A

COURSE

OF

MATHEMATICS,

&c.

GENERAL PRINCIPLES.

1. QUANTITY, or MAGNITUDE, is any thing that will admit of increase or decrease; or that is capable of any sort of calculation or mensuration; such as numbers, lines, space, time, motion, weight, &c.

2. MATHEMATICs is the science which treats of all kinds of quantity whatever, that can be numbered or measured.-That part which treats of numbering is called Arithmetic; and that which concerns measuring, or figured extension, is called Geometry.-Not only these two, but Algebra and Fluxions, which are conversant about multitude, magnitude, form, and motion, being the foundation of all the other parts, are called Pure or Abstract Mathematics; because they serve to investigate and demonstrate the properties of abstract numbers and magnitudes of all sorts. And when these two parts are applied to particular or practical subjects, they constitute the branches called Mixed Mathematics.— Mathematics is also distinguished into Speculative and Practical: viz. Speculative, when it is concerned in discovering properties and relations; and Practical, when applied to practice and real use concerning physical objects.

The peculiar topics of investigation in the four principal departments of pure mathematics may be indicated by four words: viz. arithmetic by number,geometry by form,—algebra by generality,—fluxions by motion.

3. In mathematics are several general terms or principles; such as, Definitions, Axioms, Propositions, Theorems, Problems, Lemmas, Corollaries, Scholia, &c.

4. 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.

5. 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.

6. 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.

7. 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

answers.

8. 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.

9. A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it.

10. A Corollary, or Consectary, is a consequence drawn immediately from some proposition or other premises.

11. A Scholium is a remark or observation made upon some foregoing proposition or premises.

12. 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.

13. A Postulate, or Petition, is something required to be done, which is so easy and evident that no person will hesitate to allow it.

14. 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.

15. 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.

16. A Direct, Positive, or Affirmative Demonstration, is that which concludes with the direct and certain proof of the proposition in hand.

17. 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.

18. 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.

19. 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.

20. 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 found out by the Analytic method, it may be demonstrated by a process in the contrary order, by Synthesis: and in the solution of geometrical propositions, it is very instructive to carry through both the analysis and the synthesis.

ARITHMETIC.

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, which depends on the place they stand in when joined together; as in the following table:

Here, any figure in the first place, reckoning from right to left, denotes only its own simple value; but that in the second place, denotes ten times its simple value; and that in the third place, a hundred times its simple value; and so on the value of any figure, in each successive place, being always ten times its former value.

Thus, in the number 1796, the 6 in the first place denotes only six units, or simply six; 9 in the second place signifies nine tens, or ninety; 7 in the third place, seven hundred; and the 1 in the fourth place, one thousand so that the whole number is read thus, one thousand seven hundred and ninety-six.

As to the cipher, O, though it signify nothing of itself, yet being joined on the right-hand side to other figures, it increases their value in the same ten-fold proportion: thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hundred; and so on.

For the more easily reading of large numbers, they are divided into periods and half-periods, each half-period consisting of three figures; the name of the first period being units; of the second, millions; of the third, millions of millions, or bi-millions, contracted to billions; of the fourth, millions of millions of millions, or tri-millions, contracted to trillions, and so on. Also the first part of any period is so many units of it, and the latter part so many thousands.

The following Table contains a summary of the whole doctrine.

NUMERATION is the reading of any number in words that is proposed or set down in figures; which will be easily done by help of the following rule, deduced from the foregoing tables and observations-viz.

Divide the figures in the proposed number, as in the summary above, into periods and half-periods; then begin at the left-hand side, and read the figures with the names set to them in the two foregoing tables.

NOTATION is the setting down in figures any number proposed in words; which is done by setting down the figures instead of the words or names belonging to them in the summary above; supplying the vacant places with ciphers where any words do not occur.

EXAMPLES.

Set down in figures the following numbers :

Fifty-seven.

Two hundred and eighty-six.

Nine thousand two hundred and ten.

Twenty-seven thousand five hundred and ninety-four.

Six hundred and forty thousand, four hundred and eighty-one.

Three millions, two hundred and sixty thousand, and one hundred and six. Four hundred and eight millions, two hundred and fifty-five thousand, one hundred and ninety-two.

Twenty-seven thousand and eight millions, ninety-six thousand, two hundred and four.

Two hundred thousand and five hundred and fifty millions, one hundred and ten thousand, and sixteen.

Twenty-one billions, eight hundred and ten millions, sixty-four thousand, one hundred and fifty.

OF THE ROMAN NOTATION.

The Romans, like several other nations, expressed their numbers by certain letters of the alphabet. The Romans used only seven numeral letters, being the seven following capitals; viz. 1 for one; v for five; x for ten; L for fifty; c for a hundred; D for five hundred; м for a thousand. The other numbers they expressed by various repetitions and combinations of these, after the following manner: