ENNEADES ARITHMETICAE THE NUMBERING NINE. OR, PYTHAGORAS His TABLE Extended to All Whole Numbers under 10000 AND The Numbering RODS Of the Right Honourable JOHN LORD NEPEER, Enlarged With 9999 Fixed Columns or Rods, of Single, Double, Triple and Quadruple Figures, and With a New Sort of Double and Movable Rods, for the much more sure, plain and Easy performance of Multiplication, Division, and Extraction of Roots.  
The Whole being very Useful for most Persons, of whatsoever Calling and Employment, in all Arts and Sciences.  
All having frequent Occasions of Accounts, Numbering, Measuring, Surveying, Gauging, Weighing, Demonstrating, etc. The Divine Wisdom having from the Beginning Disposed all things in Measure, Number and Weight, Sap. 11. 21  
LONDON. Printed for Joseph Moxon, at the Sign of Atlas in Ludgate-street. Where also these Numbering Rods, (commonly Called Napiers Bones) are made and Sold. 1684.   

To the READER.  
Courteous Reader,  
THe End and Use of the Ensuing Table will be the better understood, if something in Brief, by way of Preface, be premised concerning these Three Points. First, the Table of Pythagoras; Secondly, the Extensions of the same; and Thirdly, the Numbering Rods. Wherhfore be pleased to peruse the following Preamble concerning the said Points; The Reading whereof will not Cost thee much above an Hour or Two. But First, vouchsafe to hear, what is meant by the New Title of Enneades Arithmeticae, or Numbering Nine.  
The whole following Table containing Gradually all the whole Numbers from 1. to 9999▪ Inclusive, Viz. 9 of One single Figure, 90 of Two Figures; 900 of Three Figures; And 9000 of four Figures: Again, every one of the said whole Numbers (multiplied severally by all the 9 Unites, 1, 2, 3, 4, 5, 6, 7, 8, 9) making so many Columns, as there are Capital Numbers, to wit, 9999. each Column, consisting of 9 distinct Numbers, and these 9 Numbers being the products of 9 distinct Vnits, I think we may with good Reason call the said Columns (yea and the Numbering Rods also; For in effect they are the same thing) by the Name of Enneades Arithmeticae, that is, the numbering Nine, or more expressively, the numbering Ninities. For the Greek word Ἐννεας being a Noun Substantive, signifies properly the Number of nine in abstracto, which may as well be called a Ninitie, as Ἐνας signifying the Number of One in abstracto, is interpreted, Unity, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 signifying the Number of Three in abstracto, is Translated Trinity. But though there be but Nine Cells expressed in every Rod, yet one other Negative Cell of Ciphers is ever to be understood, which if you please, may be sufficiently insinuated by putting so many Points over the Vertical Cell, as there are Figures in the Vertical; see Figure 10. whose Vertical is 1̇ 2̇ 3̇ 4̇ and may be pointed, as you see here, to insinuate a Supposed Cell of Ciphers.   

