Arithmetic, Part 3
THE THIRD PART Conteining INSTRUMENTAL ARITHMETICK.
THe Arithmetick, of which we now come to treat, and which I call Inſtrumental Arithmetick, is not any new kind of Arithmetick, but is indeed the ſame with Decimal Arithmetick before taught; onelyonly, whereas in Decimal Arithmetick, there were certain Tables made of Money, Weight, and Meaſure, by help of which the Decimal of any fraction of Money, Weight or Meaſure might be ſet down (as it were) in whole numbers, here in this Inſtrumental part, we have contrived certain Scales of Money, Weight, and Meaſure, equally divided into the ſeveral Denominations into which the ſeveral Weights and Meaſures, can be equally divided, unto every of which Scales there is joyned a Scale of 100, 1000, or 10000, equal parts, according to the length of the Scale, ſo that by inſpection onelyonly you may readily and exactly without addition (as in uſing the fore mentioned Tables you muſt neceſſarily do) ſet down the Decimal fraction of any part of Money, Weight or Meaſure, with great celerity and exactneſſe, if the Scale be any thing well divided, though the Scale be but of a reaſonable length.
Now the Scale, which I have chiefly made choice of in this Work, as being of moſt uſe with Engliſh men; (though other Scales may be made for the Coyns, Weights or Meaſures of any other Countrey as well, and upon the ſame ground) are chiefly theſe, viz.
1 Of Money.
2 Of Troy Weight.
3 Of Averdupois great Weight.
4 Of Averdupois little Weight.
5 Of Liquid meaſures.
6 Of dry meaſures.
7 Of Long meaſures.
8 Of Dozens.
Unto every of theſe Scales, is joyned another Scale of 100 or 1000 equal parts, theſe Scales are made to face one another, ſo that if you look upon any one Diviſion, in the one, you ſhall alſo diſcern plainly what Diviſion or part of a Diviſion anſwereth thereunto in the other.
English Coyn, Two ſhillings being the Integer
Troy weight, Two peny weight being the Integer
Averdupois great weight, 28 lib: or one quarter of an hundred being the Integer
Averdupois little weight, 16 ounces, or one pound being the Integer
Place this betweene pag 246 & 247
Deliniavit Antonius Thompſon
Dry meaſure, S Buſhels being the Integer
Liquid meaſure, 36 Gallons, or one Barrell being the Integer
Long meaſure, one Ell or one Yard being the Integer
Theſe Scales being thus diſpoſed, as they may eaſily be upon any Ruler of Silver, Braſs, or Wood; but beſt of all upon a ſquare Ruler, made in form of a Parallelepipedon, will by inſpection onelyonly give you any Decimal fraction required without Addition, or (on the contrary) reduce the fraction into the know parts of the Integer, by inſpection alſo, without ſubtraction.
Let thus much ſuffice for a general deſcription of what I mean by Scales, the particular deſcription of them, will more plainly appear, when we treat of Numeration upon the Scales, unto which we ſhall now proceed. But firſt take a view of the Scales as they are here diſpoſed, and as they may be ſet upon ſuch a Ruler as I have here mentioned.
Numeration upon the Scales.
THe Scales here to be deſcribed are in number eight, as hath been already ſhewed, and as by the figure of them appear. Now Numeration upon a Scale is to find upon what part of the Scale, any number upon the ſame Scale will fall.
We will begin with the firſt, and ſo proceed till we have given an Example in every one.
1 The firſt Scale is of Engliſh money, and is divided into 24 equal parts, which repreſent 24 pence or 2 ſhillings, theſe parts are numbred with Arithmetical figures, from the beginning thereof, by 1, 2, 3, 4, 5, &c. to 24, each diviſion repreſenting one peny, and the whole 24 diviſions repreſenteth 24 pence or 2 ſhillings, So that where the figure 1 ſtandeth, that part of the Scale repreſenteth one peny, where the figure 2 ſtandeth, it repreſenteth two pence, where the figure 18 ſtandeth, it repreſenteth 18 pence, or one ſhilling ſix pence, and ſo of any other figure of the ſame Seale.
Then becauſe there are four farthings contained in a peny, each of theſe pence (or diviſions) is ſub-divided into 4 other equal parts by ſhort lines, every one of theſe repreſenting one farthing, ſo is the whole Scale divided in all in 96 equal parts, which are the number of farthings conteined in two ſhillings. This if you look into the Scale of Money for 8 pence 3 farthings, you ſhall find it at the letter a, which letter is here put onelyonly for example ſake, Alſo if you would find in the Scale the place of 18 d. half peny, you ſhall find it at c, and thus may you find the place of any number of pence and farthings under two ſhillings upon the Scale.
Unto this Scale of money (as to all the reſt of the Scales,) there is joyned another Scale which I ſhall alwayes hereafter call the Scale of 1000, the uſe of which Scale is this. When you have found any Number of pence or farthings upon the Scale of Money, you ſhall find upon the Scale of 1000, what parts of a thouſand is the Decimal of thoſe pence and farthings: Thus when in the Scale of money you find at the letter a 8 pence 3 farthings, if you caſt your eye directly croſſe to the Scale of 1000, you ſhall find 364 to ſtand directly againſt 8 d. 3 q. which 364 is the Decimal of 8 d. 3 q. Alſo, if you find upon the Scale of money 18 d. half penny, which is at the letter c, you ſhall find againſt it in the Scale of 1000, this number 771, which is the Decimal of 18 d. half penny. And in this manner may the Decimal of any number of pence or farthings under two ſhillings be moſt eaſily and exactly obteined.
