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Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	divide_v	equal_a	line_n	1,364	5	9.1106	5	false

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A48329	The art of numbring by speaking-rods, vulgarly termed Nepeirs bones by which the most difficult parts of arithmetick, as multiplication, division, and extracting of roots both square and cube, are performed with incredible celerity and exactness (without any charge to the memory) by addition and subtraction only / published by W.L.; Art of numbring by speaking-rods, vulgarly called Nepeirs bones	Leybourn, William, 1626-1716.	1667 (1667)	Wing L1904; ESTC R21895	21,349	105

The ART of NUMBRING BY SPEAKING-RODS Vulgarly termed Nepeirs Bones By which The most difficult Parts of ARITHMETICK As Multiplication Division and Extracting of Roots both Square and Cube Are performed with incredible Celerity and Exactness without any charge to the Memory by Addition and Substraction only Published by W. L. LONDON Printed for G. Sawbridge and are to be sold at his House on Clerkenwell-Green 1667. THE ARGVMENT TO THE READER THe Right Honourable John Lord Nepeir Baron of Merchiston in Scotland In the Composure of those ever to be admired Tables of his Invention called Logarithms finding his Calculations so laborious in long and tedious Multiplications Divisions and Extracting of Roots that his Invention to him must needs render it self very unpleasant had he not known that the Labour when finished will crown both Him and his Work He advised with divers Learned men studious in the Sciences Mathematical and to them and amongst them especially to Mr. Henry Briggs who by a Learned and able Divine was styled and not without due respect our English Archimedes to him I say this honourable Lord imparted his Invention who joyning issue with him in this Herculean Labour brought them to that perfection to which they are now to the admiration of all Europe arrived In the tedious calculation of these Numbers the Author finding his Work to go on but very slowly at length studying out for some help by Art to assist him in this his Noble Enterprise thinking upon several helps at last by the blessing of God he hapned to finde out this which I here intend to describe and shew the use of with some Additions and variation from what he hath himself done in his Treatise in Latine Published and Printed at Edinburgh in Scotland in Anno 1617 Entituled Rabdologiae seu Numerationis per Virgulas The uses whereof I shall in the following Tractate endeavour to render so plain and easie that he that can but Add and Substract shall be made able in a days time and less to Multiply and Divide any great Numbers nay and to Extract both the Square and Cube Roots I have begun this Treatise with the Fabrick and Inscription of these Rods according to the Authors Description which being not so convenient either for Portability or Practice as some others which I have seen and used I have described them I think in the best manner they possible can be contrived For their Vse I am sure I have done more than hitherto I have seen done and if I mistake not to as good and effectual purpose I do not publish it as a Novelty neither do I attribute much in it to my self besides the Method for had I not been desired I should hardly have thought upon it however it being done Accept it and Vse it till I direct something else to thee which may be more acceptable till when I bid thee heartily Farewel Place this Figure at the begining of the Book Fig 2. Fig 3. Fig 4. Square Cube CHAP. I. Concerning the Fabrick and Inscription Of these RODS IN the foregoing Argument I told you That the Author and Inventer of this kind of Instrument of which I intend to shew the Use called it RABDOLOGIA and the Word he thus defines RABDOLOGIA est Ars Computandi per Virgulas numeratrices That is RABDOLOGIE is the Art of Counting by Numbering Rods. I. Of the Fabrick of these Rods according to the Inventors Description of them These Rods may be made either o● Silver Brass Box Ebony or Ivory of which last substance I suppose the● were at first made for that they ar● for the most part by all that know or use them called NEPAIRSBONES But let the matter of which they are made be what it will their form according to this description i● exactly a square Parallelepipedon 〈◊〉 the length being about three Inches and the breadth of them about One tenth part of the length But the length of these Rods are not confined to three Inches but let the length be what it will the breadth must be a tenth part thereof but that may be accounted a competent breadth that is capable of receiving of two numerical Figures for there is never upon one Rod required more to be set on the breadth thereof The breadth of these Rods being exactly One tenth part of the length thereof when 10 of these are laid together they do exactly make a Geometrical square and if 20 of them be tabulated or laid together they will make a right-angled Parallelogram whose length is double to its breadth If 30 be tabulated the Figure will be still a Parallelogram whose length will be three times the breadth and so if 40 four times the length 65 si● 650. The Rods being thus prepared of exact length and breadth let each of them be divided into 10 equal parts with this Proviso that Nine of the Ten parts stand in the middle of each Rod and the other tenth part must be divided into two parts half whereof must be set at the one end and the other half at the other end o● the same Rod. Then from side t● side draw right Lines from division ●● division so is your Rod divided into Squares on every side thereof Lastly from corner to corner of ever● of these Squares draw a Diagona● Line and that will divide ever● Square into two Triangles Th● Rods being thus prepared and line● first into Squares and then into Triangles they are then fit to be numbered The Figure 〈◊〉 at the beginning of the Book shews the Form of one of these Rods lined as it ought to be CHAP. II. How these Rods are to be Numbred IN the two half Squares which are at the ends of each Rod on every side there are set one single Figure on each side of every Rod one in the division at the end thereof so every Rod containing four sides Ten Rods will contain 40 sides and so consequently will have 40 single Figures at the ends of every of them that is there will be upon the ten Rods amongst them four Figures of each kinde that is four Ones 1111. four twos 2222. four threes 3333. four fours 4444. four fives 5555. four sixes 6666. four sevens 7777. four eights 8888. four nines 9999. four Cyphers 0000. And here it is to be noted That what Figure soever it be that standeth at the top of the Rod alone the Figure that standeth alone on the other side of the same Rod maketh that figure up the number 9. As for example If 1 stand on one side 8 will stand on the other side so 2 and 7 be As in this Table where If 1 stands alone at the top of any side of any of the Rods then 8 standeth on the other side of the same Rod. 2 7 3 stands alone 6 standeth on 4 at the top of 5 the other If 5 any side of 4 side of the 6 any of the 3 same Rod. 7 Rods then 2 8 1 9 9 0 This also is