Concerning the Table of PYTHAGORAS.  
THE known Arithmetical Table, invented by Pythagoras, (such as you see expressed in Figure 1.) is not only an easy 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 Concourse. you will found the Number 56. the just Product of 8 multiplied 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 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 straight Column, parallel to the side A C or B D. The Columns are 9 answerable to the 9 Vnits 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, etc. But what is worthy of Observation, these two different sorts of Columns, Capital and Transverse, though most cross one to another, do most punctually agreed 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 Ten, Hundreds, Thousands, 10000ds, 100000. 1000000▪ 10000000 and 100000000. in great variety, and all actually and orderly Tabulated, showing at the same time, their true Product and Dividends, with multiplyers, Divisors, and Quotients. As for Ten you see 12 Tabuated on the 1. and 2 Col. Than 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. Than 234 345. 456. etc. As for Thousands, you have 1234. Tabulated on the four first Col. Than 2345. 3456. 4567. etc. and so of ●he rest, till you come to 123456789. a multiplicand or Divisor of all the Capital Vnits, 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 Vnit, which you choose for multiplyer to be counted from the Right Hand to the left. For example, if you multiply 123456˙789 by 2. the product will be the second transverse col. gathered from the right hand to the left, viz. 246913˙578. If you multiply it by 9 the product will be the 9th. transverse column, viz. 1111111˙101▪  
The second thing very observable is, that if you turn Pythagoras his Table in such manner, that all the Numbers remain unchanged in their cells, and yet the Figures 9 8. 7. 6. 5. 4. 3. 2. 1. ●ying between C▪ and A. become Vertical, which 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 found another great Variety of Multiplyers, Divisors, Products, Dividends and Quotients, and of greater Numbers than before, all differing from the former, and all Tabulated on contiguous columns. As for ten, you see 98 Tabulated on the two last transverse columns, than 87. 76. 65 etc. As for Hundreds, you have 987 876. 765. etc. 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˙975308˙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 show the product 8˙888888˙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, than 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 Neighbouring Columns, and makes but one Number of them both. If the two Figures added should make 10. than put down a cipher, and carry one to the next Number on the left Hand: If they make more than 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 than is Universal, whatsoever under-cell 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 Ten, 100ds. 1000ds. etc.  
The 4th. Observable thing is, and of chief 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 divides nor multiplys any Number) and the Dividend and multiplyer never so great. For in Division it performs the work, or gives the Quotient, by mere Substraction of its own Numbers out of the Dividend: and in multiplication it gives the Product, by setting down in due order it's 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 
a  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 showing its cell, and the cell the Quotient. Again, take the column 25. for a Multiplicand. and multiply it by 315861. the product will be 7896525, which before was the Dividend. The Operation ended, will appear, as underneath at 
b  , 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 selfsame 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 reproduce the Dividend as Product of Multilication.  
The fifth thing observable is, That every ennead or column, be it never so little or great, 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 margin 
c 61 122 183 244 305 366 427 488 549 2745 . By this means you may examine 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. 1 R 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 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 Error 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 intent 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 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 gins its work with the lest figure, and Ends it with the greatest: but Division quite contrary, gins with the greatest and Ends with the lest.  
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 lesle Number is made the Muitiplicand, and the greater the Multiplier; Than in Division given, the Divisor is the lesle 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 Vertical, and those either pure integers or mixed with Fractions) contain three different Numbers, which are exact Roots of three exact Square Numbers, the two lesle 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 Angles proxime 90. 53. 8. and 36. 52. For example, take the Rod of 4. whose third, four●● and fifth Cells contain these three Number's 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 lesle squares 〈◊〉 added together, make exactly 400. the greatest 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 endeavour 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 Ten, 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 to Col. 1111 inclusive; All which 127 Columns or bones are voided of all collat. add. And therefore all the 8 Under cells in them are marked with Stars, as Signs of non-addition. Note, that no Vertical cells have any collat add. nor stars before them. Note also, that not 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 etc. 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 multiply 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 expressed in Figure 13. far different from Figure ●. the Table of Pytha. whose collat. add. it wholly ●akes away, and yet in substance is the same with ●he Table.  

concerning 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 breadth way with as many lateral Numbers: The lesle extends it only 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 like it adds to the 9 Lateral Units, viz. 90 more L●terals. As all the cells with their enclosed Num● in Pytha. T. are known to be 81. by multiplying the two greatest Units, 9 Capital and 9 Lateral together; so by multiplying 99 by 99 the two greatest Capital and Lateral Numbers of two places, yo● will found the Cells of this first greater Extensio● to be 9801. The first lesle Extension adds to the● Capital Units (as did the first greater Extensio● 90 Numbers of two places from 10 to 99 inclusive, 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 greatest 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, containing 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 other 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●● ●he one of 5 greater Extensions, the other of 5 lesle 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 Margins, one page whereof will contain 112 square Inches, In every page reckon ●000 cells: In every Tome a 1000 pages.  

Five greater Extensions of Pythagoras his Table:  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˙999800˙001 99˙9998000001 Pages. 11110˙888 801/900 1˙111108˙888 801/900 Tomes. 11110. 889/1000 1111108 889/1000  

Five lesle Extensions.  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 expressed by 5. oblong Quadrats, one under another. In the first Quadrat is the Number of the Extension, First, second or third, etc. 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● 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˙998000˙001: and consequently, according to Allowances , page's ●1˙111108˙888 801/900, and Tomes in folio 1111˙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 straight line, they would make a rank of books above 52 English miles long. Or if all the aforesaid pages, their Margins cut of, 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 ourselves with the third lesle Extension, in which as the Table shows, 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, second and third lesle 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. lesle 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. Wherefore there being 9999 Capitals in this present Table, there must be also 9999 Columns, which in substance and in effect are the Numbering Nine, 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 double Rods of Ten, 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 than your Multiplicands and Divisors may be any Number under 100000˙000. But let us proceed to the third point of the Numbering Rods.   