Now on the contrary, ſuppoſe a Decimal fraction were given, repreſenting ſome part of Engliſh Coin if you look in the Scale of 1000 for your Number given, right againſt it in the Scale of Money you ſhall find what Number of pence and farthings is repreſenteth thereby. As for Example. Suppoſe .364 were a Decimal given, and it were required to find what part of Coin it doth repreſent. Look in the Scale of 1000 for the Number 364, and right againſt it you ſhall find 8 pence 3 farthings. Alſo if .771 were a Decimal given, if you look in the Scale of 1000 for .771, you ſhall find againſt that number 771, 18 pence 2 farthings. And thus of any other.
By what hath been already ſaid, it may be eaſily diſcerned of what exceeding expedition theſe Scales thus diſpoſed are of, for I dare affirme, that I will ſet down 2 (if not 3) numbers by the Scale, as ſoon as one by the Tables, and if the Scale be but of any reaſonable length, altogether with as much exactneſſe, but if I ſhould vary an unite in the laſt place, in my eſtimation in the Scale of 1000, it is not any thing material.
I have been very tedious in ſhewing the uſe of theſe Scales to finde the fraction parts of money, but the reaſon is becauſe I intend to be the briefer in the reſt, for weight and meaſure, the manner of working (when the diviſion of the Scale is known,) being the ſame in all reſpects without the leaſt alteration.
2 The ſecond Scale is of Troy weight, two Peny weight being the Integer, which Scale is divided into 48 equal parts or diviſions, each of which diviſions contains one grain, and are numbered by Arithmetical figures at every three grains by 3, 6, 9, 12, &c. to 24, and at the place where 24 ſhould ſtand, there ſtandeth PW, which ſignifieth one peny weight, or 24 grains, this PW ſtandeth in the middle of the line. Then is the ſame Scale continued farther by Arithmetical figures 3, 6, 9, 12, &c. as before to 24, and there is written PW again, repreſenting two peny weight, or 48 grains.
The Scale being thus divided, it is eaſie to finde the place where any number of Grains under 48 ſhall be upon the Scale; As for example, if it be required to finde where 8 peny weight ſhall fall, look upon the Scale of Troy weight, from the beginning thereof, and count the figures 3 and 6, then count alſo two of the ſmaller Diviſions, and that makes 8 grains, which you ſhall finde to ſtand at the letter d, which is the place of 8 grains; Alſo if upon the Scale you would finde the place of one peny weight 10 grains, you ſhall finde it at the letter e, and ſo of any other number of grains under 48, or two peny weight.
But if you had a Decimal given, and would know what number of grains it repreſenteth, if you ſeek your Decimal given in the Scale of 1000, right againſt it in the Scale of Troy weight, you ſhall finde the number of grains repreſented thereby.
Example. Let .167 be a Decimal fraction given, If you look in the Scale of 1000, for 167, right againſt it in the ſcale of Troy weight, you ſhall finde 8 peny weight.
Alſo if .708 were a Decimal given, if you ſeek 708 in the ſcale of 1000, right againſt it you ſhall finde 1 peny weight 10 grains.
3 The third ſcale is of Avoirdupois great weight, 28 pounds, or one quarter of an hundred, being the Integer, this ſcale is numbered by 1, 2, 3, 4, &c. to …8, which 28 repreſenteth 28 l. or a quarter of a hundred, and each of thoſe is ſubdivided into four ſmall parts, each repreſenting one quarter of a pound.
Now if you would know what is the Decimal of any number of pounds or quarters under 28, if you ſeek the number of pounds in the ſcale of Avoirdupois great weight, right againſt it in the ſcale of 1000, you ſhall finde the decimal thereof.
Thus if it were required to find the Decimal of 8 pound and an half, if you look upon the Scale for 8 pound and an half you ſhall find it at the letter g, and right againſt it in the Scale of 1000 you ſhall find 304, which is the Decimal of 8 pounds and an half.
On the contrary, ſuppoſe 304 were a Decimal given, and it were required to find what part of Avoirdupois great weight were repreſented thereby, if you look in the Scale of 1000 for 304, right againſt it in the Scale of Avoirdupois great weight, you ſhall find 8 pound and an half.
4 The fourth Scale is of Avoirdupois little Weight, 16 ounces or one pound being the Integer, This Scale is firſt divided unto 16 equal parts, and numbred by 1, 2, 3, 4, &c. to 16, each Diviſion repreſenting one ounce. Then again, each of theſe ounces is ſub-divided into 8 other ſmaller parts or diviſions, each of which diviſions repreſenteth two Drams; but if your Scale be large enough, you may have each ounce divided into 16 equal parts, or diviſions, each diviſion repreſenting one Dram.
Now to find the Decimal belonging to any number of Ounces and Drams, repaire to the Scale of Avoirdupois little weight, and on it find the quantity of ounces and drams required, and right againſt it in the Scale of 1000, you ſhall have the Decimal thereof.
Thus if it were required to find the Decimal of 6 ounces, and 6 drams, if you look this in the Scale of Avoirdupois little weight, you ſhall find it at the letter h, and right againſt it in the Scale of 1000, you ſhall find 398, which is the Decimal of 6 ounces and 6 drams.
But if 398 were a Decimal given, and it were required to find the value thereof in Avoirdupois little weight, if you look for 398 in the Scale of 1000, right againſt it in the other ſcale you ſhall find 6 ounces 6 drams.
