Numeration.

NUmeration is accounted the first part of Arithmetick, and it is to know how to read a Summe of fi­gures expressed in writing; or to write down any Summe to be ex­pressed.

To the doing of which there are four things necessary.

First, to know their number, which is Nine.

Secondly, their shapes, which are 1 2 3 4 5 6 7 8 9. Of which the first toward the left hand ever signifieth One, the second Two, &c.

Thirdly, to know the value of their places.

Lastly, How their proper signification i [...] al­tered thereby.

The value of their places is thus, When two, three, or more figures stand in one Summe, tha [...] is, without any Point, Line or Comma betwixt [Page 2]them, as 321, that place next the right hand where the figure 1 standeth, is called the place of Unity, or Unities, and the figure 1 standeth in that place onely for one, and the figure 2 when it is found in that first place, stands only for two; and the like of the rest.

But in the Summe 321, above expressed, the figure 2 is in the second place, and every place conteines the value of that place before towards the right hand ten times; and therefore the fi­gure 2 doth not signifie two, but (in this second place) ten times two, that is Twenty. And so the figure 3 if it had been in that place had sig­nified ten times three, that is Thirty; but be­ing here in the third place it signifies ten times thirty, that is Three hundred. And so the whole Summe 321, is to be read, Three hundred twenty and one.

Fourthly, It is hereby seen, how their proper significations, which were Three, Two and One, are altered by being thus placed, and the Sum, which otherwise had been but Six, is Three hun­dred twenty one, as before.

In like sort, if their had been more places, as seven, the value is quite through increased ten times by being a place behinde towards the left hand; as in the Summe 1̇111̇111̇, The fi­gure 1 in the second place stands for ten times one (that is ten,) in the third for ten times ten (which is one hundred,) in the fourth for ten hundred, (which is called one thousand,) in the fifth for ten thousand, in the sixth for ten [Page 3]times ten thousand (which is one hundred thou­sand) in the last (here the seventh) place for ten hundred thousand, which is called a Milion: and so on, if there were more places, observing the same order.

Now to read this readily, make a prick over the place of Unity, another over the third from it, and over every third still towards the left hand, for so those points will be over the places of Unites, Thousands, and Milions; and so be­ginning at the last, that is, at the left hand, read one Milion, and because the three following to­wards the right signifie properly one hundred and eleven, but the prick belonging to them is in the place of thousands, call them one hundred and eleven thousand, and three remaining be­ing under the point over Unity, signifie onely one hundred and eleven; and all three points read together in one summe is, One Milion, one hundred and eleven Thousand, one hundred and eleven.

In like manner, if this number 73598624 were given to be read (according to former directions) make a prick over every third figure, beginning with the first figure towards the right hand (which is the place of Unity) and then will your number stand thus.

73̇598̇624̇

Then for the ready reading thereof (because the third prick signifieth Millions) call all the fi­gures towards the left hand, standing from that prick, Milions, which in this example are 7 and 3, [Page 4]so then this number conteins 73 Millions, 598 Thousand, 624 which in words at length we read Seventy three Millions, five hundred ninety eight thousand, six hundred twenty four.

Let thus much suffice concerning the planing of large numbers, for the ready reading of them, only take these foure Tables following, for illu­stration of what hath been hitherto delivered in words, the very sight whereof is better then a whole Chapter of information.

The first Table is thus to be read] One in the first place signifies One. One in the second place signifies Ten. One in the third place sig­nifies a hundred, &c. as in the Table.

The second Table is thus to be read] In this Table you shall find the last number thereof to consist of these figures, 357.846.903. with a point or comma betwixt every third figure, for di­stinction sake, and also every three figures in their order are connected together with this

brace, which denominate the places of Millions, Thousands, Hundreds, so that the last number of this Table will evidently appear to be 357 Millions, 846 Thousands 903.

The third Table is only certain rowes of fi­gures set together and orderly disposed, having the signification or reading of the same numbers in words at length to them annexed, and is only inserted for the better satisfaction of such as shall doubt whether they perfectly understand what hath been before taught.

The fourth Table is much like the second, on­ly it consisteth but of one number and extends [Page 5]three places farther then the greatest number in the second Table doth: viz. to twelve places which figures are thus to be read. 736 Millions of Millions, 842 Millions, 708 Thousand, 645.

[Page 6]

Addition.

ADdition is the collecting or gathering to­gether of two or more sums, either of one or of diverse denominations, into one sum, which is called the (Aggregate) (Total) or Grosse sum.)

In addition of numbers of one denomination, the order is to set the numbers to be added one directly under the other, that is to say Unites un­der Unites, Tens under Tens, Hundreds under Hundreds, &c.

The Rule.

Having placed your numbers to be added in due or­der, one under another, draw a line under them, and begin at the lowermost figure towards your right hand, and adde that to the next figure above, and the summe of them to the next figure above that, proceeding in this or­der till you have added the whole line together, which when you have done, consider how many tens are contain­ed in that line, and for every ten keep one Ʋnite in your mind, to be added to the next row, but if there be any odde digits, you must set them beneath the stroake, just under the line you added together, having thus [Page 8]finished the addition of one line proceed to the next, and from thence to the third, and so forward, be there never so many. The Examples following will make this plain,

Example 1.

Let the numbers given to be added together be 7832, 5609, 376, 8547, having thus placed them in order one under another as in the Margine is done, draw a line under them, then be­gin your addition, at the lowermost figure towards your right hand, say­ing, 7 and 6 is 13, and 9 is 22, and 2 is 24 now because in 24, there is two tens, and 4 remaining, I place the 4 under the line, and carry the two zens to the next row, saying, 2 which I carryed and 4 makes 6, and 7 makes 13, and 3 makes 16, in which row there is but one ten conteined and 6 remaining, which 6 I set under the line, and carry the ten to the next row of hundreds, say­ing, 1 that I carryed and 5 makes 6, and 3 makes 9, and 6 makes 15, and 8 makes 23, in which 23, ten is conteined two times, and 3 re­maining, the three I place under the line, and carry the two tens to the next row, saying 2 which I carryed and 8 makes 10, and 5 makes 15, and 7 makes 22, in which ten is contein­ed two times, and 2 remaining, which 2 I set under the line, and because there is never ano­ther [Page 9]row to be added (to which I should carry the two tens) I therefore set it down also under the line towards the left hand, as you see done in the margine, so the total or grosse sum of these numbers being added together is 22364.

