« PreviousContinue »
The alterations made in this edition have caused the paging of the work to be altered ; and as
by detached coefficients 127 129
Tables of weights and measures 18 18 Greatest common measure...
Comparison of French and English... 25 23 Least common multiple
41 39 Involution and evolution
42 40 Powers and roots of mono-
52 49 Reduction of monomial surds... 150 147
Decimals, or decimal fractions... 54 49
55 50 Arithmetical progression.. 159 155
55 51 Geometrical progression 162 162
57 53 Circulating decimals
59 54 Geometrical proportion
Rule of three
62 57 Harmonical progression 167 165
63 58 Equations,
Involution and Evolution
65 60 Simple equations.
Extraction of the square root...
-, simultaneous 174 183
cube root 69 64
problems ... 181 191
73 69 Quadratic equations, problems 185 195
Progressions and ratios.....
Geometrical progression 79 82 Quadratic equations, problems 193 206
82 85 Cubic and biquadratic equations.... 198 209
83 86 Cardan's solution of the cubic 198 209
84 88 Simpson's, of the biquadratic... 200 212
85 89 Solution of equations by trial
89 92 The numerical Solution of Equations 206 219
90 93 Definitions and notation 206 219
93 96 Calculation of an expression.... 208 223
93 97 To increase or diminish the roots 209 224
96 100 To change the signs of the roots 212 227
To multiply or divide the roots 213 227
To form an equation from given
Expression for the transformed
Definitions, notation, and
Equations with equal roots...... 216 228
102 106 Imaginary roots are in pairs..... 217 228
112 115 Irrational roots are in pairs 218
115 119 Harriot's rule of signs ...... 219 221
117 121 Effect of successive substitutions 221 222
by detached coef-
Limits of the roots ......
120 123 Theorem on approximation ..... 223 233
Construction of plane problems...... 372 354
232 233 A line nearly equal to the circle 400
Recapitulation of processes...... 234 To measure an angle by the com- 402
240 158 To find the diameter of a sphere
triangular pile.. 240 158 Field problems........
240 159 Application of algebra to geometry.. 413 377
rectangular pile....... 241 161
Method of working .. 245 172
245 173 Definitions and notation.....
Approximation to the
Relations of the functions of an arc 423 388
247 242 Particular relations of arcs... 427
Definitions and properties... 248 244 Trigonometrical tables, their con-
Computations of logarithms 251 247 Expansion of sin x, cos x....... 435 386
252 248 Euler's and Demoivre's theorems ... 437
Description of tables. 254 248 Change of the radius
256 248 Inverse functions.....
Logarithmic operations. 257 251
264 258 Right-angled triangles....... 445 399
267 261 Oblique-angled triangles..
Method of finite differences..... 273 267
273 268 App. to heights and distances......
General term of, with order
General term of a series..... 277 277
To find the several orders... 278 270
To continue a given series .. 279 279
To transform powers into
Areas of rectilineal figures ....... 471 415
279 Length and area of the circle......... 475 420
Integration of general term 281 271 Problems on plane surfaces
To find the sum of a series., 283 273 Surfaces and volumes of solid figures
285 279 with plane boundaries.
Interpolation of series 287 281 The cone, cylinder, and sphere 487 427
Definitions in plane geometry.
290 286 Slaters' and tilers'
296 292 Plasterers'
Theorem independent of ratio...
296 292 Painters'
Ratios and proportions
322 317 Glaziers'
322 317 Pavers'
Principle employed (note)
Theorems depending on ratio... 324 320 Timber measuring
Exercises in plane geometry...
343 331 NOTES
Definitions of solid geometry..... 347 334 TABLES OF SQUARES, CUBES, AND
Theorems on lines and angles 349 336 ROOTS
Theorems on solid angles...... 360 349
on volumes of solids 364 343
1. Quantity, or MAGNITUDE, is that which admits of increase or decrease. Those kinds of magnitude only which are capable of estimation in comparison with some unit of the same kind, are the proper subjects of mathematical study.
2. Arithmetic is conversant with numbers only in their abstract state. Algebra contemplates the subjects of arithmetic in a more general form; and generally (among other objects) furnishes the rules for the more complex arithmetical operations. Geometry treats of space, as of the forms, magnitudes, and positions of figures. The differential and integral calculus, the calculus of functions, &c. are also branches of Algebra, but of which no definite idea could be conveyed till the student's progress is considerably extended.
3. The sciences of Arithmetic and Geometry are styled the Pure Mathematics : whilst all applications of them to physical, civil, or social inquiries, (as Mechanics, Astronomy, Optics, Life Insurance, Population, &c.) constitute what is termed the Mixed Mathematics.
4. In mathematics are several general terms or principles; such as, Definitions, Axioms, Propositions, Theorems, Problems, Leminas, Corollaries, Scholia, &c.
5. A Definition is the explication of any term or word in a science ; showing the sense and meaning in which the term is employed. —Every Definition ought to be clear, and expressed in words that are common, and perfectly well understood.
