Concerning the Table of PYTHAGORAS.
THE known Arithmetical Table, invented by Pythagoras, (such as you see expressed in Figure 1.) is not only an easie and sure Rule to multiply and divide by, but is also those very Operations themselves, Multiplication and Division, done to your Hands, and known by inspection, comprehending three distinct Numbers, proper to them both, viz. Multiplicand, Multiplier and Product, proper to Multiplication, Dividend, Divisor and Quotient, proper to Division. For if you take any one of the Numbers, Seated in their several Cells between A and B. for a multiplicand; for example 8. and another Number of those that are Seated in their several Cells between A and C for a multiplier, for example 7. in the Angle of their Concurse. you will find the Number 56. the just Product of 8 multiplyed by 7. Again, the said Product, 56. is also a Dividend, whose Divisor is 8 in the highest Cell, above the Dividend and Quotient [...] 7 in the 7th. Lateral Cell, over against the Dividend, the better to distinguish the 9 Units, Figures, Numbers and Cells Seated between A and B▪ from the like Seated between A and C. call th [...] first Capital Units, Figures, Numbers, Cells, as [Page 2] being placed in the Head of the Table: but the two Lying between A and C call Lateral, as occupying the side of the Table on the left Hand.
Every Capital number in the Pythagorean Table hath under it 8 other numbers lodged in 8 several quadrats or Cells, as you may see in Figure 1. all which 9 numbers make a kind of a little streight Column, parallel to the side A C or B D. The Columns are 9 answerable to the 9 Ʋnits or Capital Numbers in the Head of the Table. But observe also, that there are other 9 Transverse Columns, parallel to the side A B. or C D. which cross the former at Right Angels, and meet one another in a common Cell, ever containing a perfect Qu [...]drat Number, whose Root appears in the Heads of the two Meeting Columns: For Example, the Column of 8 Capital meets with the Transverse Col. of 8 Lateral in the Cell of 64. a square Number. So 9 Capital Meets with 9 Lateral in the Cell of 81. a square Number, &c. But what is worthy of Observation, these two different sorts of Columns, Capital and Transverse, though most cross one to another, do most punctually agree in all their Numbers, without any difference, as is manifest to the Eye.
There are yet many things more, very Observable in the Pythagorean Table. The first is that not only the 9 Units, are multiplicand and Divisors in it, but Tens, Hundreds, Thousands, 10000ds, 100000. 1000000▪ 10000000 and 100000000. in [Page 3] great variety, and all actually and orderly Tabulated, shewing at the same time, their true Product and Dividends, with multiplyers, Divisors, and Quotients. As for Tens you see 12 Tabuated on the 1. and 2 Col. Then 23. 34. 45. 56 67. 78. 89. each number Tabulated on two Contiguous Columns. As for Hundreds, you see 123. Tabulated on the first 3 Col. Then 234 345. 456. &c. As for Thousands, you have 1234. Tabulated on the four first Col. Then 2345. 3456. 4567. &c. and so of [...]he rest, till you come to 123456789. a multiplicand or Divisor of all the Capital Ʋnits, of the Table, whose multiplyer is one (or more, as you please) of the Lateral Units, and the Product is the Transverse Column of that Ʋnit, which you choose for multiplyer to be counted from the Right Hand to the left. For example, if you multiply 123456˙789 by 2. the product will be the second transverse col. gathered from the right hand to the left, viz. 246913˙578. If you multiply it by 9. the product will be the 9th. transverse column, viz. 1111111˙101▪
The second thing very observable is, that if you turn Pythagoras his Table in such manner, that all the Numbers remain unchang'd in their cells, and yet the Figures 9. 8. 7. 6. 5. 4. 3. 2. 1. [...]ying between C▪ and A. become Vertical, which [Page 4] before were Lateral, and 1. 2. 3. 4. 5. 6. 7. 8. 9. lying between A. and B. become Lateral, which before were Vertical, you may find another great Variety of Multipliers, Divisors, Products, Dividends and Quotients, and of greater Numbers than before, all differing from the former, and all Tabulated on contiguous columns. As for tens, you see 98 Tabulated on the two last transverse columns, then 87. 76. 65 &c. As for Hundreds, you have 987 876. 765. &c. And so for the rest, till you come to 987654321 a Multiplicand or Divisor made of all the Lateral unites from C. to A. which Number multiplied by 2. will have for product 1˙975308˙642. to be found in the second Capital Column, and gathered thence from the right Hand to the left. If you multiply it by 9. the capital Column of 9 will shew the product 8˙888888˙889.
The third observable thing is, That whatsoever under-cell of any Column, hath more figures or places in it, than are in the capital cell of that Column, then infallibly the Figure which is outmost on the left side of that under-cel; is to be added to the next Figure of another Column, if another Column be Tabulated by it on the left Hand. This Addition may be called Collateral, because it adds together two Figures on the sides of two Neigbouring Columns, and makes but one Number of them both. If the two Figures added [Page 5] should make 10. then put down a cypher, and carry one to the next Number on the left Hand: If they make more then 10, put down the surplus and carry one. Take this example of Collateral Addition. If you Tabulate 12. with two Rods or Columns, viz. the column of 1 and the column of 2. in the 2d. Cell of both Rods together, is 24. in the 3d. Cell is 36. in the 4th. Cell is 48. but in the 5th. Cell is 510. which make not five Hundred and 10. but 60 only, because 1 and 5 (the Neighbouring Figures of 2 Columns) are to be added into one number 6, by reason that the 5th. Cell of the Column 2 hath 10. in it, a Figure more than in the Capital of the Column two. This Rule then is Universal, whatsoever undercell of any column hath more Figures in it than are in the capital number of that column, there must be collateral Addition, if any other column be Tabulated on the Left Hand with it. Note that this Rule holds good, not only in columns of single Units, but of Tens, 100ds. 1000ds. &c.
