GRAMMELOGIA Or, the Mathematicall Ring.

Extracted from the Logarythmes, and projected Circular: Now published in t [...] inlargement thereof unto any magnitude fit for use: shewing any reasonable capacity that hath not Arithmeticke, how to resolve and worke, all ordinary operations of Arithmeticke:

And those that are most difficult with greatest facilitie, the extract [...] on of Rootes, the valuation of Leases, &c. the measuring of Plaines and Solids, with the resolution of Plaine and Sph [...]ricall Triangles applied to the Practicall parts of Geometrie, Horo [...]ogographie, Geographie, Fortification, Navigation, A [...]ronomie, &c.

And that onely by an ocular inspection, an [...] a Circular motion, Invented an [...] first published, by R. Delamain, Teacher, an [...] Student of the Mathematicks. Naturae secreta tempus [...]aperit.

# To the High and Mighty King, CHARLES King of Great Britaine.

HAd not your gracious favour given life unto my first birth, I should not have dar'd thus boldly to rush into your sacred presence, but onely in humble confidence I am incouraged, that, as your gracious Majestie pleased to accept of my imperfect endevours, the same blessed eye will not now reject that worke brought to its fuller growth and perfection. It was the great invention of that Noble Lo. Nepeir Baron of Marcheston his ever-worthily admired discovery of the Logarythmes in cutting off the toylesome working by Sines, Tangenis, Secants, &c. My paines formerly was conversant about to bring them likewise into a shorter way for practicall uses, by an instrumentall Projection of these numbers in Circles, of which I composed my Mathematicall Ring. Every thing hath his beginning, and curious Arts seldome come to the height at the first; It was my promise then to inlarge the invention by a way of decuplating the Circles, which I now present unto your sacred Majestie as the quintessence and excellencie there of, whereby Mathematicall operations which tend to ordinary and usuall affaires and such of a higher nature, are performed accuratly, without tedious extraction of Roots, operating by naturall Sines, Tangents, secants, or the Tables of Logarythmes themselves, and that by an inspection of the eye alone and a Circular motion. It is now againe your gracious acceptance and the publicke good, which my life & paines are studious of: the one I hope will acknowledge, the other defend these mine honest labours which some endevour to rob me of. I have no better thing as yet to expresse a loyall subjects heart, and affections by: but onely my selfe, which with my poore endeavours in all humble submission I cast downe at your sacred feete, accounting it my cheefest Happinesse to be

# By a Friend upon the Mathematicall Ring.

# The same to the inventer of the Logarythmes projected in Circles.

# By a Lover, and Student in the Mathematicall Arts.

# In mirificam Logarythmorum projectionem Circularem.

# To the courteous and benevolent Reader that affects this instrumentall practise of the Logarythmes projected circular Intituled Grammelogia, or the Mathematicall Ring.

AS that Honorable Lord Nepeir (the first inventer of Logarythmes) did call his Rods by reason of their facility in operation Rabdologia, the speech of Rods, and his numbers, Logarythmes, the speech of numbers: So this Instrumentall Projection as springing from that invention, I called Grammelogia, the speech of lines; for being so projected as they are, these lines graduated, doe promptly teach one what to speake in proportionall operations; which invention of projection of Logarythmes Circular, I promised to enlarge at the end of my first publication, with this intituled Grammelogia, or the Mathematicall Ring, Printed, Anno. 1630. notwithstanding it was there sufficiently perspicuous, how it might bee augmented unto any magnitude assigned, even to operate unto minuts, and seconds in Trigonometrie, and to finde Rootes, and proportionall numbers in common Arithmeticke unto sixe or more places: a diagramme of which projection I now here deliver, as fully sufficient to shadow out to the more learned the quintessence of this Logarythmall projection in Circles unto 6. 10. 40. 100. or 1000. yea, to as great a capacity as one desires, (a project by Instrument never yet produced, though long desired) which plurality of Circles in this projection, must be conceived to bee the parts of Circles, and so there may bee a quadruplicitie of them, [Page] one of which Quadruplicities, may co [...]ine the Loga [...]th [...]es [...]f [...] bers, another may comprehend the L [...]arythmes of sines and the other two may bee for the inserting of the Logarythmes of Tangents: so in this Scheme here delivered noted with the letter B. there are 16. Circles in view, but 4. in effect, each of which 4. must be conceived to be broken into parts is before: in the first of which Quadruplicities of Circles, there are graduated the Logarythmes of [...]mbers, (as before) from 1. unto 100000. noted with the letter N. and so may bee properly called the Circle of numbers: in the next quadruplicitie of Circles there are graduated the Logarythmes of sines, from 34. m. 22. se. unto 90. gr. noted with the letter S. containing two revolutions, that is, two perf [...]ct Circles, & therefore is divided & figured double, beginning his first revolution at 34. m. 22. se. and ending at 5. gr. 44. m. 21. se. which are graduated upon the inner edge of the Circle the second revolution begins at 5 gr. 45. m. 21. se. and ends at 90. gr. which are graduated and figured on the outward edge of the Circle, which graduations and divisions of sines, may be therefore called the Circle of sines. Lastly, the next two quadruplicities of Circles which are noted with the letters T. T. do containe foure revolutions or Circles, because also of their double divisions, in which are inserted the Logarythmes of Tangents, from 34. m. 22. se. unto 89. gr. 25. m. 38. se. beginning the first revolution at 34. m. 22. se. and ending at 5. gr. 42. m. 38 se. which are divided and figured upon the outer edge of the Circle; the second revolution begins at 5. gr. 42. m. 38. se. and ends at 45. gr. and are graduated and figured also upon the outerside of the Circle. The third revolution begins at 45. gr. and ends at 84. gr. 17. m. 21. se. and are graduated and figured upon the innerside of the Circle. Lastly, the fourth revolution begins at 84. gr. 17. m. 21. se. and ends at 89. gr. 25. m. 38. se. all which graduations of Tangents may be called in like manner the Circle of Tangents. And as that famous of memory (if not most injuriously late made infamous) and worthy Mathematitian, Mr. Gunter, the first that gave light to this Invention, did call his lines the lines of Numbers, Sines, and Tangents, or the li [...]es of proportions; so this whole projection as adherent unto it, may not unfitly be called the Circles of Numbers, Sines, and Tangents, or in respect of operation (as some others lately called the Circles of my King singly projected on a Plate or plaine) the Circles of proportion, in which may be noted, that if this projection be made of 10. Circles, then may there be a quadruplicitie of them, which in all make 40. Circles, one quadruplicitie of which may serve for the inserting the Logarythmes [Page] of numbers, another for the Logarythmes of Sines, and the other two for the Logarythmes of Tangents. If the projection be of 50. Circles, then the whole projection will be of 200. Circles, whose graduations, beginnings, and terminations in each whole revolution is the same with the former; and as there is a conformity in the Projection by a greater number of Circles, as there is by a lesser; so is there the same facilitie and agreement in operation by many Circles, as by few, and the way how I have delivered in this Tractate, which is either by motion in a double projection, or in a single projection, by helpe of a thred & lead, or a single or double Index at the Center, or peripheria: upon which I deliver severall wayes in this following Treatise. How it may please some mens affections I know not, my intentions and desires are free. Now since the publication of the first kinde of Logarythmes Projection, Anno 1630. or the originall of this enlarged, it hath pleased many about this City and Kingdome to take liking thereunto, some contenting themselves with the double projection, with a moveable and fixed Circle, some with the single having an Index at the Center: but generally the most part have beene by some invited if not forced to that which carries, as they say, with it the aplause and vote of men by a comparative attribution delivered by the assumed Author of that with an Index on a Plate, that the way on the plate in a single Projection with an Index at the Center, is a better way than that of my Ring, or that of a double Projection on a Plaine: The authority of whose words to some ignorant mechanick composors of that Instrument, was a sufficient motive ever since to crowne his words with a divulged rumour out of their borrowed knowledge to maintaine his assertion, to put off their commodities, howsoever if the saddle (as the proverbe is) were put on his right place, this vote and attribution belongs to another, who fitted the Instrument so as it is now used, & yet modestly would not appeare in it, and not our supposed Author, which assumption of fitting of it so with an Index on a Plate, had beene enough if not too much, without such a divulgation of endeavouring what in him lyeth, and in others, to annihilate and beate downe the way which I write upon, and to glory in the raising up of his supposed owne: thereby not onely possessing men with an untruth, but making me also ignorant in my choyse, that I should give unto the world the weakest and imperfectest part of the projection of Logarythmes, and leave the best for another to write upon: Of which single Projection with an Index at the Center, had I first writ upon and left the other way, which is the double Projection [Page] for some other to write upon: then might he have used not indirectly that comparative aspertion of better. But before I writ of the Naturall sympathy of this projection, I was not unadvised which projection to present unto the King, and to the publike view of the world first, but considered intentively with my selfe the excellency of both wayes, and the more copious performance of the one, in respect of the other; And why I delivered it first in a Ring, was, for the aptnesse and gentile-forme (as I may call it) it naturally might be cast into. Secondly, for the excellent harmony, facility and expedition that the Logarythmes so projected did afford: having no secondary assistance to helpe it in operation, but the motion of the Circle it selfe, for there was nothing to do but to move one number to another in a proportion assigned, either in a double Projection in single Circles, or the projection inlarged; and instantly there was presented all other numbers in the same proportion; By an Index on a Plate in a single Proiection it was grosse and course for the forme (in respect of a Ring) and for operation there must bee besides extending the feet of the Index to the members, upon every severall question a new search of numbers with a new motion (which extending of the feet of the Index, was the same with M^{r}. Gunters Invention of his Ruler and no new invention) Besides if the single Projection bee inlarged there doth necessarily adhere unto it sundry & manifold observations in the way of operation by it, which cannot be avoided, which to a learner at the first seemes not a little harsh & difficult; all which the Ring or the way of the Ring on a plate, in the double Proiectiō inlarged, doth naturally avoyd, and not only caries a facility in its operation, but retaines in it also a speciall advantage in its performance, once rectified, for the eye and the hand may worke together, and what the eye finds in proportion, the pen may presently expresse in writing without a second trouble to search out another number as before, and then to bring the edge of the Index to it. But some envious detractors would not admit of this forme and facility (though perhaps the succeeding times may) eyther as before to disanull the worke, or for the difficultie that was found in an unexpert workeman in the true composing and making of the worke, for if the Circles on the Ring, or double Proiection on a plaine being not exactly composed and graduated may cause some small error in operation (which is onely from an excentricke motion) the single Proiection hath not onely the same defect, but also a second to helpe it, to wit the Index, for by how much the legs of it are long and the Instrument large, by so much the more [Page] is it subject to errour, which is from a continued augmentation of an error in the fitting of it to a lesser Circle which hath reference to a greater. But to passe by the errors that are subject to the best kinde of Instruments that can be made, let us a little examine the Authors & others comparison of Better, why the way of the Index in a single Proiection is better then a mooveable and fixed Circle, which I conceive to have reference unto foure generalls about the Instrument. First, either in the forme of the Instrument; Secondly, in the ordering of the Circles thereon; Thirdly, the expedition that is found in the practise thereof; And fourthly and lastly, the copiousnesse of the uses of the Instrument: and other c [...]uses I conceive not, why this single Projection with an Index is better then a double, except it be in the magnitude that is now usually made, or for the price of the Instrument: in the first there may bee an equall extendure of magnitudes unto both Instruments, and so as a thing common unto them, and no wise different. And for the price it may bee made as cheape, if not cheaper hereafter as I shall order it for these that aflect them: Now in the first place as touching the forme, upon that I have spoken somewhat alre [...]dy, (as afore said) and may be sufficient: as for the second generall touching that of the ordering of the Circles on this double projection, to have one Circle mooveable and one Circle fixed, that is agreeable to the projection and dividing of these Circles in the first direction following, according to the great scheme in the Booke noted with the letter A, for if the Circle of numbers noted with N, N, be cut through, and the Center of that Circular plaine be fastned, so that it may move upon the Center of the other Circle, it shall fully represent the projection of my Ring upon a Plaine, to which may bee placed a small single Index as in the scheme of the title page B, to helpe the eye for the finding of opposite numbers, these Circles of the mooveable, or fixed Circles on the plaine are inserted on both sides of a Ring as it is specified at the end of the dividing of these Circles. Now to have all the Circles placed upon one side of the Ring (as is according to the second direction of making the Ring) were to leave the other side naked, without one would patch and peece some other thing on the other side, therefore to avoyde mixture, part of the projection (as an ornament) is placed on the other side, by which occasion the whole projection at once is not visible to the eye, as it would be in the second way in accommodating the Circles into a Ring, as is in the double projection on a Plaine before mentioned, noted with A. But perhaps it may be objected that [Page] the Circles are easily continued on a Plaine, and the Index being at the Center the edge of it, doth accuratly cut each Circle in the proportionalls, which intersections in a Ring are defective and difficult to finde by the eye alone. To all which I answer, that the Circles in a Ring are as easy to be continued as on a Plate, allowing the mooveable and fixed Circle a sufficient breadth, and here by the way I would have the Reader to understand that the Circle of Tangents being projected from 1. gr. unto 45. gr. is sufficient for operation (these degrees being their complements to 90. gr. but for greater expedition in working they may be continued as is seene in the great scheme A, which continuation I learned not from another (as may bee suspected by some) seeing I now published it after another, but long before that publication I instructed sundry persons upon that continuation by way of facility. As for the second clause in such Circles which are not upon the edge of the mooveable and fixed Circle, where the eye seemes to bee troubled to point out some opposite numbers, a small edge of metall may easily supply that (as many use to doe) but the graduations being so nere the edge of the Circle, the proportionalls are sufficiently given by the eye alone without such an edge. Now if in these respects the single projection with an Index, is better then that of a mooveable and fixed Circle being easily supplyed as aforesaid, it is but a poore one, in common sense. But if the way with an Index on a single Projection bee not better then that of a double for the former respects, then it may bee in the third generall to wit the Instrumentall expedition: in which there needes little declaration to prove the double Projection to have a greater expedition then the single projection with an Index, seeing it appeares so obvious, that what can be quicker; then having mooved one number to another in proportion assigned, that all other numbers are opposite one to another in the like projection (which a single Index doth point out easily to the eye as before.) By the Index in a single projection there is first putting the one foot to one number, and extending the other foot to another number, then must the eye have reference to one of the feete that it fall upon his third number, and afterward to looke for the second foote for the fourth number, and so to move it to another number, and still to have a double respect with the eye as before in every new operation: in this regard also I see not why the way of an Index in a single Projection is better then that which I have delivered in a double projection. But to passe by all the former generalls as triviall and of small consequence, let us weigh seriously things more [Page] materiall touching both wayes of these Instruments. If the said single projection on a plate with an Index at the Center bee not better in respect of its expedition in operation, then must it necessarily bee (to prove the Authors assertion) in the Instruments fourth generall, to wit, in its copious performance, which I hold eyther to be in the generall, or particular use; In the generall I considered the finding of proportionalls, and thats agreeable to the way of operation in either Instruments as is afore specified; In the particular I regard also what propositions offer themselves to the eye, eyther by motion, or without motion: by motion it is impossible for the single projection without an Index being opened at pleasure to give any more then one kind of proportionalls, the Ring, or a mooveable and fixed Circle on a Plaine, scorning as it were such lamenesse, or such an injurious tie from its naturall propertie, sheweth by motion infinite operations in various proportionalls, even through the whole body of the practicall part of Mathematicall Art, which would be too copious for me to declare, or for the Readers patience to peruse, onely some common uses by such motion I will deliver, somewhat to prove my assertion, that the single proiection with an Index, is not better then that with a mooveable and fixed Circle, therefore.

1. First, the moveable being moved about at pleasure, as 1. in the moveable passeth by any multiplier in the fixed, so doth any multiplicand in the moveable, point out its product in the fixed, or contrarily, as any divisor in the moveable doth passe by, 1. in the fixed, so doth any dividend in the moveable point out his quotient in the fixed, so as 12. (the months in a yeare) or 52. (the weekes in a yeare) or as 365. (the dayes in a yeare) in the moveable in motion doth passe by 1. in the fixed, so any somme of money in the moveable, doth point out its monthly, weekely, or dayly expences in the fixed.

2. Secondly, as 7. in the moveable doth passe by 22. in the fixed (Archimedes Proportion betweene the diameter of a Circle and its Circumference,) so doth any diameter in the moveable, point out its Circumference in the fixed, vel contra.

3. Thirdly, as a hundred waight of any commodity in the moveable, (or any other waight or measure) passeth by its price under 100. pound in the fixed, so right against 1. in the moveable, is the price of a pound waight of that commodity amongst the decimals in the fixed vel contra.

4. Fourthly, the moveable being moved about at pleasure, the Interest of all summes of money according to any rate in the hundred is given; [Page] for as 100, in the moveable passeth by its interest in the fixed, so every summe of money in the moveable, doth point out its interest in the fixed, vel contra.

5. Fiftly, as 1. in the moveable passeth by any sum of money in the fixed, so any number of yeares in the Circle of yeares, doth point out the amount of that money in the fixed, according to the terme of yeares that th [...] money was forborne.

6. Sixtly; as the measure of a side, of any dimension, of a Building, of a Fortification, of a whole mixture, or the weight of it, &c. in the moveable passeth by a greater, or lesser measure, or waight in the fixed (in homogeniall things) so the measures of the parts of any of these wholes in the moveable will point out in the fixed the proportionall parts of any other whole by way of augmentation, or diminution.

7. Seaventhly, as 1. in the moveable passeth by the square of the side of any of the tenne regular Plaines, so doth each plaine note in the moveable point out right against it, its Area in the fixed; and as any kinde of measure to the Pole in the moveable, passeth by its quantity in the fixed, so doth any other kinde of Pole point out its quantity or Area, being measured by that Pole, &c. and whatsoever may be attributed to the use of this Circle of numbers may be given by motion.

Further, if we consider the Circle of Sines and Tangents conjoyned with the Circle of Numbers in operation, or the Sines with themselves or joyned with the Tangents, then by motion you have the sides and Angles of infinite plaine and Sphericall Triangles for practicall uses, either in Geometrie, Astronomie, Navigation, Fortification, Horot ogographie, Geographie, &c.

1. First, as the sine of 90. passeth by 60. in the fixed amongst the numbers, so the sine complement of any degree in the mooveable will point out the miles answerable to any degree of Longitude in the Latitude; and as the said 90. passeth by the Tropicall point, in the fixed, so the sine of any degree of the Sunnes distance from the Equinoctiall points will point out the sine of the sunnes declination answerable to that distance.

2. Secondly, as the sine of any Rumbe in the mooveable from the East or West, sayled upon, passeth by the measure of a degree in leagues or miles in the fixed, so 1. in the mooveable pointeth out in the fixed the number of miles, or leagues to raise or depresse the pole a degree in that Latitude.

3. Thirdly, as the sine of any Latitude in the mooveables passeth by the [Page] sine of the Tropicall point in the fixed, so the sine of the Sunnes distance from the Equinoctiall points in the mooveable, that passeth by the sine of 90. in the fixed, doth point out the sine of the Sunnes greatest degree of the distance from the Equinoctiall points that the Sunne will bee due East in that Latitude.

4. Fourthly, as the sine of 90. in t [...]e mooveable passeth by the sine Complement of any Latitude in the fixed, so right against the sine Complement of all Declining plaines in that Latitude in the mooveable, are the sines of the degrees of the stiles heights in Horologographie agreeable to these declining plaines in the fixed.

5. Fiftly, as the sine Complement of any Latitude in the mooveble passes by the Tropicall point in the fixed, so the sine of the Suns distance from eyther of the Equinoctionall points, will point out right against them the sines of the Suns Amplitude.

6. Sixtly, as the Tangent complement of any Latitude in the mooveable passeth by the sine of 90, in the fixed, so the Tangent of the Tropicall point in the mooveable, doth point cut in the fixed the sine of the greatest difference of ascention for that Latitude

7. Seventhly, as the sine of the Suns position at his setting or rising, or the sine of the houre from 6. at that instant in the mooveable passeth by the sine of 90. so the sine of the Suns declination in the former, & the Tangent of that declination in the latter, will point out the sine of the height of the Equinoctiall in the former, but the Tangent of the same in the latter.

In this nature you have infinit operations performed by motion in this double Projection of a mooveable and fixed Circle; which by a single Proiection with an Index cannot as before possibly be performed, therefore if in this regard the way of the Index on a single Proiection bee not better but is farre inferiour to that of a mooveable and fixed Circle; to prove the Authors and others divulged assertion, then must it be better in the last clause, which was the Instrumentall performance without motion, in which the single projection with an Index, comes very short of other Instruments which by a single inspection of the eye shewes many pleasant, and usefull propositions. But this none, or very few at all, as onely the Logarithmes [Page] of numbers, the naturall sines and the Tangents of the Logarythmall, &c. But the double projection with a mooveable and fixed Circle doth not onely shew that, but being at any position carries with it such an excellencie that it assumes unto it selfe a prehemenencie above any Instrument never yet produced in regard of its copious use, & manifold performances which it affords without motion as by an inspection of the eye onely: a touch of w^{ch} I will unfould & unvaile which never yet came to a publik view.

1. First, the Instrument lying upon a Table open to the eye and being at any position, marke what numbers in the mooveable and fixed are opposite one unto another, according to which proportion there is represented infinite other proportionalls, in the same proportion, for one number is opposite to another through the whole Circle of numbers, sines, and Tangents, from which one might apply the proportionalls in numbers to the use of things, to expences, to proportions in Buildings, to fortifications, measurations, but too great a prolix discovery would tyre the Reader in that which he may easily from it apply hereafter unto himselfe.

2. Secondly, marke what number in the fixed, (in the Circle of numbers) is against any number of yeares in the mooveable, which suppose a Legacie to be payd for so many yeares to come or a summe of money forborne so long time; and it were to be sold for present money, right against 1. in the mooveable, is the worth in present of that Legacie or summe of money in the fixed.