Example, 2.

A man hath in his Orchard 136 Apple trees, 76 Pear trees, 107 Cherry trees, and 36 Plum trees, and he desires readily to know how many trees he hath in all, place your uum­bers one under another as in the margine, and then begin to adde them together, at your right hand, saying, 6 and 7, make 13, and 6 make 19, and 6 make 25, place 5 under the line, and carry 2 to the next row, saying 2 and 3 is 5, and 7 is 12, and 3 is 15, place 5 under the line, and carry 1 to the next row, saying 1 and 1 is 2, and 1 is 3 which 3 I set under the line, and because there was no tens conteined in that line therefore the totall is 355, and so many Trees are in the Orchard.

Other Examples for Practice. [...]

Addition of numbers of diverse Denominations. 1 Addition of English Money

The most usual Coyns used in England are Pounds, Shillings, Pence, and Farthings, of which Coyns

• 4 Farthings make 1 Peny thus charactred d.
• 12 Pence make 1 Shilling thus charactred s.
• 20 Shillings make 1 Pound. thus charactred li.

for a farthing we use q.

THE RƲLE.

In the addition of numbers of divers deno­minations this order is to be observed, viz. Place all numbers of the same denomination one direct­ly under another, as Pounds under pounds, shillings under shillings, pence under pence, and farthings un­der farthings. Then draw a line under them, and begin your Addition with the least denomination first, observing how many times the next greater denomina­tion is contained in the least, and for every time carry one unite to the next denomination, as before you did the tens, setting down the remainder if any be, then adding the next denomination together, take notice how many times the next greater denomination is contained in that lesser, carrying for every time one to the next, [Page 11]thus proceeding till you have gone over all the denominations, be they never so many.

Example 1.

Let the numbers to be added toge­ther be 37 li. 16 s. 9 d. 3 q. 21 li. 9 s. 8 d. 1 q. 13 li. 12 s. 9 d. 2 q. Place the numbers as in the margine, draw a line under them, and begin with the least de­nomination (which in this example is farthings,) first, saying, 2 q. and 1 q. is 3 q. and 3 q. is 6 q. which is one peny and 2 q. remaining, which 2 q. I place under the line, and carry the one peny to the next row which is the place of pence, saying, one peny and 9 d. is 10 d. and 8 d. is 18 d. which is 1 s. and 6 d. (Now against the 8 I make a prick with my pen for my better remembrance to signifie that there is one shilling to be carried to the place of shillings,) then go on and say 6 d. and 9 d. is 15 d. which is 1 s. and 3 d. therefore against 9. I make a prick with my pen, and (because that is the last num­ber) I set down the odde 3 d. under the place of pence, and (because I finde two pricks in the line of pence, therefore) I carry 2 s. to the place of shillings saying 2 s. which I carryed, and 12 s. is 14 s. and 9 s. is 23 s. which is one pound, and 3 s. remaining, I make a prick against 9, and go­ing on, say 3 s. and 16 s. is 19 s. which (being there is no more numbers to be added, and being also lesse then 20 s.) I set under the line, and finding one prick in the line of shillings, I therefore car­ry [Page 12]one to the place of pounds, saying, one which I carryed and 3 is 4, and 1 is 5, and 7 is 12. set down the 2 under the line (as in addition of num­bers of one denomination) and carry one to the next row, saying one that I carried and 1 is 2, and 2 is 4, and 3 is 7, which being the last I set down, and so the total or grosse sum is 72 li. 19 s. 3 d. 2 q.

Example 2

Let the numbers to be added be 29 li. 16 s. 8 d. 32 li. 17 s. 9 d. 81 li. 13 s. 11 d. and let it be required to find the totall or grosse sum. Here in this Example the least denomination is pence, therefore I begin with them, and say, 11 d. and 9 d. is 20 d. which is 1 s. and 8 d. make a prick a­gainst the 9 and say 8 d. and 8 d. is 16 d. that is 1 s. and 4 d. make a prick against the 8 and set down the odde 4 d. then (because there are two pricks in the line of pence) you must car­ry 2 s. to the place of shillings, saying 2 s. which I carry and 13 s. is 15 s. and 17 s. is 32 s. which is 1 l. 12 s. make a prick agianst 17, and say 12 s. and 16 s. is 28 s. make a prick a­gainst 16, and (because there is no more num­bers to be added) set down the odde 8 s. under shillings, and (being there is two pricks in the line of shillings) carry 2 to the place of pounds, saying 2 and 1 is 3, and 2 is 5, and 9 is 14, set down 4 and carry 1 to the next line, and say 1 and 8 is 9, and 3 is 12, and 2 is 14, which (be­cause [Page 13]it is the last) you must set down, so is the totall or grosse sum 144 li. 8 s. 4 d.

2 Addition of Troy Weight.

Troy weight is a Weight used in England, by the which is weighed, Bread, Gold, Silver, Pearl, &c. the most usual denominations of which weight are, Pounds, Ounces Peny weights, and grains, of which

• 24 Grains make 1 Peny weig. thus charactred pw.
• 20 Peny weight make 1 Ounce thus charactred ou.
• 12 Ounces make 1 Pound thus charactred lib

for a grain we write gr.

The Addition of Troy weight (and conse­quently of any other weight or measure what­soever either Domestique or Forreign) differeth nothing at all from the addition of Coine last [Page 14]taught, if the affinity of one denomination to an other be first known, for whereas in money be­cause 1 [...] d. make 1 s. you therefore observe how many twelves there are in the addition of your pence, and for every 12 you adde one shilling to the place of shillings, so in the addition of Troy weight, knowing that 24 gr. make one pe­ny weight, you must therefore in the addition of Grains of Troy weight observe how many times 24 you finde in your line of Grains, and for every 24, carry one to the place of peny weights, likewise in the addition of peny weights Troy, you must consider how many times 20 is con­tained in your line, and for every 20 carry one to the place of ounces, (because 20 peny weights make an ounce.) Also in the addition of Ounces Troy, you must observe how many times 12 you finde in your line of ounces, and for every 12 carry one to the place of pounds then lastly adde your pounds together, as numbers of one denomination.

