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SInce that Arithmetike is abso∣lutely necessary, and required in diuers and many Geometri∣call operations, I haue added the ex∣amples following onely to renew and refresh the memorie of those who haue alreadie studied it; and not those who are quite ignorant therin, (commending them to large and ample Treatises together with ma∣sters of the scien••e) but for such as by a weake memory haue let slip the
habitude which by practise they had in times past obtaind, who may easi∣ly recouer by this short Treatie suffi∣cient knowledge to performe any ordinary operation beginning with Addition, which is a collection of many numbers in one, as hee who would adde together the numbers ABC, following to haue the summe D, must begin with the first colon: 〈 math 〉〈 math 〉 And say 2 and 5, or 7 and 1 is 8, and set 8 vnder the line, as appeares by the example aboue, then in the second colon, say 3 and 4 or 7, and pose them as before; afterward say for
the 3. colon 6. are sixe, & posing or setting the numbers collected dire∣ctly vnder the figures not collected as 8 vnder 2, and 7 vnder 3, and sixe vnder sixe, but if any should bee a∣boue the number of 9, then to set 0 in the place, and set forward all the tens.
MAy be made in casting all the 9 away, though many times fals but by substraction is the way most certaine, as thus, begin with the last collon and say 6, cut of 6, rest no∣thing, and giue the 6 vnder the line a flash, then 4 and 3, or 7 and 7, out
of 7 rest nothing, and as before giue it a flash, then 1, 5 and 2, or 8, which as before shall be made nothing, the 8 vnderneath.
SVbstraction is to take away a little number from, or out of a greater, as if one would from 8 6 4 2 take away 4 3 2 1, then must the numbers bee disposed as fol∣loweth, the greater number vpper∣most, and the lesser vndermost, then draw a line:
| Great number | 8642. A | |
| Lesser number | 4321. B | |
| rest | 4321. C |
IS made in adding the lesser num∣ber B with the rest C as follow∣eth.
| Little nummer. | 4321 | |
| rest | 4321 | |
| Great number. | 8642 |
MVltiplication is the as∣sumption of one num∣ber, as many times as the other, containeth in it selfe vnities. 〈 math 〉〈 math 〉 As if one would multiply 6. by 3, that is, to take as oft 6 as 3 containes vnities, as in the example aboue,
where 2 taken 4 times makes 8, and 5 taken 4 times makes 20. so that the number A multiplied by the nū∣ber B multiplicator 4 times compre∣hended, make the product 2608.
IS made onely diuiding the pro∣duct 2608 by the multiplication 4 and the quotient will be 652. if the multiplication were well made, as appeares in the example following. 〈 math 〉〈 math 〉
DIvision is the separation of a number into aliquot partes thereof, as to diuide 5689 by 25. the figures must be disposed, as followeth, viz. the diuisor vnder the first figures of the number, which is to be diuided, as for example. 〈 math 〉〈 math 〉 And after the last figure of the num∣ber, shall bee drawne a halfe circle to separate the quotient. The num∣bers
being thus disposed, then say 2 in 5 how many times, and it shall be found twice; then set 2 behind the halfe circle as in the first example following; and then say twice 2. are 4 of 5 rests 1, dashing out the 2 and the 5. and set the 1 ouer 5, and say a∣gaine twice 5 are 10. and strike out the 5 diuisor vnder the 6. and also the 1. ouer the 5, then set forward your divisor one figure more as in the second example, and say 2 in 6. how how many times, and it shal be found 2, and say twice 2 are 4, and 4 out of 6. rests 2. which shall be set ouer the 6, then say twi••e 5 are 10. and 10. out of 18. rests 8 and 1 out of 2. rests 1. then dash out the 2. and set 1. ouer it; and also dash out the 5 and 2. di∣uisors, and set more forward the di∣visors, as in the third example, and
say 2 how many times 18. and it shall be found 7. times, and having set the 7, say 7 times 2 is 14. and 14 out of 18 rests 4, and set the 4 ouer the 8; and to conclude, say 7 times 5 are 35, and 35 out of 39 rests 4, and 3 out of 4 rests 1, and so your di∣uision is ended, as appeares here fol∣lowing by three examples one of each operation. 〈 math 〉〈 math 〉
Multiplie the quotient 227. by the diuisor 25, and to the product partiall, adde the numbers which rest, viz. 14. if any rest, and then adde the whole together, and the product shall be the first number if the diuision hath beene well made as in the example following. 〈 math 〉〈 math 〉
A Fraction is a number no∣ting the partes aliquot of an entire or whole num∣ber, whereof it is said to be a fraction as a penny, the twelfth part of a shilling, one inch the foure and fortieth part of an Ell, &c.