3. Thirdly, in Horologographie, you have the distance of the houres in a poler, and Meridi [...]nall plaine, without operation, for marke what number in the Circle of numbers in the fixed is against the Tangent of 45. gr. in the mooveable (which number in the fixed may be supposed the measure in inches, &c. of the stiles hight) so right against the Tangent of the equall houres in the mooveable you have the houre distances in the Circle of Numbers in the fixed.

4. Fourthly, the houres of a Horizontall or verticall dyall, for some one Latitude or other is shewne, for marke what degree amongst the sines in the fixed (which represents the Latitude) is against the sine of 90. in the mooveable, so the Tangent of the equall houres in the mooveable, doth point [Page] out the houre distances (for that Latitude) amongst the Tangents in the fixed, and these houres serve for a verticall Dyall in the complement of that Latitude.

5. Fiftly, note what number in the fixed in the Circle of Numbers, is against 1. in the mooveable (which suppose to be the given side of any of the ten regular figures) then against the circumscribing and inscribing notes of these regular figures in the mooveable is the Circles circumscribed and inscribed diameters of those regular figures: But if the said number in the fixed against 1. in the mooveable be taken for the square of the side of the diameter, of the side of any of the ten regular figures, then against the Regular notes in the mooveable, is the Area of these figures in the fixed.

6. Sixtly note what numbers in the Circle of numbers in the fixed, are against the regular figurative notes of equality in the moveable, such are the sides of those figures whose quantities are equall the one to the other: In like manner the numbers in the fixed against the notes of the Regular bodies in the mooveable representeth the sides of these bodies which have equall solidities the one unto another.

7. Seventhly, you have infinite oblique angled plaine Triangles represented, and such who have equall Altitudes but different Basis the sides of severall parallelograms, equall unto one and the same square or the quantitie of a Triangle given in Acres, the perpendicular and Base is also given: For first the sines of the Angles on the mooveable will point out their sides of the Triangle amongst the numbers in the fixed, vel contra.

And secondly, marke what number in the mooveable in the Circle of numbers is against 1. in the fixed that suppose to be the Altitude of a Triangle, then the equall distances from 1. on both sides of it taken at pleasure in the fixed will point out in the mooveable the segments of the Basis of the Triangle, or the sides of a parallelograme equall unto the square made of the Triangles perpendicular. And thirdly, the quantitie of the Triangle in the fixed amongst the numbers doth point out the Basis of the Triangle in the mooveable, and that number in the fixed which is against AC, (in the mooveable) is the Triangles halfe [Page] perpendicular according to the Area given.

8. Eightly, marke what number in the fixed, is against 1. in the mooveable, which suppose to be the side of any of the Regular bodies, then right against the solids inscribed notes in the mooveable, are their sphears Circumscribing diameters, but if the said number in the fixed be supposed to be the semediameter of a spheare, the numbers in the fixed (in the Circle of numbers) against the solids circumscribed notes in the mooveable shewes the sides of these regular bodies that will circumscribe that spheare.

9. Ninthly, marke what number in the fixed amongst the Numbers is against 1. in the mooveable which may be supposed the diameter of a Circle, the Axis of a spheare, the side of a plaine figure, or that of a solid body: so the numbers in the fixed in the Circle of numbers against the potentiall notes in the mooveable shall represent the diameter, Axis, or side of its homogeniall figure, or solid, according to the proportion of these potentiall notes in the mooveable.

10. Tenthly, marke what numbers in the Circle of numbers in the fixed are against the notes of the regular figures, such shall bee their Areas, and the numbers in the fixed against the regular Bodies convexities, such is their superficiall convexitie: and the number in the fixed against 1. in the mooveable, is the square of one of the sides of the regular figures, or the sides of one of these bodies, and what numbers in the fixed are against the notes of the solid bodies, such shall b [...] the severall solidities, or contents of these regular bodies, and the number in the fixed against 1. in the mooveable is the Cube of the sides of these bodies.

Lastly, most courteous Reader, (not in any braving flourishes or branding any of the Nobilitie or Gentry with the attribute of jugling, against the simple modestie of the Author,) I have in some measure supported their honours in that particular in the Epistle at the end of this Booke: and that wee may say something more upon the excellencie of this Instrument without multiplying of tautologized and needelesse prefixed graduall numbers, or Circuitions, if not Circumlocutions, in the naked truth of this Instrumentall proiection, according to its naturall propertie. The Roots of all square and cubicke [Page] numbers without partition are given; and that by an inspection of the eye onely.

Thus I might have extended my selfe more copiously in the excellent use of this my mooveable and fixed Circle, and even from the Instrumentall position by an inspection of the eye onely without motion, compile a large Booke of its ample performance, but in that which I have delivered I have onely but scatteringly glanced upon things, as making way for many occasions, and as a motive to a further inquiry: Its an ancient proverbe amongst us, good wine needs no Bush, but the wine must not be fast lockt up then, that none can come by it, if so it wants both bush and key, and to some such needelesse expressions might be avoided, the Instruments owne excellency will to the more learned easily present it selfe that which I have published: concerning it, I glory not in, but onely desire to satisfie those who would see the difference of both wayes, with, and without a mooveable Circle, & to let others know the truth of things which are conceated, and carryed away with opinion onely, that the way of the Index on a single proiection is better then the way of a mooveable and fixed Circle, which both in regard of expedition as also copiousnesse of the Instrumentall use, by motion or without motion comes short of the other. What meanes the Authors divulgation then, that the way of the Index is better then the way of a mooveable and fixed Circle, I know not, whose knowne skill in the whole Systeme of Mathematicall learning will easily free him from the suspition, that the way can be made, or the subject unvailed for him.

But I have now a little more made bold to unvaile the subiect for some, in the copious declaration of the excellent use of this Logarythmall projection Circular by a mooveable and fixed Circle, and also in its inlargement, which hitherto lay in obscurity, and as a generall benefit to those that affect the way of this Instrumentall practise. It were good that the divulgers would prove their aspertions, touching the word better, that others might participate substancially of their better way by the Instrumentall performance, eyther by motion, or without motion, and not to allure the world by a bare exhortation, unto the affection of the one Instrument, and by a dehortation to beate downe [Page] the use of the other, which savours of too high a conceite of the one, and too great a detraction from the other: Too great & too loose an aspertion hath bin cast upon me about these things, which I never thought in the least title when I first writ upon this Invention, or my name so to come to the worlds rumor as it hath since the last publication of this Logarythmall projection Circular; howsoever, here is my comfort, the guiltlesnesse and innocency of my cause, which may teach me, and others carefulnesse hereafter, how and what we publish to the world, seeing there are such carpers, and maligners even of the most usefull and best things, yea, such busie bodies who marre that which others make, who scorne to have a second, knowing all things and admiring nothing but themselves, such who have stings like Bees, and Arrowes alwayes ready to shoot against these whom they dislike, such who while they will needs have many callings neglect their owne; sharpe wittie cryticks, Diogenes like, snarling at others, and not looking home unto themselves, but by all meanes endevouring to take away the mantle of peace, and rent the seamelesse coate of love and amitie. If things be not done well by others then they triumph and send forth their invectives, if well, they professe it nothing, and cannot passe without their censure. To speake ill of a man upon knowledge shewes want of Charity; but to raise a scandall upon a bare supposition, & to act it in Print, argueth little humanity, lesse Christianity: but enough of this if not too much, I am sure some have casted too much already, perhaps others hereafter may helpe to bare a share, for my owne part I desire no favour but the truth and equitie of my cause, and the due waighing of things with their reall circumstances. Ʋeritas non quaerit Angulos. I desire no shifting, or pretences, but if I have done others wrong let me suffer; If I have beene wronged by others let me have truth, and right done me, thats all I require. Who am

# To the Reader.

SInce my first publication of the uses of my Mathematicall Ring or the Logarythmes projected Circular, I have beene oftentimes invited by sundry persons for the way of the projecting, and dividing of the Circles of my Ring upon a Plaine, so that it might be made in Pastboard to avoyde the charge of the Instrument in metal, for such which have not abilities to buy, and for others, who would first see the practise on it, before they would be at the cost of the Instrument in metal: for whose sake, and use, desiring to satisfie the affectionate, and for a publike benefit, (rather then ayming at mine owne particular profit) I have caused two Plates of metal to be cut & ingraved, the one containing the Circles of the Projection of my Ring, to bee used on a Plaine as it is there described, noted with the letter A. and the other comprehending that Projection inlarged, noted with the letter B: that so such may make use of them more readily, to avoid the labour of dividing the Circles; [Page] which schemes being pasted on a Pastboard are ready for use.

And yet further to satisfie those that are desirous, I have delivered also in the first place insuing, how those Circles are projected & divided, that so they may be made according to any magnitude. In the second place how severall wayes they may bee framed in a Ring: In the third place J shew the inlarging of the Instrumentall Invention in these Circles to as great a magnitude for use as may be desired. Jn the fourth place J deliver severall wayes how these Circles inlarged may bee accommodated for Practicall use. Jn the fift place, I make a description of the Grammelogia, or Instrument in the particular Circle of my Mathematicall Ring, projected on a mooveable and fixed plaine, to wit, of the former scheme A. And in the sixt and last place, J will declare the excellent uses of both these Instruments, in the Practicall parts of Arithmeticke, Geometrie, Astronomy, Horolographie, Navigation, &c,

## Of the projecting and dividing of the Circles of the Mathematicall Ring, and of the inlargement of the Invention, either in a single projection, or in a double, and that severall wayes.

FOr the first, according to any semediameter describe severall Circles concentricall, as here are represented by the figure A, the outmost of which may be noted with the letter E, serving for the Circle of equall parts, and be divided into 100. 1000. or 10000. equall parts according to the capacity of the Circle, and noted with figures thus, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

The next two Circles may be noted with the letters T T. which may be for the Circle of Tangents; unto the next Circle may bee set the letter S, which may represent the Circle of Sines, and the inner Circle may bee noted with the letter N, which may represent the Circle of Numbers:

These Circles of Numbers, Sines and Tangents, may bee divided out of the Table of the Logarythmes of Numbers, [Page] of Sines, and of Tangents, in this manner following.

### 1. Of the graduation of the Circle of Numbers.

First, to divide the Circle of Numbers, marke what number there is against 2. in the Table of Logarythmes, which is 3010. and account it from E, in the Circle of equall parts, (if the Circle be divided into 10000. equall parts, but if the Circle be divided but into 1000. equall parts, then account onely 301. parts) which is against 2. in the Table of Logarythmes (as aforesaid) and will bee at α, lay a Ruler upon it and the Center, and intersect the Circle of Numbers N, in 2. then marke what number is against 3. in the Table of Logarythmes, which is 4771. and account it from E in the said Circle of equall parts, viz. at β, and lay the Ruler upon the Center, and to the said 4771. or at β, and intersect the Circle of Numbers N, in 3. Againe from E, account 60, 20. viz. at y which is against 4. in the Table of Logarythmes, and lay the Ruler thereto, and intersect the Circles of Numbers N, in 4. and so proceede untill you come to 10. so the intersections in the Circle of Numbers, N, noted thus, 1, 2, 3, 4, 5, 6, 7, 8, 9. shall be the capitall divisions of that Circle.

Then may the spaces betweene 1, and 2. betweene 2, and 3, &c. be subdivided by the consequent numbers in the Table of Logarythmes thus.

Marke what number is against 11. in the Table of Logarythmes, which is 413. account this number from E. in the Circle of equall parts, viz, at δ, and laying a Ruler upon the Center A, and the said δ, or 413. note the Circle of numbers N, in ε, then marke what number is against 12. in the Table of Logarythmes, which is 791. account it from E. in the Circle of equall parts, viz. at ζ. and lay a Ruler upon it and the Center, cut the Circle N, in n. and so goe on untill you have divided the whole space betweene 1, and 2. which will be when you come to 20. against which in the Table of Logarythmes is 13010, &c. the same with the Logarythme against 2, viz. 03010. the figure of 1. (the Index) being rejected, which 1, represents one whole revolution: In like manner you may divide the space betweene 2, and 3. in the Circle of Numbers N. from the [Page] Logarythmes which are against the Numbers betweene 20. and 30. and so of the rest of the spaces betweene 3. and 4. betweene 4, and 5, &c. untill you come to 1, which represents 100. then if you will subdivide the space betweene N, and ε, marke what number is against 101. in the Table of Logarythmes which is 43, account this from E, viz. at [...], and lay a Ruler upon it and the Center and cut the Circle N, in ι. then marke what number is against 102. which is 86. account it also from E, in the Circle of equall parts; and lay a Ruler upon it and the Center, and cut the Circle of Numbers N, in λ, and so goe on, untill you come to ε, which will be at 110. and the Logarythme against which is 413. the same which before was against 11. (the Index in each being rejected) which represents as before onely the severall Revolutions. In like manner may be divided the space betweene ε, and n, in the Circle of Numbers N out of the Logarythme betweene 100, and 200. and so of the rest: But if the Instrument be large then it is requisite to divide the space betweene N, and ι, betweene ι, and λ, &c. which will bee from the Logarythmes which are betweene 1000. and a 10000. Therefore in a small Instrument you may begin to divide it at the Logarythme of 101. untill you come to 1000. but in Instruments more large begin from the Logarythme of 1001, untill you come unto 10000. and if the divisions fall large at the beginning, they may be divided successively from 10001. untill you come unto 100000.

And thus for the graduating and dividing of the Circles of Numbers; which I advise any one (that intends to graduate the other Circles, viz. that of the Sines, and Tangents,) that he fully understand & conceive this former direction, otherwise he will not so easily apprehend the graduations of the said Circle of Sines, and Tangents nor how to divide the projection inlarged; therefore I have indeavoured to be very plaine in the opening of the originall of the worke, that so the ensuing more compendiously may be conceaved.

### 2. Of the graduation of the Circle of Sines.

2. To divide the Circle of Sines, you ought to have reference [Page] to the Table of Sines, and the worke for the graduation of it is in a manner nothing different from the former, onely here may be noted, whereas all the Logarythmes of Numbers are comprehended under one absolute revolution of a Circle, the Sines (which are sufficient for practise) will comprehend or require two revolutions; The first of which begins at the Line of Conjunction, E N, from S, at 34. m. 24. s. and ends its revolution at 5. gr. 44. m. 22. s. and then begins againe at the said 5. gr. 44. m. 22. s. and ends at 90. gr. and is divided thus: In the Table of Sines; marke what number is against 1. gr. under the title Sines, which is 8. 2418. (the Index 8. being rejected) of which account onely but 2418. from E, viz. at a, and lay a Ruler upon the Center, and upon the said a, or the Number 2418. and intersect the Circle of Sines upon the inside in 1, which represents 1, gr. then marke what number is against 2. gr. under the title Sines, which is 5428. account this from E, in the Circle of equall parts, viz. at b, and lay a Ruler upon it and cut the Circle of Sines upon the inside, as before in 2, which stands for 2, gr, and so proceede untill you come to the Line of Conjunction E, N. according to which direction you may subdivide those degrees out of the said Table of Sines, into minuts, 5ths. of minuts 10ths of minuts, &c. according to the greatnesse or smalnesse of the Instrument: The inner side of the Circle of Sines being thus divided, marke what number is against 6, gr. under the title Sines, which is 192. (the Index being rejected) account this number from E. in the Circle of equall parts, viz. at e and lay a Ruler thereto and the Center, and intersect the Circle of Sines S. on the outside in 6. gr. then marke what number is against 7. whith is 858, account from E, in the Circle of equall parts, viz. at f, and lay a Ruler thereto and intersect the Circle of Sines S, in 7. gr. and so proceede untill the whole Circle of Sines be divided into its degrees and parts, unto 90.

### Of the graduation of the Circles of Tangents.

To divide the Circle of Tangents, you must have reference also to the Table of Logarythmall Tangents, as followeth.

Looke in the Table for 1. gr. and marke what Number is against it under the title Tangents, viz. 2419. (the Index being rejected) then account this number 2419. from E, viz. at a, and lay the Ruler upon it, & the Center, & intersect the lower Circle of Tangents T, on the outside in 1, which is opposite to 1. gr. in the Sines, and representeth the Tangent of 1. gr. then from E, account 5430. which is the Tangent of 2. gr. & lay a Ruler upon it & the Center, & intersect the Circle T, in the outside which shall represent 2. gr. in that Circle, and so goe on untill you come to the line of Conjunction, which will bee at 5. gr. 42. m. 40. s. then marke what number is against 6. gr. under the title Tangents which is 216. account this Number from E, in the Circle of equall parts, viz. at e, and lay a Ruler upon it and the Center, and cut the Circle of Tangents T, on the outside of it in 6. gr. then looke what number is against 7. gr. under the title Tangent which is 891. account it also from E, in the Circle of equall parts, viz. at f, and lay a Ruler upon it, and intersect the Circle of Tangents on the upper side in 7. gr. in like manner goe on in dividing the rest of the space in the degrees and parts, unto the line of Conjunction which will be at 45. gr. Then marke what Number belongs to the Tangent of 46. which is 100151. (reject the Index 10.) and the number 151. account from E, in the Circle of equall parts, viz. at g, and lay the Ruler upon the Center, and intersect the Circle of Tangents T, on the inner side in 46. gr. then looke for the Tangent of 47. gr. which is 100303. but (the Index being rejected) it is but 303. account this Number from E, in the Circle of equall parts, viz. at h, and lay a Ruler upon it, and the Center, and cut the Circle of Tangents T, on the inner side in 47. gr. and so goe on in dividing the inner side of this Circle untill you come to the line of Conjunction, which will be at 84. gr. 17. m. 21. s. then seeke for the Tangent of 85, gr. which is 110580 reject the Index which is 11. and account 580. from the line of Conjunction in the Circle of equall parts, from E. viz. at 1. and lay a Ruler upon it, and the Center, and cut the lower Circle T, upon the inner side in 85. gr. and so proceede [Page] with the rest of the graduations untill you come to the line of Conjunction, which will be at 89. gr. 25. m. 40. s.

☞These last graduations of Tangents from 45. gr. unto the said 89. gr. 25. m. 40. s. are not out of any necessity to be inserted, but for expedition, in operation, and are onely the compl [...]ments of the former, so the complement of the Tangent of 40. gr. is 50. gr. the middle space betweene which is 45. againe, the complement of the Tangent of 35. is 55. the middle of which is at 45. Now the distance betweene 50. and 45. is the same betweene 40. and 45. and so of the rest.

### How the Ring may be framed or composed.

Thus for the single projection of the Circles of my Ring, and the dividing and graduating of them: which may bee so inserted upon the edges of Circles of mettle turned in the forme of a Ring, so that one Circle may moove betweene two fixed, by helpe of two stayes, then may there be graduated on the face of the Ring, upon the outer edge of the mooveable and inner edge of the fixed, the Circle of Numbers, then upon the inner edge of that mooveable Circle, and the outward edge of that inner fixed Circle may be inserted the Circle of Sines, and so according to the description of those that are usually made.

If you bring 1. in the mooveable amongst the Numbers to 1, in the fixed, you may on the other edge of the mooveable and fixed see the Sines noted thus, 90, 90. 80, 80. 70, 70, &c. unto 6, 6. and each degree subdivided, and then over the former division and figures, 90, 90. 80, 80. 70, 70. &c. you have the other degrees, viz. 5, 4, 3, 2, 1. each of those are divided and subdivided by severall points.

Secondly, (if the Ring be great) neare the outward edge of the side of the fixed, against the Numbers, are the usuall divisions of a Circle, and the points of the Compasse: serving for observations in Astronomy, or Geometrie, and the sights belonging to the Ring may bee placed on the mooveable Circle.

Thirdly, opposite to these Sines on the other side are the Logarythmall Tangents, noted alike both in the mooveable; and fixed, thus, 6 6. 7 7. 8 8. 9 9. 10 10. 15 15. 20 20. &c. unto 45. 45. which numbers or divisions serve also for [Page] their complements to 90. as before: so 40, gr. stands for 50. gr. 30. gr. for 6 [...], gr. 20. for 70. &c. and each degree heere both in the mooveable and fixed is also divided into p [...]rts; as for the degrees which are under 6. viz. 5, 4. 3, 2, 1. they are noted with small figures above this divided Circle from 45. 40, 35, 30, 25, &c. and each of those degrees is divided into parts by small points, both in the mooveable and in the fixed Circles.

Fourthly, on the other edge of the mooveable on the same side is another graduation of Tangents, like to that formerly described. And opposite unto it in the fixed is a graduation of Logarythmall Sines, in every thing answerable to the first description of Sines on the other side.

Fiftly, on the edge of the Ring is graduated a part of the Equator numbred thus, 10, 20, 30. unto 100, and thereunto is adjoyned the degrees of the Meridian inlarged and numbred thus, 10, 20, 30. unto 70. each degree both in the Equator, and Meridian, are divided into parts, and these two graduated Circles serve to resolve such Questions, which concerne Latitude, Longitude, Roumbe, and distance in Nauticall operations.

Sixtly, to the concave of the Ring may be added a Circle to be elevated, or depressed for any Latitude, representing the Aequator, and so divided into houres and parts with an Axis, to shew both the houre, and Azimuth, and within this Circle may be hanged a Box and Needle, with a socket for a staffe to slide into it, and this accommodated with screw pinnes, to fasten it to the Ring, and Staffe, or to take it off at pleasure: Thus for the first way of inserting of the Projection, on the face and backside of the Ring, a second way followeth.

#### 2. How the Projection may be formed in a Ring, so that all the graduations may be upon one side onely.

THis may be done by a double projection, if the mooveable Circle be so fitted that it moove upon a Plate, and [Page] be of sufficient breadth to containe all the Circles of the single projection, and that one of the fixed Circles retaine the same breadth with the mooveable: as for the innermost fixed Circle, that may be but as an edge onely, then may there be a small channell in the innermost fixed Circle, in which may be placed a small single Index, which may have sufficient length to reach from the innermost edge of the Mooveable Circle, unto the outmost edge of the fixed Circle, which may be mooved to and fro at pleasure, in the Channell, which Index may serve to shew the opposition of Numbers; then upon the other side of the Ring may be placed what the fancie may allude unto.