Concerning the Numbering Rods or Bones.   rhombus 
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, than ●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 lesle Numbers than 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 lesle Number yet of four places; so is 1001. 1002. 1003. and so on till we come to 1110. all lesle Numbers than 1111. but all of four places, and requiring Stars in all their undercells. But whatsoever Number of four places is greater than 1111. as is 1112. 1113. 1114. and so on till 9999. than 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 last division. See Fig. 8. 9 10. and 11. according to this Model of placing Stars before certain 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 Vnits. which Table being directly the first lesle 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 lesle Extensions (2d. and 3d. of Pithy Table) which soon were completed, and that very readily, by the help of the double Rod● (whereof I had made some Sets) and the Table of the 1 lesle 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 itself 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 lesle 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 lesser Rods. 2dly they Tabulate the sel● same number with great variety of Rods, differing ●n Specie one from another. See Fig. 12. 3dly they ●ake away more than the half of collateral Additions, the chief trouble of numbering Rods. 4thly ●hey more readily show the product of multiplication and Quotient of Division in great numbers ●nd lesser Rods. Two of the double Rods reach ●o any number under 
a Collat. Additions 1 at the most  10˙000 three of them to ●ny under 
b Collatine Additions 2 at the most  100˙0000. Four of them to any under 
c Collat. Additions 3 at the most  100000000. etc. This and more the double Rods perform by themselves. But join 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 
d Collat. Additions 1 at the most  1000˙000. Two to 
e Collat. Additions 2 at the most  100000˙000. Three ●o 
f Collat. Additions 3 at the most  10˙000000000. etc. To use them with the Table of 9999 columns, it is necessary, that the Rods be of the same length with the columns, though the same breadth is not precisely required.  
The Rods having on them all the Capita● Numbers from 1. to 99 they will require either 50 thin two-faced Tallies, or 25 square-side● Parallelopipedons of four faces. It will be convenient to have every Rod twice over, (though 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 designs 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 practic 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 ● 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 Nine, or ●ights, or Sevens. etc. And in what multiplicity ●ou please, of the same Figures, as 3. 4. 5. Nine, ●ea 10 Nine, 20 Nine, 100 Nine: And so of ●ights, Sevens, six etc. 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 tabulating of Rods, or collateral Addition. See ●e Scheme of the said special Rods pag. ult. Fig. ●4 where observe that the nine single Units occupy 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 middle, a 3d. ending on the Right Hand. Some few ●nder cells (not above 8 in 72) have 4 Figures in ●hem, and than the two last on the Right Hand are ●nding Figures. The middle figure is most remarkable, and more than 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 supposed Vertical, abating one: for example, Suppose ●he Vertical to be Ten Nine, or 9˙999999˙999. In the 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 lesle, than Ten nine in the Vertical: So that the said 2ds. cell 198, is in operation 19˙999999˙99● or the Vertical multiplied 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 prefixed before it, according to what is abovesaid concerning Asterisks, than 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˙444444˙444. be given for a Multiplicand, and 279 for a multiplyer, than 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 lesle, than 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 lesle than 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 lesle than Ten of the Vertical, by reason of the Star prefixed before the second cell. The Work ended would appear thus in Multiplication.  
Multiplicand 4˙444444˙444 39999999996 cell. 9 Multiplier ╌279 31111111108 cell. 7  8888888888 cell. 2  Product 1239˙999999˙876   
Product 1239999999876 In Division, Multiplicand the Divisor, Product ●he Dividend, and Multiplier the Quotient.  
 Dividend  Divisor 4˙444444˙444) 1239999999876 (279 Qu.  2╌ 8888888888   35111111107   7╌ 31111111108   39999999996   9╌ 39999999996   0000000000   
The square and cube Rod aught to be once o●er in every Set, with three or four cypher-Rods, ●s you see expressed in Fig. 11, All are to be so orderly 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 itself in a page, (with its columns, 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, merely by cutting out the capital columns, and passing them on Tallies of Wood or other matter (too faced or four faced) of th● same length and breadth with the columns. Besides, as is abovesaid, this little Table of the first lesle Extension, with one double Rod applied unto it at a time, performs all the whole Work of th● great Table of 9999 fixed 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 lesle 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 advices 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 page's before your eyes, will show you a just Hu●dred of Columns. To found the Number yo● seek for, more readily, you may Tack to the Margin little outstanding Labels, or Indices, she 〈◊〉 before you open the Book, where every 100 ●nd 1000 gins, 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 lie Flat without any uprising; ●or so it will be more easy to Contabulate the Rods with the Table, when occasion requires. perhaps it would be better to print it in a little ●olio, for use at home in your Closet or Library; ●or than every page would contain its Hundred ●f Columns, easy to be found out by their own natural order.  
As well the forementioned Table of 9999 fixed Columns, as the single and double movable Rods ●erve equally in Decimal and common Arithmetic; 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 Ionas Moor, Mr William Leybourne and others in their Arithmetical Treatises. Wherhfore I shall say no more of them but only show 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 Tabulate the greater Number as multiplicand when one is greater than the other. For example. Tabulat● 4628 to be multiplied by 72. Place Unit unde● Unit, as in the Margin 