Example.

Let the numbers to be added to­gether be 7 li. 11 ou. 13 pw. 19 gr. 6 li 7 ou. 16 pw. 19 gr. 3 li. 7 ou. 9 pw. 6 gr. Place your numbers as in ad­dition of money, each under other according to their respective de­nominations, as in the margine then draw a line under them and begin your Addition with the least denomination first, viz. grains, [Page 15]saying 6 gr. and 19 gr. is 25 gr. which is one pe­ny weight and one grain, make a prick against 19, and carry the odde grain to the number a­bove, saying 1 gr. and 19 gr. is 20 gr. which (because it is lesse then one peny weight) I set under the line, then finding one prick in the line of grains, I (therefore) carry one to the place of peny weights, saying 1 and 9 is 10, and 16 is 26, which is one ounce, and 6 pw. make a prick against 16, and say 6 and 13 is 19, which (being lesse then an ounce) set under the line, then for the one prick, carry 1 to the place of ounces, saying 1 and 7 is 8, and 7 is 15, which is one pound and 3 ou. make a prick at 7, and say 3 and 11 is 14, which is one pound and 2 ounces, set down the 2 ounces, and for the two pricks carry 2 pounds to the place of pounds, saying 2 and 3 is 5, and 6 is 11, and 7 is 18, which set under the place of pounds, so is your addition ended, and the sum is 18 li. 2 ou. 19 pw. 20 gr.

3 Addition of Avoirdupois little Weight.

There is another kind of weight most com­monly used in England called Avoirdupois little weight, by which is weighed all sorts of wares or merchandise Garbable, as Suger, Pepper, Cloves, Mace, &c. This weight is commonly di­vided into these denominations, Pounds, Ounces, and Drams, of which

• 16 Drams make 1 Ounce thus charact. cu.
• 16 Ounces make 1 Pound thus charact. li.

for a dram we write dr.

In the addition of Avoirdupois weight, you must observe the very same method and order as in Money and Troy weight, having due re­spect to the quantity of the denomination, as in the addition of drams to make a prick at every 16, setting down the remainder and for every prick carrying a unite to the next place. The pre­ceding rules being so copious in this particular I shall forbear to make any verball illustration, but only give you some examples ready wrought, to­gether with the most usual parts into which the Weights and Measures now used in England are divided into: which to the ingenious will be of most validity.

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4 Addition of Avoirdupois great weight.

There is a weight commonly used in England, by which is weighed all commodities that are sold by the hundred, as Corents, Wool, Flesh, Butter, Cheese, and the like, the which hun­dred weight containeth 112 pounds, and the hundred weight is divided into quarters, pounds, and ounces, so that

• 16 Ounces makes 1 Pound thus chaeactred li.
• 28 Pounds makes 1 quarter of a C. thus chaeactred qr.
• 4 quarters makes 1 Hundred weig. thus chaeactred C.

for an Ounce we write oz.

[Page 18]

I might further proceed to shew you Exam­ples of addition of common English measures, viz. of long measures, Liquid measures, and dry measures, as also of Time, Motion, &c. but the preceding Examples being of sufficient ex­tent, I shall forbear to trouble either my selfe or the Reader with that which I conceive superflu­ous: Only, before I leave Addition, I will give you a brief view of the most usual measures in England, which take as followeth. And

The Proof of Addition.

Having placed your numbers in order, and added them together, and set the Total under the line, cut off the upper number by drawing a line with your pen betwixt that and the others, then adde all the numbers together except the uppermost, and set the Total of them under the Total before found, then adde this last Total, and the first number which you cut off with your pen together, and if the sum of those two num­bers be equal with your Total sum first found, then is your work right, otherwise not.

Example. In the first example of whole num­bers, the sums to be added were 7833, 5609, 376, and 8547, these numbers placed in due or­der and added together, the total or grosse sum of them is 22364, now to prove whether this [Page 22]Total be true or not, I cut off the uppermost number, (to wit 7832,) with a dash of the pen, and I adde the other three numbers together, name­ly, 5609, 376, and 8547, and the Total of them is 14732, which number be­ing added to 7832 (the number cut off) the sum of them is 22364 exact­ly agreeing with the To­tal first found, cleerly evi­denceth that the addition was truly performed; but if they had disagreed then the work had been erroneous. The like course must be taken for the proof of those sums which have different denominations as in Mo­ney and Weight, as by the examples following will appear.

Subtraction.

SUbtrrction is the taking of one or more small sums out of a greater, as 7 s. out of 12 s. or 37 li. out of 100 li. or 137 foot out of 983 foot, and the like.

As in Addition the sums to be Added may be either of one or of diverse denminati­ons, so likewise they may be in Subtraction, and the manner of placing them is the same, for you must set Unites under Unites. Tens under Tens, Hundreds under Hundreds, &c.

Example 1

Of Subtraction of numbers of one denomination, let it be required to subtract 234, out of 986, place the numbers one unde the o­ther as you see done in the Margine, draw a line under them and begin with the first figure to­wards your right hand, which is 4, saying take 4 from 6, and there remains 2, place 2 under the line, and go to the next figure which is 3, say­ing, take 3 from 8, and there remains 5, place 5 under the line, and go to the next figure which is 2, saying take 2 out of 9 and there re­mains 7, place 7 under the line, and your sub­traction is ended, and it is evident by the work, [Page 25]that if you take 234 out of 986 there will remain 75 [...], which you may thus prove, for if you adde the 234 to 752, you shall find the sum of that addition to be 986, which is equall to the whole sum from which 234 was subtracted,

Example. 2.

Let it be required to subtact 2976 out of 96527, place the numbers one under ano­ther as in the margine you fee done, then draw a line under them, and beginning with the first figure towards your left hand, say take 6 out of 7, and there remains 1, place 1 under the line and pro­deed to the next figure, saying 7 out of 2 I can not (wherefore you must alwayes adde 10 to the number above which in this example is 2, and it makes it 12,) therefore take 7 out of 12, and there remains 5, place 5 under the line, and (be­cause you added 10 to the 2 to make it 12, you must) carry a unite to the next figure, saying, one which I carryed and 9 is 10, take ten out of 5, which I cannot, therefore I must adde 10 to 5 and it makes 15, and say 10 out of 15, and there remains 5, place 5 under the line, and because you added 10 to 5 to make it 15, you must therefore carry a unite to the next figure saying, one which I carryed, and 2 is 3, take 3 out of 6 and there remains 3, place 3 under the line, and because there is no more figures to be subtracted from the number above, you must, say nothing fom 9 and there remains 9, set the 9 under the line, and your Subtraction is ended.