#### 3. How to fit the Circles of the Ring by a single Projection into a Ring, so that all the Circles may be upon one side onely.

THis is done by having a narrow Circle turned, to moove about the outward edge of a broad Circle, so that being framed together, it may to the eye seeme but as one Circle onely, then may all the Circles of the Projection be described and graduated on the broad Circle, and to the narrow mooveable Circles may bee placed two like edges of Mettle as the parrs of a double Index, which may extend unto the largenesse of the broad Circle, which may move somewhat straight in a Chanell made in the narrow Circle, for then these edges being placed unto any two proportionalls in the Projection, if you shall also moove that Circle which carries these two edges; as the one edge passeth by any Number in that projection, the other will shew the fourth proportionall, in the same projection, and this way doth avoyd the mooving of an Index at the Center; for that is supplyed by the motion of the Peripheria of the Circle, and so according to this direction you may retaine the forme of a Ring, and have all the Circles to the eye upon one of the sides of the Ring onely; which forme is more Gentile, and Mathematicall, then if the projection were [Page] placed on a Plate or Plaine; Many other formes might be deliverd, about this single projection, but these may serve for the present.

### Of the enlarging, or augmenting, of the Projecttion of the Circles of the Mathematical Ring, to worke accuratly Trigonometrie.

THis augmentation of the projection is very facil, being but onely the doubling, tripling, quadrupling, quintupling, decupling, centupling of Circles in the single projection, and so to conceive such decuplation, centuplation, &c. onely to be the parts of one of the single Circles, and may be divided and graduated out of the former Tables of Logarythmes, as though these Circles were onely one continued Circle, and the way how, we will somewhat open by this generall Rule, Divide 10000. (which is the Radix of numbers) by that number which you intend to have the projection enlarged upon, according to a Ratio, or proportion assigned, and the Quotient shall shew the number of parts that one single Circle shall be divided into; which Circle so divided is the ground of the whole projection. As for example, admit the Ratio, or proportion of the augmented projection were required to bee Quadruple to that which is single, that is, foure times greater, then, having described foure Quadruplicities of Circles, according to the Scheme B, and one single Circle noted with E, divide the said 100000. by 4. the Quotient is 25000. which signifieth that the outmost Circle E, (being the ¼ part of each Quadruplicitie of the other foure Circles) must bee divided into 25000. equall parts, therefore let it first bee divided into 25. equall parts, each of those parts divided into halfes, and every one of them divided into 5. parts, so the whole Circle shall be divided into 250. equall parts, then may each of those divisions bee divided againe into halfes, and every one of these halfes may bee divided also into 5. other equall parts, and then the whole Circle shall be divided into 2500. parts, and if the [Page] divisions be great enough, let every one of them bee divided againe into 10. other equall parts, so the whole Circle shall bee divided into 25000. equall parts; if they cannot bee divided into so many parts, yet wee may conceive them so to be divided; and so of other divisions for this projection, which Circle of equall parts may be noted at each capital division thus, 1, 2, 3, 4, 5, 6, &c. unto 25.

How to divide the projection inlarged.

Now to graduate these quadruplicities of Circles from the said Circle of equall parts, you ought to keepe the same method that was used in the dividing of the single Circles in the single projection, onely this by the way ought to be observed, that when you have divided one revolution and come to the line of conjunction, the tabular numbers will exceed the Radix of the Circle, viz. the former 25. and Ciphers, and then how much the tabular number is above it, account beyond the line of conjunction: so to graduate 17. upon the Circle of numbers which falls not beyond the line of Conjunction seeke the Logarythme of the said 17. which is 123044. (the Index 1. being rejected) account from E, in the Circle of equall parts in the scheme B, onely 23044. and laying a Ruler upon it and the Center, intertersect the lowermost Circle of the former, which belongs to the projection of numbers in 17. but if 18. were to bee inserted into the projection, seeke for it in the Table, so right against it is 125527. account beyond the line of conjunction onely the 25527. and lay a Ruler upon it and the Center, and intersect the next higher Circle in 18. moreover, to note out 19. upon the same Circle, against which in the Table of Logarythmes is 27875. Now because the Radix as before is but 25. and seeing 27. in this number exceeds it by 2. reckon that 2. for 27. and account from 2. in the Circle of equall parts, the other part of the number, viz. 875. and lay a Ruler thereto, and to the Center, and intersect the former Circle in 19. and so proceede for the dividing of the other numbers untill you come to the line of conjunction; then will the Radix be there 50. because of two revolutions, therefore at the figure of 1. beyond the line of conjunction, you may account it 51. at the figure of 2. account [Page] it 52, &c. and comming to the line of Conjunction let the 25. be accounted 75. because of three revolutions, then at the next 1. account it as 76. at the figure of 2, account it as 77. and so proceede untill you come to the first Radix, viz. 100000. or to the line of Conjunction at the point E. this being fully conceaved, and having a scheme here already graduated, it shall be easie from this to divide any other projection which is to bee augmented in a different manner: But if the projection inlarged bee of a Decuple, or Centuple proportion, then the outmost Circle is to be divided onely into 10000. equall parts according to the Circle of equall parts E. in the single projection, and then the Logarythmes in the Table will divide them without any consideration (the Index being onely reiected) and as the whole Circle of Numbers in its quadruplicities of Circles may bee divided by the former directions, so in like manner the circle of Sines, and Tangents, in their quadruplicities of Circles may also be divided: and so of any proiection in this kinde. Thus for the augmenting of the Circles of the proiection of my Mathematical Ring, and of the dividing of them.

### Of severall wayes how the Circles of the Mathematicall Ring (being inlarged) may be accommodated for Practicall use.

FIrst the Circles being proiected singly upon a Plaine, an Index with two feete or a flat Compasses may bee placed at the Center to open at any two tearmes assigned: and so to move it about as occasion requires, but in stead of it a semicircle may be fastned there, and a single Index to move upon it, so that the Radius, or edge of the semicircle being placed to any one number, the Index may be placed to the other number: and then if the semicircle be mooved Circular, as the Radius of the semicircle passeth by any number in the proiection, so the edge of the Index in motion shall [Page] shew the proportionalls in that projection; or a paper edge, and a thread, with a beade may bee sufficient, seeing the Circle of equall parts will shew the equall distances of the proportionalls, &c. But the way of operation upon this single proiect [...]on with an Index, &c. to a learner will bee somewhat troublesome for the severall observations that necessarily depend upon that way, and cannot bee avoyded, But a double projection makes it very facil.

#### 2. A second way of accommodating this inlarged Logarythmall Projection, for practicall use, with a single Index onely.

THis is done by having a double Projection inlarged on a Plaine, the one to be fixed and the other to be moveable, (agreeable to the scheme A, in the title page) for being so fitted it shall operate with the same facilitie and expedition, as though it were a single Projection, the proportionall numbers being alwayes by opposition, & in like differences of Circles, one caution considered; and the Index at the Center, is onely but to helpe the eye in finding them.

#### 3. A third way of fitting of the Circles of the Projection for Practicall use.

THis is performed by helpe of a double proiection, the one to be fixed, and the other to bee mooveable: But so that the Circles of the mooveable being described, every other Circle may bee cut out, that is, that there may bee a vacuitie betweene each Circle, then let the edge of the mooveable Circle, be divided and graduated answerable unto the graduation and divisions of the proiection on the fixed, but so that the whole mooveable Circle being placed at the Center, the divisions, graduations, and figures on the said fixed Circle, may be seene conspicuous [Page] through these Channells, and vacuities which are cut out in the mooveable, so this Projection shall also shew, or give the proportionalls by opposition, as the former.

#### 4. A fourth way, how the Logarythmall projection of my Ring inlarged, may be fitted in an instrumentall forme for practicall use in Calculation.

THis may bee done according to my great Cylinder which I have long proposed (in which all the Circles are of equall greatnesse,) and it may be made of any magnitude or capacity, but for a study (hee that will be at the charge) it may be of a yard diameter and of such an indifferent length that it may containe 100. or more Circles fixed parallell one to the other on the Cylinder, having a space betweene each of them, so that there may bee as many mooveable Circles, as there are fixed ones, and these of the mooveable linked, or fastned together, so that they may all moove together by the fixed ones in these spaces, whose edges both of the fixed, and mooveable being graduated by helpe of a single Index will shew the proportionalls by opposition in this double Projection, or by a double Index in a single Projection.

### Of the description of the Grammelogia or the Circles of my Mathematicall Ring on a Plaine, according to the diagramme that was given the King (for a view of that projection) and afterwards the Ring it selfe.

THe parts of the Instrument are two Circles the one mooveable and the other fixed, the mooveable Circle is that unto which is fastned a small pin to moove it by, the [Page] other Circle may bee conceived to bee fixed; upon the mooveable and fixed there are described thirteene distinct Circles, nowithstanding, on the mooveable [...]nd fixed there are 24. several Circles considerable, [...] which there are foure in the mooveable, answerable to foure in the fixed, which double Circles are divided & noted with letters both in the mooveable and fixed, as followeth.

The Circles of the fixed are noted with these letters, viz.

- ♋ The Circle of degrees and Calender.
- E. Represents the Circle of equall parts, and part of the Equator, & Meridian.
- T The Circle of Tangents.
- T The Circle of Tangents.
- S. The Circle of Sines.
- D. The Circle of Decimals.
- N. The Circle of numbers.

The Circles on the mooveable are noted with these letters, viz.

- N. The Circle of Numbers.
- E. The Circle of equated figurs, & bodies.
- S. The Circle of Sines.
- T The Circle of Tangents.
- T The Circle of Tangents.
- Y. The Circle of time, yeares, and monethes.

#### A more particular description of each Circle, and first of the Circle of equall parts.

FIrst, the Circle of equall parts on the fixed, is that which is next to the Circle of degrees, and noted with the letter E. and is figured thus, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. which figures doe stand for themselves, or such numbers unto which a Cypher, or Cyphers may be conceived to be added, or taken away: so 10. stands for 1. or 10. 100. 1000. &c. and so of the rest, and here note that the space betweene any two figures, must bee conceived to comprehend the difference in denomination betweene them, that is, so much as one number is greater than another, so many divisions must be contained betweene these numbers, so if the 20. stands for 20. the 30. then shall stand for 30. and because the 30. doth exceede the 20. in [Page] its denomination by ten, there must therefore bee conceived to be 10. divisions betweene the said 20. and 30. but if the 20. had stood for 200. and the 30. for 300. then the distance betweene 20. and 30. must bee conceived to containe 100. divisions, and so of the rest: which Circle of equall parts, in its divisions represents such numbers as one hath occasion to use, and is of speciall and singular use if it were particularly applyed, (though some one said upon the description of it, it was scarce of any use, but onely that by helpe thereof the given distances of number may be multiplyed, or divided as neede requires) had I the way made for me, and the subject unvayled to helpe my fight, I should see its use better then now I doe, & would not conceat my selfe to be so sharpe sighted, but bee thankefull to any one that would unvaile it for me yet (acording to my weak sight) had I time, and place I could not slober over such a point, but take up much of both, to dilate my selfe in the ample declaration of the uses of that Circle; howsoever I have said somethings in its use, to illustrate it, and to prove my assertion.

#### Of the Equator, and Meridian.

SEcondly, the Circle of equall parts doth not onely represent it selfe, but also a part of the Equator, or Equinoctiall Circle, containing 100. equall degrees, each degree being divided actually into 10. parts, so that each part doth containe 6. minuts, and is numbred as before at every tenth degree, thus, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. upon the inner side of which Circle are graduated the degrees of the Meridian inlarged, or the unequall degrees of Latitude, according to Wrights projection, figured & noted thus, 5. 10. 15. 20. 25, &c. unto 70. gr. each degree being divided into 10. parts, so that each part containeth 6. minuts also as before, which divisions may be called the Meridian Circle, or the Circle of Latitudes.

#### Of the Circle of time.

THirdly, the innermost Circle on the mooveable which is noted with the letter Y. figured thus 1, 2, 3, 4, &c. unto [Page] 30. and each of the spaces is subdivided into 12. equall parts and is called the Circle of time, or yeares, the spaces noted with figures being yeares, and the subdivisions, moneths, and serves for to operate such questions as concernes interest and valuation of Leases, but the outmost Circle, which is noted with letters upon the inner side, is the Calender of time, divided into moneths & daies, and is adjoyned to the degrees of a Circle, noted with the Signes that so readily knowing the day of the moneth, the Suns place may bee found, and so contrarily; which Circle of degrees serves also to take the Sunnes higth, and for the observation of Angles.

#### Of the Circle of Tangents.

FOurthly, the next Circles to that of equall parts on the fixed are the two Circles noted with T T. which are but one Circle in effect, and is called the Circle of Tangents, which comprehends neare a Quadrant or 90. gr. and is graduated and numbred on both sides, having its beginning at the line of conjunction E N, at 34. m. 22. s. the numbers that are minuts are noted with a touch over them thus, 35 [...] that so they may bee distinguished from these numbers which are degrees that are not noted at all. So the Circle of Tangents which is next to the Circle of equall parts is figured upon the outward part of it, in its divisions from 35. m. unto 5. gr. 32. m. 48. s. and the degrees are noted somewhat greater then the rest thus, 1, 2, 3, 4, 5. In the next undermost Circle these degrees from the said 5. are continued and are noted upon the outside of the same Circle, from 6. gr. unto 45. gr. which 45. gr. are continued upon the inner side of the same Circle unto 84. gr. 17. m. 21. s. and these are further continued on the inner side of the uppermost Circle unto 89, gr. 24. m. 38. s. These two Circles are called the Circle of Tangents; And the Circle of Tangents noted in like manner with T T, in the mooveable, is the same with this Circle of Tangents on the fixed: every degree in each of which Circles unto 10. gr. is divided actually into 60. divisions, each of which represents one minut, and from 10. gr. unto 80▪ gr. every degree is divided into 30. divisions, [Page] each division being 2. minuts then from 80. degres unto 89 gr. every degree is divided into 60. parts or minutes as before: This for the description of the Circle of Tangents, both on the fixed and mooveable.

#### Of the Circle of Sines.

FIftly, next unto the Circle of Tangent, is the Circle of Sines noted with the letter S. as before, having its beginning at the line of Conjunction at 34. m. 22. s. and is figured and divided upon both sides, upon the inner side it is numbred from 25. m. unto 5. gr. 44. m. 21. s. which are almost opposie to the like degrees in the Circle of Tangents, which degrees are continued upon the outward side of the same Circle unto 90. gr. and are numbred thus, 6. 7. 8. 9. 10. &c. unto 90. every degree in this Circle of fines is divided into parts according to the greatnesse of degree, and the Circle of sines on the moveable is the same with this on the fixed.

#### Of the Circle of Decimalls of money.

SIxtly, next unto the Circle of sines, on the fixed, is the Circle of the Decimals of money, noted with the Letter D. and is divided on the innerside, and figured thus, 1. 2. 3. 4. 5. &c. unto 23. and each space subdivided into 4. parts: which divisions so numbred are the decimals of pence, (or the divisions or notes of the whole Circle) & are to be conceived to be the 8/10 of a pound of money. In like manner the said Circle is farther divided on the outer side, and figured thus, 1. 2. 3. 4. 5. &c. into 19 parts & the spaces betweene each figure is divided into 12. parts, (or as many parts as each space will containe) which whole Circle so divided (is supposed to represent a pound of money,) is the Decimall of shillings, or of a pound of money: the like may be done for the Decimals of waight.

#### Of the Circle of Equated figures, and Bodies.

SEventhly, the Circle which is a like situated unto this on the moveable that is noted with the letter E. is the Circle of equated figure [...] and Bodies, serving excellently to facilitate and expedite such operations which concerne regular figures, and the five platonicall Bodies, with other occurrents, a touch upon which was showne in the Epistle. But more at large in its place is specified: which Circle of equated figures and Bodies, is divided out of this Table ensuing, as the Circle of numbers was out of the Table of Logarythmes w^{ch} containes 100. notes, 50. of them serving for Superficies, and the other 50. for Solids.

Notae figurarum. | |||

Hep. m. | 01639 | Hex. | 41465 |

De. e. | 05692 | 7. | 42254 |

Hep. cir. | 06159 | 8. | 45154 |

Cir. e. | 07954 | Tri. in. | 46040 |

Oc. in. | 08174 | 9. | 47712 |

P. 18. | 09151 | Qu. e. 10 | 50000 |

No, e. | 10444 | Cir e. | 55245 |

Oc. cir. | 11613 | Hep. | 56046 |

No. in. | 13790 | Tri. | 63650 |

2. | 15051 | Tri. e. | 68175 |

Oc. e. | 15809 | Oc. | 68380 |

F B. | 15836 | Qu. in. | 69897 |

No. cir. | 16491 | M. ac. | 75867 |

P. 16 ½ | 16749 | Tri. cir. | 76143 |

De. in. | 18719 | No. | 79111 |

A. c. | 20411 | Pen. in. | 83770 |

De. cir. | 20898 | Qu. cir. | 84948 |

Hep. e | 21976 | De. | 88616 |

Pen. | 23565 | Cir. D. | 89509 |

3. | 23856 | Pen. cir. | 92975 |

Hex. e. | 29267 | Hex. in. | 93753 |

4. | 30103 | P. 21. | 95762 |

5. | 34948 | P. 20. | 100000 |

Pen. e. | 38217 | ♃ C. s. | 100000 |

6. | 38907 |

D. cir. | 04669 | O. cir | 38907 |

T. S. | 07133 | Cil. c / [...] | 42542 |

T. in. | 08804 | D. in. | 44754 |

2. | 10034 | Cyl. G | 46852 |

I. cir. | 12161 | S. x. | 49714 |

S. in. | 15051 | O. x. | 53959 |

3. | 15904 | I. e. | 55340 |

Cyl. c / f | 16837 | ☉ | 59050 |

4. | 20068 | f. t | 61876 |

5. | 23300 | C e. | 66666 |

Gag. | 23426 | Cyl. D / f | 67117 |

F. T. | 23754 | O s | 67339 |

T. x. | 23856 | T. cir. | 69010 |

C. in. | 23856 | S. s. | 71899 |

Gag. | 24349 | ☿ | 73663 |

6. | 25938 | S. e. | 76033 |

I. in. | 27923 | O. e. | 77553 |

7. | 28170 | C. x. | 77815 |

C. cir. 8 | 30103 | ♄. | 80855 |

D. x. | 31483 | ☽. | 85497 |

9. | 31808 | D. s. | 88440 |

10. | 33333 | ♀. | 91497 |

I. s. | 33879 | I. x. | 93753 |

Cil. d / f | 34245 | ♂ | 96614 |

D. e. | 37186 | T. e. | 97612 |

#### A description of these Tabular notes.

These notes do represent the

- Regular figures.
- Cir. stands for Circle.
- Tri. stands for Triangle
- Qu. stands for Square.
- Pen. stands for Pentagō.
- Hex. stands for Hexagon
- Hep. stands for Heptagon
- No. stands for Nonogon
- De. stands for Decagon
- these letters joyned with the former, as
- e. signifie equall.
- in signifie inscribed.
- c. signifie circū scribed.

- these letters joyned with the former, as

- Regular Solids.
- T. stands for Tetraedron.
- O. stands for Octaedron.
- C. stands for Hexaedron.
- D. stands for Dodecaedron
- I. stands for Icosaedron.
- S. stands for Spheare.
- these adjoyned,
- s. signifie solidity.
- e. signifie equall.
- x. signifie convexity
- in. signifie inscribed.
- cir. signifie circumscribed.

- these adjoyned,

These notes,

- P. 16. ½ feete to a Pole.
- P. 18. feete to a Pole.
- P. 20. feete to a Pole.
- P. 21. feete to a Pole.
- Ac. represent Acres:
- F.B. represent Foote of Board.
- F.T. represent Foote of Timber.
- f. t. represent Foote of Timber equated.
- Cir. D. represent Circles Diameter.
- Cir. c. represent Circles Circumference.
- Cyl. D / s represent Cylindricall. solidite.
- Cyl. e / [...]
- Cyl. d / f represent Cylindricall foote measure.
- these notes represent mettals, viz.
- ☉ stands for Gold.
- ☽ stands for Silver.
- ☿ stands for Quiksi.
- ♀ stands for Copper.
- ♃ stands for Tinne.
- ♄ stands for Lead.
- ♂ stands for Iron.

- these notes represent mettals, viz.
- Cyl. [...]/f
- Gag. represent Gage for wine.
- Gag. A represent Gage for ale.

#### Of the Circle of Numbers.

EIghtly, the Circle next to the decimals of money, noted with the letter N on the fixed is the Circle of Numbers, and is divided into unequall parts, charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures do represent themselves or such numbers unto which a Cypher or Cyphers are added, and are varied as the occasion falls out in speech of numbers.

First, if the figure of 1. stands but for 1. then all the divisions in the Circle by deminition are the parts of 1. so the figure of 4. stands for the fourth part of 1. (if 1. be divided into 10. parts,) or the 4. stands for 40. parts of 1. (if 1. be divided into 100. parts,) 8. stands for eighth part of 1. &c. but if 1. stands for 1. by augmentation, then the figure of 2. shall represent two, the figure of 3. shall stand for three, &c. and the space betweene each figure shall be the parts of 1.

Secondly, if the figure of 1. stands for 10, then the figure of 2. stands for 20. the figure of 3. for 30. &c. hence it followeth that betweene the figure of 1. and the figure of 2. or betweene 2. and 3. must be 10. divisions to represent the intermediate numbers, the middle of those divisions is noted thus [...] as if the 2. be 20. and the 3. be 30. then at the next great division you may account 21. two divisions beyond the figure of 2. to be 22. at the next great division you may account 23. and so on, numbring till you come to 30. or any other number, which divisions noted thus [...] is onely to helpe the eye in numbring.

Thirdly, if the figure of 2. stand for 200. then the figure of 3. is 300 the figure of 4. is 400. hence there must be 100. betweene 200. and 300. and 100. more betweene 300. and 400. and so of others. Now seeing that ten tens make an hundred, there must be 10. divisions betweene the said 200. and 300. and every one of those divisions do represent 10. so the said note [...] shall be halfe of the said 100. therefore at that note you may read 250. and the middle betweene 300. and 400. viz. at [...] reade 350. &c.