The 4th. Observable thing is, and of cheif moment, that all and every column, Ennead, or Rod (Synomical words in the present matter) not only of Pythag. his Table, but of all following Tables to 9999 and much more, is singularly useful both in Division and multiplication though the Column be never so little (except the Column of 1 the first Unit, which in Rigour neither devides nor multiplys any Number) and the Dividend and multiplyer [Page 6] never so great. For in Division it performes the work, or gives the Quotient, by meer Substraction of its own Numbers out of the Dividend: and in multiplication it gives the Product, by setting down in due order its own numbers, and afterwards adding them into one Sum. For example, take the Column 25 and divide by it 7896525 the Quotient will be 315861. and the work ended will appear as underneath [...] where note that the Numbers 75. 25. 125. 200. 150. 25. all marked with this mark-are taken out of the cells of the column 25. to be substracted out of the partial Dividends: 75. is taken out of the third cell, and gives you 3. to beset in the Quotient; 25 is taken out of the 1 cell, and gives one for the Quotient, and so of the rest, the Number shewing its cell, and the cell the Quotient. Again, [Page 7] take the column 25. for a Multiplicand. and [...]ultiply it by 315861. the product will be 7896525, which before was the Dividend. The Operation ended, will appear, as underneath at [...], Where note, that the Numbers 25. 150. 200. [...] 25. 25. and 75. are all taken out of the cells of the column 25. to be placed as you see, and added into one sum for product of Multiplication. Here also you may observe, that the self-same cells or Numbers are added together in Multiplication, which were substracted in Division, only their Order inverted: what was first substracted in Division, is last taken and added in Multiplication, which always happens when the Divisor and Quoient become Multiplicand and Multiplier and [...]eproduce the Dividend as Product of Multilication.
The fifth thing observeable is, That every [...]nnead or column, be it never so little or great, [Page 8] that is, of one, or more, or many Figures in its Capital Cell) by multiplying its Capital Number with 45. will produce a sum equal to all the Figures, as they stand in that col. added into one sum. For example, take the col. of 6. and multiply 61. by 45. the product will be 2745. which is just the sum of all the column added together, as appears in the margent 61 122 183 244 305 366 427 488 549 2745. By this means you may exam [...]ne any col. whether it be right or wrong.
Add unto the former or fifth Observation another, not much unlike, to be seen [...] in this little Table of Roots, squares and cubes, or rather of the ending Figures or Units of all Roots, Squares and Cubes whatsoever; where you see the sum of each column by Addition, to be severally 45.
| 1R | sq. | cub. |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 7 |
| 4 | 6 | 4 |
| 5 | 5 | 5 |
| 6 | 6 | 6 |
| 7 | 9 | 3 |
| 8 | 4 | 2 |
| 9 | 1 | 9 |
| 45 | 45 | 45 |
The first Column is of the first ten Roots, from 0 to 9 inclusive, but all following Roots have the same ending Figures, and in the same order, as in the first column. The second Column is of the first ten ending Figures of Squares; the first ten Squares, and all the following Squares have [Page 9] the same ending Figures, and in the same Order as in the second column. The third Column is of the first ten ending Figures of the first ten Cubes, and all the following Cubes have the same ending Figures, and in the same order as in the third column. By the ending Figure of any Root, you may know the ending figure both of the square and cube by this Table: in which the square and cube stand right over against the Roots. Hence may you know, whether a Table of Roots, squares and cubes be well made or no: for if any ten ending Roots, or Squares, or Cubes lying next one under another do not make the sum 45, or that the squares and cubes do not answer the roots, as in this Table, there must necessarily be an Errour committed.
The 6th. thing very remarkable, and indeed admirable is, that multiplication and Division being two very distinct and different Operations, yet they so inseparably and essentially accompany one another, that the one, for example, Multiplication can never be wrought or Finished by its proper Rules, but that Division at the same time shall be given you without working by any Rules of Division: yea, when the Operator did neither intend Division, nor so much as think of it. That they are two different Operations, it is clear. For
1. Multiplication, by two Numbers given (multiplicand and multiplier) seeks a third, Viz, the Factum or Product: But Division by [Page 10] two Numbers given, different from those of Multiplication, (Divisor and Dividend) seeks a third, viz. the Quotient, different from the product of multiplication.
2. Multiplication begins its work with the least figure, and Ends it with the greatest: but Division quite contrary, begins with the greatest and Ends with the least.
3. Multiplication requires Addition only, without Substraction: But Division requires Substraction only, without Addition. Notwithstanding these differences of the two Operations, it is impossible to work a Multiplication but a Division will be at the same instant given you, without working or dividing. So is it also impossible to work a Division but a Multiplication shall be given you without working or multiplying. And the reason is manifest, because the self same three Numbers which constitute the Essence of Multiplication constitute also the Essence of Division, though under different denominations. The three Numbers in Multiplication are called Multiplicand, Multiplyer and Product. In Division, Divisor, Quotient and Dividend. And observe, that by how much a Multiplcand exceeds or comes short of his Multiplyer, by so much the Divisor will exceed or come short of his Quotient. The Product of Multiplication is ever equal to the Dividend in Division. See the following example.
| Multiplication wrought | Division Given | ||
| Multiplicand | 144 | Divisor | 144 |
| Multiplier | 12 | Quotient | 12 |
| Product | 1728 | Dividend | 2728 |
| Division wrought | Multiplication Give [...] | ||
| Divisor | 7324 | Multiplicand | 7324 |
| Dividend | 4789896 | Product | 4789896 |
| Quotient | 654 | Multiplier | 654 |
Observe, that when in Multiplication the less Number is made the Muitiplicand, and the greater the Multiplier; Then in Division given, the Divisor is the less Number, and the Quotient the greater. Example.
| Multiplicand | 12 | Divisor | 12 |
| Multiplier | 144 | Quotient | 144 |
| Product | 1728 | Dividend | 1728 |
The 7th. thing observable is, That the third, fourth and fifth Cell of every Ennead (whether it hath one, or more or many Figures in its Verticall, and those either pure integers or mixt with Fractions) contain three different Numbers, which are exact Roots of three exact Square Numbers, the two less being exactly equal to the greatest, according to the 47. Prop. l. 1. Euclides, and the Sides or Roots making the perfectest sort of right angle triangles, keeping proportion one to another, as 3, 4 and 5. and having constantly these [Page 12] Angles proxime 90. 53. 8. and 36. 52. For example, take the Rod of 4. whose third, four [...] and fifth Cells contain these three Numbers 1 [...] ▪ 16. and 20. the sides of a right Angle triangle [...] and true Roots of these three square Numbers 14 [...] 256 and 400. Now the two less squares [...] added together, make exactly 400. the greate [...] square of the greatest root. Other right ang. Triang. that have not the said proportion of Sides, and aforesaid Angles, must necessarily have one o [...] more defective Roots for their Sides, which will either come short or overshoot the truth, when we endeavor to square the unsquareable Numbers.
