whereas my Privilege is granted in Scotland on the first of December 1698. Yet his comes so far short of mine that I Verily believe had he seen or gotten a perfect Account of mine before he proposed his own he would have spared the pains of Publication I must Confess that for any thing I yet know his Tables or Circles for Multiplication and Division which indeed are very Ingenious and haue cost him much Thought are his own as also his Tables for the Reduction of Pence to shillings shillings to Pounds But in that Part which is common with his Roue and my Rotula he seems to haue got some hint of mine and this I am the more apt to belive because about the time that I was bussie in contriving the Rotula there was a very smart Gentleman a near friend of his Scholar with me at Stirline But that which gives me greater evidence in this particular is some expressions in his own Book which makes me fancy that he hath at least got some imperfect Description of mine before he contrived that part of his which serves for Addition and Substraction For whereas there is on my fixt Plate 3. Circles he speaks of three and yet Immediatly he takes away two of his turns them into Tables for Reduceing of Pence into shillings and shillings into pounds these not exceeding the limits of 120. and instead of the third on the fixt he gives us nothing but a little segment about a fifth part divided into parts begining at 0. and ending at 24. which he calls his fixt Index as also whereas my Circle is divided into 100. parts he Chuses to make his differ from mine 120. as being a common Product of 10. 12. and 20. These Numbers as he alleadges in the begining of his 1st Chapter being preferable to all other Numbers whatsoever and yet near the close of the same Chapter he acknowledges that it would be better to divide the Circle for Addition and Substraction into 100 parts or some Power of 10. and so the Instrument would become universall All which give me suspicion that in this part he hath goten at least some lame account of mine Moreover his Instrument is Defective and comes far short of mine even in Addition For in his the Practitioner is obliged to mind or mark down how many Revolutions his moveable Plate makes and every one being 120 he hath 120 to Multiply by the Number of Revolutions which is not only troublesom but likeways dangerous especially in reall Bussiness where a man whose mind is bussied both about the figures of his Columne and the points of his moveable plate is obliged at the same time to mind the severall Revolutions of his moveable Plate of which for every one he forgets or overlooks he loses 120 for his Pains whereas in mine a man is not tied to any such intention above once for 1000. which is more than any Columne does ordinarly contain the moveable Plate at every Revolution both marking and giving notice of the number of Revolutions But besides this in my Rotula the same Circles that serve for Addition and Substraction serve likeways for Multiplication and Division but in his Roue he hath one for Addition and Substraction and ten or eleven for Multiplication and Division and yet tho' the Circles were twice as large and tho' they contained near twice as many figures as they do they would be no more than what is necessary to do what I am able to perform by mine Lastly whereas his Tables are confined only to shillings and pence and these of limited Number not exceeding 120. There are on the waste on the middle of my moveable Plate Tables for the Reduction of shillings Pence Farthings Weights and Measures be the Numbers never so large Besides the Decimall Tables for Money weights Measures and the most ordinary common Fractions By the help of which six last sort of Tables the Multiplication and Division of Complex Numbers does become just as easie as that of Integers without all that tediousness which Mr Glover proposes in his Book To conclude what I have said here is no more than was necessary for the Vindication of my own Invention and to satisfie those who already are or hereafter may be misinformed either by the Story of Delamain or Mr. Glover's Roue Arithmetique who for what is Peculliarly his deserves a good degree of commendation and Encouragement CHAP. I. Concerning the Rotula and the Rectification thereof ALbeit in Books of this Nature it be usuall to prefix a Scheme of the Machine of which they treat Yet I have thought fit in this to omit that because such as have a Rotula need not a Scheme and such as want one have no use for a Book I shall therefore as briefly as I can describe the Rotula and then shew You how to use it The Rotula Consists of two Principall Parts to wit a Circular Plain moving upon a Center-pin this we call the Moveable Plate and a Ring whose Circles are described from the same Center this we call the Fixed Plate Because it is fixed to the Box to secure it from moving about the Center as the other does The Fixed Plate is divided into three parts by two Circles the Innermost of which is doubled with a little Interstice for Peg-Holes Near the Circumference of the Moveable there is another Double Circle with a small Interstice also betwixt them for Peg Holes The space without the double Circle on the Moveable and within that on the Fixt are both of them equally divided into 100. Parts and both are Numbered beginning at 0. 