5 The fifth Scale is of Dry Meaſures, one Quarter or 8 Buſhels being the Integer, This Scale is firſt divided into 8 equal parts, and numbred by 1, 2, 3, &c. to 8, each of which diviſions repreſenteth a Buſhel, & each of thoſe parts is again ſub-divided, firſt into 4 equal parts or diviſions each repreſenting one peck, and then thoſe again ſub-divided into 4 other ſmaller parts, repreſenting quarters, halves, and three quarters of a peck.
Now if you would know the Decimal belonging to any number of Buſhels (under 8 Buſhels or one quarter) Pecks and parts of a Peck, if you ſeek the number of Buſhels, Pecks, and parts of a Peck in the Scale of Dry Meaſures, right againſt it in the Scale of 1000, you ſhall have the Decimal required.
As for Example, if it were required to find the Decimal belonging to 5 Buſhels 2 Pecks, and half a Peck, if you look into the Scale of dry Meaſures you ſhall find 5 buſhels, 2 pecks, and an half, to ſtand at the letter k, and right againſt it in the Scale of 1000, you ſhall find 702, which is the Decimal anſwering to 5 buſhels, 2 pecks and an half.
But if 702 were a decimal given, and it were required to find what number of buſhels, pecks and parts of a peck it repreſenteth, if you look in the ſcale of 1000 for 702, you ſhall find againſt it in the ſcale of Dry meaſures, 5 buſhels 2 pecks and an half.
6 The ſixth Scale is of Liquid Meaſures, the Integer being 36 Gallons or one Barrel, this Scale is divided firſt into 36 equal parts or diviſions, and numbered by 5, 10, 15, &c. to 36, then every of theſe diviſions is again ſub-divided into 4 other ſmal diviſions, each repreſenting a quart, but (if the Scale be large enough) you may ſub-divide each Gallon into 8 parts, ſo will every part repreſent one pinte.
Now to find the Decimal belonging to any number of Gallons (under 36 Gallons or one barrel) quarts or pintes, repaire to the Scale of Liquid meaſures, and ſeek there upon the Scale, the number of gallons, quarts, or pintes, and againſt it in the Scale of 1000, you ſhall find the Decimal thereunto belonging.
So if it were required to find a Decimal repreſenting 10 gallons and two quarts, or 4 pintes, which is all one, if you ſeek in the ſcale of Liquid meaſures for 10 gallons 2 quarts, you ſhall find it at the letter m, againſt which in the ſcale of 1000, you ſhall find 292, which is the Decimal of 10 gallons 4 pintes.
Likewiſe if 292 were a Decimal given, and it were required to find what number of gallons, quarts or pintes were repreſented thereby, if you look in the ſcale of 1000 for 292, right againſt it in the ſcale of Liquid meaſures, you ſhall find 10 gallons, 2 quarts, or 4 pintes.
7 The ſeventh ſcale is of Long Meaſures, the Integer being Yards or Ells, this ſcale is divided into 4 equal parts, and numbred by 1, 2, 3, 4, repreſenting 1 quarter, 2 quarters, 3 quarters or 4 quarters of a Yard or Ell, theſe are again ſub-divided firſt into 4 other equal parts, repreſenting nails, and thoſe may be again ſub-divided at pleaſure if need be.
Now if you would know what decimal belongeth to any number of quarters or nailes of a yard if you ſeek the number of quarters and nailes in the ſcale of Long meaſure, the ſcale of 1000 will give you the Decimal thereof.
Thus if it be required to find the Decimal belonging to 1 quarter and 3 nails, if you ſeek this in the ſcale of long meaſure, you ſhall find it to ſtand at the letter o, againſt which in the ſcale of 1000 you ſhall find 437, which is the decimal anſwering to 1 quarter & 3 nails of a Yard or Ell.
Alſo if 437 were a decimal given, & it were required to find what quantity of yards or ells were repreſented thereby, if you look in the ſcale of 1000, for 437, you ſhall in the ſcale of long meaſure find againſt it one quarter and 3 nails.
8 The eighth and laſt ſcale is of repreſentative inches, the whole ſcale being divided into 12 equal parts and numbred by 1, 2, 3, &c. to 12, and thoſe parts are again ſubdivided into halves, quarters, and half quarters, as Carpenters-Rules are uſually divided.
Unto this Scale, (as unto all the other) there is joyned a Scale of 1000, this Scale will readily diſcover what is the Decimal belonging to any number of Inches, halves or quarters, and the uſe is the ſame with the Scales before mentioned
Thus have I given you a briefe deſcription of theſe Scales and the uſes of them, and do now ſuppoſe my Reader to be perfectly acquainted with the way of numbring or counting upon them, wherefore I intend onelyonly to give you a queſtion or two in the moſt uſual Rules of Arithmetick, and ſo conclude; for Decimal Arithmetick being already ſufficiently explained, I ſhall not need to repeat the Rules (or the manner of working them) again, but give you one Example, by which the exactneſſe and expedition of theſe Scales may the more evidently appear, for when we work by Scales, it is ſuppoſed that we do not uſe Vulgar, but Decimal Arithmetick, and Addition, Subtraction, Multiplication, Diviſion, the Rule of Three, and indeed, all the other Rules of Arithmetick, are to be performed, as is before taught, the Scale ſerving only to avoid Reduction.
Addition.
WHat Addition is, and the manner of working of it hath been already taught, both in the firſt and ſecond Parts, we will now come to an Example, which let be in Addition of Engliſh Coin, and let the ſums to be added be 36 l. 8 s. 8 d. 29 l. 0 s. 2 d. 31 l. 16 s. 9 d. and 6 l. 2 s. 5 d.
Firſt,
| 36 |
| 29 |
| 31 |
| 6 |
ſet down 36 l. 29 l. 31 l. and 6 l. one under another, in ſuch order as you ſee here in the margent, drawing a line by the ſide of them as you ſee here done, and alſo a line under them.