Every one of the divisions which stand for 10. is divided againe into 10. other divisions, the middle of which hath its division a litle higher than the rest, to helpe the eye to number more readily.

☞ And here generally is to be noted, that what denomination you give unto any of the figures, the next great division is the next subdenomination, and the next lesser division to that greater is the second subdenomination, as if I should speake 243. here the denominations are Hundreds, Tens, and Ʋnites: therefore the figure of 2, shall stand for 200. the foure great divisions next the [...]. shall be 40. and the next three small divisions shall represent 3. (which is within seaven small divisions of [...]) and so of others.

Thus for the description and numbring upon the Circle of numbers on the fixed. The numbers & divisions on the moveable Circle, are the very same with that on the fixed; for if you move 1. in the moveable to 1. in the fixed, there is represented to every number or division his opposite, not onely in the Circle of numbers, but also in the Circle of Sines, and Tangents.

And thus in these two Circles of numbers, and in the other Circles, there is a great body of numbers, the one standing alwayes fixed, and the other to be moved; and if any number in the moveable be moved, all other numbers move with it: so if you move 25. in the moveable in the Circle of numbers, unto 30. in the fixed on the Circle of numbers, right against 26. in the moveable, is 31. and 2. tenths in the fixed, and right against 27 in the moveable, is 32. and 4. tenths in the fixed, right against 30. in the moveable is 36. in the fixed, against 46. in the moveable, in 55. and 2. tenths in the fixed.

Againe, if 108. in the moveable be brought to 15. in the fixed, right against 16. in the fixed is 115. in the moveable, and right against 12. in the fixed, is 86. and 4. tenths in the moveable.

Thus what denomination you give unto the numbers in the Circle of numbers in the moveable, you are successively to keepe the same denomination, and the like is to be conceived touching the progressive denomination of numbers, in the Circle of numbers in the fixed. Thus for the description of the Instrument; the use followeth.

## How to performe the Golden Rule, or to finde a Proportionall Number unto another Number, as two other Numbers are in proportion amongst themselves.

THis Rule of all other is the most excellent and the most generall, as well in Mathematicall Calculations, as in Arithmeticall Computations, and therefore may not unfitly be so called, and the Instrumentall operation is rather more facile in this Rule, than in Multiplication or Division; hence it is that I have disposed it in the front of the worke, because of expedition and facility: and the way of operation is thus:

Seeke the first number in the moveable,Constructio. and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed.

### Example 1

If the Interest of 100. li. be 8. li. in the yeare, what is the Interest of 65. li. for the same time.

Bring 100. in the moveable to 8. in the fixed,Constructio. so right against 65. in the moveable is 5. 2. in the fixed, and so much is the Interest of 65. li. for a yeare at 8. li. for 100. li. per annum.

The Instrument not removed,☜ you may at one instant right against any summe of money in the moveable, see the Interest thereof in the fixed: the reason of this is from the Definition of Logarithmes.

### Example 2.

If a Troope of 50. Horse have for their pay 140. li. how much shall suffice to pay a Troope of 64. Horse.

Constructio.BRing 50. to 140 then right against 64. in the moveable is, 179. 2. in the fixed, the monthly pay of the said 64. horse. And there immediatly may you see the monthly charge of any number of Horse, for, the number of Horse given in the moveable, right against it, is their pay in the fixed.

### Example 3

It is said that the proportion betweene the circumference of a Circle to his Diameter is [...]. 22.

Bring therefore 7. in the moveable to 22. in the fixed, then immediatly at one instant may you have the Diameter or Circumference of any Circle, only by an ocular inspection: for right against the Diameter in the moveable, is the Circumference in the fixed; or right against any Circumference in the fixed, is his Diameter in the moveable: Thus for the simple Rule.

### Example 4

Further uses of the Golden Rule in ordinary service in propo [...]tionating of things.

Let FLX. represent the Perimeter of a Pentagonall Fort, and let the distance betweene the points of the Bastines, FL. be 926. foot, or KL the square side of a Building 470. foot, and the other dimentions, both of the Fort, and the Building according to the here under inscribed Tables.

- The distance betweene the points of the Bulwarke. FL. 926.
- A perpendicular C R. 617.
- The Cottine A B 662.
- The side of the Fort D N 425.
- The gorge line A D. 119.
- The Flanke D E. 100.
- The line of defence D L. 700.
- The face of the B [...]stine E F. 264.
- The cap [...]tall li [...]e A F. 224.
- The distance from the Center to the Bastine A C. 564.
- From the Cottine to the Center C I. 456.
- The bredth of the Bulwark. G E. 310.

- [Page 3]The greatest square side of the Building K L. 470. foo [...]
- Q. A court within the middle of the Building.
- The distance betweene the middle of the Court and any out angle, as K A. 236.
- The least inner square of the Court E F. 200.
- Betweene any out corner of the Building, as RX. 180.
- 0. 0. 0. &c. a stone Gallery in bredth 36.

### And so of other under roomes to other uses.

NOw admit another like Fort, or another like Building is to be erected, whose greatest distance betweene the aforesaid points of the Bastines, can be but 750. foot, or the greatest side of the peece of ground where the Building is to bee made, is but 400. foot, what shall the severall measures of this new Structure be, so that the Fort to the Fort, or the Building to the Building, in all parts be proportionall?

This is performed with much facility and expedition by this Grammelogia. Constructio.

For if you move the whole to the whole, viz. 926. to 750. or 470. to 400. right against the severall knowne measures in the moveable, you have the severall required measures in the fixed. I bring therefore 926. unto 750.

So right against

- 637
- 662
- 425
- 119
- 100
- 700
- 264
- 224
- 564
- 456
- 310

In the moveable is

- 515. 9.
- 536. 1.
- 344. 2.
- 96. 4.
- 81. 0.
- 566. 9.
- 213. 8.
- 181. 4.
- 456. 8.
- 369. 3.
- 251. 1.

In the fixed So right against

- 236
- 200
- 180
- 36

in the moveable is

- 285. 9.
- 170. 2.
- 153.1.
- 8

in the fixed.

These numbers found out by the ordinary way of Arithmeticke may trouble a nimble Arithmetician a whole houre or more, and therein subject to much error, but others 6. or 8. houres at the least, if not more; but by this Grammelogia, they are found out in lesse time than halfe a quarter of an houre: for so quicke is its operation in any question, to him that hath the way of working by it, that it gives the Answer before a man can distinctly write downe the numbers propoed in the question.

### Further uses of the Golden Rule, in matters of combination of Numbers, how to part a number into parts, as another number is already parted.

- LEt A. B. C. D. E. be five men which adventure money in a Plantation or otherwise: A. adventures 84. li. B 72. li. C 48. li. D. 54. E 42. li. by which in the returne is gotten 50. li. how much shall A. B. C. D. and E. have, according to their severall disbursments.
- Or admit F. borroweth of A. 84. li. of B. 72. li. of C. 48. li. of D. 54. li. and of E. 42. li. F. dies, and his whole estate is worth but 50. li. how much shall every Creditor have of this 50. li. according to his money lent.
- Or suppose A. B. C. D. E. were five severall metals, alotted to make a Statue, Vessell, Bell, &c. A Gold, B Silver, C Coper, D Latten, and E Tin; now when the Metals were melt and cast, there was left a peece which weighed 50. li. how much Gold, Silver, Coper, Latten, and Tin doth it containe, that so the worth of that peece may be knowne.
- Or if there were 5. Companies, or 5. Captaines, A. B. C. D. E. who expect their Pay, to A was owing for his service 84. li. to B. 72. li. to C. 48. li. to D. 54. li. and to E. 42. li. Now to keepe them from mutiny, the Generall sends them 50. li. to be parted amongst them proportionally according to each others dues, what shall A. B. C. D. E. have?
- Or admit A. B. C. D. and E. should load a ship of 300. tuns, A layes in 84 tuns, B. 72. C. 48. D. 54. and E. 42. tuns; in the voyage by reason of tempest, for safegard of their lives and Ship, there was cast over boord 50. tuns of the loading, how much shall A beare of the losse, as also B. C. D. and E.
- Further, in a Shire there is to be raised of 5. men, A. B. C. D. and E. 50. li. proportionally according to their estates; A is worth yearely 84. li. B. 72. li. C. 48 li. D. 54. li. and E. 42. li. how much shall each one pay, &c.

THus I might infinitely dilate my selfe upon one subject, tending to admirable uses, I onely in this glance by things, making but way to the occasions: The resolution of which, and all othets of this kinde, is drawne from this ensuing Axiome.Axiome.

There is such proportion betweene any whole, and his parts, as betweene the like whole, either greater or lesser, and his parts: or betweene the parts and the parts, as betweene the whole and the whole.

So in the first example,Declaratio. Adde the money of A. B. C. D. and E. together makes 300. li. this is the whole, the parts are the former: now 50. li. is another whole number, which must be broken into parts proportionall to the former; and this differeth nothing in the operation from that of the last, in proportionating the Fort to the Fort, or the Building to the Building: for such proportion as 300. li. the whole money disbursed hath unto 50. li. the whole money gotten, so shall A 84. have to his part, and so of any other.

Bring therefore 300. in the moveable unto 50. in the fixed,Constructio. so right against any particular part in the moveable is his part proportionall in the fixed, as there apparantly is seene, and from thence they are taken and placed in a Table, as here under appeares.

As 300. to 50. so

- A. 84.
- B. 72.
- C. 48.
- D. 54.
- E. 42.

to

- 14.
- 12.
- 8.
- 9.
- 7.

### More uses upon the Golden Rule, in the division of Lines.

Propositio. 1 TO finde a Line that shall keepe any proportion assigned unto another line given.

Declaratio.As, let a Line be found which shall keepe proportion to the line A. as 3. to 5.

Constructio.Measure the line A by a scale of equall parts, then bring 3. unto 5. so against the measure of the line A in the moveable, you have the measure of the line required in the fixed, viz. B. so the lines A and B are in proportion as 3. to 5. &c.

### Propositio. 2 To diuide a Line into any number of equall parts.

Declaratio.Let it be required to divide the Line A into 23. parts: first, by a seale of equall parts measure

the Line A, Constructio. which admit to bee 51. parts, bring then 23. in the moveable unto 51. in the fixed. So right against 1. 5. 10. 15. 20. in the moveable, is 2. and 2. 10. 11. and 1. 10. 22. and 2. 10. 33. and 2. 10. 44. and 3. 10. in the fixed: if these numbers be taken from the same scale, and applied to the line A, it will be divided in the points of 1. 5. 10. 15. and 20. then may those parts be easily sub-divided.

### Propositio. 3 To divide a Line in such sort or proportion as another Line is already d vided.

Declaratio.Let the Line B. C. bee divided in the points, D. E. F. G. and H. as the Line R. is divided.

Constructio.Measure the Line R. 58. and his divisions R. 12. R 15 R. 20; R. 30. R. 50. then let BC. be measured, which admit it containe 37. parts, bring 51. unto 37. so against the parts of R in the moveable, you have the parts of B C. in the fixed, viz. B D. B E. B F. B G. and B H.

Propositio. 4 To finde a line in continuall proportion unto two given lines, or a proportionall line to 3. lines, it differeth nothing from that of Numbers, and therefore wrought accordingly.

## Notions or principles touching the disposing or ordering of the Numbers in the Golden Rule in their true places upon the Grammelogia, and the congruity of those Numbers one unto another.

NOte that in any question of the Golden Rule, there are three numbers to worke upon, whereof two of them are of one denomination, the one of them hath his answer, and the other doth require an answer, and those two numbers of like denominations must be alwaies accounted or sought out upon the moveable Circle.

Example. 1 As if 30. li. doe rent 45. Acres of Land yearely, how much doth the yearely Rent of 84. Acres come to.

Here the denominations alike are 45. Acres and 84. Acres, 45. Acres hath his answer, 30. li. and 84. Acres requires his answer.

For the working of this and all others, Let the numbers in the moveable be brought to his answer in the fixed: that is, bring 45. to 30. so, right against the thing demanded in the moveable: that is, against 84. shall be the answer in the fixed, viz. 56. and so many pounds will rent yearely the said 84. Acres.

Secondly, note further, that those three numbers as 45. Acres, 30. li. and 84. Acres, are distinguished by numerall attributes, as first, second, and third. Hence of some it is called the rule of three, and the answer to 84. Acres is called the fourth number, which is ever of the same denomination that the second number is of: and the fourth number sought for hath alwaie such proportion to the third number, as the second is to the first: Vel contra.

From which by a more generall name, it is called The Rule of proportion, for that it proportionateth things unto any proportion assigned; so is the said 56. a proportionall number to 84. as 30. is unto 45. for 56. is two third parts of 84. and so is 30. two third parts of 45.

Therefore these foure numbers, 45. 30. 84. 56. are proportionall numbers one unto another:☞ And here note generally in direct proportion, if the third number be greater than the first number, the fourth number shall bee greater than the second number.

Contrariwise, if the third number be lesse than the first number, the fourth number is lesse than the second number.

But in Reciprocall proportion this fourth number is inverted; so if the third number be greater than the first, the fourth number is lesse than the second.

Example 1 So if 45. men in 30. daies, will doe a service, in how many daies shall 270. men doe it.

Here the denominations alike are 45. Men and 270. Men, the answer to 45. men is 30. daies, the answer to 270. men is required.

Constructio.If you move 45. in the moveable to 30. in the fixed, right against 270. in the moveable is 180. daies in the fixed: which answer is absurd, seeing there is more men allotted to doe the worke, there must most necessarily be lesse time.

Therefore in all Questions of Reciprocall proportion, let the demand bee sought out upon the moveable, viz. 270. and brought to the first numbers answer in the fixed, viz. 30. so right against the first number in the moveable, viz. 45. is the answer in the fixed, viz. 5. and in so many daies will 270. men doe that service, if 45. men doe it in 30. daies.

Quest. 2 Againe, if 3840. souldiers are victualed for 10. moneths, how many men may it serve that the said provision may last 12. moneths.

Constructio.In this and all others (as before) bring the third number 12. in the moveable, to the other numbers answer in the fixed, viz. 3840. so against the first number 10. moneths in the moveable, is 3200 men in the fixed, and so many men will the same provision serve for 12. moneths.

From which direction, those ensuing questions, and the like, may be resolved.

Ques. 3 If I lend 140. li. for 7. moneths, if I should borrow of him 200. li. how long might I keepe it; facit 4. moneths and 9. 10.

Quest. 4 According to the Statute, if Wheat beat 50. s. the quarter, the penny loafe should weigh 6. ounces and a halfe, what shall it now weigh, if in case wheat be at 3. li. 12. s. the Quarter; the numbers changed into decimals will be thus, if 2. li. 5. 10. give 6. ounces, & 5. 10. what shall 3. li 6. 10. give: facit 4. ounces and 5. 10.

Quest. 5 A Gallery is found to containe in the walls 380. yards, how many yards of Tapestry shall hang that Gallery of 7. quarters broad: facit 217 yards.

Quest. 6 25. Ounces of 7. yards to an ounce will serve to lace a vesture, how many ounces of 5. yards to an ounce will doe the same, &c. facit 35. ounces.

### How to proportion a Fraction that is not Decimall, into a decimall.

So if 8. & 12. 40. were to be used, 12. 40. must be changed into a Decimall, thus: bring 40. in the moveable to 10. in the fixed, so right against 12. in the moveable is 3. in the fixed, so the fraction 12 40. is changed now into 3. 10. so for 8. & 12. 40. you have now 8. and 3. 10. which may be easily found out.

Againe, let 63. 84. bee a fraction which is to be used, this cannot be found out upon the Grammelogia: change it therefore into a Decimall.

Bring therefore 84. (the Denominator) to 100. in the fixed, so 63. (the Numerator) in the moveable, gives 75. in the fixed; so 63. 84. is now changed into a Decimall 75. 100. the same in value with 63. 84. and so of any other Fraction that is not decimall.

This for Lineary Proportion.

### Of the Golden Rule, or Rule of Proportion, in respect of Lines and Quantities in plaine Figures.

Pro. 1 IF the demand be of the quantity, As if [...]he Diamiter of a Circle be 7. and the Area 38. and 5. 10. what is the Area of another Circle whose Diameter is 18. Foot.

Constructio.Bring the line knowne to the other line, that is 7. to 18. so right against 38. and 5. 10. in the moveable is 99. in the fixed, which looked out in the moveable, right against it in the fixed is 254. and 5. 10. the Area of that Circle.

In like manner consider of Squares, Triangles, and other plaine Figures.

Pro. 2 If a peece of Land of 20. Pole square be [...]worth 30. li. what is a peece of Land of the same goodnesse worth, which is 35. Pole square every way.

Constructio.Bring 20 to 35. so right against 30. in the moveable you have 52. and 5. 10. in the fixed; and right against this 52. and 5. 10. in the moveable you have 91. and 8. 10. in the fixed, the worth of that land.

Pro. 3 If a peece of ground of 50. paces square is sufficient to lodge an Army of 1600 men, how ma [...]y men shall there be ledged in a peece of ground which is 40. paces square

Constructio.Bring 50. to 40. so right against 1600. in the moveable is 1280. in the fixed,

Pro. 4 Our English land measure is 16. foot and a halfe to the Pole, the Irish Pole hath 21. foot, how many Engl sh Acres doth 30. Irish Acres make.

Constructio.Bring 16. and 5. 10. to 21. then right against 30. in the moveable is 38. and 2. 10. in the fixed, and right against this 38. and 2. 10. in the moveable is 48. and 6 10. in the fixed, and so many English Acres is contained in 30. Irish Acres, &c.

☞Our usuall measures in England to the Pole are 16. foot and a halfe 18. or 20. foot, the proportion of their squares are 68. 81. 100. I have set their measures to those numbers in the Grammelogia.

Now if the quantity be given and his measure, and the quantity be required according to another measure, you may have it with greater expedition: for bring the measure whole quantity is required to the other measure, so against the quantity knowne in the moueable, you have the quantity required in the fixed.

### Of the Golden Rule, or Rule of Proportion in respect of Lines, and the quantity of Solids.

Pro. 1 SO if in some stately structure the Columes were to bee supported with Cubes of Silver, or other rich Materiall, differing in their quantity, an estimate of their charge might be quickly had; As admit the side of the least Cube were 4. Inches, and could not be made under 12. li. what might a Cube of the same mettall be worth that is but one inch more in the side, viz. 5. inches.

Bring 4. to 5. so right against 12. in the moveable,Constructio. is 15. in the fixed, and right against this 15. in the moveable is 18. and 75. 100. in the fixed, and right against this 18. and 75. 100. in the moveable is 23. li. and 4. 10. in the fixed, and so much will the second Cube cost: this might bee applied to the weight, worth, or quantitie of other Solids.

Pro. 2 A Pe [...]ce of 5. Inches boare or Diameter, requires for her charge 16. pound of Powder, what quantity of Powder will serve another Peece of 4. Inches in the boare.

Bring 5. to 4. so right against 16. in the moveable is 12. and 8. 10. in the fixed,Constructio. and right against 12. and 8. 10. in the moveable is 10. and 24. 100. in the fixed; and right against this 10. and 24. 100. in the moveable is 8. and 2. 10. in the fixed: the answer of Powder according to Cubick proportion, but Canoniers doe somewhat qualifie this proprotion.

### To finde what Proportion in Quantity there is betweene two or more Solids.

Pro. 3 There are two Bullets, Globes, or Cylenders, the Diameter of the one is 10. inches, and the other the Diameter is 4. inches, what proportion is there betweene the Solids, or how often doth the greater containe the lesser.

Bring 10. to 4. so right against 100. in the moveable is 40. in the fixed,Constructio. against this 40. in the moveable is 16. in the fixed; and right against this 16. in the moveable, is 6. and 4. 10. in the fixed; so the proportion betweene the Solids are as 100. to 6. and 4. 10.

But how often the greater doth containe the lesser, the Rule ensuing doth teach.

### Pro. 1 How to divide one number by another.

Constructio. MOve the Divisor to 1. so right against the Dividend in the moveable, is the quotient in the fixed.

Declaratio.So if it were demanded, how many daies there is in 216. houres, because a day naturall containes 24. houres, that therefore is the Divisor. Move then 24. to 1. and right against the said 216. in the moveable is 9. in the fixed, and so many daies is 216. houres.

☞Here note that in all Divisions, by how many figures or places the Dividend exceeds the Divisor, so many places or figures shall the Quotient have. But if the figures of the Divisor may be taken from as many of the first figures or places towards the left hand of the Dividend, then the Quotient shall have one place more.

Example 2 So if it were further required, how many daies there were in 360. houres, or any other number: the Instrument not moved from his first setting, they are all given at one instant: for right against the number in the moveable, is the answer in the fixed, so right against 360. in the moveable is 15. in the fixed, and so many daies are there in 360. houres.

This note serves onely to know the number of Figures or places in the Quotient, by which the denomination of the first figure of the Quotient may be had.

Example 3 So if it were demanded how many yeares there is in 14600. daies, there being 365. daies in the yeare: this therefore is the Divisor. Bring then 365. to 1. so right against 14600. in the moveable is 4. in the fixed, but by the former note ☞ it must be 40. and so many yeares is there in 14600. daies, the Instrument not moved, right against any number of daies, as 5000. 10000. 20000. &c. in the moveable, is the yeares in the fixed. With the same expedition and facility may you divide by fractionall numbers.

#### Further uses upon Division.

Example 4 IN a yeare are 52. weeks or 365. daies. If J would know the weekly expences of any yearely summe of money.

Bring 52. to 1. then right against any summe of money in the moveable, you have the weekly expences in the fixed:Constructio.

But if you move 365. to 1. then right against any summe of money in the moveable, you have the daily expences in the fixed.

So if the expences yearely were 1000. li. or the charge of a certaine Company of S [...]uldiers:Declaratio. right against it according to the note ☞ of Division is 2. li. 7. 10. the daily charge: the Instrument not removed, you may see at one instant the daily charge of 20000. li. a yeare, 50000. li. or 100000. li. a yeare: for right against the charge or expence in the moveable, is the answer in the fixed.