1. 2. 3. and so proceeding in a Naturall Order to 99. all the Divisions being drawen streight from the Center On the Fixt many of these Divisions are protracted some only to the middle part and others run over both for confining the severall single Coefficents of the Respective Tabular Numbers to which they are Prefixed with this Caution when the Coefficients are the same they are set down in the uttermost part and when any Number admits of two pair of Coefficients the one Pair is set in the Midmost and the other in the Out-most Part. Thus against 18. on the Fixt You will find in the midmost Part 2 × 9 that is two times Nine or Nine times two This × cross signifieing the word times and 3 × 6 in the outmost Also on the Moveable there is a Segment of a Circle within the Peg-hole Circle beginning at 9. of the Naturall Numbers and ending at 72. This Segment is likewise Divided by the same lines that Divide the outmost Circle of Naturall Numbers into equall parts On the Fixt Plate at the Division betwixt 99 and 0 there is a little bit of Metal Screwed or Rivited reaching likeways a lit le further than the Peg-hole Circle on the Moveable this piece of Metall we call the Stop and must always be placed next Your left hand with the Number 25 or 30. toward your
yeeld the total Product I shall subjoyn another Example and so end with Multiplication In this last Example the two Cyphers of the Multiplier are set to the right of the Unites of the Multiplicand and then multiplying by 9. I set the two Cyphers behind the Product and so what was but 9 times before does now become 900 times the Multiplicand You see also the Unites of the Product made by 7 set under 7 of the Multiplier and the Unites of that made by 3 under 3 of the Multiplier all the rest duely observing Rank and File To conclude this Chapter and make You prompt in finding your Coefficients Observe that all the Products of any Coefficients are contain'd within Ten Times the least of the two so that all the Products of 2 are within 20 and all of 3 within 30 c. CHAP. V. Concerning Division SECTION 1. DIvision serves to find a Number shewing how oft the greate of the two given Numbers contain's the lesser The greater of the given Numbers we call the Dividend The lesser the Divisor and the Number demanded or found the Quotient When as many Figures taken from the left of the Dividend as there are Figures in the Divisor are aequivalent to the Divisor or better than it then we set a Point over the last of these to Determine the first particular Dividend which for Brevity I shall call the first Dividual But if as many taken from the left of the Dividend be less than the Divisor the Point must stand over the next Subsequent Figure of the Dividend for Determining the first Dividual Having Determined your Dividual you must refer the first of the Divisor when they are equall in Number of places to the first of your Dividual but if they are unequall to the first two of the Dividual and so forward the second third and fourth Figures of the Divisor to the Subsequent Figures of your Dividual as they ly in Order So that in subduction where you begin at the last of the Divisor you must refer or Substract the Product of it from the last of the Dividual The Remainder of the first Dividual with the next following Figure of the Dividend Yeilds you a 2d Dividual So Soon as you have Determined your first Dividual you presently understand how many Figures you are to have in the Quotient to wit one for the Point or first Dividual and one for every subsequent Figure of the Dividend Wherefore if the 2d 3d or any other Dividual should happen to be less than the Divisor you must put a Cypher in the Quotient for that Dividual And so as if it were but a new Remainder bring down another Figure from the Dividend to wit the next following for a new Dividual I shall first shew you how to Divide by one Figure and then by two and after that by as many as you please In Division by any one Figure you have nothing to do but to bring the Dividual on the Moveable to the first Cell that occurres in which the Divisor is a Coefficient the other Coefficient in the same Cell is the Quotient and that having first drawn a Line below the Dividend You must set down under the last Figure of your Dividual and the Figure at the Stop on the Moveable