This done, ſeeing that your firſt number to be ſet down to 36 l. is 8 s. 8 d. you muſt for the 8 s. (becauſe two ſhillings, which we called a Decade, or the tenth part of a pound, is made the Integer, in the Scale of Money) ſet down 4, which is done by memorie, and after it make a comma. Then your next number to be ſet by 29 l. being 0 s. 2 d. for the 0 s.
| 36 | 4, |
| 29 | 0, |
| 31 | 8, |
| 6 |
ſet down a Cypher, thirdly, for your number to be ſet by 31 l. being 16 s. 9 d. for the 16 s. ſet down 8 decades, with a comma after it, and laſtly, the number to be ſet by 6 l. being 2 s. 5 d. for the 2 s. I ſet down 1 decade with a comma after it, and then will your work ſtand, as here you ſee.
Then take your ſcale in hand, and ſeeing your firſt number of pence are 8 d. look in your ſcale of money for 8 d. and againſt it in the ſcale of 1000, you ſhall finde 333, which ſet to 36 l. 8 s. behinde the comma, then your next number of pence being 2 d. look in your ſcale for 2 d. and againſt it in the ſcale of 1000, you ſhall finde 082, which ſet to 29 l. 0 s. behinde the comma.
| 36 | 4, | 333 |
| 29 | 0, | 083 |
| 31 | 8, | 376 |
| 6 | 1, | 208 |
Then your third number of pence being 9 d. look in your ſcale for 9 d. and againſt it in the ſcale of 1000, you ſhall finde 376, which ſet to 31 l. 16 s. and laſtly, your laſt number of pence being 3 d. look in your ſcale for 5 d. and againſt it you ſhall finde 208, which ſet to 6 l. 2 s. and then will your whole work ſtand, as here you ſee.
Your ſums being thus ſet down, which is done with more facilitie then you can imagine, till you make trial and be ſomething perfect therein, you muſt then adde all the numbers together, as in Addition of Decimals, and you ſhall finde the ſum of them to be 103|4,000, Now to know what this is in money, is as eaſie as it was to ſet the ſeveral ſums down, for the figures 103, which ſtand behinde the down right line, are 103 l. and the figure 4 which ſtands between the down right line and the comma,
| l. | ||
| 36 | 4, | 333 |
| 29 | 0, | 083 |
| 31 | 8, | 376 |
| 6 | 1, | 208 |
| 103 | 4, | 000 |
are 4 decades or 8 s. and being the reſt to the right hand are all Cyphers, they ſignifie neither pence nor farthings, ſo is the total of this addition 103 l. 8 s. 0 d.
That the manner of working may appear more plain. I will give you another ſhort Example as difficult as I can invent, which I performed by a Scale of Wood but of 8 inches long. Let the ſums to be added together, be theſe following.
| li. | t. | d. | q |
| 332 | 17 | 4 | 1 |
| 159 | 6 | 8 | 1 |
| 217 | 5 | 3 | 3 |
| 709 | 9 | 4 | 1 |
Firſt, ſet down your ſeveral ſums of pounds one under another as before, and draw a line by the ſide of them,
| 332 |
| 159 |
| 217 |
and another under them. So will they ſtand as here you ſee.
1 Your ſums of pounds being thus orderly placed and lines drawn, repair to your Scale and ſeeing your firſt number of ſhillings, pence and farthings is 17 s. 4 d. 1 q. for your 17 s. ſet down 8 Decads, which is 16 s. with a comma after it, then will there reſt to be ſet down 1 s. 4 d. 1 q. or 16 d. 1 q. which if you ſeek in your ſcale of money, you ſhall find to ſtand againſt it in the ſcale of 1000 this number 677, which is the Decimal of 1 s. 4 d. 1 q.
2 Your ſecond number of ſhillings pence and farthings is 6 s. 8 d. 1 q. for your 6 s. ſet down 3 decads, which is 6 s. and then there will remain 8 d. 1 q. which if you ſeek in your ſcale of money, you ſhall find to ſtand againſt it in the ſcale of 1000 this number 344, which is the decimal of 8 d. 1 q.
3 Your third number of ſhillings, pence, and farthings, is 5 s. 3 d. 3 q. for your 5 s. ſet down 2 decads, which is 4 s. with a comma after it, then will there reſt to be ſet down 1 s. 3 d. 3 q. or 15 d. 3 q. which if you ſeek in your ſcale of money, you ſhall finde to ſtand againſt it in the ſcale of 1000,
| l. | |
| 332 | 8,677 |
| 159 | 3,344 |
| 217 | 2,656 |
| 709 | 4,677 |
this number 656, which is the Decimal of 15 d. 3 q. or 1 s. 3 d. 3 q. and the three ſums to be added together will ſtand as here you ſee.
Theſe ſums being added together, according to the rule for Addition of Decimals, you ſhall finde the ſum of them to be 7094,677, now to know what this is in money, take notice that the 709 which ſtands to the left hand of the downright line are 709 pounds, and the figure 4, which ſtands between the downright line and the comma, are 4 decads or 8 s. but (becauſe the firſt figure next after the comma is above 5, viz. 6) you muſt add 1 s. to the 4 decads, making them 9 s.; then will there remain 177, wherefore if you look in the Scale of 1000 for 177, you ſhall find againſt it in the Scale of Money 4 d. 1 q. So is the whole ſum of this Addition 709 l. 9 s. 4 d. 1 q. as by the preceding work doth appear.