a  Than 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 
b  Add the tw● cells together, and the total Sum or product o● multiplication will be 333216 as you found in 
c  But if you Tabulate 72 to be multiplied by 462● the operation will appear as in 
d  .  
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 than a Triple, a Triple than a double, ● double than 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 Margin 
e  with a Semilune, for the Quotient, and Tabulate the Divisor. Than inquire how often can the Divisor 4628. be taken out of the first partial Dividend, viz. 5678. Only once; Therefore 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 dispatched) as in 
f  . Next bring down 5 out o● the Dividend, and place it on the right Hand of the remainder, as you see in 
g  separated with a Comma. This done, seek in the cells of the Divisor for 10505 (the next partial Dividend) or for the next lesle Number than 10505. In the 2d. cell is 9256 the next lesle. 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 dispatched. Than will the work stand, as in 
h   
In like manner proceed to found the 3d. and 4th. figure of the Quotient, and the work finished will appear, as in 
i  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 
k  When in one Number ther● hap to be two columns, there will be a cipher in the Quotient; when 3 Commaes, than 2 Ciphers in the Quotient. The like Method serve● 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 of 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 
l  The 4 points foretell that in the operation there will be 4 partial Dividend●, and as many Roots. Than seek in the cells of the Square Rod for 70 (the 1 partial Dividend) or the nearer lesle number to 70. 64 is the nearest lesle 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 
m  . For the finding the 2d. 3d. or any other Square Roots following, observe these Rules: First, bring down the next parttial Dividend and join 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 lesle) ●● your last remainder joined to the next partial Dividend 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 join it to 6 to make 647.  
Secondly, Double the Root 8 and Tabulate 16 with the Square Rod.  
Thirdly, seek 647 (or the nearer lesle) in the Tabulated Rods the third cell gives 489, the next lesle, which Subtracted out of 647, leaves the remainder 158. The 3d. cell gives 3 for the 2d. Root: see the Margin 
n  For the finding o● the 3d. and 4th. Root, proceed as before. Th● whole Operation ended stands as underneath a● 
o  or according to the Method, mentioned in division, underneath at 
p  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 Numerator 2. 3. or more Couples of Ciphers, and wor●●s before, and you will found 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. Figure, and a Semi-Lune for the Roots, as underneath at 
q  how many points, so many parti●● Dividends and Roots will be in the Operation  
2ly, Seek in the Cube Rod for 94 or the neare● lesle number: In the 4th cell you found 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 
r   
For the finding of the 2d, or any other following Root, observe these Rules.  
1st. Bring down the next partial Dividend 81● and join it to the remainder 30, on the Right Hand, as in 
s   
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 lesle number, in the cells of the left Hand Rods. In the 6th cell you found 29016 the next le●● yet indeed too much, 2ss will after appear. Set th●● this number down apart, and draw a line abo●e ●t as you see in 
t  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 
u  than 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 
w  adding all the numbers under the line into one Sum, viz. 33336, as you see in 
x  which being too great to be taken ●ut of 30818. you must go back and take a lesle ●ell, than 6.  
Take therefore the cell 5. which hath in it the ●e number 24125. writ 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 margin 
y  ●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 
z  to make 27125 which taken out of 30818 there will remain 3693 and the work stand as in the margin. 
a  For the 3d Roo● do as you did for the 2d.  
1st, Bring down the nex● Dividend and join 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 
b   
Note first, that scarce can you give any prece● in writing concerning Extraction of roots, clear, but that they shall confounded or puzzle young Student of Arithmetic, who will be ab● to learn more in an Hour of a Master showing him the practice, than in a day or week by his own reading of precepts.  
2ly, note, that in Cubick Extractions it is not easy to foresee or prevent the taking of too great a number out of the left Hand and Cube-Rods. We may probably conjecture that it will hap 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 cipher is to be put in the Quotient as a Root, and the next partial dividend is to be brought down and joined 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 roots, and of the triple of the Square of all the roots, and an Unity. For example. The roots being 456 the Denominator would be 625177 See the Margin 
c Triple of Roots. 1368 Triple of Square. 623808 Unity 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 Numbering Rods.  
And here I might (had I not been too long already) exemplisy in a few instances; and there by show, that whatsoever is performed by Logarithmes in Problems of Trigonometry, Sins Tangents, Secants, Questions of Interest, etc. may be also performed by this Table of the third lesle Extension, and the double Rods, or by the double Rods alone: whether with more readiness and clearness, Practice and Experience must show.  
What also can be performed by Mr. Brigges' 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 terms of Feet, Yards, Acres, Gallons, Barrels, Bushels, etc. For Wine-Gallon-digits proceed thus, Take the Cube Root of 231, (the solid inches in 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 lesle parts, and you have the Wine-Gallon-digits.  
You measure by them, for example, a cylindrical capacity, and found 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 come. inc. For example, you measure a tetragon pyramid, whose one square side is 50 Foot-digits, and height 60. The square of 50 is 2500, and gives the area of the pyramids 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 mere 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. than you may divide it into more than 10 equal parts, as 100 1000, etc. Note 2. That in working by the aforesaid Root-dig, the contingent Fractions will be decimal and clear.  
FINIS.    