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Questions performed by Addition and Subtraction.

Question 1. What number is that which added to 376 shall make 1000? Subtract 376 from 1000, the remainder is 624, the number sought.

Question. 2. What number of pounds, shil­lings and pence must be added to 36 l. 17 s. 3 d. to make that sum up 100 li. subtract 36 li. 17 s. 3 d. from 100 li. the remainer is 63 li. 2 s. 9 d. which added to 36 li. 17 s. 3 d. makes 100 li.

Quest. 3. In the year of our Lord 1440, the famous art or mystery of Printing was invented, I would know how long it is since that time to this yeer of our Lord 1655. From 1655, subtract 1440, the remainer is 215. and so many years are expired since printing was invented.

Question 4. An Armie consisting of 13721 horse, and 26850 foot, in an engagement there were slain 3760 hors, & 7523 foot, the question is how many were slain in all, & how many horse and how many foot escaped. From the 1372 horse which went out, subtract the 3760 that were slain, there remains 9961, and so many horse escaped, Also from the 26850 foot which went out, subtract the 7523 which were slain, and there remains 19327, the number of foot [Page 32]which escaped, and by adding the 3760 horse which were slain, to the 7523 foot that were slain, their Total is 11283, and so many were slain in all.

Multiplication.

MUltiplication is that part of Arithme­tick which teacheth how to encrease one number by another, so that the number produced by their Multipli­cation shall contain one of the numbers multipli­ed so many times as there are unites contained in the other. Multiplication may fitly be ter­med a Compendium of Addition, for that it performeth at one operation the same which to effect by Addition would require many. For instance, if it were required to know how much 7 times 5 is, to perform this by addition, I must set seven fives, or five sevens, one under another, and adding them together, I shall finde that either of their Totalls shall contain 35, but this by multiplication is performed with far more brevity, as by examples hereafter shall appear.

Before you enter upon the practice of Multi­plication, it is necessary to remember the pro­duct produced by the multiplication of any one of the nine digits, by any other of the same, as readily to know; that 4 times 5 is 20, 6 times 7 is 42, 2 times 9 is 18, 7 times 9 is 63. 8 times 9 is 72, &c. Which this Table following, will[Page 34]plainly declare, and must be perfectly learned by heart before you attempt to multiply great numbers.

• 2 times 2 makes 4
• 2 times 3 makes 6
• 2 times 4 makes 8
• 2 times 5 makes 10
• 2 times 6 makes 12
• 2 times 7 makes 14
• 2 times 8 makes 16
• 2 times 9 makes 18

• 3 times 3 makes 9
• 3 times 4 makes 12
• 3 times 5 makes 15
• 3 times 6 makes 18
• 3 times 7 makes 21
• 3 times 8 makes 24
• 3 times 9 makes 27

• 4 times 4 makes 16
• 4 times 5 makes 20
• 4 times 6 makes 24
• 4 times 7 makes 28
• 4 times 8 makes 32
• 4 times 9 makes 36

• 5 times 5 makes 25
• 5 times 6 makes 30
• 5 times 7 makes 35
• 5 times 8 makes 40
• 5 times 9 makes 45

• 6 times 6 makes 36
• 6 times 7 makes 42
• 6 times 8 makes 48
• 6 times 9 makes 54

• 7 times 7 makes 49
• 7 times 8 makes 56
• 7 times 9 makes 63

• 8 times 8 makes 64
• 8 times 9 makes 72

• 9 times | 9 | makes 81.

Compendiums in Multiplication.

1 If the Multiplier consist of cyphers in the last place or places, [...] you may omit the multipl­cation of them, and place the former figures of the Multiplier under the Multiplicand, thus if it were required to multiply 3257 by 2600. place the numbers as you see in the margent, then multiplying 3257 by 26, the Product will be 84682, to which if you add two cy­phers, (because there were two cyphers in the Multipli­er) it will be 8468200, which is the true product of the multiplication.

2 If it be required to multiply any number by 10, 100, 1000, 10000, &c, You have no more to do but to add so many cyphers to the multi­plicand as there are cyphers in the multiplier, thus if you were to multiply 365 by 10, the pro­duct will be 3650, or by 100, it would be 36500, or by 1000, it would be 365000, or by 10000, it would be 3650000, &c.

3 If any number given were to be multiplied by 5, you may abreviate your worke thus, adde a cypher to the Multiplicand, take halfe that number, and it shall be the product required, thus if it were required to multiply 8627 by 5, adde a cypher to the multiplicand, then it is [Page 42]86270 the halfe whereof is 43135, which is the product required.

The Proof of Multiplication.

The most certain proof of Multiplication is by Division, but because Division is not yet known, I will here shew a neer way by which Multipli­cation may be proved. Which is thus,

THE RƲLE.

First,

take a Crosse as in the Margine, then, any sums being multiplied, you may prove the truth of your work in this manner, (1) Cast away all the nines which you can find in the multiplicand, what re­maineth set on the right side of the Crosse. (2) Cast away also the nines in the Multiplyer, and what re­mains set on the left side of the Crosse. (3) Multiply the figure on the right side of the Crosse by that on the left side of the Crosse, and out of that product cast away the nines, setting the figure remaining over the Crosse, then (4) Cast away all the nines in the product, and if the figure remaining be the same with that which standeth over the Crosse, then is your multipli­cation truly performed, otherwise not.

Example 1 [...]

Let it be required to prove the Sum in the margine.