The eighth Point observable is, that though some Columns or Enneads refuse all Collateral Addition, (because they have no more Figures in the 8. under cells, than in the Vertical) yet others far more in Number require it. For in the whole following Table of Columns, from 1. to 9999. there are only 127. that refuse collat. addition whereas 9872 require it, in one or more of their under cells. In the single Columns of the 9 Units, only the first or column of 1. refuseth Collat. Add. In the double columns of Tens, only the two first, viz. column 10 and 11. In the triple columns of Hundreds, only the twelve first, that is, all from column 100 to 111 inclusive. In the Quadruple columns of Thousands, only the first 112 Columns, that is, all from column 1000 [Page 13] to Col. 1111 inclusive; All which 127 Columns or bones are void of all collat. add. And therefore all the 8 Under cells in them are marked with Stars, as Signes of non-addition. Note, that no Vertical cells have any collat add. nor stars before them. Note also, that no Ennead, be it never so great, or have many Figures in the Vertical cell, can have any collat. add. in any one Under cell, if the two first Figures of the Vertical begin with 10 or with these three 110 or these four 1110 &c. Though all the following Figures be never so great, as 9999 in infinitum.
The ninth thing Observable is, that by how much any Ennead contains more Figures in its Vertical cell, by so much is it the better to multi [...]ly and divide by, since it takes away all collat. [...]dd. the chief trouble in gathering the products [...]n multiplication, and finding readily the Quo [...]ients in Division. For example, if you turn all the Vertical Units of Pytha. Table into one Sum, viz. [...] 2 3 4 5 6 7 8 9. and multiply it severally by 1. [...]. [...]. 4. &c, it would make an Ennead such, as you [...]ee exprest in Figure 13. far different from Figure [...]. the Table of Pytha. whose collat. add. it wholy [...]akes away, and yet in substance is the same with [...]he Table.
[...]oncerning the Extensions of Pythagoras his Table.
The Extensions of the Pytha. Table may be di [...]inguished into two sorts, the greater, and the [...]s. The greater extends it two ways; length [...]ay by Capital Numbers, and bredth way with as [Page 14] many lateral Numbers: The less extends it onl [...] length way by Capitals and not by any more Laterals, than are in Pytha. Table; which are the [...] Units. For example, the first greater Extensio [...] adds to the 9 Capital Units of the Table 90 mor [...] Capitals: that is, all the whole Numbers of tw [...] places between 10 and 99 inclusive : And the lik [...] it adds to the 9 Lateral Units, viz. 90 more L [...] terals. As all the cells with their inclosed Num [...] in Pytha. T. are known to be 81. by multiplyin [...] the two greatest Units, 9 Capital and 9 Lateral t [...] gether; so by multiplying 99 by 99 the two gre [...] est Capital and Lateral Numbers of two places, yo [...] will find the Cells of this first greater Extensio [...] to be 9801. The first less Extension adds to the [...] Capital Units (as did the first greater Extensio [...] 90 Numbers of two places from 10 to 99 incl [...] sive, but adds not any one Number to Pythago [...] his 9 lateral Units. The cells of this Extension [...] multiplying 99 its greatest Capital by 9, the grea [...] est Lateral, are found to be 891 which is not t [...] 10th. part of 9801 the cells of the first grea [...] Extension. A Table of this kind of extent, co [...] taining 9801 cells would be very useful, and be [...] of a Moderate largeness, occupying about 10 [...] 11 pages in Folio, might be easily made, as fo [...] have heretofore done: Mr. Joh. Darling and othe [...] But in this present Table we forbear to place [...] and all other Extensions of the greater sort, [...] reason of their Vast largeness and labour in [...] king and using them. In the following Tab [...] [Page 15] [...]he one of 5 greater Extensions, the other of 5 less Extensions, you may see their differences and how many cells, pages and Tomes in Folio, each one would contain. But first observe, that we allow a Folium to be 14 Inches long, and 8 broad, prescinding from Margents, one page whereof will contain 112 square Inches, In every page reckon [...]000 cells : In every Tome a 1000 pages.
| Extensions | 1st. | 2d. | 3d. |
| Multiplicand | 99. | 999. | 9999: |
| Multiplier | 99 | 999 | 9999. |
| Cells. | 9801. | 998001. | 99980001. |
| Pages. | 10 801/900 | 1108 801/900 | 111088 801/900 |
| Tomes. | 0. | 1 109/1000 | 111 089/1000 |
| Extensions: | 4 th. | 5th. | |
| Multiplicand | 99˙999. | 999˙999. | |
| Multiplier. | 99˙999. | 999˙999 | |
| Cells. | 9˙999800˙001 | 99˙9998000001 | |
| Pages. | 11110˙888 801/900 | 1˙111108˙888 801/900 | |
| Tomes. | 11110. | 889/1000 | 1111108 889/1000 |
| Extensions | 1st. | 2d. | 3d. | 4th. | 5th. |
| Multiplicand | 99 | 999 | 9999. | 99999 | 999999 |
| Multiplier | 9 | 9. | 9. | 9. | 9▪ |
| Cells | 891 | 8991. | 89991 | 89999 | 8999991. |
| Pages | 0 891/900 | 9 891/900 | 99 891/900 | 999 891/900 | 9999 891/900 |
| Tomes | 0 | 0 | 0 | 1- | 10. |
In these Tables, every Extension is exprest by 5. oblong Quadrats, one under another. In the first Quadrat is the Number of the Extension, First, second or third, &c. In the second Quadrat are two Numbers, a Multiplicand, and Multiplier, being the greatest capital Numbers; and the greatest lateral Number of that present Extension. In the third Quadrat is the Product of the abovesaid multiplicand and multiplier, or Number of cells of the Extension. In the fourth Quadrat is the Number of pages in folio, which that Extension would make. Divide the cells by 900 and the Quotient will give the pages. In the fifth Quadrat is the Number of Tomes which that Extension would make. Divide the Page [...] by 1000. and the Quotient will give the Tomes▪ The Extensions both of the greater and lesser sort may be made in infinitum, though these two Tables exhibit only five of either sort. It is incredible to our first apprehensions, what a vas [...] [Page 17] space would be taken up by a Table of the fifth greater Extension, wherein, as you see, 999˙999. capital Numbers are supposed to be multiplied by so many Laterals, and to produce the Number of cells 999˙998000˙001: and consequently, according to Allowances above-mentioned, pages [...]1˙111108˙888 801/900, and Tomes in folio 1111˙108, each Tome having 1000 pages, and (with its cover) 3 inches in thickness. These Tomes, if they were set on end, contiguous one to another in a streight line, they would make a rank of books above 52 English miles long. Or if all the aforesaid pages, their Margents cut off, should be laid close one to another on a plain, they would cover more than 30 square English miles, or 19200 square Acres.
