Arithmetic, Part 1 · Section 18 – Section 20
A Fourth way of Diviſion.
There is a fourth way of diviſion uſed by ſome, not inferiour to any of the preceding, for that it is no burthen to the memory, and it is alſo proved by Addition.
The manner of placing the figures is the ſame with the third kind of diviſion laſt taught; And for the performance of the work, this is
THE RƲLE.
Firſt write down the dividend, and on the left hand thereof the diviſor, with a crooked line betwixt them, and on the right hand of the dividend make another crooked line wherein to place the figures of the quotient, then draw a line under the dividend, and alſo make a prick under that figure of the dividend under which the laſt figure of the diviſor would fall, were it to be placed as in the firſt kind of diviſion. This done, demand how often the diviſor may be found in thoſe figures of the dividend, and place that digit in the quotient, then by this digits multiply the diviſor, and ſet the product of this Multiplication directly under the dividend, beginning at the place where you made the prick, then ſubtract this product from the figures of the dividend, and place the remainder over the dividend, cancelling the figures of the dividend as you proceed, ſo is the firſt figure of the quotient finiſhed, then make a prick under the next figure of the dividend, and demand how often the diviſor may be found in the laſt remainder, and the other figure being added thereto, which place in the quotient, and proceed in all reſpects as before, till you have pointed all the figures of the dividend.
Example, … Let it berequiren to divide 763258 by 2345, place your numbers as you ſee in the margent, and becauſe there are 4 figures in the diviſor, therefore make a prick under the fourth figure of the dividend, which is 2, and draw a line, then begin your diviſion in this manner: ſaying,
Firſt, … how many times 2345 can I have in 7632, (or how many times 2 can I have in 7) ſay 2 times, place 3 in the quotient, by which 3 multiply the diviſor, ſaying 3 times 5 is 15, place 5 under the prick, then 3 times 4 is … and 1 is 13, place 3 under the line, then 3 times 3 is 9, and 1 is 10, place a cypher under the line, then 3 times 2 is 6, and 1 is 7, place 7 under the line, then ſubtract 7035 from 7632, ſaying 5 from 12, and there remains 7, place 7 over 2. (cancelling the 5 and the 2,) and bear one in mind, then 1 and 3 is 4, out of 13, there remains 9, place 9 over 3, and cancel 3 and …, then 1 which I carried, from 6, and there remains 5, place 5 over 6, and cancel 0 and 6, laſtly, 7 from 7 there remains 0, which you need not ſet down, but cancel the two ſevens, then will the work ſtand as above, and the remainder will be 597.
Secondly, make a prick under the next figure of the dividend, namely 5, and ſay how many times 2345 can I have in 5975, anſwer two times, place 2 in the quotient, by which multiply the diviſor, ſaying 2 times 5 is 10, place 0 under 5, and carry 1, then 2 times 4 is 8 and 1 is 9, place 9 under 7, then 2 times 3 is 6, place 6 under 9, laſtly, 2 times 2 is 4, place 4 under 5, ſo is the product of this multiplication 4690, which you muſt ſubtract from 5975, ſaying 0 from 5 and there remains 5, place 5 over 5, and cancel 0 and 5, then 9 out of 17, there remains 8, place 8 over 7, and cancel 9 and 7, then 1 carryed and 6 is 7, from 9 there remains 2, place 2 over 9 and cancel 6 and 9, … laſtly, 4 from 5 reſts 1, place 1 over 5, and cancel 4 and 5, ſo have you finiſhed your ſecond figure, and your work will ſtand thus, and your remainder will be 128.
Thirdly, make a prick under the next figure of your dividend (namely under 8,) and aske how many times 2345 can I have in 12858, (or how many times 2 can I have in 12,) ſay 5 times, place 5 in the quotient, by which multiply the diviſor, ſaying 5 times 5 is 25, place 5 under 8 and carry 2, then 5 times 4 is 20, and 2 is 22, place 2 under 0, and carry 2, then 5 times 3 is 15, and 2 is 17, place 7 under 9, and carry 1, then 5 times 2 is 10, and 1 is 11, place 11 under 4 and 6, ſo is the product of this multiplication 11725 to be ſubtracted from 12858, ſaying 5 from 8 reſts 3, place 3 over 8, and cancel 5 and 8, then 2 from 5 reſts 3, place 3 over 5, and cancel 2 and 5, then 7 from 8 reſts 1, place 1 over 8 and cancel 7 and 8, then 1 from 2 reſts 1, place 1 over 2, and cancel 1 and 2, laſtly, … 1 from 1 reſts nothing, ſo is your work ended, which you ſhall find to ſtand as in the margent, the remainder being 1133.
