If the Work of this Example be confidered and compared with the Rule (Page 22.) the whole Bufinefs of Divifion will be eafy; for indeed the only Difficulty (as I faid before) lies in making choice of a true Quotient Figure, which cannot well be done according to the Common Method of Divifion, without Trials, yet thofe Trials need not be made with the whole Divifor (as appears by this last Example) for by the two First Figures of the Divifor all the reft are generally regulated; except the Second Figure chance to be 2, 3, or 4, and at the fame time the Third Figure be 7, 8, or 9, then indeed refpect must be had to the Third Figure, according as the Rule directs. However, if thofe Trials are thought too troublefome, they may be avoided, and the fame Quotient Figure may both easily and certainly be found by help of fuch a fmall Table made of the Divifor, as was of the Multiplicand in Page 20. EXAMPLE 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, Dividend. 1 79863) 70251807402 Thus, Divifor. 2159726 638904 3239589 4319452 636140 _559041 5 399315 6479178 7 559041 8638904 779997 522304 Quotient. The Work of this Operation I prefume may be eafily underftood. For thofe Figures in the Table are the Product of the Divifor into all the 9 Figures; confequently thofe Figures in the fmall Column do fhew what Figure is to be placed in the Quotient; without any doubtful Trials of the Divifor, with the 319452 Dividend, as before. 319452 (000000) 9718767 479178 10798630 431260 399315 This Method of tabulating the Divifor may be of good Ufe to a Learner; efpecially until he is well practifed in Divifion; yea, and even then if the Divifor be large, and a Quotient of many Figures be required; as in refolving of high Equations, and calculating of Aftronomical Tables, or thofe of Intereft, &c. Hitherto 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 compofed of the Divifor and Quotient. But it moft ufually falls out, that the Divifor will not exactly measure the Dividend; in which cafe the Re-. mainder (after Divifion is ended) must be fet over the Divifor, with a small Line betwixt them adjoining to the Quotient. EXAMPLE 5. Suppose it were required to divide 379 by 5. the Remainder. 5) 379 (75 the Divifor. 35° 29. 25 Remains (4) EXAMPLE 6. Again, Let it be required to divide 43789 by 67. 402 358 335 239 Remains 201 (38) How fuch Remainder's thus placed over their Divifors (which are indeed Vulgar Fractions) may be otherwife managed, fhall be fhewed 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. Divifion is truly performed by cutting off with a Point or Comma, fo many Figures of the Dividend as there are Cyphers in the Divifor; then are thofe Figures fo 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. EXAMPLE 7. Let it be required to divide 57842 by 100. The Work may ftand thus, 100) 578,42 the Quotient required; or thus 100) 57842 (578,4 the fame as before. Hence it follows, that if any Divifor have Cyphers to the Right-hand of it, you may cut off fo many of the laft Figures in the Dividend, and divide the other Figures of the Dividend, by thofe Figures of the Divifor that are left when the Cyphers are omitted. But when Divifion is ended, thofe Cyphers fo omitted in the Divifor, and the Figures cut off in the Dividend, are both to be restored to their own places. EXAMPLE 8. Suppose it were required to divide 675469 by 5400. 54.. 135 108 274 270 Remains (4) But the true Remainder is 469. Confequently the true Quotient is 125. As to the manner of proving the Truth of any Operation, either in Multiplication or Divifion, I prefume it may be easily understood, by what is delivered in Page 21, compared with the three first Examples of Divifion; for from thence it will be easy to conceive, that if the Divifor and Quotient be multiplied together, their Product (with what Remains after Divifion being added to that Product) will be equal to the Dividend. As in the Fifth Example, where the Dividend is 379, the Divifor is 5, the Quotient is 75, and the Remainder is 4. I fay, 75×5=375, to which add the Remainder 4, it will be 379: Again, in the Sixth Example, the Divifor is 67, the Quotient is 653, and the Remainder is 38. Then 653x67=43751, and 43751+38=43789 the Dividend, &c. There are feveral ufeful Contractions, both in Divifion and Multiplication, which I have purposely omitted until I come to treat of Decimal Arithmetick, Alfo I have omitted the Bufinefs of Evolution or Extracting of Roats, until further on; and fo fhall conclude this Chapter with a few Examples of Divifion unwrought at large, leaving them for the Learner's Practice. 579) 43800771 (75649. Or 75649) 43800771 ( 579 45007) 23884044718 (530674.. CHA P. III. Concerning Addition and Subtraction of Numbers of different Denominations, and how to reduce them from one Denomination to another. TH SECT. I. 1. Of English Coin. HE leaft Piece of Money ufed in England is a Farthing, Note, When 1. s. d. q. are placed over (or to the Right-hand of) Numbers, they denote thofe Numbers to fignify Pounds, Shillings, Pence, and Farthings. 1. S. d. 9. As 35 ΙΟ 6 2. Or 35. 10 s. 6d. Either of these do fignify 35 Pounds, 10 Shillings, 6 Pence, 2 Farthings. The fame must be understood of all the following Characters, belonging to their respective Tables, viz. Of Weights, Meafures, &c. 2. Troy Weight. 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 51 Hen. III. 31 Edw. I, 12 Hen. VII. But 1 But in later Times it was thought fufficient to divide the aforefaid Penny Weight into 24 equal Parts, called Grains, being the leaft Weight now in common Ufe; and from thence the reft are computed as in this Table. Gr. Grain. 24 1 P. W. Penny Weight. 480 20 10%.Ounce, 17560=240=12=1 lb Pound. Note, By Troy Weight are weighed Jewels, Gold, Silver, Corn, Bread, Land all Liquors. Befides the common Divifions of Troy Weight, I find in Angliæ Notitia, or, The Prefent State of England, Printed in the Year 1699 that the Moneyers (as that Author calls them) do fubdivide the Grain. 20 Mites 1 Grain, &c. as before. 3. Apothecaries Weights. The Apothecaries divide a Pound Troy, as in this Table. Gr. Grain. 20= 13 Scruple 60= 3= 13 Dram 480 248= 13 Ounce 5760=288=96=12=1 tb Troy, the fame as before. By these Weights the Apothecaries compound their Medicines : but buy and fell their Drugs by Averdupois Weight. 4. Averdupois Weight. When Averdupois Weight became firft in Use, or by what Law it was at firft fettled, I cannot find out in the Statute Books; but on the contrary, I find that there fhould be but one Weight (and one Measure) ufed throughout this Realm, viz. that of Troy, (Vide 14 Éd. III. and 17 Ed. III.) So that it feems (to me) to be firft introduced by Chance, and fettled by Custom, viz. from giving good or large Weight to thofe Commodities ufually weighed by it, which are fuch as are either very Coarfe and Droffy, or |