¶ Here note, that when you had ſet down your 709 l. 4 decads or 8 s. there remained beyond the comma 677, which if you had ſought in your Scale of 1000, you ſhould have found againſt it in the Scale of money 15 d. 1 q. or 1 s. 4 d. 1 q. (which is all one) as before, for it appeareth plainly by the Scale that 500 in the line of 1000 is equal to one ſhilling.
I might proceed farther in giving you Examples in Weight and Meaſure anſwerable to the Scales, but that would only make the Reader ſpend his time to little purpoſe, for being before acquainted with Decimal Arithmetick, and (as by this time I ſuppoſe he is) with Numeration upon the Scales, he cannot be deficient in the applying of the other Scales of Weight and Meaſure to the ſame purpoſe for which they were contrived, I having ſo largely exemplified the uſe of the Scale of money.
Subtraction.
SUbtraction (as hath been before ſaid) is the taking one or more ſmaller ſums out of one greater, I ſhall onelyonly give you an Example or two, as I have taken the numbers from a Scale.
Example.
Delivered to a Gold-ſmith of old plate 297 ounces 13 peny weight, 19 grains.
Received of the ſame Gold-ſmith firſt 165 ounces, 11 peny weight and 7 grains, and after that received of the ſame Gold-ſmith 32 onnces 19 penny weight, and 23 grains, what plate remains in the Gold-ſmiths bands?
Take your numbers out the ſcale of Troy weight, and ſet them down, as here you ſee.
| ounces | ||
| Delivered | 165 | 5,646 |
| Received more | 32 | 9,979 |
| Received in all | 198 | 5,625 |
| Reſes in the Goldſmiths hands | 99 | 1,271 |
or 99 ounces — 2 peny w. — 13 gr.
Then adde the ſeveral weights of plate received, together, and they make 1985, 625, or 198 ounces, 11 peny weight, 6 grains, which if you ſubtract from 2976,896, or 257 ounces, 13 peny weight 19 grains, which was the quantity of plate delivered, there will remain 991271, or 99 ounces 2 peny weight, 13 grains, and ſo much plate is ſtill in the Gold-ſmiths hand. And let thus much ſuffice for ſubtraction.
Now we ſhould proceed to Multiplication and Diviſion, but being when the numbers are taken from the ſcale and ſet down, the manner of working doth not at all differ from Multiplication and Diviſion of Decimals before taught. But that we are now a treating of Inſtrumental Arithmetick (becauſe I will not, alter the method before uſed, in the firſt and ſecond parts of this Book;) I will therefore here ſhew you how Multiplication and Diviſion, may be Inſtrumentally performed by Nepers Bones.
Multiplication by Nepers Bones.
MUltiplication and Diviſion are accounted the hardeſt parts of Vulgar Arithmetick; but by the invention of the Right Honorable John Lord Nepeir Baron of Merciſton, they are both of them made very eaſie, and more certain then in the ordinary way, the memory not being charged at all, but as in Addition and Subtraction, what this Artifice is, is ſhewed by himſelf in his Treatiſe of Rabdologia, as alſo in that of Mr. Seth Patridge and others, and therefore it will not be neceſſary to make any large diſcourſe of it now, it will be ſufficient here, onelyonly to give you a deſcription of them, and how to uſe them.
Theſe Rods or Bones are nothing elſe, but the ordinary Multiplication Table, commonly called Pythagor as his Table, cut into pieces, with Diagonal lines between the figures of every product in ſingle multiplication; as by the figure following it doth appear.
A Figure of the Bones.
The Bones are uſually made of ſmall pieces of Box, containing in length about 1 inch and 8/10 of an inch, and in breadth about 2/10 of an inch, and in thickneſſe neer 1/10 of an inch. One ſet of theſe Bones conſiſteth of five pieces of Box, and are graduated with Diagonal lines and figures, as you ſee here in the figure of them. On one of the five pieces there is 0 and 9, the 0 being on one ſide of the Wood, and the 9 on the other ſide, another hath 1 and 8. the third 2 and 7, the fourth 3 and 6, the fifth 4 and 5. Now theſe five pieces of Box contein one Table of Multiplication cut in pieces, but ſo few are not ſufficient for uſe, for you cannot well have leſſe then ſix ſets of them, that is 30 pieces, and ſo many are uſually made.
For the placing of theſe Bones, when you uſe them, there belongeth a piece of Wood about four inches in length, and two inches in breadth, having 2 ledges of the ſame thickneſſe with the Bones, on the thin legde which is at the end of this board, there is figured the nine Digits, 1, 2, 3, 4, 5, 6, 7, 8, 9. This board is called a Tabulat, becauſe we uſually ſay when the bones are laid thereon for any the Bones are Tabulated. See the figure thereof.