But setting aside all Extensions of the greater sort, we will content our selves with the third less Extension, in which as the Table shews, 9999. is the greatest capital Number (Multiplicand and Divisor) and 9. the greatest lateral Number (Multiplier and Quotient. The product of cells is 89˙991. The pages in folio are 99 891/900 which scarce make the 10th. part of a Tome in folio. And observe that 9999 contains all the capitals, both of the Pythagorean Table, and of the first, [Page 18] second and third less Extensions. For 9 (the Unit on the right hand) counts the 9 Units of the Pythagorean Table; the next 9 counts 90 Numbers of two places, from 10 to 99 exclusive, and makes the first Extension. The third 9. counts 900 num. of 3 places, from 100 to 999. and makes the 2d. Extension: The 4th. 9 counts 9000 num. of 4 places, from 1000 to 9999. and makes the 3d. Extension observe also, that the foresaid numbers, of 990. 900. and 9000. added together make just the number of 9999: and being multiplied severally by 9 do produce severally these num. 81. 810. 8100. 81000. all which added together, make up the just num. of cells of the 3d. less Extension viz 8999. Observe lastly, what we touched before speaking of Pytha. Table, that every Capital number from 1 to 9999 being multiplied by all the 9 Units or single figures, produces 9 distinct numbers, one greater than another, which being orderly placed and perpendicularly one under another, make a certain column, whose length is divided into 9 equal parts or cells, the Seats of the 9 Numbers produced, the Capital being the highest. Wherfore there being 9999 Capitals in this present Table, there must be also 9999 Columns, which in substance and in effect are the Numbring Nines, Enneads, Rods or Bones, or what else you please to call them: and not only the single Rods of Units (as they were first invented, and hitherto too commonly used) but [Page 19] double Rods of Tens, Triple Rods of Hundreds, and quadruple Rods of Thousands: So that whatever Operation can be performed in matter of Multiplication or Division, by 1. 2. 3. or 4. of the single Rods, the same may be performed by one Rod or column of this Table, and with far greater expedition, without any collateral Addition. For here are actually Tabulated to your Hand all and every whole Number (Multiplicands and Divisors) under 10000, and ever with one column or Rod alone. Nay, it will not be hard to work by two columns of this Table at the same time, and then your Multiplicands and Divisors may be any Number under 100000˙000. But let us proceed to the third point of the Numbring Rods.
Concerning the Numbring Rods or Bones.
These Arithmetical Rods (described by most Authors, who have writ of Arithmetick, since they were first found out) own for their first Inventor, the Right Honourable and Learned John Lord Nepeer, Baron of Marchiston, who put forth a Latin Treatise concerning them, Intituled, Rabdologia, that is, a Discourse or Treatise of Rods, calling them Virgulae and Laminae: others have or may call them, Columellae, Tesserae, Enneades, adding the Epithets, Numerales or Arithmeticae. To the same Noble Lord, Posterity is obliged for [Page 20] another Ra [...]e Invention of Logarithmes, both them aiming at, and attaining the same en [...] which is, to facilitate and perform with greate [...] dispatch ease and certainty the harder parts o [...] Arithmetick, viz. Multiplication, Division an [...] Extraction of Rootes. These late years past Si [...] Samuel Morland most ingeniously invented two Arithmetical Instruments to the purpose abovesaid, for which he deservs singular praise. Thoug [...] the Instruments in themselves be excellent and useful, yet they have been hitherto more sparingly used for these two reasons First, because few Artificers are found, who have Hand and Head sufficient to make them so exactly as is requisite▪ 2ly, because the Vulgar sort either wants Head [...] to comprehend them, or purses to purchase them▪ being somewhat chargeable: whereas the Rods of the Lord Nepeer are plain, easie and Cheap. H [...] happily fell upon the conceiving and devising o [...] them by throughly considering the Pythagore [...] Table, in which as before I mentioned, an incredible variety of great and little numbers is found orderly Tabulated, multiplyed and divided, with apparent Multiplyers, Divisors and Quotients. For example, He said 1234. Tabulated by the first 4 capital Columns of the Table, which he multiplyed by 9 and found the product in the 9 cell to be (Collateral additions being observed) 11106▪ This product he perceived to be also a Dividend [Page 21] whose Divisor appeared 1234. and Quotient 9. o [...] vice versa Divisor 9. and Quotient 1234. Further he Noted that 1234. inverted was 4321. This Number he also found Tabulated together by the [...]irst four Transverse Columnes, which multiplied by 9. gave 38889 in the 9th. Capital Column. This being also a Dividend shewed 4321 for Divisor, and 9. for Quotient, or viceversa, &c. But [...]aking a middle Number between 1234. and [...]321. For Example, 3142. or 2413. he was at a [...]oss, not finding them Tabulated together, nor [...]he Product lying together, but was to be picked [...]ut here and there, not without trouble of the Head and delay of time. This inconvenience hapned as he well perceived, because the Table was always made with its Columns fixt in the [...]ame gradual Order of Unites, encreasing from [...]. to 9. But the Remedy of this inconvenience [...]oon occurred, which was to unfix the fixt co [...]umns by cutting them asunder, and making them moveable, apt to be placed in what order he [...]leased, as occasion required. Thus were the famous Numbring Rods extracted and dissected [...]ut of the Pythagorean Table, and in reality are [...]othing else but the Table it self cut out into its [...]olumns, adding thereunto 3 more for the square, [...]ube and cypher-Rods; such as you see in Fig. 2. [...] is true, the Lord Nepeer, to mind the Operator [...]f [...]c [...]llateral Addition, drew Diagonal Lines [...]hrough all the 8. undercells of every column or [Page 22] Rod, whereby frequent Rhombes of this shap [...]
appeared in the Rods Tabulated; And whatsoever 2. Figures should be found in one Rhombe [...] they were to be added and made one Number See Fig. 5. Others by making a cross Diagonal i [...] the undercells, included the 2. Figures to be added, in this kind of Diamond form [...] See Fig. 6. Others again included them in a round Circle. See Fig. 7. But because all these three Wayes seem to offend the Eye, and breed Confusion by so many Lines; Others with much less ado note all the addend Figures with this mark ▪ or this ▪ declaring that every Figure so Noted require [...] collateral addition, if Tabulated on the right Hand with another Rod or column on the left. See Fig▪ 3. [...]et because the far greater part of the undercells in the Pythagorean Table, (having one Figure more in them then is in the capital) would require this mark▪ to wit, 58 cells, whereas only 14 cells refuse it : Again, because thi [...] mark▪ and one Figure more then is in the capital require a greater bredth of Rod, I have rather chosen to put an Asterisk *, as a sign of non-addition to a few cells and lesser Rods, then thi [...] Sign of Addition ▪ to four times more cells an [...] larger Rods, declaring that the Star in an [...] undercell hinders collateral Addition; an [...] where the Star is not in an undercell, ther [...] must ever be collateral addition, if another Ro [...] be Tabulated on the left Hand with it.
