tained in the Dividend, is called the Divifor (Divifor, Lat.)

105. The Number fought (viz. that fhewing how often the Divifor is contained in the Dividend) is called the Quotient, (Quoties, Lat.)

106. As the Divifor is not always contained exactly a certain Number of Times in the Dividend, there will, fometimes, after the Divifor has been taken out of the Dividend as often as poffible, be a Number remaining; which we therefore call the Remainder (from Remain, Eng. from Remaneo, Lat.)

F07. A Sub-multiple, (from Sub and Multiplex, Lat.) or Aliquot (Aliquot, Lat.) Part, is a Number greater than an Unit, and which is contained in another Number a certain Number of Times. Thus, 2 is a Sub-multiple of 6, for 2 is contained in 6, exactly 3 Times. Or, in other Words, a Sub-multiple is a Number greater than an Unit, that will measure another Number, exactly, without a Remainder.

108. An Axiom. If equal Things be divided by equal Things, their Quotients will be equal.

109. To divide one Number by another, when both the Divifor and Dividend are fingle Digits; or the Divifor a fingle Digit, and the Dividend not confifting of more than two Figures; we may fubtract the Divifor as often as poffible out of the Dividend: But if the Learner be perfect in his Pythagorean, or Multiplication-Table, he will be able readily, by Memory, to tell how often the Divifor is contained in the Dividend; that is, he will be able to take a Digit, by which multiplying the Divifor, the Product will be (either) equal to, or the next lefs than the Dividend; and the Digit, fo taken, will be the integral Part of the Quotient; and the Remainder (which must be always lefs than the Divifor, because, if at any Time we fhould bring out a Remainder greater than, or equal to, the Divifor, we can take the Divifor out of the Remainder, and therefore have not taken it out as many Times as poffible) be

ing put over the Divifor as a Fraction, (fee Art. 6.) compleats the Quotient.

Examples. Divide 10 by 5; here 5 is contained in 10 two Times, for 5x2=10; ·.∙ 10÷5=2. Also, 13523, for 5x2=10, and 13-10-3; . the integral Part of the Quotient is 2, and the fractional Part is 3 out of 5, or 3 Fifths; fo that 5 is contained in 13 two Times, and 3 Parts of 5, of a Time

more.

110. To divide any Number by another. First, fee how many Times the Divifor is contained in as many Places of the Left-hand of the Dividend, as the Divifor confifts of; (but, if you cannot go once, then take in one Figure more, and try how often the Divifor is contained in that Number, which cannot be more than nine Times;) and place the Figure expreffing how many Times you can go, in the Quotient; (but remember, first, to fee that the Dividual Minus the Product of the Figure which you go and Divifor be less than the Divifor ;) then multiply the Divifor by that Quotient Figure, and fubtract the Product from the before-mentioned Number, and to the Remainder bring down the next Figure of the Dividend (proceeding to the Right) and try how many Times the Divifor is contained in this Number, and multiply, fubtract, &c. as before, 'till all the Figures of the Dividend are taken down. We must here observe, that, when any Remainder with one Figure of the Dividend annexed, as juft now mentioned, is less than the Divifor, then, as we cannot take the Divifor out of it, we must put o in the Quotient, and take down another Figure of the Dividend, and then try how many Times we can go. But many Times the greatest Difficulty is in finding how many Times to go, which is to be found only by Trials; however, the following Obfervations may be of fome Use in nearly determining the Times, viz. 1. By Art. 132. we can never go more than 9 Times. 2. If the Number, out of which the Divifor is to be taken, is of the fame Number of Places, fee how

many

many Times the first Figure of the Divifor is contained in the first (Left-hand) Figure of that Number; but, if that Number confifts of one Place more than the Divifor, then fee how often the first Figure of the Divifor is contained in the two first Figures of that Number, (called by fome Authors the Dividual (Dividuus, Lat.) or Partial (Partial, Fr.) Dividend;) and the Number of Times we can take the Divifor out of the aforefaid Number, may be equal to, but cannot exceed the Times thus found. Note, It will be convenient, in order to prevent Mistakes, to make a Dot (.) under each Figure of the Dividend as we use it.

III. Example 1. Divide 2371 by 5.

Remainder.

23,

First, See how many Times 5 in which is 4 Times, (because 5×5=25 is too much; and 5×3 =15, and 23-15=8, which is greater than the Divifor, and... 3 Times is too little, and confequently we must go 4 Times). place 4 in the Quotient, and, fubtracting 5x4-20 from 23, there remains 3; to which take down the next Figure of the Dividend 7, and the first Dividual will be 37: Then the Divifor 5 is contained in 37 feven Times, in the Quotient, and subtract 5×7=35, from the firft Dividual 37, the Remainder will be 2; to which having brought down the next Figure of the Dividend I, the fecond Dividual is 21; in which the Divifor is contained 4 Times; ·. put 4 in the Quotient, and then 5×4-20, and 21-20-1, the Re

place 7

mainder.

mainder.

As we have now taken down all the Fi

gures of the Dividend, the integral Part of the Quotient is 474, and the fractional Part 1 out of 5; fo that 5 is contained in 2371, 474 Times and (one Fifth) of another Time.

25)800(32

112. Example 2. Divide 800 by 25. First, In order to know how many Times 25 in 80, try how many Times 2 in 8, which is 4 Times, because 25x4= 100; '.' try 3 Times, which we can go, for 25x3=75, and 80-75=5, to which bring down the next Place

75

50
50

of the Dividend, which is o; then we have, how many Times 25 in 50? Which is 2 Times. See the Operation itself.

113. In long Operations the Method of tabulating the Divifor (as fhewn in Multiplication, in tabulating the Multiplicand) is very useful, as we may fee by Inspection the Times we can go, and the Product of each Time; fo that there is very little Difficulty in performing Divifion by Help of a Tariffa; take an Example.

Divide 88350768 by 25476.,

It is fuppofed that, if the Reader compares the Work

of this Divifion with the Tariffa, he will want no other

Explanation.

114. If

114. If the Tariffa is made on feparate Pieces of Paper, we may fave the Trouble of writing the feveral Products under the 25476) 88350768 (3468 Dividuals; for we may apply the Slips of Paper to them, and, fo fubtracting, only put down the Remainders, and then the Work will appear fhortened thus:

119227

173236

203808

115. The Rule given in Art. 110, for finding how many Times one Number is contained in another, may be thus demonftrated: It is plain, that by Art. 110. we part the Dividend into feveral Parts for we first take a Part of the Dividend for a Dividual, and, having divided this, to the Remainder we add another Part of the Dividend; which being alfo divided, another Part of the Dividend is added to this Remainder for a new Dividual, and fo on, 'till all the Parts of the Dividend have been added, and the Number of Times the Divifor is contained in those Parts been feparately found. Therefore, if the Method here laid down will find how many Times the Divifor can be taken out of those Parts, it will be all that is required: * For, as often as the Divifor is contained in the Parts which make up the Dividend, fo often must the Divifor be contained in the whole Dividend. As to the Finding of each Figure of the Quotient fingly, as the true Quotient of the Divifor, out of the feveral (Parts of the Dividend or) Dividuals, confidered by themselves, wè need no Demonftration; because they are found by Trials, and are not written down, 'till it is found that the respective Dividual, Minus the Product of the corresponding Quotient Figure into the Divifor, is less than the Divifor, and that therefore the Figure is taken right; fince, the Remainder being lefs than the Divifor, the Divifor cannot be taken once more out of it; and, confequently, we have taken the Divifor out of the Dividual as often as poflible. Whence