Now theſe Bones thus ordered and Tabulated doth not onelyonly give you the product of every ſingle Digit, multiplyed by it ſelf, or any one of the Digits, but the ſeveral products of all the 9 Digits together by every of the 9 Digits, as if it were demanded how much 123456789, will amount unto, being multiplyed by 6, in this Table it will be found to be 740740734, for the product of 9 multiplyed by 6 is 54, the firſt figure whereof towards the right hand, viz. 4, is placed under the Diagonal line, and 5 being in the place of tens is placed above it, again 8 multiplyed by 6 is 48, the firſt figure whereof towards the right hand, viz. 8 is placed under the Diagonal line as in the other product, and theſe 2 figures 5 in the upper part of the former product, and 8 in the lower part of this muſt be added together, and being ſo they make 13, therefore next unto 4 in the firſt product, I ſet down 3, and carry 1 to 4 the figure placed above the Diagonal in this ſecond product, which together with the lower figure in the next product, viz. 2, do make 7, for 4 and 1 that I carried is 5 and 2 the figure under the Diagonal in the next product are 7, and ſo the 3 firſt figures of this product towards the right hand are 734, in like manner muſt the figures remaining between the Diagonals be added together, and their ſum if it amount not to ten muſt be ſet down in his proper place, but if they make or exceed ten, ſet down the overplus, and carry the ten in mind to the next figures between the Diagonals as in the laſt product; So likewiſe in that which followes 4 and 6 the figures between the Diagonals being 10, I ſet down a Cypher, and carry one to the next figures, which are 3 and 0, now 4 and 1 that I carried is 4, and 4 and 0 is 4 ſtill, and therefore next to the Cypher I ſet down 4 in the paper; the figures in the next Diagonal are 3 and 3. that is 7, wherefore I ſet down 7 in the paper, the next are 2 and 8 that is 10, therefore I ſet down 0 in the paper and carry 1, the next are 1 and 2 which with 1 that I carried make 4, therefore I ſet down 4 in the paper, the laſt are 1 and 6 which make 7 and ſo at laſt the whole product of 123456789 multiplyed by 6 is 740740734.
Now as the multiplication Table thus ordered doth give the product of all the 9 Digits placed orderly by every of the 9 digits, ſo being cut in pieces or ſlips, and having many of them made in Paſt-board of Wood, any other ſum may be placed on the Tabulat, and the ſeveral products thereof by every of the 9 digits will be found in the like manner; thus if it were required to multiply this number 5767 by every of the 9 digits, placing the ſlips with theſe figureson the top orderly as, they are in the following Figure, product of 5767 by 3 will be found to be 17301, the product thereof by 8 is 46136, and ſo for any other.
Alſo if it were required to multiply 5767 by 749, the product of 5767 by 9 is 51903, and the product thereof by 4 is 23068, and the product by 7 is 40369, … and theſe products being ſet down and added together, as in ordinary Multiplication, the product of 5767 multiplyed by 749 will be 4319483, as by the operation in the margine it doth appear.
Of Diviſion by the Rods.
LEt it be required to divide 4319483 by 5767 placing this Diviſor on the Tabulat, as was directed for Multiplication, the products of your Diviſor by every of the 9 Digits is given you, and may be deducted from the Dividend, as hath been already ſhewed Pag. 55, and according to the Rules there delivered, I ask how often is 5767 in 43194, and the anſwer by my Table is 7 times, and therefore I ſubtract 40369 the product of 5767 by 7 from 43194, and there reſts 2825, which with the next figure in the Dividend, viz. 8, is 28258, in which I find 5767 to be conteined 4 times, and therefore I ſubtract 23068 the product of 5767 by 4 from 28258, and there reſts 5190, which with the next figure in the Dividend, viz. 3, is 51903, in which I find 5767 to be conteined nine times, … and therefore I ſubtract 51903 the product of 5767 by 9, from 51903 the remaining figures of the Dividend, and there reſteth nothing, and therefore the quotient is 749, as by the operation it doth appear.
And thus may you doe with any other ſum (having a competent number of Bones or Rods ready for the work) without any charge to the memory, which in other wayes cannot be avoided, and by reaſon of which many miſtakes may happen, which are in this prevented. Alſo by the Bones there is this commodity, that when it is doubtfull what figure to ſet in the Quotient, the Bones will certainly direct you, ſo that you cannot poſſibly miſtake.
Theſe Rods, or Bones (as they are commonly called) are of excellent uſe in extracting the Square and Cube Roots, but that being already taught in the firſt Part, I ſhall not here inſert it again, for he that can extract the Square and Cube Roots any way, cannot be ignorant to effect the ſame by the Bones, if he can multiply and divide by them.
Having done with the four Species of Arithmetick in this inſtrumental way, I ſhould now proceed to the Golden Rule of three, and conſequently to the other Rules of Arithmetick, but having in the former parts of this Book performed that already, both in whole numbers, Fractions, and Decimals, I ſhall onelyonly give you an Example in the moſt neceſſary Rules, and ſo conclude this third part.
Example in the Rule of Three Direct.
IF 37 Ells and a half of linnen cloth, coſt 24 l. 7 s. 9 d. what ſhall 283 Ells and an half coſt?
Firſt, ſet down your 37 Ells, then if you look in your Scale for your half Ell, you ſhall find it to ſtand againſt 500 in the Scale of 1000, which 500 may be called 5 onelyonly, for the two Cyphers may be omitted. Then ſet down your 24 l. and for your 7 ſet down 3 decads, which is 6 s. and look in your Scale for the Decimal of 1 s. 9 d. which you ſhall find to be 875; Laſtly, ſet down your 183 Ells, and for your half Ell ſet down 5 as before, ſo will your numbers ſtand thus, …
…
Your numbers thus taken out of your Scale, and placed as here you ſee, if you multiply the ſecond and third together, you ſhall finde the product of that multiplication to be 4475.10625, which divided by the firſt number 37.5, giveth in the quotient 119.3361, which is 119 pounds, 3 decads, or 6 ſhillings, and 361, which reduced by the Scale giveth 8 pence 2 farthings, and ſomething more.
Example in the Rule of Three Reverſe.
IF when the price of a Quarter of Wheat is 1 li. 5 s. 6 d. the penny white loaf ſhall weigh 12 ounces 16 penny weight, I demand what the penny white loaf ſhall weigh, when the price of the Quarter of Wheat is 7 s. 6 d?
If you place your numbers according to the tenor of the Queſtion, they will ſtand as followeth.