#### More uses upon Division.

Example 5 It is said that Land is bought after the rate of 14. yeares purchase: if 14. be therefore brought to 1. right against any summe of mony in the moveable, you have the Annuall Rent in the fixed answerable to that money. And thus you have lying before you a whole Circularity of Numbers, by which at one instant, doe but speake the summe of money, right against it is his Rent.

But if the Rent were given and the Purchase required, it is the inverse of this, and is proper to Multiplication, and the Rule followeth in the next page.

Other uses upon Division to finde the Scale to divide the Meridian line in a Sea Chart, according to any bredth, & to a Latitude assigned.

Example 6 Let the bredth of the Chart extend from the Latitude of 30. unto 40. the degrees of the Equator answerable to the difference of those Latitudes, according to M. Wrights projection, are 12. & 24 100. Bring this 12. & 24. 100. to 1. so right against the bredth of the Card in the moveable, you have the Inches, or parts of Inches in the fixed to make your scale by to divide the Meridionall line.

So if the bredth or the Card were 33. Inches,Declaratio. right against it in the fixed is 2. Inches, 7. 10. the largenesse of a degree of the Equator: if the bredth were 24. & 5. 10. right against it is 2. Inches: if 20. & 8. 10. then the bredth of a degree is 1. & 7. 10. if 14. & 7. 10. then 1. & 2. 10. if 8. & 5. 10. then 7. 10. &c.

### To multiply one Number by another, or to finde the Product of two Numbers.

Constructio.MOve 1. to the multiplier, then right against the Multicand in the moveable Circle, you have the Product in in the fixed Circle.

☞Here note that the Product of any Multiplication, is ever as many figures or places, as there are places or figures contained in the Multiplicand and M ltiplier, if the two first Figures towards the left hand being multiplied together have excrescence (that is, if the Product exceed 9) otherwise the Product shall bee one figure or place lesse than there are figures or places contained both in the Multiplicand and Multiplier.

Declaratio. So if 38. be multiplied by 2. the Product will be but two places: But if the said 38. be multiplied by 5 the Product will be three places, for that 3. by 2. multiplied doth not cres [...]ere, but the said 3. by 5. doth beare excrescence, viz. more than 9.

This Note is only to give domination to the first figure of the Product towards the left hand, for if the Product have two figures, then the first figure of that Product towards the left hand is ten or tens; if the Product have three figures, then the first figure of the Product towards the left hand is hundreds, &c.

Example 2 To Multiply 18. by 5. Bring 1. to 5. then right against 18. in the moveable is 9. in the fixed, which by the former note ☞ of observation is 90. which is the Product of 18. by 5.

But if 35. were to bee multiplied by 4. move 1. to 4. so right against 35. in the moved is 140. by the last note ☞.

Example 3 To multiply Fractionall, as 40. and 5. 10. by 7. and 3. 10. Bring 1. to 7. and 3. 10. so right against 40. and 5. 10. in the moveable is 295. and 6. 10. in the fixed; the Product required.

So to multiply 8. 10. by 5. 10. Bring 1. to 5. 10. so right against 8. 10. in the moveable is 4. 10. in the fixed.

#### Vses upon Multiplication.

Example 3 12. Monetht make a yeare, bring 1. unto it, so right against any monethly expences in the moveable you have the yearely expences in the fixed according to the note ☞: So if the monethly expences were 75. li. right against it in the fixed is 9. which by the former note ☞ makes 900. if [...] 50. li. for a moneth, right against it in the fixed is 1800. li. the yearely charges or expences.

#### Other uses upon Multiplication.

Example 4 60. Minutes make an houre, bring 1. to 60. so right against any number of houres in the moveable is the minutes of those houres in the fixed.

#### Further upon Multiplication.

Example 5 Admit lands be sold at 14. yeares Purchase, bring 1. to 14. so against any Rent in the moveable you may at one instant see the purchase thereof in the fixed, having regard to the former note ☞.

#### How to square a Number.

Example 6 To square 18. bring 1. to 18. so right against 18. in the moveable is 324. in the fixed, the square of the said 18. In like manner may you square whole numbers and fractions, as to square 13. and 5. 10. facit 182. and 25. 100.

#### How to Cube a Number.

Example 7 As to Cube 6. and 2. 10. bring 1. to 6. and 2. 10. so right against 6. and 2. 10. in the moveable is 38. and 4. 10. in the fixed, and right against 38. and 4. 10. in the moveable is 228. in the fixed, the Cube of 6. and 2. 10.

Againe to Cube 6. bring 1. to 6. for right against 6. in the moveable is 36. in the fixed; and right against this 36. in the moveable is 216. in the fixed, the Cube of 6. &c.

### To finde Numbers in continuall proportion unto any two Numbers assigned.

Constructio.BRing the first number to the second, then right against the second upon the moveable, is the third number in the fixed, and against this third number in the moveable, is the fourth number in the fixed, &c.

Declaratio.So if the numbers to be continued in proportion be 2. to 4. move 2. to 4. so right against 4. in the moveable is 8. in the fixed, and 8. in the moveable gives 16. in the fixed, and those numbers, 2. 4. 8. 16 &c. are said to be in continuall Proportion.

Example 2 Againe, it I would continue a Proportion, as 2. to 3. move 2. to 3. then 3. in the moveable shall point out 4. & 5. 10. in the fixed, and 4. & 5. 10. in the moveable shall give 6 & 7. 10. i [...] in the fixed, and so on (if need were) to finde others: and those numbers are said to bee in continuall propor [...]ion one unto another.

☞The increase or interest of Mon [...]y from this ground is easily found, seeing the increase of the Money must bee in continuall Proportion to the Principall, as 100. li. is to his Interest.

Example 3 As if the Propor [...]ion were to be continued to 40. li. as 100. to 108.

Constructio.Move 100. to 108. then against 40. in the moveable is 43. li. 2. 10. in the fixed: the first yeares Interest and its Principall, & against this 43. li. 2. 10. in the moveable, is 46. li 8. 10. in the fixed, which is the seconds yeares Principall and Interest: in like manner may you proceed to other yeares.

☞ The Instrument being at this stay, the eye may denote out at one instant the Interest of any summe of Money: for right against your number in the moveable, is both Principall and Interest in the fixed.

Example 4 As if it were 27. li. 14. s. (that is, 27. li. 7. 10.) right against it is 30. li. ferè, and so much doth 27. li. 14. s. come to at the yeares end, and so all other summes of money doe offer themselves at one instant to the eye in their resolutions.

### To finde a meane proportion, or many betweene any two Numbers given.

MArk what number of equal parts in the fixed is against each of the given numbers, (which equall parts represent the Logarythmes of these numbers, if the Logaryt [...]mall Index be put unto them, which is a unity lesse then the places of any given number) and adde these Logarythmes, or equall parts together, then take the halfe of that summe, which sought out in the former Circle of equall parts, right against it in the Circle of numbers is the meane proportionall required, or the halfe difference of these two Logarythmall numbers, or equall parts being added to the lesser Logarythme will give the same, or sub. &c.

But if many meane proportionalls be required divide the differentiall Logarythme of the two numbers, or number of equall parts, by a unity more then the number of meane proportionalls, which quotient being by succession added to the Logarythme or equal parts belonging to the lesser number, doth shew the severall meane proportionalls required. So if betweene the Cubicke numbers 27. and 64. two meane proportionalls were required, the third part of the difference of equall parts betweene these numbers is 125. which being added unto 431. the equal parts against 27. makes 556. against which in the Circle of numbers is 36. the first meane proportionall unto which 556. againe adde successively the said 125. which makes 681. against these equall parts in the Circle of numbers is 48. the other meane proportionall.

According to the same manner betweene 243. and 1024. foure meane proportionalls might be found, to wit, 324. 432. 576. and 768. Or proportions may be found to a tearme assigned, betweene two numbers, either by augmentation from a greater, or diminution from a lesser.

### Of the Extracting of square, and Cubicke Roots, and others.

THe construction of this depends upon the latter, because the square Roote of any number, is nothing but a meane proportion betweene a unity and the given number, the Cubicke Roote is the first of two meane proportionalls, betweene the unity and the Cube proposed▪ the Biquadrat Roote, is the first of three meane proportionalls betweene the unity and the given number, &c. But because the former direction specified in finding of meane proportions, adheres unto the way and nature of Logarythmes, it being more facil by them, then so to apply it Instrumentally, therefore we will somewhat compendiate that labour by the Instrument alone, and avoyd the search of the Logarythmes of numbers, and their partitions, and as an ease for Radicall extractions.

### To finde a meane proportionall with more facility then is formerly delivered

Constructio. Marke what number of equall parts in the Circle of equall parts E, in the fixed is against the lesser of the two given numbers in the Circle of numbers in the fixed.

then bring the lesser number in the moveable to these parts in the Circle of equall parts noted with Q betweene

- Q A
- A Q

if these two numbers have

- like places or exceede one another even places.
- exceede one another by odde places.

so the like number of equall parts in the fixed in the Circle Q which is against the greater number in the Circle of equal parts E, being sought out betweene Q A, in the fixed shall right against it point out the meane proportionoll in the mooveable.

But here note that 10. in this Rule must be accounted to have but one place, 100. to have but two places, 1000. three places, &c.

For the extraction of square and Cubicke Roots more compendiously, thats done by an inspection of the eye onely, as is specified in the aforesaid Epistle to the Reader, at the last clause of the use of the Instrument, without motion: but hereafter more in that nature. And by the way, note that in the extraction of square Rootes, the Roote doth containe in places alwayes the just halfe of the places of the number given if it hath even places, but if it have odde places, then the Roote hath as many places as the greater halfe comes to.

Now as every two places in square numbers, affords one place for its Roote: so Cubicke numbers, affords for every third place, or ternarie one for its Roote: but if the number have any places above, the ternarie or ternaries of places, then the Roote shall be one place more then the number of ternaries.

☞ Note further, that in seeking of meane proportionalls it may bee doubtfull what denominations to give unto it when it is found on the Instrument, which may be discovered in this manner, finde the places that the two extreame numbers given would make if they were multiplyed together, which the Rule in Multiplication, Pag. 14. will shew you; then having the number of places for the product, the former Rule which doth allude to the places for the square Roote, will tell you what denomination to give the meane proportion sought for.

### To finde a meane proportion betweene two numbers.

Pro. 1 NOte if the two Numbers have like places,☜ or exceed one another by two places, move the numbers to and fro, untill 1 in the fixed be equally distant betweene them, which the divisions in the pricked Circle A B will helpe you; so right against 1. in the fixed, is the mean proportiō in the moveable.

Pro. 2 If the two n [...]mbers exceed one another by one, or three places, move the numbers to and fro, untill 1. in the fixed bee equall distance betweene them; so right against B in the moveable is the meane Proportion.

#### Some uses upon meane Proportionals.

Pro. 1 To find how much is taken in the 100. li. in Loane of mony. If 40. li. be lent for two yeares, Declaratio. and at the end thereof were received 48. li. and 4. 10. what was taken in the 100.

Finde a meane proportion betweene 40. and 48. 4. 10. which will be 44 according to the last rule; so right against 48. 4. 10. in the moveable, is 110. in the fixed, which is the Principall and its Interest; so ten pound is taken per centum. Pro. 2 Pro. 2. In warlike discipline, the weakest place opposed to danger, is supplied with strongest force.

Now there are two companies allotted for two severall services, the one containing 500. Souldiers, the other 320. Souldiers, there is a third place, neither so strong as the latter, nor so weake as the former, therefore a meane number of Souldiers is thought convenient for the defence thereof: what number shall it be?

Finde a meane proportion betweene 500. and 320. facit 400. and this is a meane proportionall number betweene 320. and 500. and the number of men required.

Pro. 3 Pro. 3. To finde the Scale that protracted a Plot or Building by.

Let the Rectangle A C be 8. Acres, Declaratio. Constructio. and let the Scale be sought for by which it was protracted or plotted: With any Scale measure the side A B. admit of 10. parts in an inch, and suppose it make 33. & 33. 100. parts, and A D. 26. & 66. 100. parts, according to which the Area of the Rectangle now is 5. Acres and 56. 100. parts; finde a meane proportion between this and the forme. 8. Acres, which is 6. & 67. 100. and this stands against 1. in the fixed, which represents 10. his scale, but 8. in the moveable gives 12. in the fixed, and such were the parts in an Inch of the scale sought for.

### Pro. 1 How to extract the Square Root by the Grammelogia.

Constructio.LEt 1. in the fixed stand toward you, and seeke that number to be extracted in the moveable, if it have 1. 3. 7. or 9. places, &c. bring the number towards the left side of 1. in the fixed: but if the number have 2. 4. 6. or 8. places, &c. bring it twards the right side of the fixed 1. and move your number to and fro, untill 1. in the moveable bee as farre distant from 1. in the fixed, as your given number is from 1. in the fixed: (the equall parts in the Circle A B will helpe you in this) so the number in the moveable right against the fixed 1. is the Root sought for.

Here note that 1. or 2. figures hath but one figure for his Root, 3. or 4. figures hath 2. figures or places for its Root, 5. or 6. figures hath 3. figures for its Root, &c.

### How to extract the Cubicke Root.

☞VPon the moveable there are those letters A. B. C. the distance betweene A. B. is divided into 10. equall parts, and each part subdivided: the distance betweene A. C. B. is also divided into 10. parts, and each part subdivided, their uses may be thus.

Constructio.Let 1. in the fixed stand alwaies betweene A and B in the moveable for the Extraction of Cubicke Roots, and move the moveable to and fro, untill that the given number and 1. in the fixed be of like number of parts distant from A in the moveable.

So if the given Number have

- 1. 4. 7. or 10.
- 2. 5. 8. or 11.
- 3. 6. 9. or 12.

Places, &c. the Cubicke Root is right against

- A
- C
- B

In the fixed.

And here note that a number of 1. 2. or 3. places hath but 1. figure for the Root; a Number which hath 4. 5. or 6. places hath but 2. figures or places for its Root; a number which hath 7. 8. or 9. places hath but 3. figures or places for its Root, &c.

### Vses upon the square Root.

#### Pro. 1. There are two square formes, the one is 12. every way, and the other 16. every way, if of those two were made one, how many should it be every way in the side.

Pro. 1 BY the first proportion,Constructio. pag. 16. finde a number in continuall proportion to 16. as 12. to 16. facit 21. & 3. 10. adde this to 12. facit 33. and 3. 10. Then by 1. Pro. pag. 17. finde a meane proportion betweene that 33. and 3. 10. and 12. facit 20. the side of the Square required.

Otherwise square [...] 12. and 16. according to the eighth example, pag. 15. facit 144 and 256. the summe of those Squares is 400. and the Root Quadrat of it by Pro. 1. pag. r 8. is 20. as before, those extractions serve to wonderfull uses in finding the Diagonals of Rectangles, the Diameters and Axis of Solids, the Area, Difference or Agregate of Figures, as well plaine as solid.

Pro. 2 Otherwise we might apply the Pro. thus, A B is the bredth of a ditch 16. foot, B C the heigh of a wall 12. foot, the length of a scaling Ladder to reach from A to C. would bee as before 20.

Pro. 3 A and C are two Townes, Alies West of the Meridian of C 16. miles, and C lies North of the Parallel of A 12. miles, the distances of the two Townes would bee as before 20. miles, &c.

#### Pro. 4. How to encampe horse or foot, according to any proportion assigned.

Pro. 4 240. men or horse are to be imbatled,Declaratio. that the Flanke to the Front shall be in proportion, as 3 to 5, how many shall be in the Front, and how many in the Flanke.

Bring 3. to 5. so against 24. in the moveable is 400. in the fixed,Constructio. the square Root of which is the Front viz. 20. divide the said 240. by the Front, 20. the Quotient is 12. the Flanke.

#### In mentall reservation of a number, to finde that number.

Pro. 5 Let the number be broken into two parts, and to the product of the parts adde the square of the halfe [...] difference of the parts, the Root, Quadrate of the Agragate is halfe th [...] number conceived, &c.

### Further uses upon the Grammelogia in the resolution of Questions, touching Interest, Purchases, valuation of Leases, and such like.

☞NOte that from 1. in the moveable, there is charactered 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. &c. all of equall distances, those serve for the number of yeares as occasion requires.

#### Pro. 1. To finde what a summe of money comes to, at the end of any number of yeares, accounting 8. li. for 100. p [...]r Annum.

Pro. 1 Declaratio.So if 20. li. were forborne 12. yeares, how much doth it come to allowing Interest compound at 8. li. for 100. li.

Constructio.Bring 1. in the moveable to 20. in the fixed, so right against 12. yeares in the moveable you have 50. li. 4. 10. in the fixed. And so much will 20. li. amount to being forborne 12. yeares.

☞The Instrument not removed, you may at one instant see the amount of the said 20. li. for any number of yeares of parts of a yeare; for right against the time in the moveable, you have the answer of the money in the fixed.

#### Pro. 2 Pro. 2. To finde what a summe of money which is due any number of yeares to come, is worth in ready mony, allowing 8. li. for 100. li.

Declaratio.So if the said 20. li. were due 12. yeares hence, what is it worth in Present.

Constructio.This is only the converse of the former: bring therefore 12. yeares in the moveable to 20. in the fixed, so right against 1. in the moveable is 7. li. 94. 100. in the fixed, which is about 7. li. 18. s. 3. d. and so much is able to buy the said 20. li. to be received 12. yeares hence.

#### Pro. 3 Pro. 3. A yearely Rent of a Lease, or a Pension to be sold for any number of yeares, to finde the worth thereof in ready money: Or the Rent for any number of yeares being unpaid, to finde what it amounts unto, a [...]counting 8. li. for 100. li. per Annum.

Declaratio.Let a Lease or Pension of 20. li. per Annum be sold for ready mony, which is in being 12. yeares, how much is it worth? Bring 8. to 100. then right against 20. li. in the moveable is 250. li. in the fixed;Constructio. unto this 250. in the fixed bring 12. yeares; so right against 1. in the moveable is 99. li. 3. 10. in the fixed, which taken out of the said 250. there remaines 150. li. 7. 10. the worth required.

### If the Rent were behind unpaid 12 yeares.

THen bring 1. to the said 250. so right against 12. yeares in the moveable is 630. li. in the fixed, take the former 250. from this 630. li. it leaves 380. li. and so much doth the said Rent of 20. li. per Annum amount to forborne 12. yeares at 8. li. for 100. li. per Annum.

#### Pro. 4. A summe of money borrowed, and a Lease ingaged for that money, to finde how long the Lease ought to be kept.

Pro. 4 Let 300. li. be borrowed upon a Lease in being 20. yeares,Declaratio. of 50. li. a yeare, how long shall the Rent be received, that neither be damaged one by the other, accounting 8. li. for 100. li. per Annum.

Bring 8. to 100. so right against 50. in the moveable is 625. in the fixed:Constructio. from this 265. li. subtract the mony borrowed, viz 300. li. it leaves 325. li. then bring 1. to this 325. in the fixed; so right against the same 625. in the fixed, is 8. yeares 5 10. in the moveable, and so long time shall the Lender of the Money enjoy the Borrowers Lease, after 8. li. for 100. li. per Annum. This may be inverted, knowing the Summe and time to finde the Rent.

#### Pro. 5 Pro. 5. A Lease to beginne for yeares to come, and then to continue for any nu [...]ber of yeares, to finde the worth thereof in present, accounting Interest Compound at 8. li. for [...]00. li. per Annum.

Let a Lease of 40. li. per Annum beginne 7. yeares hence,Declaratio. and then to continue 10. yeares after; if it were to bee sold, what is it worth in ready money?

By the third Pro. finde the worth thereof in the present for the 10. yeares, facit, Constructio. 268. li. 4. 10. then by the second Pro. finde what that 268. li. 4. 10. is worth in present if it were to be received 7. yeares hence, facit, 156. li. 6 10. and so much is the said Lease of 40. li. per Annum worth, which is to beginne 7. yeares hence, and then to continue unto 10. yeares.

Thus I might have gone further in those matters, but I intended not to be large in this Tract, onely shewing what weighty, and difficult matters in this kinde by the Grammelogia, or Mathematicall Ring, may bee easily and speedily resolved.

### Conclusion.

IF there be composed three Circles of equal thicknesse, A. B. C. so that the inner edge of D and the outward edge of [...] be [...] answerably graduated with Logarithmall signes, and the outward edge of B and the inner edge of A with Logarithmes; and then on the backside be graduated the Logarithmall Tangents, and againe the Logarithmall signes oppositly to the former graduations, it shall be fitted for the resolution of Plaine and Sphericall Triangles.

Example.So if you move the Signe of 90. Degrees vnto the Tropicall point in the fixed, you have the Declination of any Degree of the Eclipticke onely by an ocular inspection, for right against the Sunnes longitude in the moveable amongst the Signes, is the Sunnes declination in the fixed.

Againe, in the [...] of Tangents, if you bring the complement of any Latitude in the moveable to 45. in the fixed, you may at one instant have the time of Sun rising or Sun setting for any Declination required in that Latitude; for right against the Tangent of the Sans Declination, you have the fine of the Suns ascentionall differen [...]e: and in plaine Triangles the operations are performed with like facility.

Hence from the forme, I have called it a Ring, and Grammelogia by annoligie of a Lineary speech; which Ring, if it were projected in the Convex unto two yards Diameter, or there abouts, and the line Decupled, it would worke Trigonometrie unto seconds, and give propo [...]tionall number [...] unto six places only by an ocular inspection, which would compendiate Astronomicall calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability so health and time.

Whereas Richard Delamain, Teacher of the Mathematicks, hath presented vnto Vs an Instrument called Grammelogia, or The Mathematicall Ring, together with a Booke so intituled, expressing the use thereof, being his owne Invention; We of our Gracious and Princely favour have granted unto the said Richard Delamain and his Assignes, Privilege, Licence, and Authority, for the sole Making, Printing, and Selling of the said Instrument and Booke: straightly forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument or Booke within any our Dominions, during the space of ten yeares next ensuing the date hereof, upon paine of Our high displeasure.

