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If the Work of this Example be considered and compared with the Rule (Page 22.) the whole Business of Division will be easy; for indeed the only Difficulty (as I said before) lies in making choice of a true Quotient Figure, which cannot well be done according to the Common Method of Division, without Trials, yet those Trials need not be made with the whole Divisor (as appears by this last Example) for by the two First Figures of the Divifor all the rest are generally regulated ; excepe the Second Figure chance to be 2, 3, or 4, and at the same time the Third Figure be 7, 8, or 9, then indeed respect must be had to the Third Figure, according as the Rule directs.
However, if those Trials are thought too troublesome, they may be avoided, and the same Quotient Figure may both easily and certainly be found by help of such a small Table made of the Divifor, as was of the Multiplicand in Page 20.
E X AMPLE 4.
Let it be required to divide 70251807402 by 79863. See the Example of Multiplication, Page 20, and as there directed make a Table of the Divifor 79863, Thus,
Divisor. Dividend. Quotient. 1 79863) 70251807402 (879654 2759726 638904 The Work of this Operation 3239589
636140 I presume may be eafily under41319452 559041
stood. For those Figures in the 5399315
Table are the Product of the Di
770997 6479178 718767
visor into all the 9 Figures ; con71559041
sequently those Figures in the 81638904
small Column do Thew what 91718767
Figure is to be placed in the 101798630
Quotient; without any doubtful 399315
Trials of the Divifor, with the 319452 Dividend, as before. 319452 (000000)
This Method of tabulating the Divifor may be of good Use to a Learner; especially until he is well practised in Divifion; yea, and even then if the Divifor be large, and a Quotient of many Figures be required; as in resolving of high Equations, and calculating of Aflronomical Tables, or those of Interest, &c.
Hitherto I have made choice of Examples wherein the Dividend is truly measured or divided off by the Divifor, without leaving any Remainder, being exactly composed of the Divisor and Quotient. But it most usually falls out, that the Divisor will not exactly measure the Dividend ; in which case the Re. mainder (after Division is ended) must be set over the Divifor, with a small Line betwixt them adjoining to the Quotient.
67) 43789 (65384 the true Quotient required.
Remains (38) How fuch Remainder's thus placed over their Divisors (which are indeed Vulgar Fractions) may be otherwise managed, Ihall be shewed farther on.
N. B. When the Divifor happens to be an Unit, viz. 1, with a Cypher or Cyphers annexed to it, as 10, 100, 1000, &c. Division is truly performed by cutting off with a Point or Comma, so many Figures of the Dividend as there are Cyphers in the Divifor; then are those Figures so cut off to be accounted a Remainder, and the reft of the Figures in the Dividend will be the true Quotient required, because an Unit or i doth neither multiply nor divide.
E X A M P L E 7. Let it be required to divide 57842 by 100. The Work may stand thus, 100) 578,42 the Quotient required; or thus 100) 57842 (5787160 the same as before.
Hence it follows, that if any Divifor have Cyphers to the Pright-hand of it, you may cut off fo many of the last Figures
in the Dividend, and divide the other Figures of the Dividend, by those Figures of the Divifor that are left when the Cyphers are omitted. But when Division is ended, those Cyphers so omitted in the Divifor, and the Figures cut off in the Dividend, are both to be restored to their own places.
E X A M P L E 8.
5400) 675469 (125
Remains (4) But the true Remainder is 469. Consequently the true Quotient is 125 40.3.
As to the manner of proving the Truth of any Operation, either in Multiplication or Division, I presume it may be easily understood, by what is delivered in Page 21, compared with the three first Examples of Division ; for from thence it will be easy to conceive, that if the Divifor and Quotient be multiplied together, their Product (with what Remains after Division being added to that Product) will be equal to the Dividend. As in the Fifth Example, where the Dividend is 379,
the Divisor is 5, the Quotient is 75, and the Remainder is 4. I say, 75*5=375, to which add the Remainder 4, it will
Again, in the Sixth Example, the Divisor is 67, the Quotient is 653, and the Remainder is 38.
· Then 653x67=43751, and 43751+38=43789 the Dividend, &c.
There are several useful Contractions, both in Division and Multiplication, which I have purposely omitted until I come to. triat of Decimal Arithmetick, Allo I have omitted the Business of Evolution or Extracting of Roots, until further on; and so shall conclude this Chapter with a few Examples of Division unwrought at large, leaving them for the Learner's Practice.
579) 43800771 (75649. Or 75649) 4380077! ( 579.
45007) 23884044718 (530674.
356) 244572000 (687000.
79) 282016 (356995.
CHA P. III.
Concerning addition and Subtraction of Numbers of
different Denominations, and how to reduce them from one Denomination to another.
1. Of English Coin.
4= id. Pen. 48= 12= Is. Sbill. 960=240=20=il. Pound Sterling.
5 s. is a Crawn.
Note, When l. s. d. q. are placed over (or to the Right-hand of) Numbers, they denote those Numbers to signify Pounds, Shillings, Pence, and Farthings.
6 Or 35 l. 10 s. 6id. Either of these do fignify 35 Pounds, 10 Shillings, 6 Pence, 2 Farthings.
The same must be understood of all the following Characters, belonging to their respective Tables, viz. Of Weights, Meafures, &c.
2. Troy Weigbt. The Original of all Weights used in England, was a Corn of Wheat gathered out of the middle of the Ear, and being well dried, 32 of them were to make one Penny Weight, 20 Penny Weight one Ounce, and 12 Ounces one Pound Troy. Vide Statutes of si Hen. III. 31 Edw. I, 12 Hen. VII.
But in later. Times it was thought sufficient to divide the aforesaid Penny Weight into 24 equal Parts, called Grains, being the least Weight now in common Use; and from thence the rest are computed as in this Table.
Gr. Grain. 245
I P.W. Penny Weigbe. 480= 20=10z.Ounce. 17560=240=12=1 lb Pound.
By Troy Weight are Note,
weighed Jewels, Gold, Silver, Corn, Bread, Land all Liquors.
Besides the common Divisions of Troy Weight, I find in Angliæ Notitia, or, The Present State of England, Printed in the Year 1699 that the Moneyers (as that Author calls them) do subdivide the Grain,
24 Blanks = 1 Periot. Thus
20 Periots = i Droite.
24 Droites = I Mite.
3. Apothecaries Weights.
The Apothecaries divide a Pound Troy, as in this Table.
203 1 3 Scruple 60= 3= 13 Dram 480= 245 8= 13 Ounce 5760=288=96=12=1 l Troy, the same as before.
By these Weights the Apothecaries compound their Medicines : but buy and sell their Drugs by Averdupois Weight.
4. Averdupois Weigbt.
When Averdupois Weight became first in Use, or by what Law it was at first settled, I cannot find out in the Statute Books ; but on the contrary, I find that there should be but one Weight (and one Measure) used throughout this Realm, viz. that of Troy, (Vide 14 Éd. III. and 17 Ed. III.) So that it seems (to me) to be firft introduced by Chance, and settled by Custom, viz. from giving good or large Weight to those Commodities ufually weighed by it, which are such as are either very Coarse and Drofly, or