1 li. — 5 s. — 6 d. — 12 où. — 16 pw. — 7 s. — 6 d.
But being taken out of the Scales of Money an Troy Weight, they will ſtand thus, …
Here if you multiply 1.275 which is the Decimal of 1 l. 5 s. 6 d. by 12.8. which is the Decimal of 12 ounces 16 peny weight, you ſhall find the product of that multiplication to be 16.3200, which being divided by .375, which is the Decimal of 7 s. 6 d. the quotient will be 43.5, which is the Decimal of 43 ounces, 10 peny weight 3 grains, and ſo much ought the peny white loafe to weigh, when the quarter of Wheat is ſold for 7 s. 6 d.
Example in the Double Rule of Three.
IF 24 yards of ſtuffe of three quarters broad coſt 4 l. 14 s. what ſhall 328 yards of the ſame ſtuffe coſt being 5 quarters broad.
If you place your numbers according to the directions of this Rule, they will ſtand thus, If 24 yards of 3 quarters coſt 4 li. 14 s. what ſhall 328 yards coſt of 5 q.
But if you take your fraction numbers out of their proper Scales, they will ſtand thus, … …
Firſt, multiply the two firſt numbers, as 24 and 3 together, they make 72 for Diviſor, then multiply 4.7, which is the Decimal of 4 l. 14 s. by 328, and the Product is 15416, which again multiplied by 5 the laſt number, giveth 77080, unto this Product, (that there may be a competent number of figures in the quotient,) I adde two Cyphers, making it 7708000, which I divide by 72, and the quotient is 107.055, which is 107 l. 1 s. 1 d. and ſo much is 328 yards of ſtuffe worth, being 5 quarters broad.
Example in Barter.
TWo Merchants having two ſeveral commodities, are willing to barter, or exchange the one with the other. The one hath Indigo, which he will ſell at 4 s. the pound, for ready money, but in Barter he will have 4 s. 9 d. the pound; the other Merchant hath Kerſies, which for ready money he will ſell for 3 s. 6 d. the yard. Now the queſtion is, at what price he muſt rate his Kerſies in Barter, to equalize the 9 d. advance upon the pound of Indigo.
The tenor of the Queſtion is this.
If 4 s. in Barter require 9 d. what ſhall 3 s. 6 d. require?
Your numbers placed will ſtand thus, 4 s. — 9 d. 3 s. 6 d.
But being taken out of your Scale they will ſtand thus, …
Say then by the Rule of three Direct, If 2 decads or 4 s. in Barter require. 357, which is the Decimal of 9 d. what ſhall 1.57 require? which is the Decimal of 3 s. 6 d.
Firſt, multiply .375 by 1.57, the product is .65625, (but being it is a fraction I cut off the two laſt figures becauſe we require onelyonly three figures in the quotient, which divided by 2 giveth in the quotient .328, which is the Decimal of 7 d. 3 q. this 7 d. 3 q. added to his 3 s. 6 d. maketh 4 ſhillings 1 peny 3 farthings, and ſo much ought he to rate his Kerſies at by the yard in Barter to ſave himſelf harmleſſe.
Example in Fellowſhip.
THree Perſons A, B, and C bought 4000 Sheep, which coſt 48… li. 6 ſh. 8 d. of which money A paid 203 li. B paid 165 li. 15 ſh. 8 d. and C paid 114 l. 11 s.
Firſt, ſay by the Rule of three Direct.
1 If 483 l. 6 s. 8 d. buy 4000 ſheep, how many ſheep ſhall 203 l. (which is A ſhare) buy? Anſwer 1680.
2 Say, if 483 l. 6 s. 8 d. buy 4000 ſheep, how many ſheep ſhall 165 l. 15 s. 8 d. (which is B ſhare) buy? Anſwer 1372.
3 Say again, if 483 l. 6 s. 8 d. buy 4000 ſheep how many ſheep ſhall 114 li. 11 s. (which is C ſhare) buy? Anſwer 948.
Your numbers being taken out of your Scale, proceed as followeth.
Firſt for A.
…
Secondly for B.
…
Thirdly for C.
…
The manner of Work.
1 For A multiply 203 l. (which is A ſhare) by 4000, (which is the number of ſheep bought) and the product is 812000, which number ſhould be divided by 483.3333, but being it is greater then 812000, I therefore add four Cyphers thereto, that I may have four figures in the quotient, and it makes 8120000000, which divided by 483.3333 giveth in the quotient 1680, and ſo many ſheep belong to A.
2 For B, multiply 165.7833 (which is the Decimal of B ſhare) by 4000, (the number of ſheep bought) and it produceth 6631332000, which divided by 483.3333, giveth in the quotient 1372, and ſo many ſheep belong to B.
3 For C, multiply 114.55, (which is the Deeimal of C ſhare,) by 4000, (the number of ſheep bought,) it produceth 45820000, which number ſhould be divided by 483.3333, but being it is not large enough to give figures enough in the quotient, I therefore adde two Cyphers making it 4582000000, which divided by 483.3333, giveth in the quotient 948, and ſo many ſheep ought Cto have.
Now for proof,
| A | 1680 | |
| B | 1372 | |
| C | 948 | 4000 |
if you add the number of ſheep that A, B and C ſhould ſeverally have, you ſhall find them in all to make 4000 which demonſtrates the work to be true.
Examples in Loſſe and Grain.
IF one yard of Stuffe coſt 6 s. 8 d. and I ſell the ſame again for 8 s. 6 d. what ſhall I gain in laying out 100 l. upon ſuch commodity?