1 Cast away all the nines in the Multipli­cand, saying 4 and 3 is 7, and 2 is 9, which being rejected there [Page 43]remains 4, which I set on the right side of the crosse, then

2 Cast away all the nines in the Multiplier, saying 2 and 3 is 5, which (being lesse then 9) I set on the left side of the crosse, then

3 Multiply 4 by 5, saying 4 times 5 is 20 from which cast all the nines, and there remain 2, place 2 over the crosse, and

4 Cast away all the nines in the Product, say­ing, 2 and 5 is 7, and 4 is 11, cast away 9, and there remains 2, which exactly agrees with the figure over the crosse, and demonstrates that the multiplication is truly performed.

Questions performed by Multiplication only.

Question 1. If a piece of land be 236 perches long, and 182 perches broad, how many square perches are contained therein? multiply 236 the length, by 182 the breadth, the product is 42952, and so many square perches are contain­ed in such a square piece of land.

Question 2. In a year there are 365 dayes na­tural, and in every day 24 houres, how many houres be there in a year? Multiply 365 the number of dayes, by 24 the number of houres, the product is 8760, and so many houres be there in a year.

Question 3. From London to Coventry it is ac­counted 76 miles, how many yards therefore is it from London to Coventry? Multiply 1760 (which are the number of yards contained in one mile) by 79, the product is 133760, and so many yards are between London and Coventry.

Division.

DIvision is the just contrary to Multipli­cation, for that turnes small denomina­tions to greater, as Multiplication turns greater to smaller. Or (in whole Num­bers, of which only we yet speak) Division is the asking how many times one Sum is contained in ano­ther? and the number which answereth to that question is called the Quotient.

And the number containing, which is to be divided, is called the Dividend.

And the number contained, or by which the Dividend is to be divided, is called the Divisor.

And as often as the Dividend contains the Divisor, so often doth the quotient containe Unity.

The wayes of performing Division are divers I will begin with that which is most used and taught, which is as followeth.

THE RƲLE.

Place the Divisor under the Dividend, so that the figures next to the left hand stand directly one under the other, if the rest of the Divisor be not greater: or [Page 46]if all the Divisor be greater then that above it, then the said Divisor must be devolved one place further toward the right hand; having so placed them, try how many times the lower figures are contained in the upper figures, and write that figure which answereth that question within a crooked line in the margine of the work, which is called the Quotient, and by it mul­tiply the first figure of the Divisor, and take the pro­duct out of the figures directly over it, beginning the Subtraction toward the left hand; then cancel that figure of the Divisor, and also, that of the Dividend which hath been already used, with a light dash of a pen, and write the remaine (when the product of the first figure multiplied by the quotient is subtracted as before) just over the figure used and cancelled; the [...] proceed to doe the like with the second, third, and fourth figure of the Divisor if there be so many; till having cancelled it all, and set the remaine orderly a­bove the Dividend, you have finished one work.

Now if the Dividend have still some figures un­touched towards the right hand, then remove the Di­visor still toward the right hand, but one place at a time, and then againe aske or try how many times the lower may be had in the upper, and write the answer in the quotient whether it be 1 or more, (onely it can­not be above 9) or nothing, then put 0 in the quotient, and multiply the Divisor by this new figure, and sub­tract the product, setting the remainer orderly above, as before, this work must be repeated by removing the Divisor still one place towards the right hand, un­til Ʋnity in the Divisor stand under Unity in the Di­vidend, and then the work is done.

A Second way of Division.

Although (as I have said) the former way is more used, yet this may seem plainer and more natural to some, I will therefore give one exam­ple of it.

A Third way of Division.

There is another kind of division which is ve­ry much used, and is in most request with those who have most occasion to divide great num­bers, the manner of working is not much unlike the way before taught, one or two examples will make it plain.

Example 1. [...] Let it be required to divide 162483 by 1321, set down your numbers as you see them placed in the margine, viz. First, set down 162483 the dividend, then on the left hand there­of set the divisor 1321 with a crooked line between them, then on the right hand there­of make another crooked line which must serve to set the figures of the quoti­ent in, so are your numbers placed in due order, then draw a line under the dividend, and make a prick under the figure 4, because so far the fi­gures of the divisor would extend if they had been placed under-neath the dividend, according as in the other examples, this prick serves onely to shew how far you have proceeded in your work, and must at every division be removed a place further, till at length you come to the last figure of the dividend: your numbers being thus placed with a line under them, you are ready for the work, which must be performed according to the directions of the following rule.

[Page 58] THE RULE.

Demand how often the divisor may be had in the di­vidend, and place that number in the quotient, then multiply the divisor by the quotient, and place the pro­duct under the line: then subtract this product from the dividend, and set the remainer under the product, then make a prick under the next figure of the dividend, and bring that figure down to the remainer, and then proceed as before.

Example, [...] Your numbers being placed as is be­fore directed, you may begin your work in this manner, first, say how many times 1321, can I have in 1624? say once, place 1 in the quoti­ent, by which 1 mul­tiply the divisor 1321, beginning at the left hand, saying, once one is 1, place 1 under the line, then once 2 is 2, set 2 under the line, then once 3 is 3, place 3 under the line, lastly, once 1 is 1, place 1 under the line, then subtract this 1321, from 1624, and there will remain 303, to this 303 bring down the next figure in the dividend, namely 8, so will that number be 3038, under which draw a line, and repeat the same work again, saying, how many times 1321 can I have in 3038, which [Page 59]may be had two times, place 2 in the quotient, by which 2 multiply the divisor 1321, saying 2 times 1 is 2, place 2 under the line, then 2 times 2 is 4, place 4 under the line, then 2 times 3 is 6, place 6 under the line, lastly, 2 times 1 is 2, place 2 under the line, and subtract this 2642 from 3038, and there will remain 396, to this 396 bring down the next figure of the dividend, which is 3, so is this number made 3963, under which draw a line, and repeat the work once a­gain, saying, how many times 1321 can I have in 3963, which may be had 3 times, by which 3 multiply the divisor 1321, saying 3 times 1 is 3, then 3 times 2 is 6, then 3 times 3 is 9, and lastly, 3 times 1 is 3, which place under the line, and subtract it from the line above, which in this example is the same number, therefore there re­mains nothing, and the work is ended, but if a­ny remainer had been, that should have been set under the line, as by the examples following will appear.

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 [Page 61]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 [Page 62]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, [Page 63]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 [Page 64]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.

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 [Page 66]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.

Progression.