TO reduce 8¾ and 5⅔ all in∣fractions you must mul∣tiplie 8 by 4, saying 4. times 8 is 32, and the 3 being added, makes 35, which shall be set aboue a line iust vnder the first figures, and vnder that line set 4, as in the example following, to shew that the 45 are al fourth parts, and doe in the like manner by the 5⅔, and say 3 times 5 are 15, and 2 makes 17. and so you shall haue 17. thirds, as in the example following.
To reduce 8¾ and 5⅔ all into fracti∣ons. 〈 math 〉〈 math 〉
TO bring these two fractions to one deno. the 35 quarters numerators must bee multi∣plied by the other numerators 17. thirds, and set the product ouer a line as here following, then multiply the 4 denominator by the other 3.
denominator, and set the product of those vnder the line, and then will there be, 〈 math 〉〈 math 〉
TO adde ⅔ to ¼ the figures, dis∣pose the figures as followeth, and say 3 times 1 is 3, and set the 3. ouer a line aboue the head of the o∣thers, and then say 4 times 2 are 8. and set them ouer a line aboue the head of the others also, then say 3 and
8 are 11, and set them ouer a line be∣tweene the first numbers and the 11, shall be numerators, and say 4 times 3 are 12, & set them vnder the mid∣dle line, and those shall bee denomi∣nators, and so you shall haue eleuen twelfth parts as followeth. 〈 math 〉〈 math 〉 But if the numerator bee greater then the denominator, then it shall bee diuided by the denominator, and the product shall be an entire or entires, and what resteth (if any rest) shall bee fraction, which ought to be abridged, as appeareth here following.
〈 math 〉〈 math 〉 Where the numerator 38 is greater then the denominator 24. then be∣ing diuided by the denominator 24. doth yeeld 1 intire, and 14/14 the which being abridged come to 7/12, which is almost two thirds, and so of all o∣thers.
BEing proposed to 243⅔ vnto 462¼, the intiers must be added as in the first chapter of Addition and the fractions must be added as in the precedent chapter, and set product as appeares following. 〈 math 〉〈 math 〉
WHosoeuer would sub∣stract ⅔ from ¾ must dis∣pose the figures, as in the example following; and first multiply the numerators by the denominators a crosse, as 3-times 3 are 9. and 4 times 2 are 8, and set 9 and 8 ouer the lines aboue the heads of the others, and then say take 8 out of 9, and there rests 1, which must bee set ouer a line be∣tweene
both, and afterward say 3-times 4 is 12, multiplying denomi∣nators by denominators, and set them vnder the middle line, which shall bee the denominators for the rest, as appeares cleerely in the ex∣ample following.
Out of 9 pay 8 rest 1 〈 math 〉〈 math 〉
TO substract 1831 from 267 2/•• the fractions must first be mul∣tiplyed as in the example fol∣lowing; and say twice 2 are 4, and set the 4 ouer a line, then 3 times 1 are 3, and set them ouer a line, and then who payes 3 out of 4. rests 1, which must be set ouer the middle line, which done, multiply the one denominator by the other, and set the product vnder that which doth rest, as followeth.
〈 math 〉〈 math 〉 But if the fraction of the receiued should be lesse then the fraction of the summe paid, then must there be one borrowed from the whole or intire number, and count it according to the denomination of the fraction; for if the denominator bee 4, then shall the intier be 4/4, if 5, then 5/5, if 6 then 6/6, &c.