[Page 23] But here observe, that when any (9) hath a Star before it, and (1) carried to it, by reason of a Rod Tabulated on the right Hand of it, then [...]hat (9) becomes 10, and is capable of lateral addition, if another Rod follow on the left Hand. Observe also, that all these Enneads 1. 11. 111. 1111. 11111. and the like in infinitum, require Stars in all their undercells, unless when a (9) becomes 10 by (1) carried to it, as now we said. Note also, that all less Numbers then these, having equal places or Figures with them in the Vertical, require Stars in all Undercells: For example, 1111. is an Ennead of four places, and so is 1000, a less Number yet of four places; so is 1001. 1002. 1003. and so on till we come to 1110. all less Numbers then 1111. but all of four places, and requiring Stars in all their undercells. But whatsoever Number of four places is greater then 1111. as is 1112. 1113. 1114. and so on till 9999. then in fallibly it will reject the Star, and require lateral addition in one or more of the undercells. See the eighth Observable.
Moreover, to avoid Multiplicity of Lines, as much as may be, in the Rods, I reduce 8 of those lines to 2. which formerly separated the 9. cells from one another, as you may see in Fig. 8. 9. 10. and 11. For dividing the length of the Rod into three equal parts by two lines, I place the three highest cells in the first Division, three others in the second, and the three last cells in the [Page 24] last division. See Fig. 8. 9. 10. and 11. accordin [...] to this Model of placing Stars before certai [...] undercells, (viz. such as have equal Number o [...] Figures with their capital cell) and dividing every Rod or column into three equal parts by two Lines; I made a Table, wherein all Capital Numbers from 1. to 99 inclusive, were multiplied by the 9. lateral Ʋnits. which Table being directly the first less Extension of Pythagoras hi [...] Table, I caused to be cut in brass some years ago and a few Copies to be printed for my own and other Friends use. At that time I had in prospect the other two less Extensions (2d. and 3d. of Pyth [...] Table) which soon were compleated, and that very readily, by the help of the double Rod [...] (whereof I had made some Sets) and the Table of the 1 less Extension now mentioned: For laying one double Rod at a time to the Columns of that Table, you Tabulate any number from 100 to 9999. and see immediately the product of multiplication in all the 9 cells. The other Numbers from 1 to 99 the Table it self Tabulates and multiplies See a printed Copy of the Table, inserted in pag. 27. As the single Rods of my Lord Nepeer were cut out of the Pyth. Table, so both single and double Rods have been cut out of the Table of the 1 less Extension, and found by Experience of 9 or 10 years to double the usefulness of the single Rods For first they sooner Tabulate any great number with fewer Rods. 2dly they Tabulate the sel [...] [Page 25] same number with great variety of Rods, differing [...]n Specie one from another. See Fig. 12. 3dly they [...]ake away more then the half of collateral Addi [...]ions, the chief trouble of numbring Rods. 4thly [...]hey more readily shew the product of multipli [...]ation and Quotient of Division in great numbers [...]nd fewer Rods. Two of the double Rods reach [...]o any number under Collat. Additions 1 at the most 10˙000 three of them to [...]ny under Collat. Additions 2 at the most 100˙0000. Four of them to any under Collat. Additions 3 at the most 100000000. &c. This and more the double Rods perform by themselves. But joyn or Tabulate them with the Table of the 3d Extension, [...]nd they will most readily multiply and divide [...]ast numbers. For one Rod and the Table reaches [...]o Collat. Additions 1 at the most 1000˙000. Two to Collat. Additions 2 at the most 100000˙000. Three [...]o Collat. Additions 3 at the most 10˙000000000. &c. To use them with the Table of 9999 columns, it is necessary, that the Rods be of the same length with the columns, [Page 26] though the same bredth is not precisely required.
The Rods having on them all the Capita [...] Numbers from 1. to 99. they will require eithe [...] 50 thin two-faced Talleys, or 25 square-side [...] Parallelopipedons of four faces. It will be convenient to have every Rod twice over, (thoug [...] once over will be sufficient if your single Rod [...] of the 9 Units be twice or thrice over,) wherea [...] an ordinary Set of single Bones must have eve [...] Rod 4. 5. 6. 7. or more times over, according a [...] the Operator designes the working of greater o [...] lesser Numbers.
Another way of supplying the want of mo [...] Rods of one and the same Number, may be [...] the Table of 9999. Enneads, for in that Tab [...] are all Numbers of four places, and consequent [...] this Number 5757. Besides, in the double Bon [...] are all Numbers of two places, from 10. to 9 [...] inclusive, and consequently this Number 5 [...] wherefore in the Table and double Bones [...] have 57. three times over. But setting as [...] the Table, the Bone alone of 57. is in practi [...] equivalent to three or more Bones of the sa [...] Number 57. for if you set down with your [...] three times 57. thus 575757, as one Vertic [...] Number of one Ennead, you will know w [...] is the Content of every under cell by the und [...] cells of the Rod 57. thrice setting them do [...] For example, the second cell of 57. is [...] which thrice repeated, is 114. 114. 114. Or [...] [Page 27] serving lateral Addition 1151514, which is the [...]cond cell of the Ennead 575757. In this man [...]er your Operation will be as ready, as if you [...]ad had three distinct Rods of 57 a piece. There [...] yet a third way of most ready and clear work [...]g, multiplying and dividing vast Numbers of the [...]lf same Species of Figures, viz. all of Nines, or [...]ights, or Sevens. &c. And in what multiplicity [...]ou please, of the same Figures, as 3. 4. 5. Nines, [...]ea 10 Nines, 20 Nines, 100 Nines: And so of [...]ights, Sevens, sixes &c. Some 5 special Enneads, [...]r 5 two-faced Rods (or two four-faced square [...]ods) are required to this sort of Operation, [...]herein you will not be troubled, either with any [...]abulating of Rods, or collateral Addition. See [...]e Scheme of the said special Rods pag. ult. Fig. [...]4 where observe that the nine single Units oc [...]upy severally nine Vertical cells, and their under- [...]ells to contain for the most part only 3 Figures, [...]e leading on the left Hand, another in the mid [...]le, a 3d. ending on the Right Hand. Some few [...]nder cells (not above 8 in 72) have 4 Figures in [...]hem, and then the two last on the Right Hand are [...]nding Figures. The middle figure is most remarkable, and more then it appears; For in O [...]eration it is to be repeated, or taken so often o [...]er, as there are Figures of one kind in the suppo [...]ed Vertical, abating one: for example, Suppose [...]he Vertical to be Ten Nines, or 9˙999999˙999. In the [Page 28] 2d. Cell of the special Rod (9) are these thre [...] figures 198. where (9) in the middle between (1) and (8) is truly nine times nine, that is, one nine less, then Ten nines in the Vertical: So that the said 2d. cell 198, is in operation 19˙999999˙99 [...] or the Vertical multiplyed by (2) This Rule i [...] Universal, yet hath two exceptions; first whe [...] any cell hath four figures in it; 2dly. when any cell hath a Star prefixt before it, according to what is abovesaid concerning Asterisks, then infallibly the middle figure is to abate, not only one, but two of the number in the Vertical. One example will clear all. Let ten fours or 4˙444444˙444. be given for a Multiplicand, and 279 for a multiplyer, then in your special Rod of 4 Vertical, take out the ninth cell, apparently 396, but really 39999999996. the middle figure (9) requiring to be repeated nine times▪ or one less, then the number of fours in the Vertical. Next take out the 7th cell, apparenly 3108 but really 31111111108. the middle figure (1) requiring Eight repetitions, or two less then Ten of the Vertical, because this 7th cell hath 4 figures in it. Lastly take our the 2d cell, apparently * 888. but really * 8888888888. because the middle figure (8) requires 8 repetition [...] (besides the leading and ending 8) or two less then Ten of the Vertical, by reason of the Star [Page 29] prefixt before the second cell. The Work ended would appear thus in Multiplication.
