom.

fractions to whole nnmbers of lower denom.

DECIMALS,-general principles,

To reduce Vulgar Fractions to Decimals,

REPETENDS, OR CIRCULATING DECIMALS,

To reduce whole numbers to decimals of higher denominations, 176

To reduce decimals to whole numbers, of lower denom.,

DUODECIMALS.

REDUCTION OF CURRENCIES,

Table, showing comparative values of currencies,

FOREIGN COINS,

days, com. method,

days, accurately,

at any rate per cent.,

TIME, RATE, and INT. given, to find PRIN.,

PRINCIPAL, INT. and TIME given, to find RATE,

Concise rules for finding decimals of time,

PRIN. RATE, and INT. given, to find TIME,

AMT., RATE and TIME given, to find PRIN.

DISCOUNT,-by division, and by multiplication,

Mode of discounting in banks,

196

198

200

201

202

66

203

204

205

206

208

209

210

212

213

215

217

66

218

219

220

221

222

Contraction in finding int. by first obtaining it at 12 per cent., 223

To find interest on Sterling Money,

PROFIT AND LOSS, when allowance is made for TIME.

Rule of three, direct and inverse, explained,

ANNUITIES,-at simple interest,

at compound interest,

Table of multipliers for finding amounts,

present worths,

To find what annuity a given sum will buy,

Annuities in reversion,

PERMUTATION,-to find how many permutations can be

made of a given number of things,

276

277

278

279

66

• 280

To find how many, when a given number is taken at a time, 281

COMBINATION,

Combination lotteries,

OBSERVATIONS, &c.-ancient mode of calculating,

Roman ABACUS, and Chinese SWAN-PAN,

Essential and accidental properties of numbers,

Notions of PYTHAGORAS and his disciples,

MAGIC SQUARES,

MISCELLANEOUS EXAMPLES,

POSITION,- single and double, principles explained, and ex.

amples given of each,

66

285

ERRATA.

IN printing a work. on a scientific subject, entirely from manu.
script, perfect accuracy is not to be expected. Accordingly, some
errors will undoubtedly he discovered in this book; but it is believed
that the following are all, which will be likely to occasion perplexi-
ty. Page 60. Ans. ex. 10, for 328, read 378.-p. 71, in ex. 4, for 4
men read 1 man. p. 151, line 13, for 9 read 6-line 15, for multiplier,
read multiple.-p. 229, in ex. 5, insert at 5 per cent.---p. 236, ex. 2,
Ans. should be $446,985--p. 237. Ans. by Mass. method should
be $11,370.47175.-p. 244, ex. 43, for 65, read 650.
p. 249, line
6, for miles, read hours.-p. 250 1. 30, for 3, read 315.-p. 284, in
Ans. to ex. 1, for read.

31

In a small part of the impression, p. 82. 3d line in ex. 87, for 65
'read 75.-p. 261, line 3, for products, read quotients.-p. 283, line
38, for 3 read 7, three times.

In § LXXXIX, the rule called the Massachusetts rule is erroneously
stated to have been established by law in that state. It has been
made the rule of the state of New-York, by the decision of Chancel-
lor Kent. The statement was made on the authority of a number
of writers. For this correction, as well as for other important
suggestions, the author is indebted to Professor DEWEY, now Prin
cipal of the BERKSHIRE GYMNASIUM.

ARITHMETIC.

INTRODUCTORY EXERCISES.

NUMERATION.

§ I. ARTICLE I. Here is an anchor.

How many anchors do you see here?

Ans. ONE.

How many ropes are there, fastened to the anchor? Now there is a single anchor, and a single rope; and each of these single things, you tell me, is ONE. Then, what is a single thing called? Hence,

A SINGLE THING OF ANY KIND IS CALLED ONE THING, OR ONE. Also

A SINGLE THING OF ANY KIND IS OFTEN CALLED A UNIT, OR UNITY.

Thus, the anchor in the picture above is a unit.

If there were another anchor in the picture, how many anchors would there be? Ans. Two.

How many sheaves of wheat are there here?

11

How many hands have you? How many thumbs? How many eyes? How many feet?

do

When you see one thing, and one more together, what you call them? Hence,

ONE UNIT AND ONE MORE ARE CALLED TWO; that is, ONE and one ARE TWO.

If there were one more sheaf in the last picture, how many sheaves would there be? Ans. THREE.

How many pairs of snuffers are there here?

If you had another hand, how many hands would you have? If you had another eye, how many eyes would you have? When you see two things and one more together, what do you call them? Hence,

TWO UNITS AND ONE MORE ARE CALLED THREE; that is, TWO AND

ONE ARE THREE.

If there were another pair of snuffers in the last picture, how many would there be? Ans. FOUR. How many hammers are there here?

TTT T

If you count your hands and feet together, how many have you?

When you see three things and one more together, what do you call them? Hence,

THREE UNITS AND ONE MORE ARE CALLED FOUR; that is, THREE AND ONE ARE FOUR.

If there were another hammer in the last picture, how many would there be? Ans. FIVE.