ERRATA.  
Fol. 32. lin. 12. set them thus 72.  
Fol. 33 l. 21. read thus  
Fol. 39 l. 12.   

TETRASTICHON. In Enneadas Arithmeticas.  
COnditor Aen●idos peperit sibi nobile Nomen▪ Nullum Nomen habet Conditor Enneados. Si tamen Enneados, quaeratur, quis fuit Author? Baro, Refer, Neperus, Pythagorasque fuit.  
The Aeneid's Author is a Man much Famed,  
The Ennead's Author not so much as named;  
But if you are asked who th' Ennead's Author was▪  
Say Lord John Neper, and Pythagoras.   

DISTICHON. In Tabulam 10000 Enneadum.  
HAc Tabula Enneadas decies tibi Mille Ministr● Pythagorae tantum prisca Tabella decem.  
This Table gives Ten Thousand Enneades, when Pythagora's Old Table gives but Ten.   

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 * 2 * 4 * 6 * 8 10 12 14 16 18 * 20 * 22 * 24 * 26 * 28 * 30 * 32 * 34 * 36 * 38 * 40 * 42 * 44 * 46 * 48 * 50 ▪ 52 * 54 * 56 * 58 * 60 * 62 * 64 * 66 * 3 * 6 * 9 12 15 18 21 24 27 * 30 * 33 * 36 * 39 * 42 * 45 * 48 * 51 * 54 * 57 * 60 * 63 * 66 * 69 * 72 * 75 * 78 * 81 * 84 * 87 * 90 * 93 * 96 * 99 * 4 * 8 12 16 20 24 28 32 36 * 40 * 44 * 48 * 52 * 56 * 60 * 64 * 68 * 72 * 76 * 80 * 84 * 88 * 92 * 96 100 104 108 112 116 120 124 128 132 * 5 10 15 20 25 30 35 40 45 * 50 * 55 * 60 * 65 * 70 * 75 * 80 * 85 * 90 * 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 * 6 12 18 24 30 36 42 48 54 * 60 * 66 * 72 * 78 * 84 * 90 * 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 186 192 198 * 7 14 21 28 35 42 49 56 63 * 70 * 77 * 84 * 91 * 98 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203 210 217 224 231 * 8 16 24 32 40 48 56 64 72 * 80 * 88 * 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 248 256 264 * 9 18 27 36 45 54 63 72 81 * 90 * 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 234 243 252 261 270 279 288 297 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 * 68 * 70 * 72 * 74 * 76 * 78 * 80 * 82 * 84 * 86 * 88 * 90 * 92 * 94 * 96 * 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 102 105 108 111 114 117 120 123 126 129 132 135 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 189 192 195 198 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 204 210 216 222 228 234 240 246 252 258 264 270 276 282 288 294 300 306 312 318 324 330 336 342 348 354 360 366 372 378 384 390 396 238 245 252 259 266 273 280 287 294 301 308 315 322 329 336 343 350 357 364 371 378 385 392 399 406 413 420 427 434 441 448 455 462 272 280 288 296 304 312 320 328 336 344 352 360 368 376 384 392 400 408 416 424 432 440 448 456 464 472 480 488 496 504 512 520 528 306 315 324 333 342 351 360 369 378 387 396 405 414 423 432 441 450 459 468 477 486 495 504 513 522 531 540 549 558 567 576 585 594 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 201 204 207 210 213 216 219 222 225 228 231 234 237 240 243 246 249 252 255 258 261 264 267 270 273 276 279 282 285 288 291 294 297 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 402 408 414 420 426 432 438 444 450 456 462 468 474 480 486 492 498 504 510 516 522 528 534 540 546 552 558 564 570 576 582 588 594 469 476 483 490 497 504 511 518 525 532 539 546 553 560 567 574 581 588 595 602 609 616 623 630 637 644 651 658 665 672 679 686 693 536 544 552 560 568 576 584 592 600 608 616 624 632 640 648 656 664 672 680 688 696 704 712 720 728 736 744 752 760 768 776 784 792 603 612 621 630 639 648 657 666 675 684 693 702 711 720 729 738 747 756 765 774 783 792 801 810 819 828 837 846 855 864 873 882 891  