6. A Mathematical Proposition refers either to something proposed to be demonstrated, or to something required to be done ; and is accordingly either a Theorem or a Problem. 7. A Theorem is a demonstrative Proposition; in which some property is
1 asserted, and the truth of it required to be proved. Thus, when it is said that,
The sum of the three angles of a plane triangle is equal to two right angles, that is a Theorem, the truth of which is demonstrated by Geometry. A set or collection of such Theorems constitutes a Theory.
8. A Problem is a proposition or a question requiring something to be done ; either to investigate some truth or property, or to perform some operation. As, to find out the quantity or sum of all the three angles of any triangle, or to draw one line perpendicular to another. A Limited Problem is that which has but. one answer or solution. An Unlimited Problem is that which has innumerable
And a Determinate Problem is that which has a certain number of answers.
9. Solution of a Problem, is the resolution or answer given to it. A Numerical or Numeral Solution is the answer given in numbers. A Geometrical Solution is the answer given by the principles of Geometry. And a Mechanical Solution is one which is gained by trials.
10. A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it.
11. A Corollary, or Consectary, is a consequence drawn iinmediately from some proposition or other premises.
12. A Scholium is a remark or observation made upon some foregoing proposition or premises.
13. An Axiom, or Maxim, is a self-evident proposition ; requiring no formal demonstration to prove its truth; but received and assented to as soon as mentioned. Such as, The whole of any thing is greater than a part of it; or, The whole is equal to all its parts taken together; or, Two quantities that are each of them equal to a third quantity, are equal to each other.
14. A Postulate, or Petition, is something required to be done, which is so easy and evident that no person will hesitate to allow it.
15. An Hypothesis is a supposition assumed to be true, in order to argue from, or to found upon it the reasoning and demonstration of some proposition.
16. Demonstration is the collecting the several arguments and proofs, and laying them together in proper order, to establish the truth of the proposition under consideration.
17. A Direct, Positive, or Afirmative Demonstration, is that which concludes with the direct and certain proof of the proposition in hand.
18. An Indirect, or Negative Demonstration, is that which shows a proposition to be true, by proving that some absurdity would necessarily follow if the proposition advanced were false. This is also sometimes called Reductio ad absurdum ; because it shows the absurdity and falsehood of all suppositions contrary to that contained in the proposition.
19. Method is the art of disposing a train of arguments in a proper order, to investigate either the truth or falsity of a proposition, or to demonstrate it to others when it has been found out. This is either Analytical or Synthetical.
20. Analysis, or the Analytic Method, is the art or mode of finding out the truth of a proposition, by first supposing the thing to be done, and then reasoning back, step by step, till we arrive at some known truth. This is also called the Method of Invention, or Resolution ; and is that which is commonly used in Algebra.
21. Synthesis, or the Synthetic Method, is the searching out truth, by first laying down some simple and easy principles, and then pursuing the consequences flowing from them till we arrive at the conclusion. This is also called the Method of Composition ; and is the reverse of the Analytic method, as this proceeds from known principles to an unknown conclusion'; while the other
goes in a retrograde order, from the thing sought, considered as if it were true, to some known principle or fact. Therefore, when any truth has been discovered by the Analytic method, it may be demonstrated by reversing the process or by Synthesis: and in the solution of geometrical propositions, it is very instructive to carry through both the analysis and the synthesis.
Arithmetic may be viewed as a subject of speculation, in which light it is a science; or as a method of practice, in which light it is an art.
As a science, its objects are the properties and relations of numbers under any assigned hypothesis respecting their mutual relations or methods of comparison and combination.
As an art, it proposes to discover and put into a convenient form, compendious methods of obtaining those results which flow from any given methods of combining given numbers; but which results could, in the absence of these compendious methods, only be ascertained by counting the numbers themselves into one single and continuous series.
When it treats of whole numbers, it is called Vulgar, or Common Arithmetic; but when of broken numbers, or parts of numbers, it is called Fractions.
Unity, or an Unit, is that by which every thing is regarded as one ; being the beginning of number; as, one man, one ball, one gun.
Number is either simply one, or a compound of several units; as, one man, three men, ten men.
An Integer, or Whole Number, is some certain precise quantity of units; as, one, three, ten. These are so called as distinguished from Fractions, which are broken numbers, or parts of numbers; as, one-half, two-thirds, or threefourths.
A Prime Number is one which has no other divisor than unity; as, 2, 3, 5, 7, 17,.19, &c. A Composite Number is one which is the product of two or more numbers; as, 4, 6, 8, 9, 28, 112, &c.
A Factor of a composite number, or simply a Factor, is any one of the numbers which enters into the composition of that composite number.
NOTATION AND NUMERATION.
These rules teach how to denote or express any proposed number, either by words or characters: or to read and write down any sum or number.
The Numbers in Arithmetic are expressed by the following ten digits, or Arabic numeral figures, which were introduced into Europe by the Moors, about eight or nine hundred years since ; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cipher, or nothing. These characters or figures
8 were formerly all called by the general name of Ciphers ; whence it came to pass that the art of Arithmetic was then often called Ciphering. The first nine are called Significant Figures, as distinguished from the cipher, which is of itself quite insignificant as a number.
Besides this value of those figures, they have also another, or local value,