The elements of arithmeticke most methodically deliuered. Written in Latine by C. Vrstitius professor of the mathematickes in the Vniuersitie of Basill. And translated by Thomas Hood, Doctor in Physicke, and well-willer of them which delight in the mathematicall sciences

CHAP. V. Of the second kind of Numeration.

HEtherto of prime Numeration. Now fol∣loweth second Numeration, vvhich num∣breth one number vvith an other, so often as one of the two numbers giuen requireth, and that ey∣ther by multiplying or diuiding: vvhereupon it is called Multiplication or Diuision.

Second Numeration is either often Addi∣tion, as Multiplication: or often Subtraction, as Diuision. For one of the numbers giuen is so often eyther encreased or diminished, as the other, eyther the multiplyer or diuisor re∣quireth, according to the number of the vni∣ties contayned in them.

Multiplication is a seconde Numeration, vvhich ioyning together the multiplicande, so often as there bee vnities in the Multiplier,

bringeth foorth the facit.

This is the 15. definition of the 7. Booke of Euclide. It maketh no matter which num∣ber you make the multiplicande, or which number you make the multiplier, as it appea∣reth by the 16. prop. of the seuenth Booke of Euclide, which sayth: If two numbers mul∣tiplied together, the one into the other, pro∣duce any numbers, the numbers produced are equall the one to the other: that is, they make one and the same number. Notwithstanding the Scholemen set the greatest number vp∣permost for the multiplicande, and the least neathermost for the multiplier, for that it see∣meth most conuenient for young beginners.

Multiplication of single numbers is to be con∣ceiued in our mindes by this Theoreme.

If eyther of the two numbers giuen be di∣uided into certaine partes, the product com∣ming of the whole numbers multiplied toge∣ther, is equall to the product made by one of the whole numbers and the parts of the other number so diuided. The 1. proposition of the second booke, or else, if both the numbers gi∣uen be diuided into certaine parts, that which is made of the whole number, is equall to the product made of their parts.

That we may multiply easily and readily, we must haue in our mindes, a table whereby we may know what the pruduct of euery sim∣ple figure is, being multiplied one by an o∣ther: the which thing may be easily done by

the rule of whole numbers and their partes. Euery child by the direction of nature can tell how many twise foure, or foure times fiue, or thrise sixe doe make. But if you happen to sticke in greater numbers, a litle exercise will make this table very readie. As if you would knowe how much seuen times eight is: di∣uide eyther of the two numbers giuen, into as many partes as you list, as 7. into 2. 3. 2. Then multiply 8. by euery one of these parts, and adde the particular productes together, and you shall haue 56. How many are eight times 9. diuide 9. into 3. 3. 3. and multiply 8. by those partes, adding the products toge∣ther, so you shall make 72. The same may be done if you diuide both the numbers. The examples must be set downe thus.

I.

8   2. 3. 2. 7	8   3. 3. 3. 9
16 24 16 56	24 24 24 72

II.

4. 4 8 3. 4 7	5 3 8 5 4 9
16 16 12 12 56	12 20 15 25 72

The Scholemen frame this Table by this Theoreme. Two numbers being giuē which ioyntly together are more then tenne, if you multiply the difference of each of them from tenne one by the other, & then subduct cross∣wayes one of the differences out of one of the numbers giuen, the product and the remain∣der parted into diuerse degrees, shall be the product of the numbers giuen. As you see in these examples.

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Multiplication of compound figures is drawen out of the Theoreme following.

If proceeding from the first of the nūbers giuen toward the left hand, you multiply the figures of the multiplier, into euery one of

the multiplicand, and ioyne together the par∣ticular products being lesse then ten, setting them orderly vnder their multiplier, and put the ouerplus to the degree following: the to∣tall made of the partes, is the product of the numbers giuen.

For example sake, multiply, 365 by 3. Here the numeration must proceede from part to part. Hauing therefore set downe the numbers, as you did in the former kindes of numeration, multiply the neathermost figure seuerally into euery one of the vppermost thus: three times 5 are 15, I set downe 5, and keepe 1: thrise 6 are 18, and one which I kept, are 19, I set downe 9 and keepe one to be ad∣ded to the degree following: thrise 3 are 9, and one are 10, which I set downe wholy. The example standeth thus.