LEt there bee proposed a superfi∣cies in the forme of a paralello∣gram, vulgarly cald square, the sides whereof the one is in length 7/8 of a fa∣thom, and the other in breadth ¾, and these two fractions are to bee multi∣plied together to find out how much the whole superficies doth containe the figures must be disposed as follo∣weth. 〈 math 〉〈 math 〉
And multiplie one numerator by the other numerator, and then one denominator by the other denominator, saying 3 times seuen is 21, and set them ouer head, then foure times 8 is 32, and set them vnderneath, and the whole will bee 21/32 parts of a fathom, which certainely containeth the required superficies.
TO multiply ¼ by 2½, you must first of al reduce the whole into fracti∣ons, and then as here aboue multi∣ply numerator by numerator, and de∣nominator by denominator, and the product will bee 45/••8, as plainely ap∣peares by the example following. 〈 math 〉〈 math 〉
But if it were proposed to multiply greater numbers, as 20 by 15 26/29, then multiply the 15. intiers by the deno. 29. of the fraction, & then adde the numerator 26 of the same fraction; which done will mount to 461/29, then set the 461 ouer a line, and the 29. vnder it, and afterward multiply the 20 intiers by the 461. which done, diuide the product of the whole by the denominator 29. and the num∣qer required shall bee 317 27/19, as ap∣peares: 〈 math 〉〈 math 〉
TO diuide ¾ by ⅓, each numerator is to bee multiplied by each denominator oppo∣site, and set the pro∣duct ouer a line aboue them, and then diuide the greatest product by the least as followeth. 〈 math 〉〈 math 〉
TO diuide 12⅔ by 3⅙, they must first be reduced all into fractions as before, and then you must multi∣ply the numerators by the denomi∣nators acrosse as followeth, and then diuide the greatest product by the least, as this example doth cleerely demonstrate. 〈 math 〉〈 math 〉
SVppose you were to abridge 7/9 parts of a fathom, first you must consider what are the parts of the intier or whole, as 6. foot, or 72. inches: then you must multiply the numerator 7, by the denominator 72 parts, and let the product bee di∣uided by the denominator 9, and then you will finde 56 inches for the eualuation of 7/9 parts of a fathom.
〈 math 〉〈 math 〉 By this meanes any fraction may be abridged as well in Geometrie as as commerce, although they seeme not to be abridged.
YOu must consider that the fa∣thom of 6 foot in length, doth containe in superficies 36. and that
the 72 inches in length doth contain in superficies 5184 inches, and of o∣ther measures then to valuate a fra∣ction of 19/4•• parts of a fathom, square in superficies you must multiply 5184 by 19. and diuide the product by 47, and there will be 2095 inches for the square of 29/47 of a fathom square and so of other like measures.
MVltiply the second num∣ber 400 by the third 12. and product 4800 by the first number 4, and the quotient shall be the number re∣quired, and dispose your rule as fol∣loweth.
months—pounds—months. If in 4—400—12
IS to multtplie the first num∣ber 4 by the fourth number 1200, and to multiplie the second by the third, and the two products will bee equall if the rule bee well made.
FIrst all the intiers must bee re∣duced into fractions as follow∣eth.
| yards | pounds | yards. |
| If 2¼ | 12½ | 7½ |
| 9/4 | 25/2 | 15/2 |
the third number of fractions 15, & then againe multiply the product by 4 the denominator of the first num∣ber, and then say 2 times 2, or 4, and 4 times 9 is 36, which must bee set vnder the line, by which you shal di∣uide the first product 1500, and the quotient shall be the number requi∣red, as appeares, 〈 math 〉〈 math 〉
Heere followeth two examples, differing the one from the other; whereof the manner of multiplying the one, is more easier then the other the first is multiplyed as the prece∣dent, but the last is multiplied first
by all the intirres, viz. by 3, by 8 and by three, leauing the fraction •• by it selfe, and after all take the thir•• of the intier, viz. of 50000, saying th•• third part of 5 is one rest 2, for the 10 which is valuated at 20, then say the third part of 20 is 6, and so rests 2 for the second 0, and so to the end, and what shall rest at last, shall be set ouer a line, and your 3 4th or 5 vnder the line, then all being added together, you shal diuide the product cutting off the figures to the quanti∣tie of the first number, saying, by ten, by a hundred, by a thousand, by ten thousand, by a hundred thousand, & the remainder is the number requi∣red, as appeareth, 191⅔.