Given under our hand and Signet at our Palace of Westminster, the fourth day of Ianuary, in the sixth yeare of our Raigne.

## Vpon his Ring.

## By a Friend.

## To the Reader.

NOw by way of advertisment to the Reader, in this Circular projection of Logarithmes, you may make use of the Projection of the Circles of the Ring upon a Plaine, having the feet of a paire of Compasses (but so that they be flat) to move on the Center of that Plaine, and those feet to open and shut as a paire of Compasses (which some call a Sector abusively) now if the feet bee opened to any two termes or numbers in that Proj [...]ction, then may you move the first foot to the third number, and the other foot shall give the Answer; and so moving those feet along Circularly, as one foot passeth by any number in the Projection, the other foot shall shew his proportionall number in that Projection; it hath pleased some to make use of this way. But in this there is a double labour in respect of that of the Ring, the one in fitting those feet unto the numbers assigned, and the other by moving them about, in which a man can hardly accomdate the Instrument with one hand, and expresse the Proportionals in writing with the other. By the Ring you need not but bring one number to another, and right against any other number is his Answer without any such motion. But this or the former I leave to such as shall best affect them, only the latter for Construction I account most facile, and for expedition most excellent, and upon that I write, shewing some uses of those Circles amongst themselves, and conjoyned with others, in the resolution of such Questions which are ordinarily practised in Astronomy, Horolographie, in plaine Triangles applyed to Dimensions, Navigation, Fortification, &c. as a preparative ground for a more ample worke, and as a declaration of the admirable, and excellent use of this Ring in expedition, and facility. But before I come to Construction, I have thought it convenient by way introduction, to examine the truth of the graduation of those Circles which may be from the ensuing Tables and directions.

## Of the Examination of the Graduation of the Circles of the Ring, which may serve as an inducement and furtherance to the Learner, to fit and acquaint him how with promptnesse to conceive of opposite numbers in the answering of Questions following.

FIrst, to examine the Circle of Numbers, bring any number in the moveable to halfe of that number in the fixed: so any number or part in the fixed shall give his double in the moveable, and so may you trie of the thirds, fourths, &c. of numbers, vel contra.

2. Bring 2. in the moveable unto 3. in the fixed, so against 3. in the moveable is 4. and 5. tenths in the fixed, against 4. and 5. tenths in the moveable is 6. and 75. 100. in the fixed, and so may you goe on in trying the divisions of the Circle of Numbers in continuall proportion to other numbers, according to the Table A.

3. The Instrument not removed from the rectification of 2. unto 3. right against 3. in the moveable is 4. and 5. tenths in the fixed, and against 4. in the moveable is 6. in the fixed, but against 4. in the fixed is 2. and 66. 100ths in the moveable; against 5. in the moveable is 7. & 5. tenths in the fixed, but against 5. in the fixed is 3. and 33. 100ths in the moveable, and so may you proceede in examining farther, according to the Table B. in which M. at that head of the Table signifieth moveable, and F. at the head of the Table signifieth fixed, and so against the numbers under M. or F. is one anothers answer.

4. Bring 3. in the moveable unto 2. in the fixed, so right against the sine of 90. in the moveable is 41. gr. 44. m. in the fixed; [...]gainst 60. gr. is 41. gr. 02. m. against 75. gr. in the moveable is 40. gr. 05. m. in the fixed, and so you may examine further, as in the Table C.

5. To examine the sines amongst them selves in continuall proportion, as 6. gr. to 7. gr. bring 6. gr. in the moveable to 7. gr. in the fixed, so right against 7. gr. in the moveable is 8. gr. 10. m. in the fixed, and right against this 8. gr. 10. m. in the moveable is 9. gr. 32. m. in the fixed, and so may you goe on in examining other sines on the Instrument in continuall proportion, according to the Table D.

6. The Instrument being at this stay against 10 gr. in the moveable as 11. gr. 41. in the fixed, but against 10. gr. in the fixed is 8. gr. 34. m. in the moveable; against 15. gr. in the moveable is 17. gr. 34. m. in the fixed, but against 15. gr. in the fixed is 12. gr. 50. m. in the moveable, and so may you examine other sines, according to the Table E.

7. In the examination of the graduation of the Tangents: the Instrument not removed right against the Tangent of 6. gr. in the moveable is the Tangent of 7. gr. & the sine of 7. gr. 3. m. in the fixed; against the Tangent of 7. gr. in the moveable is the Tangent of 8. gr. 9 m. & the sine of 8. gr. 14. m. in the fixed, against the Tangent of 8. gr. in the moveable is the Tangent of 9. gr. 18. m. & the sine of 9. gr. 26. m. in the fixed, & so may you examine further according to the Table G.

A | |

M | F |

2 | 3 |

3 | 4. 5 |

4. 5 | 6. 75 |

6. 75 | 10. 10 |

10. 10 | 15. 2 |

15. 2 | 22. 8 |

22. 8 | 34. 2 |

34. 2 | 51. 2 |

51. 2 | 76. 9 |

76. 9 | 115. 4 |

C | |

90 | 41. 49 |

80 | 41. 02 |

75 | 40. 05 |

70 | 38. 47 |

65 | 37. 10 |

60 | 35. 16 |

55 | 33. 06 |

50 | 30. 42 |

45 | 28. 07 |

40 | 25. 22 |

35 | 22 29 |

30 | 19. 28 |

25 | 16. 22 |

20 | 13. 10 |

15 | 9. 56 |

10 | 6. 39 |

B | |||

M | F | ||

2 | F | 3 | M |

3 | 4. 5 | 3 | 2. 0 |

4 | 6. 0 | 4 | 2. 66 |

5 | 7. 5 | 5 | 3. 33 |

6 | 9. 0 | 6 | 4. 00 |

7 | 10. 5 | 7 | 4. 66 |

8 | 12. 0 | 8 | 5. 33 |

9 | 13. 5 | 9 | 6. 00 |

10 | 15. 0 | 10 | 6. 66 |

15 | 22. 5 | 15 | 10. 00 |

20 | 30. 0 | 20 | 13. 33 |

25 | 37. 5 | 25 | 16. 66 |

35 | 52. 5 | 35 | 23. 33 |

45 | 67. 5 | 45 | 30. 00 |

55 | 82. 5 | 55 | 36. 66 |

65 | 97. 5 | 65 | 43. 33 |

75 | 102. 5 | 75 | 50. 00 |

85 | 127. 5 | 85 | 56. 66 |

95 | 142. 5 | 95 | 63. 33 |

100 | 150. 0 | 10 | 66. 66 |

D | |

M | F |

6. 0 | 7. 0 |

7. 0 | 8. 10 |

8. 10 | 9. 32 |

9. 32 | 11. 08 |

11. 08 | 13. 00 |

13. 01 | 15. 13 |

15. 13 | 17. 50 |

17. 50 | 20. 55 |

20. 55 | 24. 35 |

24. 35 | 29. 20 |

29. 01 | 34. 27 |

34. 27 | 41. 15 |

41. 15 | 50. 15 |

50. 15 | 63. 41 |

E | |||

M | F | ||

6 | F | 7 | M |

10 | 11. 41 | 10 | 8. 34 |

15 | 17. 34 | 15 | 12. 50 |

20 | 23. 03 | 20 | 17. 04 |

25 | 29. 31 | 25 | 21. 15 |

30 | 35. 39 | 30 | 25. 24 |

35 | 41. 58 | 35 | 29. 28 |

40 | 48. 33 | 40 | 33. 27 |

45 | 55. 32 | 45 | 37. 20 |

50 | 63. 16 | 50 | 41. 04 |

55 | 72. 51 | 55 | 44. 38 |

60 | 47. 58 | ||

65 | 51. 00 | ||

70 | 53. 42 | ||

75 | 55. 56 | ||

80 | 57. 38 | ||

85 | 58. 42 | ||

90 | 59. 04 |

G | |||||||||||

M | F | F | M | F | F | M | F | F | M | F | F |

T | S | T | S | T | S | T | S | ||||

6 | 7. 0 | 7. 3 | 16 | 18. 29 | 19. 32 | 26 | 29. 38 | 34. 39 | 36 | 40. 16 | 57. 54 |

7 | 8. 9 | 8. 14 | 17 | 19. 37 | 20. 52 | 27 | 30. 43 | 36. 27 | 37 | 41. 18 | 61. 28 |

8 | 9. 18 | 9. 26 | 18 | 20. 45 | 22. 16 | 28 | 31. 48 | 38. 19 | 38 | 42. 20 | 65. 23 |

9 | 10. 28 | 10. 38 | 19 | 21. 52 | 23. 40 | 29 | 32. 52 | 40. 15 | 39 | 43. 22 | 70. 46 |

10 | 11. 37 | 11. 51 | 20 | 23. 00 | 25. 06 | 30 | 33. 57 | 42. 18 | 40 | 44. 22 | 78. 03 |

11 | 12. 46 | 13. 06 | 21 | 24. 07 | 26. 35 | 31 | 35. 00 | 44. 28 | |||

12 | 13. 55 | 14. 21 | 22 | 25. 14 | 28. 06 | 32 | 36. 04 | 46. 46 | |||

13 | 15. 04 | 15. 37 | 23 | 26. 20 | 29. 40 | 33 | 37. 09 | 49. 15 | |||

14 | 16. 12 | 16. 54 | 24 | 27. 26 | 31. 16 | 34 | 38. 11 | 51. 51 | |||

15 | 17. 21 | 18. 12 | 25 | 28. 52 | 32. 56 | 35 | 39. 14 | 54. 33 |

### A Type of the Ringe and Scheme of this Logarithmall projection, the use followeth.

These Instruments are made in siluer or Brasse by Iohn Allen neare the Sauoy in the strand

### In Astronomie.

#### Pro. 1. The sunnes place or distance from the Aequinoctiall points knowne, to find his Declination.

Constructio. BRing the sine of 90. in the moveable unto the sine of the Tropicall point, viz. 23. gr. and a halfe in the fixed, so right against the sine of the degree of the Suns neerest distance from ♈ or ♎ in the moveable is the sine of the Suns declination of that degree in the fixed.

Declaratio. So if the Suns place where in the beginning of ♊. ♌. ♐. or ♒. which is 60. gr. of distance from the Aequinoctiall points; right against this 60. gr. in the moveable, is the Declination in the fixed, viz 20. gr. 12. m. if the d [...] stance were 15 gr. 12. m. the Declination would be 6. gr. if 10. gr. of distanceth [...] in the moveable gives 3. gr. 58. m. in the fixed, if the Sunne have 3. gr. of distance this in the moveable gives 1. gr. 12. m. in the fixed.

Otherwise you may turne the 3. gr. or such which are lesse) into minutes by allowing 60. minuts to a degree which 3. gr. makes 180 m. minuts, this sought out in the moveable amongst the Numbers gives 72. minuts in the fixed, which is 1. gr. 12. m. of Declination as before: But if you make a degree to containe 100. minuts or parts, then the 3. gr. will be 300. minutes or parts, so right against this in the moveable amongst the Numbers is 1 [...]0. in the fixed, which is 1. gr. & 20. hunderth of a gr. of Declination. The Instrument being not removed, you may have the Declination for any other parte of the Ecclipticke, as for 1. minut for right against 60. seconds in the moveable (which is answerable to 1. minut) is 24 seconds in the fixed, the declination belonging to 1. minute, &c.

#### Pro. 2. To finde the Sunnes amplitude, or distance of rising or setting from the East or West knowing the Sunnes place and Latitude.

Constructio. Bring the sine of the Complement of the Latitude in the moveable unto the sine of 23. gr. 30 m. in the fixed, so against the sine of the Sunnes distance from ♈ or ♎ in the moveable is the sine of the Suns Amplitude

in the fixed.

Declaratio. So the Latitude being 51. gr. 30. m. the Complement is 38. gr. 30. m. Bring this to the sine of 23. gr. 30. m. in the fixed: now if the Sunne have 90. gr. of Longitude: right gainst this 90. in the moveable is 39 gr. 50. m. in the fixed, the greatest Amplitude of the Sun in that Latitude.

If the Longitude were 70. 60. 50. 40. 30. 20. 10. or 5. right against any of these numbers in the moveable (or any other) is the Suns Amplitude in the fixed, viz. against the Longitude of | 70 | is | 37. 00 |

60 | 33. 42 | ||

50 | 29. 23 | ||

40 | 24. 19 | ||

30 | 18. 38 | ||

20 | 12. 39 | ||

10 | 6. 23 | ||

5 | 3. 12 |

As for any Longitude which is neere the Equinoctiall point, the Amplitude of it may be had on the Numbers, as in the former Example.

### In Astronomie. But if the Declination of the Sunne, or a Starre be knowne; the Amplitude may be found thus.

Constructio. BRing the Sine of the Complement of the Latitude in the moveable unto the sine of 90. in the fixed, so right against the sine of the Sunne or Starres declination in the moveable, is the sine of the Sunne or Starres Amplitude in the fixed.

Declaratio. So if th [...] Latitude were 51. gr. 30. m. the Complement

is 38. gr. 30. m. bring this in the moveable unto the sine of 90. in the fixed, and if the Declination of the Sunne or a Starre were 20. gr. right against this 20. in the moveable is the Amplitude in the fixed, viz. 33. gr. 20. m.

The Instrument not removed you may for any Declination have the Amplitude of it: for right against the Declination in the moveable is the Amplitude in the fixed, &c. and there may you see what Declination such Starres have, which never rise, or set in that Latitude.

This Proposition may be inverted and applied to practice in Navigation to finde the Latitude, by knowing the Sunnes place and Amplitude, for if you bring the degree of the Sunnes Amplitude amongst the Sines in the moveable, unto the degree of the Sunnes place in the fixed; right against 23. gr. 30. m. in the moveable, is the degree of the Complement of the Latitude in the fixed.

#### Pro. 3. In any Latitude to finde what higth the Sunne most have to be due East or West knowing the Sunnes place.

Constructio. Bring the sine of the Latitude in the moveable unto the sine of 23. gr. 30. m. in the fixed; so right against the sine of the Sunnes distance from ♈ or ♎ in the moveable, is the Sine of the Sunnes he [...]ght in the fixed:

when he is due East or West.

Declaratio. So if the Latitude were 51. gr. 30. m. bring the Sine 51. gr. 30. m. in the moveable, unto the Sine of 23. gr. 30. m. in the fixed: and if the Sunnes place were from ♈ or ♎ 90. gr. 80. 70. 60. 50. 40. 30. 20. or 10. gr.

G. M. | ||||

Right against this | 90 | In the moveable is | 30. 38 | in the fixed, the Sunnes height answerable to his place. |

80 | 30. 07 | |||

70 | 28. 37 | |||

60 | 26. 11 | |||

50 | 23. 00 | |||

40 | 19. 08 | |||

30 | 14. 46 | |||

20 | 10. 02 | |||

10 | 5. 04 |

### In Astronomy.

HEre note that if the Zenith be betweene the Tropickes, that then the Sun sometimes will not be East or West, & what degrees those are you may easily try, for any Latitude, by moving the moveable softly along, that as the sine of any Latitude in the moveable passeth by the sine of the Tropicall point in the fixed, so any degree in the moveable that passeth by the sine of 90. in the fixed, doth shew that degree to be the greatest degree of Longitude in the Eclypticke from ♈ or ♎ that the Sun will be due East or West in that Latitude, and if the Instrument stay at that Elevation, any degree of Altitude in the fixed doth shew his degree of Longitude in the moveable, or any degree of Longitude in the moveable will shew at what Altitude the Sunne is due East or West in the fixed, for the one is opposite to the other. So if the Latitude were 23. gr 30. m. every degree of Longitude in the moveable would give the same degree of Altitude in the fixed to bee due East or West; so if the Sunne be in Longitude 20. gr from ♈ or ♎ then the Sunnes A [...]titude would be also 20. degrees when it comes due East or West: if the Suns place be 50. from ♈ or ♎ then the Altitude would be 50. to make due East, &c. But if the Latitude were lesse than 23. degrees, 30. m. as admit 15. degrees bring the sine of this 15. in the moveable unto the sine of 23. degrees, 30. in the fixed: so right against 90. degrees in the fixed is 40. degrees, 28. m. in the moveable: so when the Sun commeth to be in Longitude, from ♈ or ♎ above 40 degrees 28. m. that is beyond the 10. degrees, 28. m. of ♉ the Sun will not be East or West for more than 100. daies, untill hee come to the 59. degree, 32. m. of ☊, and then every day after the Sunne will crosse the Verticall Circle of East or West, untill the S [...]nne passe the other Equinoctiall point of ♎.

#### If the declination of the Sunne or the Starre bee knowne, you may finde the Altitude thereof at the point of East in any Latitude by this Rule.

Bring the sine of the Latitude in the moveable unto the sine of 90. in the fixed: so right against the sine of any Declination in the moveable is the sine of the Sunnes Altitude in the fixed.

Hence you may conceive, that if the Zenith bee betweene the Tropicks, then the Declination that is equall to the Latitude, shewes the greatest Altitude to bee East: and any greater Declination shewes you that the Sunne will not be East: but if the Declination bee lesse than the Latitude, then right against it in the moveable is the Sunnes Altitude in the fixed.

### In Astronomie.

#### Pro. 4. Knowing the height of the Sunne at the point of East or West, in any Latitude to finde the houre of the day.

Construct. BRing the Tangent of 45. in the moveable unto the Sine of the complement of the Latitude in the fixed; so right against the Tangent of the Suns Altitude in the moveable, is the Tangent of the houre from six

Declaratio. Let the Latitude be 51. gr. 30. m. and the Suns Altitud [...] bring the Tangent of 45. gr. in the moveable unto the Sine of 38. gr. [...] ed, so right against the Tangent of 14. gr. 45. m. in the moveable is 9. gr. 18. m. in the fixed, which reduced into Time, by allowing to every degree 4. minutes makes 37. minutes of time: so the Sunne was due East that day 37. minutes after 6. in the morning, or due West 37. minutes before 6. in the afternoone.

The Instrument not removed, you have the time of the Sunnes being East or West for any other Altitude: for right against the Suns Altitude in the moveable is the degree of time that the Sunne comes due East or West in the fixed.

Or if the declination of the Sunne bee knowne. Bring the Tangent of 45. in the moveable unto the Tangent of the Latitude in the fixed (if it bee greater than 45. gr.) so right against the Tangent of any Declination in the moveable

is the degree of time amongst the Sines in the fixed, that the Sunne or Starre will bee due East or West according to that declination; but if the Latitude be lesse than 45. gr. then bring the Tangent of the Latitude in the moveable unto the Tangent of 45. gr. in the fixed, so right against the Tangent of the Declinat [...]on in the moveable is the degree of time in the fixed amongst the Sines. Thus if the Latitude were 70. bring the Tangent of 45 unto the Tangent of 70. in the fixed, so right against the Tangent of the Suns Declinat [...]on in the moveable admit 23. gr. 30. m. is the sine of 9. gr. 6. m. the Instrument not removed, you have the time of the Sunnes comming East or West, for any other Declination, for right against the Declination in the moveable are the degrees of time in [...]he fixed.

But if you move the moveable softly along, as the Tangent of 45. in the moveable passeth by the Tangent of any Latitude in the fixed, so the degrees of Declination in the moveable passeth by their degrees of time in the fixed, amongst the Sines at what houre any such degrees are due East or West, untill the Tangent of 45. in the moveable bee opposite unto 45. gr. in the fixed: for then as the Tangent of any Latitude lesse than 45. in the moveable passeth by the Tangent of 45. in the fixed, so any degree of Declination in the moveable will shew his degree of time amongst the Sines in the fixed, that the Sunne is due East or West.

This Proposition of finding the houre of the Sunnes being East or West, may serve to great use both on Sea and Land, in rectifying of Glasses, Watches, or such like, to keepe and Regulate the account of Time.

### In Astronomie.

#### Pro. 5. To finde the time of the Sunne Rising or Setting in any Latitude, the Sunnes declination knowne.

Constructio. BRing the Tangent of 45. gr. in the moveable unto the Tangent of the Latitude in the fixed (if it be under 45. but if the Latitude be [...] above 45. then bring the Tangent of the Complement of Latitude in the moveable unto the sine of 90. in the fixed.) So against the degree of Declination amongst the Tangents in the moveable, is the degree of the Ascentionall difference among the Sines on the fixed.

Declaratio. So if the Latitude were 20. gr. and the Declination were 10. gr. Bring the Tangent of 45. gr. in the moveable unto the Tangent of 20. gr. in the fixed, so against the Tangent of 10. gr. in the moveable is 3. gr. 41. m. amongst the Sines in the fixed, which being reduced into time, makes neere 15. m. and this added unto 6. (if the Sun have South declination, gives the Sunne Rising to be a quarter past 6. and that 15. minutes taken from 6.

gives the Sunnes setting: But if the Sunne have North Declination; in stead of Addition, use Substraction, &c. The Instrument not removed, you may at one instant have the difference of Ascention for any Declination, for right against the Declination in the moveable is the difference of Ascention in the fixed. Or contrarily, the difference of Ascention knowne, you have the Declination answerable unto it: In like manner may you worke for any Starre. Or you may speedily compare the longest day of any two, three, or more places by knowing their Latitudes, for having rectified the Instrument according to the former directions, right against the Tropicall point in the moveable, viz. the Tangent of 23. gr. 30. m. is the Sine of the greatest ascentionall difference in the fixed for that Latitude; by which according to the former directions you may have the Sunne setting, this doubled as before gives the lengest or shortest day for that place. Hence for Practice, if you move the moveable softly along, as the Tangent of 45. gr. in the moveable, passeth by the Tangent of any degree of Latitude in the fixed, so the Tropicall point in the moveable, will passe by the Sine of the greatest difference of Ascention in that Latitude.