The Proof of this Diviſion.
This kind of diviſion is alſo proved by Addition, for, if you draw a line under the work, and adde all the figures between the two lines together, (in order as they there ſtand) taking in the remainder (if any be) the Total of this addition will be equal to the Dividend, if the work be true.
Other Examples for Practice proved.
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Queſtions performed by Diviſion only.
Queſtion 1. If a ſquare piece of Land contained 42952 ſquare perches, and one of the ſides thereof be 236 perches long, how long muſt the other ſide be? Divide 42952 by 236, the quotient will be 182, and ſo many perches long muſt the other ſide be.
Queſtion 2. In a yeaar there are 8760 houres, and in every natural day there are 24 houres. I demand how many dayes be there in a year? Divide 8760 by 24 the Quotient will be 365, and ſo many dayes be there in a year.
Queſtion 3. The diſtance from London to Coventry is 133760 yards, and in one mile there is contained 1760 yards, now I would know how many miles it is from London to Coventry; Divide 133760 by 1760, the quotient will be 76, and ſo many miles is it from London to Coventry.
Theſe Queſtions performed by Diviſion only, are the converſe of thoſe that were performed by Multiplication, which I the rather make choice of, that the reader might ſee how Multiplication and Diviſion prove each other.
There are one or two more kinds of Diviſion, ſomething like theſe laſt, but I ſhall forbear exemplifying them; for much variety helps to make a Book rather great then fit.
¶ Here is to be noted that in the following Rules, where there is continual uſe of Diviſion, I ſometimes uſe one kind of Diviſion, and ſometimes another, for variety ſake, but the practicioner may uſe which he is beſt skill'd in, for they all produce the ſame effect.
Reduction.
IS twofold, firſt, that which turns greater denominations into ſmaller, as pounds into ſhillings or pence, this is done by Multiplication as followeth.
Example 1.
Let it be asked how many pence are contained in 729 li. 11 s. 7 d?
Firſt, a ſhilling is contained in a pound 20 times, therefore multiply 729 by 20, or (which is the ſame, but ſhorter) by 2, and put o to the product, as in the margine, … this ſhews that in 729 l. there are 14580 ſhillings.
To which adde 11 s. it makes 14591 ſhillings.
Again, becauſe one peny is conteined in one ſhilling 12 times, multiply 14591 by 12, it produceth 175092, to which adde the 7 pence, ſo the ſumme will be 175099, and ſo many pence are contained in 729 li. 11 s. 7 d.
Example 2.
Let it be asked how many pintes are contained in 4 Tun, 3 Hogſheads, and 27 Gallons?
Firſt, 1 Tun is equal to 4 Hogſheads, therefore 4 Tun is equal to 16 Hogſheads, to which adde the 3 Hogſheads, ſo there is 19 intire Hogſheads.
Againe, becauſe one Hogſhead containes 63 Gallons, multiply 19 by 63, … it produceth 1197 Gallons, to which add 27, it gives 1224 Gallōs.
Laſtly, becauſe every Gallon contains 8 pintes, multiply 1224 by 8, it produceth 9792, and ſo many pintes are contained in 4 Tuns, 3 Hogſheads and 27 Callons.
After the ſame ſort might drie Meaſures be reduced, as quarters to buſhels, pecks, or gallons, and likewiſe all weights and outlandiſh Coins, of which the proportion of the greater to the leſſer is (before) known, or given.
Secondly, it is often requiſite to turn ſmaller denominations to greater: this is done by Diviſion, as followeth.
Example 1.
Let it be asked how many pounds are contained in 80976 ſhillings?
Divide 80976 by 20, … the quotient is 4048 li. and 16 s. remaining, which is the true anſwer.
Example 2.
Let it be asked how many pounds are in 109754 d?
Becauſe a pound contains a ſhilling 20 timesand a ſhilling containes a peny 12 times, there fore if 109754 be divided firſt by 12, the quotient 9146 ſhillings, and 2 pence over, then if 9146 be divided by 20, the quotient is 457 pounds and ſix ſhillings remaining; ſo that 109754 pence is equal to 457 li. 6 s. 2 d.
Or if 109754 had been at firſt divided by 12 times 20, that is by 240, (which is the number of pence contained in a pound) the quotient had been 457 pounds, and 74 pence remaining, which is all one with the former; for 74 pence is equal to 6 ſhillings 2 pence.
More inſtances ſhall not need herein, becauſe the thing of it ſelf is very clear.