IS also of two sorts, the first is of certain num­bers in Arithmetical Proportion from 1, that is, such as differ equally, as 1, 2, 3, 4, 5, 6, where the common difference is 1, (as is easily seen,) or 1, 3, 5, 7, 9, 11, where the common difference is 2, or any other, as 1, 8, 15, 22, 29, 36, where the common difference is 7, this is called Arithmetical Progression.

2 Secondly, of certaine numbers in Geometri­cal proportion from 1, that is such as increase by a common Multiplication, as 1, 2, 4, 8, 16, 32, where the common multiplier is 2, that is the first by 2, produceth the second, and the second multiplied by 2, produceth the third, and so on.

Or as 1, 3, 9, 27, 81, 243, where the common Multiplier is 3, this is called Geometrical Progres­sion.

Both the common difference (in the first) and the common Multiplication (in the later) shall for shortnesse hereafter be called the common ex­cesse.

First, now of the first sort, or Arithmetical [Page 71]Progression, the principal use of this is,

1 If the number of places, and common ex­cesse be given, to find the last number.

2 When the number of places, and the last number is given, to find the aggregate, or total summe of all the numbers.

3 When the last number, and the total summe is given, to find the number of places.

4 The Number of places, and total summe being given to find the last number.

5 The last number, and number of places gi­ven, to find the common excesse.

6 The last number, and common excesse be­ing given; to find the number of places.

I will instance in no more, few of these ever hapning to be used.

THE RƲLE.

Multiply the number of places lesse by 1, by the common excesse, and to the product adde the first number; the summe is equal to the last number.

So here multiply 99 by 1, the product is 99, (for 1 neither multiplies nor divides) to this adde the first number 1, it gives 100 for the last number.

Or let the numbers be 1, 7, 13, 19, 25, 31, [Page 72]where the common excesse is 6. and the number of places also 6.

Now (if the number of places lesse by 1, that is) 5 be multiplied by the common excesse, which is 6, the product is 30, to which adding the first number which is 1, the last number 31, is thereby composed. This is so casie that it needs no proof.

2 For the second, which is, The last number, and the number of places given, to find the total summe of all the numbers.

THE RƲLE.

Adde the first and last numbers together, and mul­tiply the summe by half the number of places, the pro­duct is equal to the aggregate or summe of all the num­bers added together.

So if to the first number 1 be added the last number 100, it gives 101, which multiplied by 50 (which is half the number of places) produ­ceth 5050, which is equal to all the hundred num­bers added together.

And hereby may that vulgar question be an­swered, which is,

If a man take up 100 stones placed a yard one from another all in a right line, by one at once, and bring them back one by one to his first standing, how many yards doth he goe backwards and forwards?

It is shewed before that he goes forward 5050 yards, and he must needs come back just as much; that is, in all 10100 yards. which is 5 miles and 3 quarters; wanting 20 yards.

[Page 73]Or secondly, suppose the numbers were 1, 9, 17, 25, 33, 41. Whereof the common excesse is 8, the first and last added gives 42, which multiplied by 3, (half the number of places) the product is 126, which is the summe of them all.

3 For the third thing, that is, by the last num­ber and the total, to find the number of places.

THE RULE.

Adde the first and last numbers, and by the summe of them divide the total, [...] the quotient will be equal to half the number of places.

This is so plain it needs no cleering.

4 For the fourth, if the total, and number of places bee given, to find the last number.

THE RULE.

Divide the total by half the number of places, the quotient is a number, from which if 1 be taken, the rest is the last number.

As let the number be 1, 3, 5, 7, 9, 11, 13, 15, or any other (in arithmetical proportion) whatso­ever. The summe of these is to be 64. And the number of places is 8, the half of it 4. Now if 64 be divided by 4, [...] the quotient is 16, from which if 1 be ta­ken, there remains 15 for the last number.

5 Now for the fist variety, If the last number [Page 74]and number of places be given, to find the com­mon excesse.

THE RƲLE.

From the last number take 1, and the remaine shall be the Dividend; then from the number of places also take 1, and make this later remaine the Divisor▪ then the quotient of this Division shall be the common excesse.

Example, Let the numbers be 1, 4, 7, 10, 13, 16, from 16 take 1. remains 15, for the dividend, then from 6, (which is the number of places) take al­so 1, remains 5 for the Divisor.

Now when 15 is divided by 5, the quotient is 3. And 3 is also the common excesse, or dif­ference between 1 and 4, or 4 and 7, &c.

6 Lastly, let the last number, and the com­mon excesse be given, to find the number of places.

THE RULE.

From the last number take 1, and divide the re­maine by the common excesse; then to the quotient adde 1, the summe is the number of places.

As let the numbers be 1, 5, 9, 13, 17, 21, 25, 29, from 29 take 1, remains 28, which divide by 4, (which is the excesse) the quotient is 7, to which adde 1, the summe is 8, which is the number of places, as the reader may easily count.

Geometrical Progression.

I shall not be so large in this as in the former, because these things are of little use to the A­rithmetician, except where a number is to be many times doubled, tripled, or the like, which cannot be so easily abridged here, as in the other, because there the last number arising of many additions of the excesse to 1, was easily found by one multiplication: but here the last number being made by many Multiplications of the ex­cesse, is therefore many times harder then the other

The varieties here shall be but two.

1 The common excesse and number of pla­ces being given, to find the last number.

2 The excesse, and last number being given, to find the total summe.

The first of these may thus be found. Let the numbers be 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, the exceesse is 2, the places 10, find out the fifth number (which is easily done, for any one may reckon so farre by heart,) that is here 16, and multiply 16 by 16, it produceth 256, which is the ninth number, lastly, multiply 256 by the excesse 2, thence ariseth 512, the number desi­red.

So if the places had been more, as 72, having found the 9 number 256, multiply it by 256, thence comes 65536 for the 17th. number, which mul­tiplied [Page 76]by the excesse 2, gives 131072 for the 18th. place, which multiplied by 131072, gives 17179869184, for the 35 place; and that multi­plied againe by the excesse 2, gives 34359738368, for the 36 place, that multiplied by 34359738368, then the product will be 1180591620717411327|424, for the 71 place, which lastly, multiplied by the excesse, gives 2361183241434822654848, for the 72 place, which is the last number of the pro­gression required to be found.