365

3

1095

Now take an example of multiplication to be wrought by the parts of both the num∣bers giuen. Multiply 1568 by 54: Item mul∣tiply 3508476 by 2509. In both these num∣bers you shall haue so many rowes of num∣bers, as there be figures in the multiplier. Wherefore we must take diligent heede, that we confound not the particular products, and beware that in distinguishing the products, we set the first figure of euery one of them

vnder his particular multiplier, as you see here. 〈 math 〉〈 math 〉

An abridgement of multiplication.

Neither are the shortest wayes to be neg∣lected: this therefore is an especiall abridge∣ment of multiplication.

If one, or both the numbers giuen, haue cyphers in the beginning, thē if multiplying only the signifying figures together, you put the cyphers to the totall, you shall haue the product of the numbers giuen.

The correllary of this abridgement may be this.

If the last figure of the multiplier be an v∣nitie, and the other cyphers, then setting the cyphers before the multiplicand, you shall haue the product of the numbers giuen. The exsamples.

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Diuision is a numeration, vvhich drawing one number from another, as often as may be, findeth out the quotient of the greater.

In diuision there are three numbers to be considered, the Diuidende, the Diuisor, & the Quotient, the which must be so placed, that the Diuidend may stande aboue, the Diuisor beneath, and the Quotient at the side, or be∣twene them both. The vse of diuision is to declare how many times the lesse is contained in the greater. The Artisicers terme diuision by a Geometricall phrase, calling it Compari∣son, which is the applying of a measure giuen to any right line, as here,

A—|—|—|—|—B

C—D the line CD, is compared with the line AB to see how many times it may be cōtained in it. Or it is the ioyning of one side to an other side, giuen to make a rightangled parallelogram. As therfore in multiplication, the multiplying of two sides togither maketh a right angled figure, if the lines meete per∣pendicularly, euen so in diuision, the diuiding of the area or platforme of the right angled fi∣gure giuen by the length as it were by the di∣uisor findeth out the breadth which is repre∣sented by the quotient, as you see here.

[illustration]

Multiply the side A C, by the side C D, the product will be the area or platforme of the right angled figure A B C D. Likewise diuide the platforme of the right angled fi∣gure E F G H, by the side G H which is the length, you shall find out the other side E G which is the breadth which answereth to the quotient in diuision. Hereupon it is called the breadth of the comparison or of the com∣pared figure.

A quotient is a part of the diuidende, hauing the same denomination vvith the diuisor.

This definition is taken out of the 39 prop. of the seuenth booke of Euclide which sayth: If a number measure any number, the number measured shall haue a part of the same Denomi∣nation with the number which measureth it. And contrariwise, as it is in the 40. prop. If it haue a part, the number vvhereof the part taketh his denomination, shall measure it. For Euclide ta∣keth measuring here for diuiding: as for ex∣ample, because 48 may be diuided by 6 into 8 the quotient, I say that 8 being the quotient, is the 6 part of 48, and taketh his denomi∣nation of the Diuisor. And contrariwise, be∣cause 8 is the 6 part of 48, therefore 6 diuide 48. Likewise if 6 diuide 54, the quotient will be 9, and 9 therefore is the 6 part of 54, and because 9 is the sixth part of 54, therefore 6 diuide 54, &c.

A part of a number is a lesse number in re∣spect of a greater, when the lesse measureth the

greater.

This is the first definition of the 5 and the third def. of the seuenth booke. Euclide will not haue euery number that is lesse then a greater number, to be a part of the greater, but onely that which being taken oftentimes by it selfe alone, measureth the greater, that is, eyther maketh it, or taketh it cleane away: As two is the third part of 6, fiue is the halfe part often. For that 2 being three times taken, eyther maketh or taketh away 6, and 5 twice takē make 10 or destroyeth ten. This kind of part is called commonly, pars metiens, or men∣surans. A measuring part and of the barba∣rous it is called pars aliquota, an aliquot part, that is, a quotient: but the number which by it selfe alone being oftentimes taken, maketh not a number without the helpe of some other part or number, is the parts of the whole, v∣sing the worde parts in the plurall number for distinction sake. As 2 are the parts of fiue, because two make not fiue alone, without the helpe of three, or some other numbers. This kind of part they commonly call, pars consti∣tuens, or componens: of the barbarous it is cal∣led, pars aliquanta.

Diuision of a simple quotient may be taken out of the table of Multiplication. For by the same numbers that a number is made, by the same it may be diuided.