you must set over the same last Figure of the Dividual for a Remainder And so proceed Rectifying every time before you apply to the next Dividual Example Here you see that 8 the foremost Figure of the Dividend being less than 9 the Divisor the Point for Determining the first Dividual stands over the second fiigure of the Devidend So that my first Dividual is 88 Which being thus Determined I understand that I am to have in my Quotient 7 Figures to wit one for the first Dividual and one for every Subsequent Figure of the Dividend These things considered and the Rotula Rectified I bring the first Dividual 88. on the Moveable to the first Cell that occurres on the Fixt in which 9 is a Coefficient and because the other Coefficient in the same Cell is 9 I set that down under 8 the last Figure of my Dividual and having 7 on the Moveable at the Stop I set 7 over the same last Figure of my Dividual for a Remainder then I Rectifiie Now the first Remainder and next Subsequent Figure being 76 I bring 76 to the first 9 Coefficient there I find 8 for my Quotient and 4 at the Stop for my Remainder these I set down as before the one under the other over the last Figure of the 2d Dividual and then Rectifie The 3d. Dividual being 45 and having without any Motion a Cell of 9 directlie against it I sind 5 for my Quotient and 0 for my Remainder So that the 4th Dividual becomes 04 which being less than 9 I set 0 in my Quotient and then The 4 still Remaining with the next Subsequent Figure of my Dividend making 43 I bring 43 on the Moveable to the first 9 Coefficient and there finding 4 for my Quotient and 7 at the Stop for my Remainder Having set down these and Rectified I find my next Dividual 72 against a Cell of 9 in which I have 8 for my Quotient and 0 for my Remainder So that my last Dividual being oniie 01 which is less than my Divisor I set nought in my Quotient and 1 the the last Remainder I set over 9 the Divisor at the end of the Quotient with a litle Line betwixt them for a Fraction Thus 1 9 If any Divisor consist only of one signifying Figure Cyphers you must Divide only by the signifyng Figure from the Quotient cut off as many Figures towards the right-Hand as there are Cyphers in the Divisor observing that if the signifying Figure be only an Unite You have no use for the Rotula or any other Instrument but meerly to write down the Dividend below the Line in the Quotient then cut off from it conform to the Number of your Cyphers Example first Divisor 1000. 976583 Dividend 976583 Quotient In this Example you see the Figures in the Quotient are the same with those in the Dividend because the signifying Figure of the Divisor is but an Unite But because there are three Cyphers to the Right of the Divisor I have cut off three Figures from the Right of the Quotient where you see that as your Dividual Point Intimates You have only three Integers in your Quotient namely those to the left-Hand and the Remainder is a Decimal Fraction or if you will the Numerator of a common Fraction whose Denominator is the Divisor thus Example 2d In this Example I first Divide as if my Divisor were only 8 so that I have 5 Figures in the Quotient just as if the Dividual Point had stood over 7 the Second of the Dividend But because of the two Cyphers in the Divisor I cut off two from the Right of the Quotient and so I understand that if 800 Men had to
Chap. 3. Of Spherical Triangles pag. 13. Chap. 4. Of the projection of the Sphere in plano pag. 20. Chap. 5. Of Planometry and the Centre of Gravity pag. 23. Ch. 6. Of solid Geometry p. 29. Chap. 7. Of the Scale of Ponderosity aliàs the Stilliard p. 43 Chap. 8. Of the 4 Compendiums for quadratique Equations pag. 45. Chap. 9. Of recreative Problems pag. 47. Dary's Miscellanies CHAP. I. Of the Inscription and Circumscription of a Circle 1. FOrasmuch as the Ratio of an Arch line to a right line is yet unknown it is absolutely necessary that right lines be applyed to a Circle for the Calculation of Triangles wherein Arch lines come in Competition 2. Right lines applyed to a Circle are Chords Sines Tangents Secants and versed Sines 3. The Chord of an Arch is a right line extended from one end of that Arch to the other end thereof The Sine is a right line drawn from one end of that Arch Perpendicularly upon the Diameter drawn from the other end of that Arch The Tangent is a right line touching one end of that Arch extended till it Concur with the Secant The Secant is a right line extended from the Center of the Circle till it Concur with the Tangent The Versed Sine is a right line being a Segment of the Diameter drawn from one end of that Arch till it be cut by a Perpendicular i. e. the Sine from the other end of that Arch. 4. It is to be noted by this Definition in Prop. 3. that the Chord of an Arch is common to two Arches one of them