Perhaps this may seem somewhat tedious but where things cannot be performed without la­bour, the Reader must content himself with such Rules as make it lesse; for it is certaine that this way is much shorter then to have multiplied still by the excesse 71 times, which else he must have done.

All this notwithstanding, he is not bound to use the same numbers, much lesse in other que­stions where the number of places is not the same, but whereas I began from the place 9, he may begin at 8, 10, or 12, or where he pleases; so as he remembers still where he is; for this is general, if the number belonging to any place what­soever, be multiplyed by it self, the product shall be the number belonging to twice so many places want one place.

Now for the second thing, which is to find the summe of all the numbers.

THE RULE.

From the last number take the first, and divide [Page 77]the remaine by the excesse want 1, then multiply the quotient by the excesse; and to the product adde the first number, the summe of them is equal to the summe of all the numbers.

So if from the last number, or 72 place be ta­ken 1 remain is 2361183241434 22654847 which should be divided by 1, (that is the excesse want 1 for the excesse is but 2) but because 1 neither multiplies nor divides that labour is saved; Now multiply this remain by the excesse, the product is 4722366482869645309696, to which adding the first number 1, by making the fi­gure 6 next the right hand to be 7, you have the total summe of all the 72 numbers.

THE GOLDEN RULE, Or Rule of Three.

THis is the most useful and most easie Rule in Arithmetique, and deserves a Golden name. It is when there are three numbers given or known to find a fourth in proportion with them.

But 4 numbers are in proportion, and called Proportional, when as the first is to the third, so is the second to the fourth.

As if there were given, 3, 4, and 6, to find a fourth which may be to 4, as 6 to 3, that is double, and that fourth number is 8; And this is called Proportion direct: and the Rule whereby it is done, The Direct Rule.

There is also another proportion called Recipro­cal; which is when as the first is to the third, so is the fourth to the second: As 3, 4, 6, and 2, this is called The Reverse Rule.

In direct proportion, the product of the two middle numbers multiplied together, is ever e­qual to the product of the first and last multipli­ed together; which serves not onely for a Proofe, [Page 81]but a ground of the Rule, which Rule shall here follow: the Reverse Rule being deferred till we have done with this.

The Golden Rule Reverse.

The Golden Rule Compound of five Numbers.

Of Fractions.

THe word Fraction signifies a breaking or breach of any intire thing into parts; and when a number is broken so, the parts (which must needs be every one lesse then the whole; and the whole is accounted but One, or Ʋnity) being lesse then Unity, are called fractions (that is fragments, or pieces) of Unity. Now the Unite, or intire number which is to be broken, may be any thing, as one pound, in re­spect of which, shillings and pence, and far­things are fractions; or one shilling, in respect of which, pence, and farthings are fractions; or one peny, in respect of which, farthings are fra­ctions; and the like of weights and measures, or any other thing to be broken into parts.

In Fractions, we shall treat first of Numeration, then of Multiplication and Division, then of Redu­ction; then lastly, of Addition and Subtraction.

The reason of this Order will soon be seen; for Multiplication and Division are here much easi­er then Addition, &c. and therefore ought to be learned before them.

Numeration.

NUmeration is nothing else but the way of writing Fractions; and that this may be done, we must consider that any Unite, or Number representing an Unite, may be broken into two parts equal; and then each of the parts is called one second, or halfe; or it may be parted in­to three equal parts, and then each part is called one third; and two of them are called two thirds; and the like may be understood if it were parted into 4, 5, 6, 7, 8, 9, 20, 50, or 100, or how many soever.

Now to write these; doe thus,

• Write one half Thus ½
• Write one third Thus ⅓
• Write one fourth Thus ¼
• Write one fifth Thus ⅕
• Write one sixth Thus ⅙
• Write one seventh Thus 1/7
• Write one eighth Thus ⅛
• Write one nine Thus 1/9
• Write one tenth Thus 1/10

[Page 98]In every one of these 10 fractions, the Num­ber below the line is called the Denominator, and it shews into how many parts the Unite is broken.

The Number above the line shews how ma­ny of those parts are taken, or contained in the fraction, and is therefore called the Numerator: So in the fraction ⅗: the Denominator [...] shewes the Unite to be broken into 5 parts: and the Numerator 3 signifies 3 of such parts to be con­tained in the fraction; which fraction therefore is called three fifths.

And here it is plaine, that as the Numerator is in proportion to the Denominator: so is the fraction to 1, for 3/3 or 5/5: or any the like, is equal to 1.

And therefore all fractions are quotients of lesser numbers divided by greater, as 4/7 signi­fies 4 to be divided by 7, and as the dividend 4, is to the divisor 7: so is the quotient 4/7 to Unity.

And therefore this line of separation which is drawn between the dividend and divisor, doth properly signifie division.

Hitherto we have spoken onely of such fra­ctions as are lesse then 1, and those are called Proper Fractions: but there are also 2½, 3¾, 5 1/7, 6⅗, and the like mixed Numbers; which so written signifie two and an half, 3 and 3 quarters, five and a seventh, 6 and 3 fifths. These by multi­plying the whole Numbers, by the Denomina­tor, and to the product adding the Numerators [Page 99]respectively are turned to 5/2, 15/4, 36/7, 33/5, which are called Improper fractions, because every one of them contains more then Unity.

These, neverthelesse may be multiplyed, divi­ded, added, or subtracted in the same way as are proper fractions. And this shall serve for Numeration of Fractions.

Multiplication.

THE RULE.

MƲltiply all the Numerators together, the last product shall be the Numerator of the product required: Likewise multiply all the Denomina­tors together, the last product shall be the Denominator of the product sought.

Division.

DIvision of one fraction by another, is but crosse multiplication of them; that is, the Numerator of the one, by the Denomina­tor of the other, and hereby the proportion of one fraction to another is seen.

Reduction.

REductioon of Fractions is threefold.

1 To reduce one fraction (which is not already in the least) to a lesser Denomi­nation.

2 To reduce many fractions of divers deno­minations, to one denomination.

3 To reduce any fraction from one denomina­tion (as neer as may be) to any other denomi­nation desired.

For the first of these, divide both the Nume­rator, and the Denominator, by the greatest com­mon divisor that you can think of; the two quo­tients being placed respectively in a fraction, that fraction shall be equal to the former fraction, and in lesser termes.