Take the difference between the price that your commodity coſt, and the price for which you ſell it, that is, in this Example, the difference between 6 s. 8 d. and 8 s. 6 d. which is 1 s. 10 d, then ſay by the Rule of Three Direct.
If 6 s. 8 d. gain 1 s. 10 d. what will 100 li. gain?
If you place your numbers according to the Rule of Three Direct, as they are here given, they will ſtand thus, 6 s. 8 d. — 1 s. 10 d. 100 li.
But being taken out of your Scale and placed, they will ſtand as followeth.
…
Your numbers being placed, multiply 425 which is the Decimal of 1 s. 10 d. by 100 li. and the Product is 42500, to which I adde two Cyphers (that I may have a competent number of figures in the quotient) and it makes 42500|…0, which divided by .3333 the decimal of 6 s. 8 d. giveth in quotient 127.5, which is 127 li. 5 livers or 10 s. ſo there is 27 li. 10 s. gained in laying out of 100 l.
I will here prove this queſtion by the converſe.
If one yard of Stuffe which is ſold for 8 s. 6 d. there was gained 27 li. 10 s. in laying out of 100 li. I demand what the ſaid ſtuffe coſt a yard at the firſt hand?
Adde 100 li. and 27 li. 10 s. together, and they make 127 li. 10 s. Then ſay by the Rule of Three Direct,
If 1 27 li. 10 s. give 100 l. what ſhall 8 s. 6 d. give?
Take your numbers out of your Scale and place them as here you ſee.
…
Here if you multiply 425, which is the Decimal of 8 s. 6 d. by 100, you ſhall have 42500, to which if you adde a Cypher, you make it 425000, this number being divided by 127.5, which is the Decimal of 127 l. 10 s, giveth in the quotient 3333, and if you had added more Cyphers to the Dividend, you ſhould have had more threes in the quotient, and no other figures, but theſe four threes are enough, and are a Decimal fraction repreſenting 6 s. 8 d. and ſo much did the yard of Stuffe coſt at the firſt hand.
Examples in Loſſe and Gain upon Time wrought by the Double Rule of Three.
IF one Ell of Lockeram, coſt me 2 s. 8 d. ready money, and I ſell the ſame again for 2 s. 10 d. the Ell to be paid at the expiration of three moneths, I demand what I ſhall gain in 12 moneths, laying out 100 l. upon that commodity.
This and ſuch like queſtions, although they may be wrought by the Rule of Three Direct, at two operations, yet they are beſt performed by the Double Rule of Three compounded of five numbers, wherefore the queſtion may be thus ſtated.
If 2 s. 8 d. in three moneths, gain 2 d. what ſhall 100 l. gain in 11 moneths?
If you take your numbers out of your ſcale▪ and place them according as was directed in the firſt part of this Book, you ſhall finde them to ſtand thus, …
Your numbers being placed according to the tenor of the queſtion, if you multiply 1.333, which is the Decimal of 2 s. 8 d. by 3 moneths, the product will be 3999, which muſt be your Diviſor, then multiply .83 which is the Decimal of 2 d. by 100 l. and it makes 8300, that again multiplied by 12 moneths, giveth for the product 99600 for your Dividend, wherefore if you divide 99600 by 3999, it will give you in the quotient 25 almoſt, which is 25 li. for the Decimal fraction remaining is ſo ſmall, that it wanteth not neer a farthing of 25 li. and therefore we call it 25 li, and ſo it is exactly, as you may try, if you reduce all the numbers to their leaſt denominations, and work as is before taught in Vulgar Arithmetick.
I will prove this queſtion by the converſe.
If one Ell of Lockerum coſt me 2 s. 8 d. ready money, for what price ſhall I ſell the ſame again to be paid at the end of three moneths. So that I may gain 25 li. in 100 li. for 12 moneths?
Say by the Rule of Three Direct.
If 100 li. in 12 moneths gain 25 li. what ſhall 2 s. 8 d. gain in 3 moneths?
If you take your numbers out of your Scale, and place them according to the Double Rule of Three, they will ſtand as followeth.
…
Your numbers being thus placed, if you multiply 100 li. by 12 moneths, you ſhall finde the product to be 1200, which is your Diviſor. Then multiply 25 li. by 1.333, which is the Decimal of 2 s. 8 d. and the product thereof will be 33325, which multiply again by 3, and the product will be 99975 for your Dividend, this 99975 divided by 1200, giveth in the quotient. 83, which is the Decimal of 2 d, which 2 d. added to 2 s. 8 d. the price which the Ell of Lockeram coſt, giveth 2 s. 10 d. and at that price muſt you ſell the ſame at 3 moneths time ſo that you may gain 25 l. in the 100 li. in 12 monethmonth….
I might further proceed to ſhew you Examples in divers other Rules; As in Alligation, Poſition, &c. but thoſe Rules being already handled in the Firſt Part of this Book, it will be eaſie to apply them to the ſcales, And (as I intimated at the beginning of this Third Part,) that although I have onelyonly made choice of the eight ſcales there mentioned and deſcribed, yet it was eaſie to contrive ſcales for the Coins, Weights, and Meaſures of other Countries, and not onelyonly ſo, but when the value of the Money, Weight, or Meaſure of one Countrey, and the Money, Weight, or Meaſure of another Countrey is known, it is eaſie to contrive two ſcales, which facing one another, ſhall immediately tell you how many pounds in one place ſhall be equal to ſo many Crowns, or other Coin in another place, but this I do onelyonly intimate, that ſuch as are deſirous, may fit themſelves with ſcales anſwerable to their moſt neceſſary occaſions. And thus ſhall I conclude this third part, referring ſome things neceſſary to ſuch a work, to the Appendix following.