| Multiplicand | 4˙444444˙444 | 39999999996 | cell. 9 |
| Multiplier | ╌279 | 31111111108 | cell. 7 |
| 8888888888 | cell. 2 | ||
| Product | 1239˙999999˙876 | ||
Product 1239999999876 In Division, Multiplicand the Divisor, Product [...]he Dividend, and Multiplier the Quotient.
| Dividend | |||
| Divisor | 4˙444444˙444) | 1239999999876 | (279 Qu. |
| 2╌ | 8888888888 | ||
| 35111111107 | |||
| 7╌ | 31111111108 | ||
| 39999999996 | |||
| 9╌ | 39999999996 | ||
| 0000000000 | |||
The square and cube Rod ought to be once o [...]er in every Set, with three or four cypher-Rods, [...]s you see exprest in Fig. 11, All are to be so or [...]erly placed in a neat pocket case, that every Rod be known what Number it hath. by a mark [...]r figure, even before you take it out of the Case. [...]f you please to reprint the Table of the first [...]ess Extension by it self in a page, (with its co [...]umns, equal to the Columns of the great Table [...]f 9999.) and give it a Varnish to last the longer▪ [...]ou may immediately thence make Sets of the [...]ouble bones, meerly by cutting out the capital [Page 30] columns, and passing them on Talleys of Wood or other matter (two faced or four faced) of th [...] same length and bredth with the columns. Besides, as is abovesaid, this little Table of the first less Extension, with one double Rod applied unto it at a time, performs all the whole Work of th [...] great Table of 9999 fixt columns, only with thi [...] disadvantage, that it will often have one collateral Addition, whereas the great Table wil [...] have none. Notwithstanding the great performances of the less Table, the great Table hat [...] many special uses, for which it deserves to b [...] published, especially not being of any great extent, nor making any great bulk. By advice [...] Judicious Friends, I thought good to put it fort [...] in a Duodecimo, as a convenient Enchiridion, [...] pocket-book, every four pages containing 10 [...] columns with their 900 cells, or every page 2 [...] columns with 225 cells. In which case the Table alone would require 400 pages in Duodeci [...] But perhaps it will be better, to contract the pag [...] to half the Number, viz. to 200. in this manne [...] Let every page have five Ranks of Columns o [...] under another, each Rank consisting of Ten C [...] lumns: so will every page contain 50 Column [...] and wheresoever you open the book, the two p [...] ges before your eyes, will shew you a just Hu [...] dred of Columns. To find the Number yo [...] seek for, more readily, you may Tack to the Ma [...] gent little outstanding Labells, or Indexes, she [Page 31] [...] before you open the Book, where every 100 [...]nd 1000 begins, such as are seen in certain ac [...]ount-books of Merchants. Let the Book be so [...]ound, that wheresoever you open it, the leaves [...]n either side, may lye Flat without any uprising; [...]or so it will be more easie to Contabulate the Rods with the Table, when occasion requires. [...]erhaps it would be better to print it in a little [...]olio, for use at home in your Closet or Library; [...]or then every page would contain its Hundred [...]f Columns, easie to be found out by their own na [...]ural order.
As well the forementioned Table of 9999 fixt Columns, as the single and double moveable Rods [...]erve equally in Decimal and common Arithme [...]ick; yea in Decimal, they in a manner take away [...]ll the trouble of Division. Neither do they re [...]ui [...]e any particular Rules in operation, different [...]rom those which have been delivered concerning [...]he Lord N [...] single bones, by himself in his Rab [...]ologia, by F. Andrew Taquet, Sir Jonas Moor, Mr William Leybourne and others in their Arithmetical Treatises. Wherefore I shall say no more of them but only shew by single examples, (one of multiplication, another of Division, the [...]d of extraction of the Square Root, the 4th of [...]he cube Root) the ordinary use of them.
Example of Multiplication.
In multiplication commonly it is best to Tabu [...]ate the greater Number as multiplicand when one [Page 32] is greater then the other. For example. Tabulat [...] 4628 to be multiplyed by 72. Place Unit unde [...] Unit, as in the Margent [...] Then for 2 (the Un [...] of the Multiplyer) take the 2d cell of the Multiplicand, viz. 9256. and for 7 of the multiplyer tak [...] the 7th cell of the Multiplicand, viz. 32396, an [...] set both cells down, as you see in [...] Add the tw [...] cells together, and the total Sum or product o [...] multiplication will be 333216 as you find in [...] But if you Tabulate 72 to be multiplyed by 462 [...] the operation will appear as in [...].
In Division the Divisor is to be Tabulated; an [...] it much imports, for the speedier dispatch of yo [...] operation, that the leading Rod of the Divisor be [...] great Rod or Column of the Table; a quadrupl [...] rather then a Triple, a Triple then a double, [...] double then a single, if the Number of your Divisor will permit.
