Iohnsons Arithmatick in 2. bookes the first, of vulgare arithma: with diuers briefe and easye rules: to worke all the first 4. partes of arithmatick in whole numbers and fractions by the author newly invented the second, of decimall arithmatick wherby all fractionall operations are wrought, in whole numbers, in marchants accomptes without reduction; with interest, and annuityes by Iohn Iohnson survaighour; practitioner in the mattiematiqu

Double Position.

The Rule of double Position.

SVppose a number at pleasure, as in the last Rule of single Position, and proceed as if you had found the right number, and if by working you find the true number, then your Position was the right number, which doth seldome happen. First, if by your working there commeth out more then the true number; then note it thus + with a crosse; if lesse, then thus − with a long line, which doth signifie lesse.

Secondly, suppose another number, grea∣ter or smaller, and worke as before, vntill you doe find the true number sought; which if you doe not find, see the difference also from the true number sought, and note it with the signe + or − as it shall bee found.

Then thirdly, set your suppositions with their errours, more or lesse, as in the exam∣ples following.

Fourthly, multiply crosse the first positi∣on

by the seconds errour, and the second po∣sition by the errour of the first, and then if the signes be both alike + or −, abate the lesser from the greater, and the remaines shall be the diuidend. Also the lesser error abated from the greater, leaues the diuisor▪ but if the signes be contrary one +, the o∣ther lesse, add both together to make the di∣uidend, and adde the two errors to make the diuisor: and lastly diuide the diuidend by the diuisor, and the quotient is the true number desired.

1. Example.

A certaine man seeing a purse in his friends hand, saith vnto him: It seemeth vn∣to me, that there is 100 Crownes in your purse▪ To whom the other answered: Nay (quoth hee) there are not 100 Crownes, but (saith he) if they were increased 1∶2 and 1∶3 ▪ and 1∶4, and lastly, one Crowne ouer∣plus, then would they be iust 100 crownes.

I suppose there were 12 Crownes in his purse, to which if I adde one half; of 12, which is 6; and one third of 12, which is 4; and one fourth of 12, which is 3; and lastly, one Crowne more, the totall will be but 26

Crownes, but they should be 100 Crownes, so that this errour is two little by 74 Crownes, which I note thus:

74 − 12

Secondly, I suppose he had 24 Crownes, to which I adde 1∶2 of 24, which is 12 and 1∶3, which is 8 and 1∶4, which is 6: and lastly, one Crowne ouerplus, the totall is 51, but it should bee 100 Crownes, so that this is an errour of 49, too little, which I al∣so note thus: 49 − 24

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The answere is; that hee had 47 pound 13∶25 parts of a pound in his purse. The proofe followeth.

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2. Example.

Twenty yards of Sattin, and 12 shillings is equall vnto 12 yards of veluet lesse, 10 shillings; the price of either sort is requi∣red.

To answere this, or any other like questi∣on, take any number for the price of a yard of the lesser number, which here is veluet, which at 20 shillings a yard, lesse 10 shil∣lings, amounteth vnto 230 shillings. Now admit a yard of Sattin at 14 shillings, so 20 yards and 12 shillings amounteth vnto 292 shillings; from which subtract 230 shil∣lings, rests 62 s. more then the truth. A∣gaine,

rate a yard at 12 shillings, so the 20 yards and 12 shillings makes 252 shillings; from which take 230 shillings, rests 22 shil∣lings more then the truth also. Now mul∣tiplying 22 by 14, and 62 by 12, the pro∣ductes are 308, and 744, and the difference of those numbers is 436; then take 22 from 62, rests 40 for diuisor, by which diuide the difference, makes 10 shillings, 9∶10 shil∣lings for the price of a yard of Sattin.

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3. Example.

Otherwaies if 40, the difference of er∣rors gaine a, the difference of positions, then 62 the first error yeelds 3 and 1∶10▪

Or if 40 yeeld 2, what 22? makes 1 and 1∶10; this taken from 12, or 3, 1∶10 from 14, leaues 10, 9∶10 for the price, as before.

4. Example.

A Carpenter was hired to work 20 daies at 12 pence a day, but euery day that hee was idle, hee was to abate 18 pence of his wages, and in the end he receiued but 8 shil∣lings: now the question is, how many daies he wrought.

First, suppose he wrought 12 daies, which commeth to 12 shillings, then must the 8 dayes that hee played, come to 12 shillings at 18 pence a day also: but this question saith, there came due to him 8 shillings? Be∣hold an error of 8 shillings too little.

Againe, I say that he wrought 14 dayes, amounting to 14 shillings▪ then 6 dayes that he played at 18 pence a day, commeth to 9 shillings; this taken from 14 shillings,

leaues 5 shillings, and it should bee 8 shil∣lings, which is an errour of 3 shillings too little. Now multiplying 12 by 3, and 14 by 8, the products are 36. and 112, and the excesse is 76; which being diuided by 5, the difference of the errours, quoteth out 15, 1∶5 for the number of working dayes, and 4 dayes 4∶5 for the number of playing dayes.

12 − 8	5
14 − 3

Otherwayes.

If 5▪ the difference of errours, yeeld 2, the difference of positions, what 8 the first er∣rour? makes 3, 1∶5 to be added to 12.

Or if 5 be 2, what is 3▪ makes 1, 1∶5 to be added to the second position 14, where∣by all three wayes the numbers of the Dayes he wrought are found out.