〈 math 〉〈 math 〉
The second number being mul∣••••plied by the third, doth mount to 57500000, and being diuided by the first multiplyed by 3, as before is taught, the quotient will be 191 2/••0.
〈 math 〉〈 math 〉
FIrst dispose your num∣bers as followeth out of which you meane to draw the roote se∣parating your figures by two and two beginning at the latter end; but first strike the halfe circle 73/21/01 (and then say the root of 73 is 8, and set 8 before the half circle, & rests 9, then double the quotient 8, and say 2 times 8 are 16, and set the 6 vnder the last figure of the second part of figures, and 1 vn∣der
the first figure of the first part a•• in this first example. 〈 math 〉〈 math 〉 Then say how many times is 1 in 9, and it shall be 5. which you shall al∣so set vnder the 1 of the second sepa∣ration, as apppeares in this second example. 〈 math 〉〈 math 〉 And then say 5 times 1 are 5, which taken out of 9 rests 4, and 5 times 6 are 30, and 30 out of 32 rests 2, and 3 out of 4 rest 1, and then againe say 5 times 5 are 25, which out of 3••
rest 6. and 3. out of 12. rest 9. and then double the quotient, and say twice 5 are 10 set 0 vnder 0 of the last sepa∣ration, and keepe •• in memorie, and say twice 8 are 16, and 1 that I keepe in mind makes 17, then set downe 7 vnder the 5, and the 1 vnder the 6 of the middle separation, as appeares in this example following. 〈 math 〉〈 math 〉 And then say how many times is 1 in 9 and it shall be 5 times, which shall be set downe for quo••ient, and also vnder the last figure 1, and then say 5 times 1 are 5, out of 9 re••ts 4, and 5. times 7 are 35, which out of
36 rests 1, and 3 out of 4 rests 1, and 5 times 0 is 0, and 5. times 5 are 25. out of 31 rests 6, & 3 out of 10 rests 7, and 1 out of 1 rests 0, and so the rule is ended as appeares following. 〈 math 〉〈 math 〉
〈 math 〉〈 math 〉 After you haue separated your fi∣gures
by two and two, and drawne 251 the square roote, there doth yet rest 268, which must bee reduced in∣to fractions, and to begin set that rest 268 ouer a line at the end of your root, and that rest shall be numerator of the fraction; and to finde the de∣nominator only, double the root 251 if it be bigger then the rest, but if less as here, adde one to the doubling of the first figure, saying twice 1 is 2, and 1 that I adde makes 3, adde only double the rest, and set it vnder the line, and that shall be the denomina∣tor of the fraction; and to haue the root of this fraction here aboue, first take the root of the numerator, and set that root ouer a line, and it shall be numerator as appeares following.
〈 math 〉〈 math 〉 Then draw roote of the denomina∣tor, and set it vnder the line, and that shall be a denominator, and so you shall finde 16/22, and what rests is vnsen∣sible. 〈 math 〉〈 math 〉 But because that this fraction 16/22 is not precisely perfect, and that there is a rest in each extraction, you may ope∣rate as followeth to haue it mooue exactly, adde as well to the numera∣tor as to the denominator two 00, or foure, or sixe, &c. and from each pro∣duct
or quotiēt out of one figure for euery two 00 which you shall haue added, and the more that you adde 00, the more precisely you shall haue the roote as appeares following. 〈 math 〉〈 math 〉 But if it were proposed to extract the square roote of 16/25, there would no∣thing rest, nor would it bee needfull to adde any 00, for the root of 16 is 4 and the root of 25 is 5, and so wee should haue 4/5, and the like in all such other accidents; and thus much for Arithmetike. Vale.