being the Complement of the other to a whole Circle and likewise the Versed Sine is common to two Arches one of them being the Complement of the other to a whole Circle But the Sine of an Arch is common to two Arches one of them being the Complement of the other to a Semi-circle 5. As the Sum of two Sines is to their difference so is the Tangent of the ½ Sum of those Arches to the Tangent of their ½ difference 6. As the Sum of two Tangents is to their Difference so is the Sine of the Sum of those Arches to the Sine of their Difference 7. As the Sine of the Sum of two Arches is to the Sum of their Sines so is the difference of those Sines to the Sine of their Difference 8. If you put R = The Radiu s of a Circle A = an Arch proposed C = the Chord of that Arch S = the Sine of that Arch T = the Tangent of that Arch and Z = the Secant of that Arch. Then 9. If twice three Arches equi-different be proposed Then as the Sine of one of the means is to the sum of the Sines of its Extreams so is the Sine of the other mean to the sum of the Sines of its Extreams 10. And hence if a rank of Arches be equi-different As the Sine of any Arch in that rank is to the sum of the Sines of any two Arches equally remote from it on each side So is the Sine of any other Arch in the said rank to the sum of the Sines of two Arches next to it on each side having the same common distance 11. Three Arches equi-different being proposed If you put Z = the Sine of the greater extream Y = the Sine of the lesser extream M = the Sine of the Mean m = the Co-sine thereof D = the Sine of the common difference d = the Co-sine thereof and R = the Radius 12. From the last before going it is evident that if two thirds i. e either the former or the latter 60 deg or the former 30 deg and the latter 30 deg of the Quadrant be compleated with Sines the remaining third part of the Quadrant maybe compleated by Addition or Subduction onely 13. If in a Circle two right lines be inscribed cutting each other The Rectangles of the Segments of each line are equal And the Angle at the point of Intersection is measured by the Half-sum of its intercepted Arches 14. If to a Circle two right lines be adscribed from a point without The Rectangles of each line from the point assigned to the Convex and Concave are equall And the Angle at the assigned point is measured by the half difference of its intercepted Arches 15. If in a Circle or an Elipsis three right lines shall be inscribed one of them cutting the other two Then the Rectangles of the Segments of each line so cut are directed proportional to the Rectangles of the respective Segments of of the Cutter 16. If a plain Triangle be inscribed in a Circle the Angles are one half of what their opposite sides do subtend 17. Therefore the Angles of a plain Triangle are equal to a Semi-circle 18. And hence if a Rectangled Triangle be inscribed in a Circle the Hypothenuse thereof is the Diameter of the Circle 19. As the Diameter of a Circle is to the Chord of an Arch so is that Chord to the versed Sine of that Arch. 20. And hence if from the right angle of a rectangled Triangle a Perpendicular be let fall upon the Hypothenuse the Hypothenuse is thereby cut according to the Ratio of the squares of the sides 21. If in a Circle any plain Triangle be inscribed and a Perpendicular be let fall upon one of the sides from the opposite angular point Then as that Perpendicular is to one of the adjacent sides so is the other adjacent side to the Diameter of the Circumscribring Circle 22. If a Circle be inscribed within a plain Triangle Then as the Perimeter is to the Perpendicular so is the Base on which it falleth to the Radius of the inscribed Circle 23. If a Quadrilateral Figure be inscribed in a Circle and Interfect with Diagonals The Rectangle of the Diagonals is equal to the two Rectangles of the opposite sides 24. If a Circle be both inscribed and circumscribed by two like ordinate Polligons Then as the Co-versed Sine of the side of the inscribed is to the Diameter so is the Area of the Inscribed to the Area of the Circumscribed 25. If an ordinate Polligon be both Inscribed and Circumscribed by two Circles Then as the Diameter of the Circumscribed is to the co-versed Sine of the side of the Polligon So is the Area of the Circumscribed to the Area of the Inscribed 26. In any right lined Figure if a Circle be Inscribed Then as the Peripheria of the Circle is to the Area thereof So is the Perimeter of the right lined Figure to the Area thereof Et Con. 27. But in all Circles as the Peripheria is to the Area so is 2 to the Radius 28. Therefore In any right lined Figure if a Circle be inscribed as 2. is to the Radius So is the Perimeter of the right lined Figure to the Area thereof CHAP. II. Of Plain Triangles 1. A Triangle is a Figure Comprehended of three sides and