the columns headed proportional parts, in the same line with the number already found, and under the given fourth figure. The natural numbers thus obtained will generally be exact to a unit in the last place, except towards the end of the

table; the number of decimal places will depend upon the characteristic of the logarithm. This table is chiefly intended to save time when many logarithms in succession are to be looked out.

Variation of quantities in direct and inverse
proportion. 137. Algebraical symbol of
variation-variation of quantities bearing
other relations. 138. Rule of single posi-
tion: example. 139. Rule of double posi.
tion: example.

Of Arithmetical Progression, p. 39.

Art. 140. Numerical examples of arith-
metical series or progressions-common dif-
ference. 141. Algebraical form of these
series, with a positive and negative com-
mon difference. 142. Method of deriving
any term of the series from the first term
and common difference. 143. General me-
thod of finding any number of arithmetical
means between two numbers. 144. Expres-
sion for the sum of any number of terms of
an arithmetic series-rule: examples. 145.
The sum of the first n odd numbers is equal
to the square of n. 146. Numerical pro-
perties of numbers, the digits of which are
divisible into periods in arithmetical pro-
gression.

Of Geometrical Progression, p. 41.

Art. 147. Numerical example of ascend-
ing and descending geometrical progressions
-common ratio. 148. Algebraical form of
these series. 149. General method of find-
ing any number of geometrical means be-
tween two numbers. 150. Numerical ex-
ample of the summation of a geometrical
series. 151. General expression for the sum
of any number of terms rule: example.
152. Sum of a descending progression going
on to infinity-illustration of this. 153.
General rule for summing descending geo-
metrical series going on to infinity: exam-
ples,

Of Decimal Fractions, p. 44.

Art. 154. Explanation of notation. 155.
Decimal fraction-vulgar fraction-decimal
point-rule for reducing vulgar fractions to
decimals. 156. Repeating or circulating de-
cimals-how they arise. 157. Of fractions
which can be reduced to terminating deci-
mals. 158. Of fractions leading by reduc-

operations of multiplying and dividing deci-
mals, when perfect accuracy is not required.
168. Rule for the reduction of a decimal, of
a lower denomination, to one of a higher:
example. 169. Rule for the reduction of a
decimal, of a higher denomination, to one
of a lower: example. 170. Repeating de-
cimals less numerous in a scale of notation,
of which 12 is the base.

Of the Square and Cube Roots, and of
Surds, p. 50.

Art. 171. The square root-a square
number. 172. The square root of a num-
ber, not a square, cannot be expressed by
means of any fractional part of a whole
number, or is incommensurable with unity.
173. Cube root-cube number-irrational
numbers, or surds. 174. Table of squares,
and cubes of the first nine numbers. 175.
Method of finding the square root of num-
bers. 176. Example of finding the square
root-explanation of the principle of the
process. 177. Algebraical analysis of the
process-general rule for finding the square
root of numbers. 178. Addition to the rule

when the number is not a perfect square.
179. Rule for finding the square root of de-
cimals. 180. Method of extracting the cube
root of numbers. 181. Algebraical investi-
gation of the extraction of the cube root:
examples. 182. Remarks upon the process.
183. General rule for extracting the cube
root. 184. Addition to the rule, when part
of the number proposed is decimal. 185.
More simple process of extracting the square
and cube root, where a perfectly accurate
result is not required. 186. Fourth, fifth,
and other roots usually found by means of
a table of logarithms. 187. In extracting
roots we may approximate as nearly as we
please to the true result. 188. What is
meant by the product of two or more
surds. 189. The product of any num.
ber of surd factors is the same in what-
ever order we arrange these factors.
190.
/abc = "/ a x /bx/c. 191.
What is meant by the division of one surd

m

P

mn

m

max

198. "/
/a
199.
Va

=

an
am

x / a = mn/

a

200. The double

sign is prefixed to an even root of any num-
ber. 201. Imaginary or impossible quanti-
ties: possible or real quantities. 202. Ge-

K 2

neral manner of representing these quanti
ties: results deduced from it. 203. Every
quantity has two square roots, three cube,
four fourth, and generally n nth roots.