Example of Division
Let 5678556 be a Dividend, and 4628 the Divisor. set them down as in the Margent [...] with a Semilune, for the Quotient, and Tabulate the Divisor. Then enquire how often can the Divisor 4628. be taken out of the first partial Dividend, viz. 5678. Only once; Therfore put 1 for Quotient in the Semilune, and Subtract 4628 out of 5678. and t [...]ere will remain 1050 the work standing (if you cash t [...]e Figures dispatcht) as in [...]. Next bring down 5 out o [...] the Dividend, and place it on the right Hand of the remainder, as you see in [...] separated with a Comma. This done, seek in the cells of the Divisor for 10505 (the next partial Dividend) or for the next less Number then 10505. In the 2d. cell is 9256 the next less. Put 2 in the Quotient, and Subtract 9256 out of 10505. and the remainder will be 1249 to which bring down the other) of the Dividend, and dash what is dispatcht. Then will the work stand, as in [...]
[Page 34] In like manner proceed to find the 3d. and 4th. figure of the Quotient, and the work finished will appear, as in [...] Others dash no figures at all, not place the Quotient on the Right Hand of the Dividend, but set every quote-figure over against the cell of the Divisor from whence it was taken. Every remainder they distinguish with a colum [...] from the figure brought down out of the Dividend▪ as you may see in [...] When in one Number ther [...] happen to be two columns, there will be a cypher in the Quotient; when 3 Commaes, then 2 Cyphers in the Quotient. The like Method serve [...] [Page 35] wel [...] Extraction of the Square Root, and much Fac [...] [...]tate the examining of the work done. If any [...]umber remain after Division ended, it will be [...]he Numerator of a Fraction, whose Denominator [...]s the Divisor. When you turn Integers into a Ra [...]ius of Decimals, Division either ceases, or is easily [...]ad by cutting off so many Figures (on the Right [...]Hand) from the product of multiplication as are in [...]he Radius, excepting one.
Example of the Square Roots Extraction
The Extraction of the Square Root is very ready and plain by the Table, or by the double Rod [...] or both of them together. Let the number given, whose Root you seek, be 70476025. Put a point under the Unit, and every A [...]tern Figure with a Semilune after the Unit as you see in [...] The 4 points foretell that in the operation there will be 4 partial Dividend [...], and as many Roots. Then seek in the cells of the Square Rod for 70 (the 1 partial Dividend) or the nearer less number to 70. 64 is the nearest less to 70 in the 8th cell. Therefore 8 is your first Root to be placed in the Semilune, and 64 is to be Subtracted out of 70. The remainder will be 6 the work appearing as in [...]. For the finding the 2d. 3d. or any other Square Roots following, obs [...]rve these Rules: First, bring down [Page 36] the next parttial Dividend and joyn it to the [...] remainder on the Right Hand.
Secondly, double the Root or Roots found, an [...] Tabulate that double on the left Hand of th [...] Square Rod or (working by the Table) carry t [...] Square Rod to the double in the Table.
Thirdly, seek for the Number (or next less) [...] your last remainder joyned to the next partial D [...] vidend in the cells of the Tabulated Rods and th [...] Cell wherein it is found, will give you the nex [...] Root.
Fourthly, Subtract the cells Number out of th [...] remainder and partial Dividend, and proceed a [...] before, wherefore in our present example to fin [...] the 2d. Root,
First, bring down 47 and joyn it to 6 to make 647.
Secondly, Double the Root 8 and Tabulate 16 with the Square Rod.
Thirdly, seek 647 (or the nearer less) in the Tabulated Rods the third cell gives 489, the next less, which Subtracted out of 647, leaves the remainder 158. The 3d. cell gives 3 for the 2d. Root: see the Margent [...] For the finding o [...] the 3d. and 4th. Root, proceed as before. Th [...] [Page 37] whole Operation ended stands as underneath a [...] [...] or according to the Method, mentioned in [...]ivision, underneath at [...] where any number re [...]ains after the work ended, it is the Numerator [...]f a Fraction, whose Denominator is the double [...]f all the Roots and one Unit more. But if you [...]esire a more exact Fraction, add to the Nume [...]ator 2. 3. or more Couples of Cyphers, and wor [...] [...]s before, and you will find the nearer Decimal [...]raction.
Example of the Cube-Roote's Extraction.
[...]st. set down the number (whose Cube-root you [...]ek) with a point under the Unit and every 3d. [Page 38] Figure, and a Semi-Lune for the Roots, as unde [...] neath at [...] how many points, so many parti [...] Dividends and Rootes will be in the Operatio [...]
2ly, Seek in the Cube Rod for 94 or the neare [...] less number: In the 4th cell you find the nearest [...] Set down 4 for the 1 Root and Subtract 64 o [...] of 94, the remainder will be 30; and the work appear as in [...]
For the finding of the 2d, or any other following Root, observe these Rules.
1st. Bring down the next partial Dividend 81 [...] and joyn it to the remainder 30, on the Right Hand, as in [...]
2ly, Tabulate the triple of Root or Roots found (Root 4 the triple 12) with Rod or Rods apart call them for distinction, Right Hand Rods.
3ly. Tabulate the triple of the Sq. of the Root [...] found (Root 4 Square 16 the triple of 48) with Rod or Rods. placed on the left Hand, of the Cube Rod: call these left Hand Rods. Or working by the Table, carry your cube-Rod to 48 in your Table.
4ly, Seek for 30818 the present partial dividend or next less number, in the cells of the left Hand Rods. In the 6th cell you find 29016 the next le [...] yet indeed too much, 2s will after appear. Set th [...] [Page 39] this number down apart, and draw a line abo [...]e [...]t as you see in [...] over the Unit; and above the [...]ine place 6 the number of the cell, out of which [...]9016 was taken: On the left Hand of 6 place [...]he Square of 6, viz. 36 as you see in [...] then take the 6th and 3d. cell (by reason of 36 the Square) [...]ut of the right Hand Rods, viz. 72 and 36 and place them as you see in [...] adding all the numbers under the line into one Sum, viz. 33336, as you see in [...] which being too great to be taken [...]ut of 30818. you must go back and take a less [...]ell, then 6.
Take therefore the cell 5. which hath in it the [...]e number 24125. write it apart with a line a [...]ove it, and an above line over the Unit: place 5 the cells number) and on the left Hand of 5 the he Square of 5 viz. 25 as in the margent [...] [...]ake out of the right Hand Rods the 5th. and 2d. [...]ell (by reason of the Square 25) viz. 60 and 24 and add them, as you see in [...] to make 27125 which taken out of 30818 there will remain 3693 and the work stand as in the margent. [Page 40] [...] For the 3d Roo [...] do as you did for the 2d.