Figure. 1   A  B 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81 
Fig. 2  S. 1 C. 01 0 4 08 0 9 27 0 16 64 0 25 125 0 36 216 0 49 343 0 64 512 0 81 729 0 
Fig. 3  A  B 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81 
Fig. 4  S. 1 C. 01 0 4 08 0 9 27 0 16 64 0 25 125 0 36 216 0 49 343 0 64 512 0 81 729 0  numerical table too complicated to transcribe 
Fig. 5   numerical table too complicated to transcribe 
Fig. 6   numerical table too complicated to transcribe 
Fig. 7  
Fig. 8  12 * 24 * 36 34 * 68 * 102 * 48 * 60 * 72 136 170 204 * 84 * 96 108 238 272 306 
Fig. 9  123 * 246 * 369 * 492 * 615 * 738 * 861 * 984 1107 
Fig. 10  1̇2̇3̇4̇ * 2468 * 3702 * 4936 * 6170 * 7404 * 8638 * 0872 11106 
Fig. 11  S. 1 * 4 * 9 C. 01 * 08 * 27 0 * 0 * 0 00 * 00 * 00 000 * 000 * 000 16 25 36 * 64 125 216 * 0 * 0 * 0 * 00 * 00 * 00 * 000 * 000 * 000 49 64 81 343 512 729 * 0 * 0 * 0 * 00 * 00 * 00 * 000 * 000 * 000 
Fig. 12  78,96,78 78,96,7,8 78,9,67,8 7,8,9,67,8 7,8,96,78 7,8,9,6,78 54,72,54 72 54,72,5,4,7,2 54,7,25,4,72 5,47,25,4,72 5,4,7,2,54,72 
Fig. 13    1 2 3 4 5 6 7 8 9 * 2 4 6 9 1 3 5 7 8 * 3 7 0 3 7 0 3 0 7 * 4 9 3 8 2 7 1 5 6 * 6 1 7 2 8 3 9 4 5 * 7 4 0 7 4 0 7 3 4 * 8 6 4 1 9 7 5 2 3 * 9 8 7 0 5 4 3 1 2 1 1 1 1 1 1 1 1 0 1. 
Fig. 14  1 * 222 * 333 2 * 444 * 666 3 * 666 999 4 888 132 5 110 165 6 132 198 7 154 231 8 176 264 9 198 297 * 444 * 555 * 008 * 888 110 132 132 165 108 170 220 204 220 27● 330 204 330 396 3108 385 402 352 440 5328 300 405 504 * 777 * 888 * 999 154 176 198 231 264 297 3108 352 396 385 440 495 462 5328 594 5439 0210 693 6216 7104 792 693 792 891 
Fig. 15  Sq. 1 * 4 * 9 C. 01 * 08 * 27 0 * 0 * 0 * 00 * 00 * 00 16 25 36 * 64 125 216 * 0 * 0 * 0 * 00 * 00 * 00 49 64 81 343 512 729 * 0 * 0 * 0 * 00 * 00 * 00