The first principle to be obserued in diui∣sion is to consider wittily, and to know rea∣dily

by what number, and what number eue∣ry one of the single figures do diuide another, as sixe times 7 are 42, therefore 7 diuide 42 by 6, and 6 diuide it by 7. Item seuen times 9 are 63, therefore 9 diuide 63, and the quotient is 7, or 7 diuide it, and the quotient is 9.

The diuision of a number that hath many fi∣gures, is wrought by the Theoreme follow∣ing.

If, beginning at the last figure of the num∣bers giuen, you multiply the particular quo∣tient of the diuisor contayned in the diuidend (setting it downe aside by it selfe) into the Diuisor, and then subtract the product from the diuidend, doing this as often as may be, by setting forwarde the Diuisor towarde the right hand, till you come to the first figure of the diuidend, the quotient of the parts will be the quotient of the numbers giuen.

Although diuision commonly be wrought by finding out the quotient by multiplica∣tion & subduction: yet the principall worke wherein the whole force or vertue of diuision consisteth, is onely the finding out of the quo∣tient, which being once found, the diuision in minde is supposed to be done. But when we haue found out the quotient, we are to thinke and consider with our selues into how many partes the diuisor cutteth the number which is set ouer it, and how often it may be drawen out of it. That this therefore may be set before our eyes, and that in remouing

forward the Diuisor, the remainder (wherein afterwarde the quotient must be sought for) may more manifestly appeare, the truth of that which we conceiued before, is proued by multiplication and subtraction.

To expresse this by examples, diuide 4936 by 4, we must seeke how often the Diuisor 4 being set vnder the last figure of the diui∣dend, is contayned in the same. I say then that 4 is contayned in 4 once: wherefore I set downe 1 in the quotient, and take 4 out of 4, there remaines nothing. Then set I the Di∣uisor one degree forwarde towarde the right hand, where 4 are contayned in 9 twice: wher∣fore I note 2 in the quotient, and multiply the Diuisor by it, the product is 8, which being subducted from 9, there remaine 1. The figures out of which the subtraction is made, must straight way be blotted out. Againe set forwarde the diuisor: there 4 may be taken from 13 three times, wherefore I set 3 in the quotient, and multiply 4 by it, the product is 12, which being drawen out of 13, there remaines 1. To conclude, I find 4 in 16 foure times, therefore I set downe 4 for the quoti∣ent, and multiply 4 by it, which being ta∣ken out of 16, there is nothing left. The ex∣ample.

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Againe deuide 1008 by 36. In this exam∣ple there be many things to be taken heede of. First, because the last figure of the diui∣dend is lesse then the last of the Diuisor, ther∣fore I set not 2 vnder 1, but one place farther towarde the right hande vnder the o. And first I consider with my sefe, how often three may be had in ten, I find it to be thrise, and there remaineth one. But for so much as I cannot subduct the Product made by multi∣plying the quotient into the whole diuisor, out of the number standing aboue it, to wit, 108 out of 100, therefore I take a quotient lesse then that by one, that is 2, whereby the diuisor being multiplied, there arise 72, which being taken out of 100, there remaine 28 to be written ouer the head. Then the diuisor being set forward, I see three to be contained in 28 nine times, but because I must haue re∣gard also of the figure following, I set but 8 in the quotient, which being multiplied into the diuisor, make 288 to be drawen out of the number set ouer it. The example is thus.

It appeareth therefore by this, that the que∣stion must be made not of the whole Diuisor, vnlesse it be a single number, but only of the last figure of the Diuisor, and euery particu∣lar must be but a single figure, as if you shold seeke how often 2 were in 21, you can not take it tenne times, but 9 times, but 9 times at the most. Moreouer the nature of the thing requireth that we find out alwayes such a quotient, as being multiplied by the Diuisor, maketh no greater a number then the diuidend is. See the examples following.

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An abridgement of diuision.

If the Diuisor end in cyphers, the worke may be wrought by the signifying figures a∣lone setting the cyphers in the meane time, vnder the vtmost figures of the diuidend, next to the right hand. But if the last figure of the Diuisor be an vnitie, and the rest cy∣phers, then setting the Diuisor as before, and

taking the figures that haue no cyuphers vn∣derneath them for the quotient, the diui∣sion is dispatched. the figures that remaine after the diuision is done, must be set downe as the partes are with the diuisor vnderneath them for their denominator.

As for example, diuide 165968 by 360. Item 6734 by 100: the example shall stand thus.

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