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A29742 An account of the rotula arithmetica invented by Mr. George Brown. Brown, George, 1650-1730.; Dary, Michael. Dary's Miscellanies.; Cooke, Francis, fl. 1669. Principles of geometrie.; Georgius, Henisschius. Tables of the astronomical institutions. 1700 (1700) Wing B5019; ESTC R4627 82,687 247

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compasse thereof doth couer all thinges like a thing without measure and ende but the elementall parte is couered within the compasse of the heauen the diameter thereof conteining the diameter of the earth 23. times Their distinction according to their motion is in that the celestiall parte hath a circulare and a sphericall motion the elementall a right motion more imperfect then the circulare Finally they are distinguished according to their office For of those thinges that are ingendred in the elementall parte the heauen working by a continuall motion is as it were the formall and efficient cause from whence life is deriued and the elementall parte which is subiect vnto passion and alteration is as it were the materiall cause from whence nourishment doth proceede The reason of the sublunare or elementall Region Chap. 9. THe Elementall region which the heauen encompasseth comprehendeth within it the elements wherin we are to consider the definition the number and the situation or order The elements are simple bodies aswell in respect of the mixt bodies which are vnderstoode to be compounded of them as of the simple and least partes as also in respect of the diuision for that they cannot be diuided into bodies of diners kindes if they be giuen pure and without mixture For the vse of liuing creatures and things growing doth make them impure The elements are 4. in number found so to be by sense and by reasons The elements are found to be soure by sense which the Physicians doe sollow First for that more simple bodies cannot be shewed 2 nature hath alotted vnto them certaine places to the end that other things might by thē be bred and nourished 3 nothing els can euidently be shewed wherof other things may be made 4 in liuing cretures there are certain parts agreable vnto the natures of the seuerall elements The Elementes are found to be foure by two reasons the former whereof is drawen from the number of the foure prime qualities and the foure folde possible knitting together of them For heate may be ioyned either with drinesse which two make fire or els with moisture which two do make vp aer and colde may be ioyned with moisture as it commeth to passe in the water or with drinesse as in earth The later reason is taken from the fower folde difference of the right motion For the elements are directly moued either vpwarde or downward Such things as moue vpward as light thinges do are said so to do either simplie as the fire which is the lightest of the rest or respectiuelie as the aer which is lighter then the water or the earth Such things as moue downeward as heanye things do are said so to do either respectiuelie as the water compared vnto the fire and aer or simplie as the earth which is the heauiest of all the rest The situation and order of the Elementes is found either by their motion or els by the communication of their qualities And first by the motion For inasmuch as the fire and the aer do naturally moue vpward the fier shall occupy the highest place the aer an vpper place and for that the water and the earth do naturally moue downward the water shall possesse a lower place and the earth the lowest Againe the order of the Elements is found out by the communication of their qualities for it were vnsit that such things as are merely contrary but such as in some sorte can agree together should be nigh one another The fire therefore shal be ioyned vnto the aer by reason of the heate common vnto them both the aer vnto the water by reason of the common moisture and the earth vnto the water by reason of coldnesse common to them both The two folde differences of the celestiall motions Chap. 10. THe whole frame of the world is caried round about with 2. motions each of them being distinguished from the other in name and in reason The one of them is called the first and vniuersall motion likewise the diurnall or worldlye motion because it bringeth the day vnto the world For in this motion the ☉ and all the celestiall bodies do euery day arise and set they call it also the violent and rapt motion because by the violent swiftnes thereof it carrieth with it the rest of the Spheres The other is called the second and particular motion altogether contrary vnto the former as by which all the particular orbes do resist the vniuersall motion They call it also Sinister motus the motion to the left hand as the former is in like sorte called dexter that is the motion to the right hand The 2. motions are also distinguished according to the reason or the substance in the which they are inherent For they differ the one from the other three waies The first difference is in respect either of the whole For the diurnal motion is common vnto all the celestiall bodies or els of the partes or starres either fixed or wandring which haue a motion peculiar and propre vnto themselues The second difference is either in regarde of the situation of the Axes For the diurnall motion is made vpon the Axe and poles of the world and therfore the Equator diuideth it in the middle but the propre motion is made vpon the Axe and poles of the Zodiake and therefore the Zodiake doth cut it in the middle Or els it is in regarde of the position of the termes inasmuch as the diurnall reuolution is made from the East vnto the west or as Plinte termeth it from the right towarde the lefte hand but the propre reuolution is from the West vnto the East or from the left toward the right hand The third difference is in consideration of the swiftnes For the diurnall motion fulfilleth his course within the space of 24. common howers but the propre motion in diuers distances of time according to the largenes of the orbes namely the orbe of the fixed starres performeth his circle in 36000. yeares of ♄ in 30 of ♃ in 12 of ♂ in 2. yeares of the ☉ in 365. dayes and about 6. houres of ♀ in 384. dayes after Plinie the orbe of ☿ in as many dayes as the ☉ and the orbe of the ☽ in 27. daies 8. howres The circulare forme and circulare motion of the heauen Chap. 11. THe Heauen is circulare in motion and in figure The circulare motion of the Heauen is proued as well by 2. experiments as by 2. argumentes The one experiment is taken from the starres of the 8. orbe which both in their rising setting do alwaies keepe one the same habitude both in regarde of the earth and one to another which thing can agree with none other then a circulare motion about the centre The other experiment is also taken from the starres of the 8. orbe alwaies appearing and retaining in diuers places the same distance from the Poles and onestom another which also agreeth with the circulare motion onelie The first