G. | G. M. | |||

So as 45. passeth by the Latitude of | 25 | The Tropicall point, viz. 23. gr. 30. m. will passe by | 11. 41 | The greatest difference of Ascension for those Latitudes. |

30 | 14. 32 | |||

35 | 17. 43 | |||

40 | 21. 24 | |||

45 | 25. 27 | |||

50 | 31. 12 | |||

55 | 38. 23 | |||

60 | 48. 51 | |||

65 | 68. 49 |

### In Dyalling.

#### Pro. 1. To finde the distance of the houres, in horizontall plaines for an oblique Sphaere.

Constructio. BRing the Tangent of 45 gr. in the moveable unto the sine of the latitude in the fixed, so right against the degrees of the houres, in a right Sphaere amongst the Tangent in the moveable under 45. gr. are the degrees of the distances of the houres, from 12. amongst the Tangents on the fixed for an oblique Sphaere, but if the degrees of the houres bee more than 45. then right against the Tangent of those degrees in the fixed, are the degrees of the houre distances in the moveable.

☞ Note, that the houres in a right Sphaere are equall, the one unto another, viz. 15. gr. for one houre, 30. gr. for two houres, 45. gr. for three houres, &c. hence here after they are called equall houres, &c.

Declaratio. So if the Latitude were 51. gr. 30. m bring the Tangent of 45 in the moveable unto the sine of 51. gr. 30. m. in the fixed, so right against 15. gr. in the moveable amongst the Tangents is 11. gr. 50. m. in the fixed amongst the Tangent, and so much is the distance of the houre of 1. or 11. from 12. in the Latitude of 51. gr. 30. m. The Instrument not removed you may have any other hou [...] distance, halfe houres, quarters, &c.

For right against those equal houres, viz. | 15 | Are the unequal houre distances, viz. | 11. gr. 50. m. | For the houre of | 1 | or | 1 [...] |

30 | 24. 20 | 2 | 10 | ||||

45 | 38. 03 | 3 | 9 | ||||

60 | 53. 35 | 4 | 8 | ||||

75 | 71. 05 | 5 | 7 | ||||

90 | 90. 00 | 6 | 6 |

By inverting this, you may trie for what Latitude ordinary pocket dialls are made, knowing the distance betweene the houre of 11. and 12. for if you bring the Tangent of 15. gr. in the moveable unto the Tangents of the distance in the fixed, right against the Tangent of 45. gr. in the moveable is the sine of the Latitude in the fixed.

#### Pro. 2. To finde the distance of the houres for a direct south Dyall in an oblique Sphaere.

Constructio. This differeth [...] little from the former, only, here bring the Tangent of 45. gr. in the moveable unto the fine Complement of the Latitude in the fixed, so right against the Tangent of the degrees of the equall houres, are the Tangents of the degrees of the houre distances in the oblique sphaere.

Declaratio. So if the Latitude were as before 51. gr. 30. m. bring the Tangent of 45. gr. in the moveable, unto the sine of 38. gr. 30. m. in the fixed; so right against the Tangent.

Of those degrees, viz. | 15 | Are the degrees of the houre distances, viz. | 9. gr. 28. m. |

30 | 19. 45 | ||

45 | 31. 55 | ||

60 | 47. 10 | ||

75 | 66. 42 | ||

90 | 90. 00 |

Now if you move the moveable softly along as the Tangent of 45. gr. passeth by the sine Complement of any Latitude, so the degrees of the equall houres, will shew the degrees of the unequal houre distances, in comparison of one Latitude to another. This may be otherwise represented in this fundamentall Diagram, if EZ, and EH be divided out of the Table of naturall Tangents, so that each Radius represent 90. gr. and so to move upon E, then may you place H to P, and as you move H from P to increase in Latitude, so the meridians passing by E H will shew the houre distances in a Horizontall plaine, and the houre distances in E Z for a verticall plaine, and this kinde of projection and motion may serve to other excellent purposes, &c.

### In Dyaling.

#### Pro. 3. Knowing the Declination of a verticall plaine, and the latitude of the place, to finde what Angle the Axis makes with the plaine, commonly called the height of the stile.

Constructio. BRing the Sine of 90. in the moveable unto the sine Complement of the latitude in the fixed, so right against the sine Complement of the Declination, in the moveable, is the degree of the stiles height amongst the sines in the fixed.

The Instrument not removed, you may at one Instant see

the stiles Altitude for all Declinations in that Latitude: for right against the Complement of the Declination in the moveable, is the stiles height in the fixed.

Declaratio. So if the Latitude were 51. gr. 30. m. the complement of it is 38. gr. 30. m. which seeke amongst the sines in the fixed, and bring 90. gr. to it, now if the Declination of the plaine were 10. 20. 30 40. 50. 60. 70. 80. &c.

The Complement of those are. | 80 | So right against these in the moveable are | 37. 49 | in the fixed the stiles he [...]ght answerable to these Declinations. |

70 | 35. 48 | |||

60 | 32. 37 | |||

50 | 28. 22 | |||

40 | 23. 35 | |||

30 | 18. 08 | |||

20 | 12. 18 | |||

10 | 6. 12 |

#### Pro. 4. Knowing the Latitude of the place, and the Declination of a verticall plaine to finde the number of degrees betw [...]ene the Meridian of the place, and the Meridian of the plaine, which may be called the difference of Meridians.

Constructio. Bring the Tangent of 45. gr. in the moveable unto the sine of the Latitude in the fixed (which admit 51. gr. 30. m.) so against this 45. [...] is the Tangent of 38. gr. 3. m. in the fixed: So if the Declination of the Plaine be 38. gr. 3. m. then the difference of meridians is 45. gr. But if the Declination be lesse then this 38. gr. 3. m. right against the Tangent of any degree for such a Declination in the fixed is the degree of the difference of meridians in the moveable.

So if the Declination were | 35 | Right against these in the fixed are | 41. 49 | In the moveable, the difference of meridians answerable to those Declinations. |

30 | 36. 25 | |||

25 | 30. 44 | |||

20 | 24. 56 | |||

15 | 18. 54 | |||

10 | 12. 42 | |||

5 | 6. 23 |

If the Declination be above 38. gr. 3. m. you may move the Tangent of 45. softly alonge by the Tangentiall degrees of Declination in the fixed, untill 45. gr. in the moveable be opposite to 45. gr. in the fixed, and as it passeth by any Declination, the sine of the Latitude in the fixed, will give the difference of Meridians amongst the Tangents in the moveable. Lastly, if the Declination be above 45. gr. then the Declination most be accounted amongst [...]he degrees of the moveable, and if you move the moveable softly along: As the degree of any Declination, passeth by the Tangent of 45 gr. in the fixed: so the Tangent of the difference of Meridians in the moveable passeth by the sine of the Latitude in the fixed, &c.

#### Pro. 5. To finde the houre distances in Dyalling in a Declining plaine by knowing the former, viz. the stiles height, and the difference of Meridians.

Declaratio. LET the Declination of a Plaine be S. W. 50. gr. according to the former Instruction, the stiles height would be 23. gr. 35. m. and the difference of Meridians 56 gr. 43. m. Now before the houre distances can be knowne which are alwayes unequall, there may be made a Table of equall houres thus.

First, place downe 56. gr. 43. m. the difference of Meridians against 12. as in the Table, then by adding 15. gr. unto this difference of Meridians, viz. to this 56. gr. 43. m. it makes 71. gr. 43. m. for the distance betweene the houre of 11. and the Meridian of the plane: unto this 71. gr. 43. m. adde another 15. gr. makes 86. gr. 43. m. for the houre distance betweene 10. and the Meridian of the Plaine: But for the houre of 1. 2. 3. that 15. gr. must be taken from the said 56. gr. 43. m. so the houre distance of 1. is 41. gr. 43. m. the houre distance of 2. is 26. gr. 43. m. the

10 | 86. 43 | 81. 50 |

11 | 71. 43 | 50. 20 |

12 | 56 43 | 31. 20 |

1 | 41. 43 | 19. 35 |

2 | 26. 43 | 11. 22 |

3 | 11. 43 | 4. 45 |

4 | 3. 17 | 1. 15 |

5 | 18. 17 | 7. 32 |

6 | 33. 17 | 14. 35 |

7 | 48. 17 | 24. 10 |

8 | 63. 17 | 38. 25 |

houre distance of 3. is 11. gr. 43 m. now because 15. gr. cannot be taken out of 11. gr. 43. m. take this out of that, so the houre distance of 4. will be 3. gr. 17. m. now unto this adde 15. gr. makes 18 gr. 17. m. for the houre of 5. then adde 15. gr. to that, and so prosecute the Table which may be called the Table of equall houres from the Meridian of the plaine.

Now to finde the true houre distances, bring the Tangent of 45. gr. in the moveable unto the sine of the stiles height in the fixed, viz. 23. gr. 35. so right against the Tangent of any equall houre in the moveable under 45. gr. is the Tangent of the true houre distance in the fixed A. But if the equall houre distance be above 68. gr. 12. m. (which is right against 45. gr. in the moveable) then right against the equall houre distance in the fixed, is the true houre distance in the moveable B. Lastly, if the equall houre distances be betweene 45. gr. & 68. gr. 12. m. then move the moveable softly along, and as the Tangent of any equall houre in the moveable passeth by the sine of the stiles [...]e [...]ght in the fixed, so right against the Tangent of 45. gr. in the moveable is the Tangent of the true houre distance in fixed.

So the equall houre distances being | A | 3. 17 | The true houre distances would be | 1. 15 |

18. 17 | 7. 32 | |||

33. 17 | 14. 35 | |||

B | 86. 43 | 81. 50 | ||

71. 43 | 50. 20 | |||

C | 48. 17 | 24. 10 | ||

56. 43 | 31. 20 | |||

63. 17 | 38. 25 |

Having gotten the true houre distances from the Meridian of the Plaine, they may be placed against the houres as in the Table, and protracted thus, draw the houre of 12. C. M. and on C. describe a semi-Circle: now seeing in the Table that the Meridian of the Plaine is from the houre of 12. in its true distance 31. gr. 20. m. protract it in the Circular Arke from D. to S. (because of West declination) otherwise contrary, and draw C. S for the Substiler; from this S. protract all the houres distances, as S. R. 4. gr. 45. m. for the houre of 3. S. Q. 11. gr. 22. m. for the houre of 2. &c. then may we draw the houre lines CN. CO. CP. CQ. CR. CT. CV. CW. CX. CΩ. and upon the said CS. place the stile A. B. C. perpendicular to the plaine, so it shall bee fitted for the casting of shadowes upon the said houre lines,

### Of Plaine Triangles. Praecognita.

Theoreme. 1. IF a right line fall upon a right line, or if the side of an Angle be augmented to make another Angle, those two Angles put together are equall to two right angles, that is twice 90. or 180. gr. by the thirtenth of the first of Eue. therefore knowing one of those Angles, the other is also knowne▪ So in the Triangle A B C augmenting the angular side B A to R. by the line R B and C A there is made two Angles, viz. C A B. and C A R. which together make 180. gr. and seeing the Angle C A B. is assumed to be 30. gr. necessarily by the said thirtenth Proposition, the other Angle C A R shall be 150. gr.

Theoreme 2. The three Angles of any plaine Triangle either Right, or Oblique by the 32. of the first of Euc. are equall to two right Angles, therefore one acute Angle knowne in a right angled Triangle, and two Angles knowne in an Oblique Angled Triangle, the third Angle is likewise knowne: hence in the right angled Triangle A B C knowing the Angle at A. 30. gr. the Angle at C most be 60. gr. And so in the Oblique Triangle A H I. the Angle at A and I put together being 135. the Angle at H must be 45. gr. &c.

Propositio. I. | ||||

In any | Right Angled | Triangle, knowing | One Acute Angle, and a side opposite to either of the Angles, or the Hypotenusae, and one side | to find the rest. |

Oblique Angled | Two Angles, and a side, or two sides and one Angle opposite to either of those sides |

#### Axiome 1. The Axiome for the resolution of this proposition is thus.

The sides in all such Triangles, beare proportion the one to the other, as the sines of their opposite Angles doe. Or the sines of the Angles are directly proportionall to their opposite sides, by the seven & twentieth of the first Booke of Regiomontanus, the thirtenth Chap. of the 1. of Copernicus, and by the second Axiome of the third booke of Pitiscus.

That is, the side A B (in the first Triangle) is in proportion to B C, as the sine of the Angle at C is to the sine of the Angle at A, or as A B. to A C. so the sine of the Angle at C. to the sine of the Angle at B. which are the opposite Angles of those sides: Againe in the oblique angled Triangle A D E, as the side A E is to E D, so is the sine of the Angle at D. to the sine of the Angle at A, or as the sine of the Angle at H, (In the fourth Triangle, is to the sine of the Angle at A, so is the side A I, to the si [...] I H, &c.

☞ And here note generally, that in the resolution of such Triangles there is three things given (as afore said) by which a fourth is found, according to the method of the Golden Rule. But the disposing of the said three termes, is primarily, and principaly to be considered.

### How to dispose the three Termes knowne in any of the former Triangles fit for operation.

FIrst, if a side be required in a right, or oblique Triangle, & a side be knowne, and the Angles opposite to those sides bee also knowne: the Angle opposite to the side knowne shall have the first place of the Golden Rule, and the Angle opposite to the side required may have the second place of the Golden Rule, and the side knowne the third place.

Secondly, if an Angle be required, and an Angle knowne with the sides opposite to those Angles, then the side opposite to the Angle knowne shall have the first place of the Golden Rule, the side opposite to the Angle required may have the second place of the Golden Rule, and the Angle knowne shall have the third place: the disposing of the Termes thus considered, the Angles have reference to the Circle of Sines upon the Ring, and the sides to the Circle of Numbers.

Example. Let the side BC. in the Rect-Angle Tri-Angle be required, knowing the side AB. 25. and the Angles at A and C. 30. gr. and 60. gr. therefore by the former. As the Sine of the Angle at C. 60. gr. to the Sine of the Angle at A. 30. gr. so the side AB. 25. to the side BC. 14. and 43. 100. and so of the rest.

#### The construction by the Ring is thus.

Bring 60. gr. amongst the Sines in the moveable to the Sine Complement of it in the fixed, viz. 30. gr. so right against AB. 25. in the Circle of Numbers in the moveable is 14. and 43. 100. in the fixed. But if the side AB. had beene any other number, you might instantly have the said BC. according to the same proportion.

☞ Here note generally, that if the two first Termes be upon the Circle of Sines, the other two termes are then upon the Circle of Numbers, vel contra. *

☞ Note further, that in all Trigonometrie, the termes or parts given either Angles or sides, are noted with a small stroke of the Pen thus — and the Termes or parts required either in the Angles or in the sides, thus O. so the sides AB. AD. AE. [...]. with the Angles at A. are given the sides AC. CB. A [...]. and [...]. and the Angles at E. and G. are demanded; now the mutuall proportion of these parts one unto another, is according to the former Axiome; by which infinite propositions may bee resolved of admirable consequence, lying under the habit of some one of those whose excellent use in ordinary Practicall things, I will illustrate in severall kinds by sundry propositions; first, In Dementions; secondly, in Fortification; thirdly, in Navigation; fourthly, in Dyalling, as followeth.

### Vpon Plaine Triangles in Dementions. How to measure an Inaccessible height scituated upon a Hill, the Practice being not upon a Plaine.

THe example upon this I will take from mine owne observations which I made upon one of the goodliest Hills in this Kingdome, which belongs to Sir Richard Newport in Shropshire, called the Wreaken, not farre from Shrewsbury, which Hill I found to be neere 6. miles and a halfe in Circuit, and in the Perpendicular height 995. foot, as followeth.

Declaratio. Let the figure P. Q. S. N. represent the body of the Hill, and A B E C D a part of the side of it, now admit in the side of the Hill as at C, be a spring of water: and let a Well be sunke from the Top of the Hill at D to be levill with the spring C, viz. at R: here it may be demanded how deepe this Well may be sunke, viz. D R, how farre it is from the spring C, to the botume of the Well R, and how farre it is from the said spring C, to the Top of the Hill D.

Constructio. First, with conveniencie I made choise of a Station at A, and there rectified my Instrument upon his Stay, or Rest, from thence I caused to be measured to B 20. Chaines, and at B I caused another Stay or Rest to be erected of the same height that the former at A was: Then loking from A to B. by the Sights of the Instrument, I found the Angle of Ascent B A X. 4. gr. 44. m. and so I had the rectangled triangle A X B. in which A B was knowne 20. Chaines, or 1320 feet with the Angle at A 4. gr. 44. m. and therefore the Angle at B by the second Theorem Pagr. 64. is 85. gr. 16. m. and the Angle at X by the same is 90. gr. and this is opposite to the side knowne, viz. A B. 20.

Now according to the Rule of operation by the Ring Page. 65. if the Sine of 85. gr. 16. m. in the moveable be brought to the sine of 4. gr. 44. m. in the fixed, right against 20. in the Circle of Numbers in the moveable is 1. Chaine, and 65. 100. of a Chaine in the fixed, which according to Page the ninth is 108 Foote, and 9. tenth. this shewes that the station at B was so many Foote higher, than that at A. By the same Rule A X. would be found to be 19. Chaines, and 93. hundreths of a Chaine, or 1315. Foote, and 5. tenths, those two dementions I first sought out.

Secondly, before I removed from A. I observed the Angle C A H. which I found to be 12. gr. 8. m. and also the Angle D A H. which was 15. gr. 55. m. those Angles I noted downe, and then going from A to B. there I rectified my Instrument as I did formerly at A. and there I observed the Angle C B Y. which I found to be 16. gr. 45. m. and the Angle D B O. to be 22. gr. 12. m. thus for the observation.

But according to the first Theoreme Page. 64. if particularly those Angles be taken from 180. gr. leaves the angle G B C. 163. gr. 15. m. and the angle G B D. 157. gr. 48. m. unto these severall Angles, Adde the Angle B A X. 4. gr. 44. m. so have you the obtuse angle A B C. 167. gr. 59. m. And the obtuse Angle A B D: 162. gr. 32. m. And seeing formerly that the Angle C A H was 12. gr. 8. m. and the Angle D A H was 15. gr. 55. m. From either of those Angles I subtract the Angle B A X, viz. 4. gr. 44. m. leaves the Angle C A B. 7. gr. 24, and the Angle D A B 11. gr. 11. m. By which in the oblique Triangle A B D, and A B C are knowne in each of them two Angles, and so consequently by the a [...]oresaid second Theoreme, Pag. 64. The other Angles are also knowne (viz.) the Angle A C B, 4. gr. 37. m. and the Angle A D B 6 gr. 16. m.

NOw by the afore going first Axiome seing that in the Triangles A B D, all the Angles are knowne, and one side to wit A B. 20. the side A D may be also knowne.

Constructio. Bring the sine of the Angle A D B 6. gr. 16. m. in the moveable unto the sine of the Angle A B D. 1 [...]2. gr. 31. m. (that is to the Sine of the Complement of that Angle, viz. 17. gr. 29. m.) in the fixed, so right against 20. in the moveable amongst the Numbers is 54. Chaines and 93. 100. the. in the fixed, which in feete is 3625. and 6. tenths. and so farre it was from the eye at A to the top of the hill at D.

Thirdly, in the Rectangled Triangle A H D knowing as afore said the Angle D A H 15. gr. 55. m. by the second Theoreme Page 64. all the other Angles are knowne, viz. A D H 74. gr. 4. m. A H D 90. and seeing by the last worke that A D was found to be 54. Chaines, and 93 100ths. the other part [...] of the Triangle by the afore said Axiome will be likewise knowne. For the Instrument not removed the Sine of 90. is against the Sine of 15. gr. 55. m. and right against 54. and 93. 100ths. is 15. Chaines, and 07. 100ths. of a Chaine, which by Page the ninth makes 995. foote the higth of the hill. In like manner may you finde the other side of the Triangle viz. A H, that is from the eye under the top of the hill, viz. 52. Chaines, and 82 hunderths, or 3486. foote, and 4. tenths.

Fourthly, in the Oblique Triangle A B C. all the Angles are knowne (as afore) with the side A B, and therefore the side B C would be found to bee 32. Chaines, or 2113. foote.

Fifthly, in the rect angled Triangle B Y C the Angle C B Y was formerly known [...] to be 16. gr. 45. m. the Complement of which is the Angle B C Y. 73. gr. 15. m. and the side C B was formerly found to be 32. Chaines. therefore by afore said first Axiome of plaine Triangles, the other sides of the Triangle, viz. B Y. will be found to be 30. Chaines, and 65. hunderths. or. 2023. foote, and 3 tenths, and the side C Y. to be 9. Chaines, and 23 hunderths, or 608. foote, and 9. tenths.

Sixthly, seeing B X. is equall unto Y Q. unto the said C Y. 608. foote, and 9. tinths, add. B X. formerly found, viz. 108. foote, and 9 tinths makes Q C. 717. foote, and 8. tenths. but this Q C. is equall to H R. which taken from the higth of the whole hill D H, viz. 995. foote leaves D R. 277. foote, and 2. tenths, the depth of the well.

Seventhly, for as much as B Y was found to be 2023. foote, and 3. tenths. which is equall to X Q and A X being as before 1315. foote, 5. tenths, those two put to [...]ether makes A Q 3338. foote, and 8. tenths, which taken from A H, which was [...]ormerly found to bee 3486. foote, 4. tenths leaves Q H. 147. foote. 6. tenths, which is equall to C R. the distance betweene the spring C, and the botome of the well R.

Eighthly, and lastly knowing D R. 277. foote, and 2. tenths, and C R 147. foote, and 6. tenths C D is found to bee 313. and 8. tenths, according to the first Proposition, Page 19ths. the example being the same with the second Proposition: thus for the Instrumentall way: such as desir [...] to examine the worke by numbers may proceede by the Notes here under specified.