1st, Bring down the nex [...] Dividend and joyn it to t [...] last remainder.
2ly, Tabulate a part on the right Hand Rod [...] the triple of the Roots found.
3ly, Tabulate the triple of the Square of t [...] Roots on the left Hand of the Cube-Rod.
4ly, Seek in the cells of the left Hand rods fo [...] the left Hand rods for the partial Dividend.
5ly, Set down apart the number required foun [...] and draw a line above it: above that line a [...] over the Unit place the Figure of the cell take [...] and on the lefthand of the figure, place its squar [...] [...]s was expressed as before above in the margen [...] (u) the whole Operation ended, will appear brie [...] ly as in [...]
Note first, that scarce can you give any prece [...] in writing concerning Extraction of rootes, clear, but that they shall confound or puzzle young Student of Arithmetick, who will be ab [...] to learn more in an Hour of a Master shewi [...] [Page 41] him the practice, than in a day or week by his own reading of precepts.
2ly, note, that in Cubick Extractions it is not easie to fores [...]e or prevent the taking of too great a number out of the left Hand and Cube-Rods. We may probably conjecture that it will happen so, when the number taken is almost as great as the partial Dividend, and yet is to be increased by adding 1 or 2 cells more out of the Right Hand Rods.
3ly, Note, that when the capital cell of the left hand and cube-rods is greater than the partial Dividend, a Cypher is to be put in the Quotient as a Root, and the next partial dividend is to be brought down and joyned to the former.
4ly, note, that if any number remain after Extraction, it must be set down as the Numerator of a common Fraction, whose Denominator is a number made of the triple of all the rootes, and of the triple of the Square of all the rootes, and an Unity. For example. The rootes being 456 the Denominator would be 625177 See the Margent Triple of Rootes. 1368 Triple of Square. 623808 Ʋnity 1 Summa 625177 But far better it is to add to to the Numerator, or the remaining number, 3 or 4 triples of cyphers thus, 000, 000, 000. and work out by the precedent rules a clear and plain decimal Fraction.
Thus much (and indeed more than I first intended) concerning Pythagoras his Table, the Extensions thereof, and the Numbring Rods.
[Page 42] And here I might (had I not been too long already) exemplisy in a few instances; and there by shew, that whatsoever is performed by Logarithmes in Problemes of Trigonometry, Sines Tangents, Secants, Questions of Interest, &c. may be also performed by this Table of the third less Extension, and the double Rods, or by the double Rods alone: whether with more readiness and clearness, Practice and Experience must shew.
What also can be performed by Mr. Brigges's Table of 20 Chiliades of Logarithmes, may be done (if I mistake not) more plainly and speedily by this Table. For though it be but the half of 20 Chiliades, (it being only 10 Chiliades), yet by applying one double Rod thereunto, it exceeds 20 Chiliades by 980˙000 Chiliades.
For Conclusion, I will here suggest certain Lines divided into certain digits, which are singularly useful in measuring most things measureable, and make your Operation quick and plain, without trouble of division, or necessity of reducing inches into other known Integers. For though you measure by digits only, and multiply them by one another, yet the Product of Multiplication immediately gives you in hundreds or thousands, the superficial or solid content, not only in digits, but in other known termes of Feet, Yards, Acres, Gallons, Barrels, Bushels, &c. For Wine-Gallon-digits proceed thus, Take the Cube Root of 231, (the solid inches in [Page 43] a Wine-Gallon) which is 6, 136 vulg. Inc. proxime. Divide this Root into ten equal parts exactly, with subdivisions of each part into other ten less parts, and you have the Wine-Gallon-digits.
You measure by them, for example, a cylindrical capacity, and find the Diameter of it to be 56 digits, and height 60 digits, the area of that circle will be 24,64 superficial digits, which multiplied by 60 digits, produces 147.840 solid digits, whereof every thousand is a just Wine-Gallon. There are therefore 147 Gallons, and 840 digits towards another thousand or Gallon: that is above three quarts.
For Beer or Ale Gallon digits, take the cube Root of 282 solid vulg. inches in the AleGallons, which is 6,558 proxime. Divide this Root into ten equal parts, with subdivisions, as above.
For Beer-barrel-digits, take the cube Root of 10152 (solid vulg. inc. in a Beer-barrel of 36 Gall.) which is 21,653 proxime, to be divided into ten equal parts with subdivisions, a [...] before.
For Foot-digits take the square-Root of 144 (a Foot square) or cube▪Root of 1728, (a Foot solid) both which is 12 common inc. Divide this Root 12 into ten equal parts, with subdivisions; measure and work by them, every hundred square will be a true Foot square, equal to 144 common inc. and every thousand solid, a Foot solid equal to 1728 com. inc. For example, you measure a tetragon pyramide, whose one square side is 50 Foot-digits, and height 60. The square [Page 44] of 50 is 2500, and gives the area of the pyramid [...] at the bottom, viz. 2500 sq. dig. or 25 sq. Feet [...] multiply the area 2500 by 20 (a 3d of the height 60) and the product will be 50000 fol. dig. that is 50 solid Feet, equal to 86400 solid common inc [...]
These 4 lines of Wine-gall. dig. Beer-gall. dig. and foot-digits, are of excellent use in Gauging, and measuring any thing by feet sq. or solid, and may be conveniently cut on a Ruler, or long measuring staff, hard by or on both sides of a line of common inches, so that by meer inspection you may see how much they differ amongst themselves, and from common inches. If you desire yard-dig. to measure by sq. or cube-vards, divide 36 the square Root of 1296 (a sq. yard in common inches) and cube Root of 46656 (a yard sol. in common inches) into ten equal parts, as in other digits above; So may you have bushel-dig. by dividing 12, 958, which is proxime the cube Root of 2176 (commonly esteemed a solid bushel in vulgar inches) into 10 equal parts. For measuring of Land, Mr. Gunter's Chain (of 100 links, equal to 4 perches or 66 foot in length, is very convenient. Every 10000 sq. links is a chain sq. or 16 perch. sq. or the 10th. part of an Acre s. 100000 of links sq. is 10 chains sq. or 160 perches sq. or an Acre sq. Note 1. That when the Root is great, as 20, 30, or more vulg. inc. then you may divide it into more than 10 equal parts, as 100. 1000, &c. Note 2. That in working by the aforesaid Root-dig, the contingent Fractions will be decimal and clear.