Column of Hundereds c. Nay the Denominator of Tens is 100. c. Also 4 is the Denominator of Farthings Lippies Firlots and Pecks in relation to Firlots but 16 is the Denominator of Bolls and of Pecks in relation to Bolls and 12 the Denominator of Pence and 20 of shillings c. This is well to be minded because we may have frequent use to speak of the Denominators of Species CHAP. III. Concerning Substraction SVbstraction finds the Difference betwixt two unequal Numbers The greater of these two Numbers is called the Charge and the lesser the Discharge In Substraction you must allways bring the several Figures of the Charge one after another together with the respective Figures of the Discharge the one on the Moveable and the other on the Fixt directly against one another and if the Figure of the Charge be equal to or greater than the respective Figure of the Discharge you have the Remainder at the Stop on the Moveable But if the Figure of the Discharge be greater than that of the Charge then against the Denominator of the Species on the Moveable you have the Remainder on the Fixt Thus were I to take 8 from 8 or 7 from 7 having set the one against the other you have 0 at the Stop So likeways if I were to take 5 from 8 having brought 8 on the Moveable against 5 on the Fixt I have at the Stop on the Moveable 3 for a Remainder But if I had been to take 8lib from 5lib the Remainder is on the Fixt against 10 the Denominator of Integers on the Moveable And 8d from 5 pennies the remainder is on the Fixt against 12 the Denominator of Pennies on the Moveable And 8sh from 5sh the Remainder is still on ehe Fixt against 20 the Denominator of shillings on the Moveable And 8 Ounces from 5 Ounces the Remainder is on the Fixt against 16 the Denominator of Ounces on the Moveable And the Peason of all this is plain because the Discharge in this case cannot be taken off the Charge but off the Denominator which is equivalent to a borrowed one of the next greater Species and the Overplus by the very Position of the Instrument is added to the Charge Only mind carefully that as often as the Remainder is found on the Fixt which allways happens when any figure of the Discharge is greater than the respective Figure of the Charge you must in that case esteem the next preceeding Figure of the Discharge an Unite more than really it is Taking 1 for 0 and 2 for 1 and 3 for 2 and so of others lib. ss d. 25123478 11 4 Charge 23254906 14 8 Discharge 01868571 16 8 Thus in this Example I bring 8 on the Moveable to 4 on the Fixt and because the pennies of the Discharge are greater than the Pennies of the Charge I look for 12 the Denominator of Pennies on the Moveable and against it I find 6 on the Fixt for my remainder These I set down under the Pennies Again because I found my last Remainder on the Fixt I esteem 14sh in the Discharge to be 15. for which cause I bring 15 on the Moveable to 11 on the Fixt and against 20 the Denominator of shillings I have 16 on the Fixt These I set down under shillings Thereafter for the same Reason esteeming the 6lib of my Discharge to be 7 I bring 8 the respective Figure of the Charge on the Moveable to it and because the Figure of the Charge is greater than that of the Discharge I have 1 on the Moveable at the Stop for the Remainder Thence because the last Remainder was found on the Moveable I must not change my 0 but bring 7 on the Moveable to 0 on the Fixt and the Remainder at the Stop is 7. And proceeding conform to these Directions with the rest I perfect the Operation finding allways the Remainder on the Moveable at the Stop when the Charge Figure is greater than that of the Discharge or equal to it but on the Fixt against 10. the Denominator of Integers on the Moveable when the Discharge Figure is greater than that of the Charge After the same M●nner and by the same Directions You may Substract any other Species whatsoever if You do but carefully mind the Denominators of the severall Species CHAP. IV. Concerning Multiplication MVltiplication supposes two Numbers called Coefficients to find a third called the Product which Product contains any one of the Coefficients as oft as the other contains an Unite Any one of the Coefficients especiallie the greater may be called the Multiplicand the other the Multiplier thus 3 times 4 is 12. of which 3 and 4 are the Coefficients and 12 the Product which Product contains 3 as oft as 4 contains 1. When one of the Coefficients is 10 100 1000. You need no Instrument for Multiplication in Integers for this is done merely by adding the Cyphers to the right hand of the other Coefficient Thus 10 × 64 is 640 and 100 × 64 is 6400 c. But when the Coefficients are all or many of them signifying Figures set the lesser Number under the greater thus Having Rectified the Rotula with the Pen in your Right Hand ready to Write and your Left Hand at the Rotula to turn the Moveable as shall be necessary because 8 is the last Figure of the Multiplier you must look for 8 × everv Figure of the Multiplieand one after another beginning at the last but you must not regard the Products on the Fixt but only upon the Moveable nameing allways the Tens of the Product first as a single Figure putting that Figure on the Moveable immediatly to the Stop and then the Unites setting them down on the Paper thus First I look for 8 × 2. and against that on the Moveable I find 1 and 6. for which cause I set 01 on the Moveable to the Stop and 6 I write on the Paper below the Coefficient 8. Then I look for 8 × 4. and against that on the Moveable I find 3 and 3. for which cause I turn 03 to the Stop and write down 3 on my Paper Then I look for 8 × 5. against which I find 4 and 3. here I put 4 to the stop and 3 again to the Paper Thence at 8 × 6. I find 05 for the Stop and 2 for the Paper And 8 × 9. I find 07 for the Stop 7 for the Paper And lastly at 8 × 7. I find 63 all for the Paper because it is the last Product After which I Rectify again Now as I have gone over all the Figures of the Multiplicand with 8 the last Figure of the Multiplier so may you do by 7 for the second Product and 5 for the third and 9 for the fourth carefully observing only to set the first Figure or that next the right hand of every particular Product under that Figure of the Multiplier by which it is produced these particular Products summed up do