Of Quadratic Equations, p. 60.
Art. 204. Solution of a question pro-
ducing an equation involving the square of
the unknown quantity. 205. General form
of a quadratic equation. 206. Solution of
a quadratic equation in its general form.
207. Roots of a quadratic equation: their
number. 208. How to form a quadratic
equation whose roots shall be any given
numbers. 209. General rule for the solu-
tion of quadratic equations. 210. Problem
producing a quadratic equation : double an-
swer. 211. Impossible result. 212. Nega.
tive result. 213, 214. Problems producing
quadratic equations involving two unknown
quantities: their solution: when the un-
known quantities are involved symmetri-
cally, the values of both are the same. 215.
Solution of quadratic equations involving
two or three unknown quantities, when the
sum of the indices of the unknown quanti-
ties is the same in every term.

Of Negative Exponents, p. 65.
Art. 216. When the exponent of any power
of a quantity is an integer greater than
unity, the division of that power by the
quantity itself is the same thing as subtract-
ing unity from the exponent: when the ex-
ponent is not greater than unity, this is a
matter of notation. 217. Result deduced
from this method of representing division;

P

9

m

m

n

Of Fractional Exponents, p. 67.
Art. 223. Method of representing the roots
of a quantity by fractional exponents;
"I am is represented by a", whether m be
or be not a multiple of n. 224. Or by
a, if be any fraction equivalent to ".
The following results are true, whether m
and n be integral or fractional. 225. am x
am+n. 226. am÷an = am-n. 227.
(am)n = amn. 228. /am = a". 229. Re-
marks on the above rules, and on the nota-
tion by negative and fractional exponents.
230. General rules, with examples, for the
multiplication, division, involution, and evo-
lution of powers and roots of the same

an

=

m

quantity. 231. Method of finding whether

a certain root of any given number is a
surd or not. 232. Operations and results
much simplified by separating surd numbers
into their prime factors. 233. A fractional
exponent may be represented by a decimal:
simplification when the exponent is a re-
peating decimal.

Of Logarithmic Arithmetic, p. 70.

Art. 234. Logarithm - base - common
logarithms. 235. Derivation of the word
logarithm, and explanation of its derivative
signification. 236. Few logarithms can be
expressed by terminating decimals, but are
sufficiently accurate when carried to six or
seven decimal places. 237. Characteristic
-rule for finding the characteristic. 238.
The logarithm of a number gives imme-
diately the logarithm of that number mul-
tiplied by any power of the base. 239. Ta-
bles of logarithms, their extent. 240. Me-
thod of finding the logarithm of a number
of six digits, from a table containing the
logarithms of numbers up to 100,000 only.
241. Method of finding the logarithm of a
number consisting of seven digits, from the
same table-how far accurate. 242. Gene
ral rule for the above-proportional parts.
243. Rule for finding the logarithm of any
number greater than unity, and consisting
of six or seven digits. 244. Method of
finding the logarithms of numbers less than
unity-negative logarithms. 245. Rule for
determining the characteristic of the loga-
rithm of a number less than unity. 246.
Rule for the logarithms of numbers less
than unity-example. 247. Of the loga-
rithms of negative numbers. 248. Rule for
finding the number to which a given loga-
rithm belongs. The operations of multipli-
cation, 249; division, 250; the solution
of questions in proportion, 251; the ope
rations of involution, 252; and of evolu-
tion, 253; may be performed by means of
a table of logarithms. 254. Exponential
equation-how solved by a table of loga-
rithms-to find a logarithm to any base
from a table of logarithms to a given base.