So (in the 3 Example of Division) to reduce 2880/360 to 8/1, divide 2880 by 360, quotient is 8, then divide 360 by 360, the quotient is 1, and the new fraction 8/1 is equal to the former fraction, 2880/360, and in lesse termes; as you may see. But to find the greatest common divisor.

The Rule is,

Divide the greater terme by the lesser (I mean by [Page 105]termes, the Numerator and Denominator) and by the remainer (if any be) divide the divisor, and if any thing still remains, by that divide the last divi­sor, continuing this course till nothing remain greater then Ʋnity; that divisor which is last of all is the greatest common measure of both termes, by which both being divided, and the quotients placed like a fra­ction, that fraction shall be equal to the former fraction, and in the least termes.

Addition.

TO adde many fractions into one sum, consi­der whether they be of one Denomination or divers; if of one, then adde all the Numera­tors together into one summe, that summe is the new Numerator: and the Denominator, in this case is not altered.

Subtraction.

IN Subtraction of one fraction from another, if they be both of one denomination; it is done by taking the Numerator of one from the Numerator of the other, the remaine is the new Numerator, and the Denominator the same as before.

So if 2/9 be subtracted from 8/9. the remaine is, 6/9 the like of all others.

But if they be not of one Denomination, they must first be reduced to be so; then that which is said before is sufficient.

The golden Rule in fractions is the same as in whole Numbers, I will give you but one in­stance.

If ¾ of a yard of tape cost ½ of a peny, what shall one inch, that is, 1/36 of a yard cost?

Multiply the second by the third, the product is 1/72, which divided by ¾, the quotient is 4/216 of a peny, for the price of 1/36 of a yard.

Otherwise,

Seeing ¾ of a yard may be turned to 27 inches; [Page 113]Say, if 27 cost ½, what 1? divide ½ by 27, it makes 1/54 for the answer: which is equal to 4/216, and in the least termes.

And wheresoever this may be done, to have the first and third Numbers fractions of one de­nomination, the best way is to work with their Numerators, not regarding their denominator at all: as if ⅔ cost ¾, what 7/3? instead thereof write, if 2 cost ¾, what 7? multiply ¾ by 7, it produceth 21/4, which divided by 2, the quotient is 21/8, and that is the answer in the least terms.

And all this while it should have been noted that the fractions are ever written in a smaller figure then the whole Numbers.

The Rule of Fellowship.

THis Rule is usefull for Merchants, and all such as trade in Companies, with a joint stock; and must share a proportional part of the gaines, or losse; every one accord­ing to his stock which he laid in.

The Rule is two-fold, with equal time; or with unequal time.

That which is with equal time, is commonly called, The Rule of Fellowship without time.

Of this we will first speak.

THE RULE.

As the whole joint stock, is to all the gain or losse; So is each mans particular stock, to his part of the gaine, or losse.

The Rule of Fellowship with time.

THis Rule is to be used when the times of the continuance of the particular stocks are un­equal, and differ; so that here the difference of time, and also the difference of stock being both to be considered; it can be done no better way then by taking the Power of them both to be the particular stock; and all those Powers added, to be the whole stock, that which I call the Power, is the product of the money of every one multi­plyed by his time; And then

THE RƲLE.

As the summe of those Products, is to the whole gain; so is each particular product, to its part of the gain.

The Rule of Alligation.

THis hath its name from binding, tying, or uniting many particulars in one Masse or Summe, the nature of it will be understood in working some Questions or Examples.

The Rule of False Position.

THis Rule serves to resolve questions, which are not presently fit for the Golden Rule; and therefore in stead of the true Number which is sought: Suppose any number great or small, and make trial of it, whether it resolve the question without any er­rour; if so, it is the true Number: if not note what errour is at the end of the worke, and whe­ther it be too much or too little: if too much, mark it thus + but if too little, thus −

Then suppose again another Number, (it im­ports not whether it be neerer or further off) and try as before, and mark that errour also;

And, then

THE RULE.

Multiply the first Position by the second Errour, and the second Position by the first errour, and (if the errours be both + or both −) Subtract the lesser product from the greater, and keep the remaine for a Dividend, and the difference of the errours for the [Page 132]Divisor: the quotient of that division is the true Number required.

But if the errours had been one +, the other −, then the summe of the products added toge­ther had been the Dividend: and the summe of the errours, the Divisor; the rest of the work the same as before.

The Rule of Ceres and Virginum.

THis is the most uncertaine, and unneces­sary Rule in Arithmetick; being seldome used except in sporting questions to puz­zle young beginners, with easie pro­blems; such as follow.

Extraction of Roots.

ANd first for the square Root, that is, a square Number being given, to find the root or side of it in a Number: which root or side, being multiplyed into it self, must therefore pro­duce that square Number.

The doing of this is shewed in (almost) all books of Arithmetick; and the reason of it (in some) which is taken from the fourth proposition of the second book of Euclide; which saith,

If a right line be divided by chance; the squares made of the parts, together with the Rectangle made of the parts twice, is equal to the square of the whole.

Here followeth a Table of Roots and their Squares from 1, to 1000.

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Extraction of the Cube Root.

ACube is a solid, or Body contained with­in six equal squares; and may be fitly re­presented by a Die.

When a Cube is given, as is 435̇519̇512̇; point the Number as you see in the place of U­nity, and every third figure after; then see what is the root of the greatest Cube contained under the first point toward the left hand; that is, in 435, it will be found 7, (for 8 times 8, taken 8 times is 5 12, which is too much) put this 7 for the first figure in the quotient, having taken the Cube thereof, which is 343, out of 435: thus, [...]

[Page 163]And so the first work is done.

So the quotient might be 6, but must be but 5, because the Cube of the new quotient, and 210 times the square of the said quotient, must be allowed in this worke, as followeth.

Which taken from the second remaine, there remaines now thirdly nothing. And therefore the last quotient 8, put to the former two, it is 75.8 for the whole Root, as may be tryed by multiplying 758 into it selfe; and the product a­gaine by 758, then the last product shall be equal to the whole Cube, which was given at first to be resolved.