has but one answer or solution. An Unlimited Problem is that which has inAnd a Determinate Problem is that which has a certain

numerable answers.

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 an 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 on some foregoing proposition or premises.

12. An Axiom, or Maxim, is a self-evident proposition; requiring no formal demonstration to prove the truth of it; but is 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 show 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.—This kind of Demonstration is most satisfactory to the mind; for which reason it is called sometimes an Ostensive Demonstration.

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 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. And therefore, when any truth has been found cut by the Analytic method, it may be demonstrated by a process in the contrary order, by Synthesis.

ARITHMETIC.

ARITHMETIC is the art or science of numbering; being that branch of Mathematics which treats of the nature and properties of numbers.-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 called one; being the beginning of number. As one man, one ball, one gun.

Number is either simply one, or a compound of several units.

three men, ten men.

As one man,

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 three-fourths.

NOTATION AND NUMERATION.

NOTATION, or NUMERATION, teaches 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. Also the first nine are called Significant Figures, as distinguished from the cipher, which is quite insignificant of itself.

Beside this value of those figures, they have also another, which depends upon 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 0, it stands for nothing of itself, but being joined on the right hand side to other figures, it increases their value in the same tenfold proportion : thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hun dred; 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:

Periods. Quadrill. Trillions; Billions; Millions; Units.

Half-per. thun. th. un. th. un.

th. un.

th. un.

Figures. 123,456; 789,098; 765,432; 101,234; 567,890.

NUMERATION is the reading of any number in words that is proposed or set down in figures, which will be easily done by the help of the following rule, deduced from the foregoing tablets 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, 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. I for one; V for five; X for ten; L for fifty; C for a hundred; D for five hundred; M for a thousand. The other numbers they expressed by various repetitions and combinations of these, after the following

EXPLANATION OF CERTAIN CHARACTERS.

THERE are various characters or marks used in Arithmetic and Algebra, to denote several of the operations and propositions; the chief of which are as follow:

signifies plus, or addition.

......

.........

.........

minus, or subtraction.

multiplication.

division.

proportion.

equality.

square root.

cube root, &c.

7 x 3, denotes that 7 is to be multiplied by 3.
84, denotes that 8 is to be divided by 4.
2:34:6, shows that 2 is to 3 as 4 is to 6.

6410, shows that the sum of 6 and 4 is equal to 10.
√3, or 32, denotes the square root of the number 3.

ADDITION is the collecting or putting of several numbers together, in order to find their sum, or the total amount of the whole. This is done as follows:

Set or place the numbers under each other, so that each figure may stand exactly under the figures of the same value; that is, units under units, tens under tens, hundreds under hundreds, &c; and draw a line under the lowest number, to separate the given numbers from their sum, when it is found. Then add up the figures in the column or row of units, and find how many tens are contained in their sum.-Set down exactly below, what remains more than those tens, or if nothing remains, a cipher, and carry as many ones to the next row as there are tens. Next add up the second row, together with the number carried, in the same manner as the first. And thus proceed till the whole is finished, setting down the total amount of the last row.

TO PROVE ADDITION.

First Method.-Begin at the top, and add together all the rows of numbers downwards, in the same manner as they were before added upwards; then if the two sums agree, it may be presumed the work is right.—This method of proof is only doing the same work twice over, a little varied.

Second Method. Draw a line below the uppermost number, and suppose it cut off. Then add all the rest of the numbers together in the usual way, and