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THE alterations made in this edition have caused the paging of the work to be altered; and as
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 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 answers. And a Determinate Problem is that which has a certain number of
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 immediately 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 Affirmative 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, O cipher, or nothing. These characters or figures 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,