serveth for the Square Root having the word Square written over head that for the Cube Root hath Cube written over head Thus having given you the Fabrick and Inscription of these Rods I will now shew you their use And first Concerning the Extracting of the Square-Root In Extracting of the Square-Root you must as in common Arithmetick when you have set down your Number make a Prick under the first Figure towards your right hand and so successively under every second Figure then under those Pricks draw two Lines parallel whereinto set the Figures of your Root as you finde them Your Number being thus placed and pricked as before is directed and as in the following Example you see done you may proceed to Extract the Root thereof as followeth Example 1. Let it be required to finde the Square-Root of this Number 12418576 first make a Prick under 6 another under 5 another under 1 and another under 2 under which Points draw two Lines in which you must place your Root and then will your Number stand thus Take the Rod for Extracting of the Square-Root and look in the first row or Colume thereof for the nearest Number you can there finde less then 12 which is as far as the first Prick in your Number reaches and you shall finde 9 against which in the third Colume you shall finde 3 set 3 under the first point between the Lines and 9 under the Line and substracting 9 from 12 there will remain 3 which set over 12 so will your Number stand thus Then in the middle Colume of your Rod between 9 and 3 there stands 6 take therefore one of your Rods which hath 6 at the top thereof and lay it upon your Tabulat by the left side of your square Rod then being there is 341 to the next Prick seek the nearest Number less upon your two Rods and you shall finde the next less to be 325 against which in the last Colume of your Square Rod stands 5 therefore place 5 under your second Prick and set 325 under 341 and substracting it from 341 there will remain 16 which set over head then will the Sum appear thus And in the middle Colume of your Square Rod against this 5 there stands 10 for this 10 you should take a Rod that hath 10 at the top but being there is no such take therefore one that hath a Cypher and place that between your Square Rod and your Rod of 6 and change your Rod 6 for one of 7 then shall you have upon your Tabulat one Rod of 7 another of 0 and your Square Rod. Thus must you always do when the Number in the middle Colume exceeds 10. Then looking upon your Sum you shall finde 1685 to your third Prick look therefore upon your Rods for the nearest less Number which you shall finde to be 1404 against which stands 2 in the last Colume set 2 between the Lines under the third Prick and 1404 under 1685 and substracting it from 1685 and there will remain 281 which place above so will your Sum stand thus And because the Number standing against in the middle Colume of your Square Rod between 1404 and 2 was 4 set 4 under your last Prick and take a Rod of 4 and put it between your square Rod and your Rod of 0 and because 28176 remains upon your Sum to the last Prick Look upon your Rods for the nearest Number thereunto and you shall finde the very Number it self to stand against the Figure 4 set therefore 28176 below and substract it from that above and there will remain nothing which denotes the Number 12418576 to be a square Number and the Root thereof to be 3524 and the work finished will stand thus This Sum had it been wrought by that second way of Division which I shewed in Chapter 7 it would stand as followeth Square Root Caution If at any time you look for the remainder upon your Rods and you cannot finde it there you must then place a Cypher between the Lines and proceed to the next Figure as by trying this other Example which I have inserted for practice will appear Another Example added for Practice CHAP. X. Concerning the Extraction of the Cube Root THere is somewhat more difficulty in Extracting of the Cube then of the Square Root Wherefore before I come to Example I will deliver the manner of the Operation together with such Cautions as are to be observed in the performance thereof All which immediately follow in this GENERAL RULE Write down the Number whose Cube Root you are to Extract and under the first Figure towards the right hand make a Prick or Point and so under every third Figure towards the left hand till you come to the end of your Number Under these Pricks draw two Parallel Lines as you did in Extracting the Square Root between which Lines you are to place the Figures of your Root as you finde them Then beginning at the Figure or Figures of the left hand Prick and going forward towards the right hand Extract by help of the Rod for Extracting tracting the Cube Root their Root or if the true Number be not on the Plate then the nearest less and placing this Root which never exceeds one Figure between the Lines and under its Point and take its Cube from the uppermost Figures which stand before or leftwards of the first Point and note the Remainder above Secondly Keep the Triple of this Root sound in the head or top of the Rods and triple the Square of the same Root and set this Triple on the head of the Rods and apply it leftwards of the Cubick Rod and the reserved Rod or Rods right-wards the Cubick Rod being in the midst between them and out of the left hand Rods and the Cubick Rod together pick or finde out the Multiple or next less Number then the Figures preceding the second Point which write apart in a Pater and note its Quotume over its utmost right-hand Figure and write the Square of that Quotume left-wards from the Quotume it self even in that order as you finde them in your Cubick Rod and under every several Figure of this Square write their Multiples found right-wards even such as the Figures themselves do shew So that every Multiple may end under its Figure or Quotume then add together these Multiples cross-wise and take their sum from the Figures foregoing the second Point and write the Remainder over them but write the right-hand Quotume before noted under the second Point between the Lines for the second Figure or Quotume of the Root And so is performed the Operation of the second Point which you must repeat through the several Points even to the last But in the practice by this Rule you may sometimes be at a stand wherefore to this GENERAL RULE that there may be no obstacle I will add these two CALLTIONS 1. CAUTION But in all Operations and Points it must be observed That if no