Of Permutations and Combinations, p. 77.
Art. 255. Combinations. 256. Permuta-

tions. 257. Expression for the number of
permutations of m different things taken n
at a time. 258. If n be equal to m, the
number of permutations is equal to the
product of all the natural numbers from 1
up to m. 259. Remarks upon the above
expression. 260. Expression for the num-
ber of combinations of 8 things taken 4 at
a time. 261. General expression for the
number of combinations of m things taken
nat a time. 262. This expression always
produces an integer. 263. The number of
combinations of 10 things taken 7 at a time,
or taken 3 at a time, is the same. 265. Ge
nerally, the number of, combinations of m

things taken nat a time, is the same as the

Of the Binomial Theorem, p. 83.

Art. 271. The definition of a coefficient extended. 272. A binomial expression-a polynomial-the binomial theorem. 273, 274. The expression for (x+a) m can be directly derived from that for the product of m binomial factors, when m is any positive and integral number. 275. This expression deduced. 276. Observations on this expression-development-expansion - general form of the mth term-and of the middle term, or terms. 277. The expression for (l + y)m deduced. 278-281. The above proof only applicable when mis a positive integer-the same development is true whether m be integer or fractional, positive or negative - demonstration - (note, extending the demonstration to the case where m is an irrational quantity). 282. Remarks on the nature of this demonstration. 283. Development of (x-a)m. 284. Application of the binomial development to the ex

ral expression for (x+a)-it goes on for ever-observation on the propriety of considering the binomial development as always going on for ever. 288. Expression for (1+1)m or 2m-its connexion with combinations. 289. Application of the binominal development to the extraction of the square root of numbers, to any degree of 290. Application to the exaccuracy. traction of other roots. 291. Investigation of the accuracy of the approximation. 292. The cube root of 31 found by this process. 293. The development of any power of a polynomial expression.

Of Interest, p. 94.

amples. 303. Present value, and discount at compound interest: examples. 304. Compound interest considered algebraically. 305. Compound interest considered as accruing at every instant-generality of the algebraical expressions: examples. 306. Use of logarithms for the calculation of compound interest.

Of Annuities, p. 101.

Art. 307. Amount of an annuity at simple interest-its present value. 308. Different theory sometimes maintained. 309. Amount of an annuity at compound interest-its present value.

Indeterminate Equations, p. 103.

Art. 310. Introductory remarks upon indeterminate equations - equations of the first degree between two unknown quantities. 311. Solution of the equation 5x+7y=81-an indeterminate. 312. Remarks upon the process-questions producing indeterminate equations. 313. Investigation of the relations of the several values of the two unknown quantities. 314. Solution of an equation of the first degree between three or more unknown quantities. 315. Solution of two independent equations of the first degree involving three unknown quantities. 316. General considerations upon a system of equations involving a greater number of unknown quantities than the number of equations. 317. Indeterminate equations above the first degree.

Of Continued Fractions, p. 110.

Art. 318. A common fraction reduced to

the form of a continued fraction-successive approximations their utility: example. 319. Other applications of continued fractions. 320. Continued fractions considered generally. 321. Converging fractions or convergents-manner of deriving each approximation from the two preceding ones. 322. Other relations of the succes

sive approximations. 323. Application of the solution of indeterminate equationsnumerators and denominators of converging fractions prime to each other. 324. The approximations are alternately too great and too small. 325. Method of estimating the degree of their accuracy.

derations upon the method of estimating it. 296. Simple and compound interest-their distinction. 297. Rule for calculating simple interest: example. 298. Discount. 299. Algebraical investigation of the subject of simple interest. 300. Discount considered algebraically. 301. Remarks on discountEquation of payments-rule. 302. Compound interest rule for calculating: ex

of Logarithmic Tables, p. 115.

Art. 326. Most logarithms must be expressed in series. 327. Introductory theorem demonstrated equality of the coefficients of the homologous terms, of two equal series, involving powers of the same variable. 328, 329. a, expressed in a series ascending by integral positive powers of x.