In Trianguli. | AXB | Data | BAX 4. gr. 44. m. | Quaeritur | A X. 1315. & 5. 10ths. |

A B. 1320. | B X. 108. and 9. 10ths. | ||||

ABD | A B. 1320. | A D. 3625. & 6. 10ths. | |||

B A D. 11. gr. 11. m. 43. s. | |||||

ABD. 162. gr. 31. m. 45. s. | |||||

AHD | D A H. 15 gr. 55. m 43. s. | D H. 995. | |||

A D. 3625. and 6. 10ths. | A H. 3486. & 4. 10ths. | ||||

ABC | C A B. 7. gr. 24. m. | B C. 2113. | |||

A B C. 167. gr. 59. | |||||

A B. 1320. | |||||

C B Y. 16. gr. 45. m. | B Y. 2023. & 3. 10ths. | ||||

B C. 2113. | C Y. 608. & 9. 10ths. | ||||

CRD | R D. 227. and 2. tenths. | C D. 383. & 8. 10th | |||

C R. 147. and 6. tenths. |

## Ʋpon his Ring.

## By a Friend.

### How to operate, in the finding of Proportionalls by my Logarythmall Projection of Circles inlarged, eyther by a mooveable and fixed Circle, or by a single Projection, with an Index at the Peripheria, or Center.

THe way of operation is drawne from the nature of Proportionall Logarythmes, that as they keepe equall differences, so in a lineary or Circular Instrumentall projection of these Logarythmes, Proportionall numbers shall alwayes have equall distances, which as a fundamentall ground may serve to the more learned both for a full demonstration, and direction in operation.

But to make things more obvious, & to remoove such scruples as may arise in working by this Projection; the numbring [...]f the Circles especially is to be considered, that is either by augm [...]ation or diminution, in continuation or discontinuation, and th [...] [...]ath relation to the line of Conjunction E. T. which sheweth the breaches of the Circle, or the uniting or continuing of the parts: which multiplicitie of Circles, must be conceived to be but the parts of one Circle (as before amply in the Epistle to the Reader was specified touching this projection) and so continued or discontinued, by ascending or descending on this or that side of the line of coniunction, as by the succession of the Graduations, or divisions in those Circles is most evident and conspicuous: this well premised:

Const [...]uctio. Bring the first number in the mooveable, to the second n [...]mber in the fixed, and marke the severall revolutions or Circles betweene them ascending or descending; for then the fourth Proportionall is had on the fixed, right against the third number in the mooveable by the same number of revolutions or Circles ascending or descending, as was betweene the first and second numbers.

So if the line of coniunction on the mooveable, be on the right or left side on the line of Coniunction on the fixed, and the first and second numbers be betweene them, and also the third, or all these three numbers be not betweene them, the fourth number or proportionall is had without any consideration, but onely by the same number of Circles, as was betweene the first and second numbers. But if the third number be on the otherside of the line of Coniunction, and that the Proportionalls [Page] did augment, or diminish, the same number of Circles or revolutions accounted ascending or descending from the third number, will likewise shew the fourth proportionall required.

### Or without considering the absolute revolutions, you may operate by the difference of Circles noted on the single Index thus.

Bring the first number in the mooveable unto the second number in the fixed, and marke the difference of Circles, eyther ascending or descending betweene them; then whensoever in operation the sayd first and second number, and also the third number are neyther of them betweene the lines of Coniunctions, or all of them are betweene them, then the fourth proportionall is had in the fixed right against the third number in the mooveable, by the same difference of Circles ascending or descending, as was betweene the first and second numbers.

But if the line of conjunction on the moveable, bee on the right side of the line of conjunction on the fixed, and the first number bee in a lower Circle then the second and

- bee not betweene the lines of Conjunctions, but the third,
- be betweene the lines of conjunctions & not the third

then the fourth proportionall is in one Circle

- more
- lesse

then the difference of Circles betweene the first and second numbers. But if the first number bee in a higher Circle then the second, then the difference more will bee lesse, and that which is lesse more.

Contrarily, if the line of Conjunction on the mooveable bee on the left side of the line of Conjunction on the fixed, the operation is the converse of the former.

The like may be observed in operation by the single Projection inlarged, with a double Index to move on the Peripheria, or Center of a Circle, if in stead of moving one number to another, you extend the feete of the Index, as followeth.

### An Example of the operation upon the Projection of the Circles of my Ring inlarged, according to the conclusion of my first Booke, in a Scheme or Instrument where the Circles of Numbers, Sines, and Tangents are decuplated, the diameter being but 18. Inches onely.

SVppose the Sunne being in the Tropicke, and his greatest Amplitude were required in the Latitude of 66. gr. 29. m. The common Rule to operate which, is thus delivered.

As the Sine complement of the Latitude, is unto the Sine of the Suns declination proposed, so is the Sine of 90. gr. unto the sine of the Suns Amplitude required; the Diagramme and Demonstration of which, see pag 57. and the Instrumentall operation may be [...]hus.

Constructio. Place one of the edges of the Index unto the sine of 23. gr. 31. m. (the Complement of the Latitude proposed) and extend the other edge unto the Sine of 23. gr. 30. m. (the Suns greatest declination) then moove the edge of the first Index unto the Sine of 90. gr. so the edge of the second Index shall give the fourth proportionall sought for.

But because the answer in this Proposition falls neare the sine of 90. gr. (where the degrees are small, and the graduations close) the answer is not so exactly discerned in the minuts as if it were larger. To supply which, or such like as may fall out in Practise, I have continued the Sines of the Proiection unto two severall revolutions, the one beginning at 77. gr. 45. m. 6. s. and ends at 90. gr. (being the last revolution of the decuplation of the former, or the hundred part of that Proiection) the other beginning at 86. gr. 6. m. 48. s. and ends at 90. gr. (being the last of a [...]ernary of decuplated revolutions, or the thousand part of tha [...] Proiection) and may bee thus used.

Lay the edge of the Ind [...]x upon the former 23. gr. 31. m. and marke what equall parts in the Circle of equall parts are intersected thereby, which will be 0099. Place also the edge of the Index upon 23. gr. 30. m. which will cut in the Circle of equall parts 0069. Then take ten times the distance of these two numbers in the Circle of equall parts betweene the feete of the Index by decuplation, that is, place one foote of the Index upon 99. and extend the other foote unto 69. so the edge of the first foote being placed on the sine of 90. gr. the edge of the second [Page] foote will point out accuratly 87. gr. 54. m. and such is the Amplitude in the Latitude of 66. gr. 29. m. the Sun being in the Tropicke aforesaid.

Againe, admit that in the Latitude of 51. gr. 32. m. the Suns place were required by knowledge of his Meridionall Altitude, which suppose was found to be 61. gr. 57. m ⅙. Now the common Axiome according to the Diagram of Pag. 56. grounded on Mathematicall doctrine to operate which is: As the sine of the Suns greatest declination, viz. 23. gr. 30. m. is unto the sine of 90. gr. so is the sine of the Suns Declination given viz. 23. gr. 29. m ⅙ unto the sine of the answer. Therefore Instrumentally open the Index unto the first two tearmes proposed, and place the edge of the first foot upon the third number, so the edge of the second foote shall give the fourth proportionall required.

But for as much also as the answer in this doth fall amongst the small divisions neare the sine of 90. gr. in the decuplated Proiection, we may supply that contraction according to the former, but truer by the largest augmented Circle of Sines, to wit, that which is the thousand part of its Proiection, thus; Lay one of the edges of the Index upon the sine of 23 gr. 30. m. and marke the number of equall parts which it cuts in the Circle of equall parts, viz. 70. lay also the edge of the Index upon 23. gr. 29. m. ⅙. and it cuts likewise in that Circle 64. then take a hundred times the distance of these two numbers in the Circle of equall parts betweene the feete of the Index, that is, place one of the edges of the Index upon 70. m the Circle of equall parts by Centuplation, and extend the other edge unto 64 (for the Decuplating, and Centuplating of the equall parts, doe naturally without trouble represent themselves in this projection) so the edge of the first Index being placed at the sine of 90. gr. the other edge will accuratly point out 89. gr. 5. m. the Suns place at the time observed, and so in like manner for other operations.

Nam quam in minoribus circulis aequalium partium distantiam proportionales habent: eandem in circulis adauctis proportionales retinebunt.

Thus I have here now produced to a publike veiw my Proiection of Logarythmes inlarged by way of decuplating, Centuplating, &c. of the Circles, as I promised at the Conclusion of the first publishing of this Invention, to the world, and have in some measure shewed the Accurate working of Trigonometry by it, neare the Sine of 90. gr. where difficulty seemes to be; for being thus inlarged the greatnesse of a degree betweene the sine of 89. gr. and the sine of 90 gr. is more then 4. Inches, and if I should have inlarged the Circle of Tangents according to the Sines, the capacity of a minute at 45. gr. would bee more than 8. Inches [Page] which inlargement amongst the degrees which falls neare the Sine of 90. gr. doth operate as true (according to the former Diameter but if 18. Inches) as if M^{r}. Gunters excellent Lines were extended or projected unto 4000. foote. And if I should frame a Ring as is specified in the conclusion of my first publication of this Invention, being of two yards diameter, and apply it to Astronomicall Calculations, no doubt I might shew a way (or others may easily) to compendiate many operations therein; and sufficiently cleare my intentions then delivered touching the Prostaphereses of the motions; though some one in contempt of my good indevours divulged after it came to the worlds view, it could not bee done, nor possibly a minute expressed at the sine of 90 gr. (as I have now produced it) thereby endeavoring to annihilate my labours and to spread an unsavory rumor, which might seeme to argue not onely his ignorance of my intentions, but also of the manner of extending and inlarging of that invention (though now given out that I had that invention from him:) But touching such applications to prove my assertion hereafter, as God shall give life, & ability of health: and let a further time bring them to maturitie, that my Iealous opposite may bee no more mistaken with a suspected untimely birth. I confesse these single Circles before, were something untimely, in regard of these which are of fuller growth, and yet may have further application, without wayting his time to perfect them.

HItherto the world hath beene abused as well as my selfe, with a false Rumor (raised by some rude ignorant tongue) that my Invention both of Ring and Quadrant was got, or borrowed, or stollen (even as they please to miscall it) from another Man, by the sight of a Letter, by some private Conference by — J know not what meanes (nor they neither) but onely by their malicious phancy: which how true, how just it is, that the world and my selfe may have our right, and they the shame, by this following just defence and future event, J hope thou mayest fully be enformed. I did not intend to take this course, but sought peace and my right by a private and friendly way; but failing of it my good intentions scorned and slighted; J desire all may now Iudge who is in fault; and let this ensuing discourse be my Plea.

# To the Reader

HOw undeservedly oft are the single and sinc [...]re endevours of some men by the malevolent disposition of envious detractors backbited, (w^{ch} sometimes rebounds backe aversly upon them) not onely by bare assertions, but also by injurious, and contumelious aspertions; In which kind I have not a little lately suffered: for having for a generall end (more then ayming at mine owne particular) published the making & use of my Horizontall Quadrant, with my new invention of the Projection of Logarythmes Circular by a former booke of the use of it, intituled with this, Grammelogia, or the Mathematicall Ring, since that time I have beene deepely glanced upon, and scandalized both about the former and latter, these detractors taking away from one, and giving to another, famousing some, and infaming others, which did not a little disturbe the quiet and Peace which formerly I injoyed, but did also disorder and slacke my intentions, in the publishing of the inlarging of the Invention of my Ring, as I promised to doe in the Conclusion of the aforesaid Booke of the use of that Ring, which would serve as a helpe for such as affect Mathematical practises for the working of Trigonometrie unto minuts, and to give accuratly Rootes and Proportionall numbers, unto 5. or 6. places, as is specified before in the description of the projection, and had ere now beene published with the excellent use thereof, (not out of any Mercinary respect nor interlased with untruths, delusions, and bumbast stuffe, by way of Illustration if not confusion) had not some envious calumniators stole away my intentions, in stealing from me my labours, by detracting from me, and assuming unto themselves the inverting of the Circles of my Ring upon a Plate or Plaine, accommodated with an Index to open and shut at the Center, when another imitating thereof, did so fit i [...], whose modesty was such that he would not derogate the invention of the projection from me, to himselfe, but ingenuously acknowledged it being so made up, and contrived, to be the same proiection with my Ring, But the manifestation of it, so first to the world, properly and soly belongeth unto him; which Circles so projected on a Plate with an Index, was not also [Page] unknown unto me, as by good testimony I can produce, before any of thes [...] things came to the worlds view, though publikely I writ not first upon them: the Learned in those Artes, and those that understand the projection, know it to bee one and the same, and are not deluded b [...] supposing a new thing, as many are, for it is but (as a learned ma [...] said) as to turne a Garment in and out; A motion must performe th [...] operation to give proportionalls with such expedition, and not otherwise, as in my Epistle to the Reader upon the use of my Ring I hav [...] delivered, howsoever the projection on the Plate or Plaine I had published ere now, not in so an unprofitable and obscure method as i [...] now delivered, had I not beene prevented by some others, whos [...] callings might have invited them to spend their houres better, tha [...] to snatch with greedinesse that out of anothers hands which wa [...] not their owne: for these Circles of my Ring projected on a Plate o [...] Plaine, so fitted with an Index at the Center as aforesaid, by cleare testimoney in its particulars I shall prove in his due place to be anothers, and not the supposed Authors, whose conscience may checke [...] him and tell him herein, that he never saw it, as he now callengeth it to be his invention, untill it was so fitted to his hand, and that he made all his practise on it after the publishing of my Booke upon my Ring, and not before; so it was easie for him, or some other to write some uses of it in Latin after Christmas, 1630. & not the Sommer before, as is falsely alledged by some one who hath made himselfe a spokeseman to another in some things by equivocations, & in other things by confessed untruths, whose ambition to be some body, hath incited him forward to deliver some supposed new stuffe, or scrambling peeces, if not confused fragments of his owne, or some others, to a publike view, in obscure and various phrases, a thing supposed to bee forged by sundry heads, rather than by one alone, seeing there is such roving from the Text; amongst whom to blow some smoake thereto, there was some grosse one, seeing the matter is so common; for to a finer element perhaps his capacitie could not assent, or ascend. A blinde Guide and a Parrats speech are not much different, the one walkes hee knowes not whether, and the other speakes hee knowes not what; and such are all precepts in Artes, which leade and make men speake without Demonstration; which doth not onely protract the studious, and frustrate the affectionate, but makes an ingenious spirit (who ever is more Rationall th [...]n practicall) [Page] to contemne such Circumlocutions, and laugh in private, if not in publike, at the learned stile of some Authors, who making themselves by their obscure kind of writing seemingly famous, sticke not to calumniate others to make them infamous. Those that are but initiated unto knowledge, for their faylings and defects of order, are worthy of some blame, but for others who would so are above all, and not onely picke holes in the coates of the living, but also vilifie the dead, tis a shame; making too great hast at the beginning to glory in that, which no doubt will prove shame in the ending. Its a common thing, that, one man having laboured, planted and sowed, with great paines, another reaps his Harvest with no industry; yet in this there was some honesty shewen, not to take the Crop but the Gleanings, houlding it easier to follow a beaten path then hazard a discovery, but the way was not made plaine, and the vaile remooved to helpe his sight: God that gave me the former invention without the advise of any, hath also reserved for me the manifestation of the latter, without the helpe of any, which I formerly mentioned in the Conclusion of the use of my Ring, to declare as is aforesaid, and I hope no envious, and insinuating detractors, will hereafter assume this also to himselfe as his owne, and say I had it from him: I have hitherto borne the injury of the infamy with great greefe of heart, (and God that is the discerner of the spirits knowes mine integritie and innocencie herein.) The window hath beene as yet close, and darkenesse possesseth the place, I will now withdraw the Curtaine that the Sunshining light may appeare, to expell those [...] mists that have beene scattered, and by a true and sincere medium remove that which was suggested from a false. Many have spurred me hereto, who suffer also with me, who wonder at my slacknesse, and long patience, others contrarily have as much gloried, I hope in the end that truth will burst forth, that God may have the glory, man the shame: and I doubt not but such as men sow unto others, such shall they reape unto themselves againe, and with what measure they meate, the same shall be meated unto them, I will therefore in the first place answer for my Quadrant, then afterwards for my Ring, and lastly for my selfe, and others.

## The Answer upon my Quadrant.

I Have not onely suffered by way of divulging of the said project in of Logarythmes Circular, 1630. but also by the late publishing of my [Page] Horizontall Quadrant, 1631. by a scandalous aspertion couched under the relation of the Author, in the words of his Translator (but rather his Transcriber, if not in the most of that Booke as is suspected, his Compactor) which are as followeth, Which whilst I went about to doe, another to whom the Author in a loving confidence discovered his intent, went about to preoccupate, and prevent, if not Circumvent. Now because his words are cautelous and subterfugious, we must a little examine them; If they be true, then that which I have produced and delivered as mine owne cannot stand, well then, this preoccupating, preventing, and circumventing, and discovering in loving confidence his intention to me, must be eyther about that of my Horizontall Quadrant in the making and use thereof, or touching that of my Ring; the former of which I lately produced before the supposed Author did write so publikely upon the same Projection, as he hath now done in his booke named the Circles of projection, which tract is so farre from being answerable throughout, to that which it promiseth in the beginning, that it seemes rather to pussle the studeous then in any wise to further them) of which I say no more, but advise the studeous Reader onely so farre to trust as is agreeable to the Text, and a true Doctrinall Method, which being therein omitted, it doth not onely surcharge the memory of the learner, but doth much more frustrate and delude the Ingenious, by a labyrinth of tedious Rules, and ambiguous precepts, when few might serve demonstratively, making them speake Parret like, which would be as little vendible as it is abstruse, were it not thrust on men; I would be loath to put an untruth upon it or him, or any other; it were unmannerly, howsoever I will prove that concerning neyther of these Instruments, to wit, that of my Quadrant, or that of my Ring, the said Author did discover his intent unto me, either in whole, or in part, in a loving confidence or otherwise. And for the making of my Quadrant, I could not circumvent him whom I knew not, for I drew the Proiection long before ever I heard of the Authors name, (as in its due place I can prove) but this aspertion no doubt was from an inveterate hate of some, who endeavoureth not onely to annihilate my interest in the Invention of that Ring, and Quadrant, but also to bring me in a disrepute, and to leave a blot of infamy upon my Name unto posteritie; But I doubt not in a publike Audience I shall cleare my selfe of it, and the disgrace that was intended to be cast upon me in the beginning, may light upon the contrivers thereof in the ending, and as there was a pit digged for another, perhaps the diggers may fall therein themse [...]ves, let them laugh on, as they have [Page] begun while I hasten the Issue of it. The extendure of Gods hand in his donations is manifould, and where his spirit pleaseth to breath, there a dore is opened; now whether the gift to the creatures bee divine or humane, we should blesse God as the first and principall Author, and giver of all, no wise d [...]spise man, as the second agent, or receiver: Its very phantasticall in some therefore, who thinke such and such things are not worthy the generall vote and allowance, if they proceede not from such, and such an one, envying they should be produced by any other; which if they be so divulged, as much as lyes in them, they will hasten to possesse the world with a contrary opinion, thereby wronging God in his dispensation, and man in his reputation, but such men wedded to a private and deluding fancie, chuse rather to abandon the Lore of sound reason, then to be divorced from their prejudicate affections. I will in the first place therefore make way for my Quadrant, that I did not circumuent the said Author for the attayning thereof, (as before:) But whereas hee taxed me with going about to preoccupate, or prevent him, it seemes somewhat soaring like, to abridge any mans free affections, in constraining them to waite on any others concealed intentions, which they never knew (in which every man hath a freedome to himselfe) It is (no doubt) somewhat too malepart, & too rigid, to ty the liberties of others in their actions, who desiring the good of others in not concealing things, if they shall by their industrious labours, shew the excellent use of such or such a thing, to a publicke view, for a generall benefit, are for it envied and calumniated: which as it smells of too great a detracting from others; so it hath its s [...]ource from the philautie, and too high conceat which some entertaine of their owne worthines: the said Author having had the Projection 30, yeares before by him (as he giveth out) wherefore did hee not publish it then, or give way to it as hee hath now done, how could it bee so long concealed, and others never so much as heare of such a thing, was it that hee would not have the same communicated to others, or that hee would not be knowne in his name by a profitable action, or that some others might challenge an Interest in it besides himselfe, or that the uses of the Projection lay hidden, and obscure; or that they were not at that time so plentifully made manifest, untill M^{r}. Gunters ingenuitie opened the mystery of it, and applyed that projection so methodically, and copiously, to Horolographie in his Booke of Dyalling in the use of the Sector, and accommodated that Projection long before in a Dyall for an honorable personage, as the first that ever was made in [Page] that kind, and the same forme with these that are now made, and therefore to give every man his due, and not to iniure the dead, it is properly [...]en M^{r}. Gunters Dyall, for that composure, and not anothers, (notwithstanding the inverting of things and detractions from him) But it should seeme he would publish it then, eyther when the way was made faire for him, or when he might catch at some one, as lately hee hath done in an unfit and uncharitable way: And as M^{r}. Gunter was copious in applying that projection to that particular of Dy [...]lling, so might he no doubt have beene in the Astronomicall uses of it also, though he delivered so few as but six observations onely to a publicke view upon that Projection, which before the publishing and also after, both in the uses and the Projection it selfe I often intentively looked upon, and extracted from them many usefull performances more, even in M^{r}. Gunters time; and since his death have published my Horizontall Quadrant, extracted from that dyagramme: in which I have abundantly supplyed the emptinesse and obscurity of that Projection, fitting it for a Pocket Instrument, or according to any magnitude, as a helpe and benefit to those that are studious of Mathematicall practices; which labours, of M^{r}. Gunters, and my owne, if they were not to unvaile the subiect and to make way for another to helpe his sight, which writes afterwards, I referre it to an eye not partiall, to be Iudge, it being rare and wonderfull for one man to see all at once, and there are farre more excellent uses yet upon that Projection, which may be also knowne if some one will open the vaile a little more; if there be any that knowes further uses upon it, let them discover them, for the present use for others, and let them not, if others, more respective of the common benefit of such who delight in those things, publish them, by scandalous detraction deliver that they are prevented, if not circumvented. Which Quadrant in the originall as before specified I did extract and compose from the fundamentall dyagramme of M^{r}. Gunters Booke of the sector, page 66. as I have specified in the Epistle to the Reader, in the booke of the use of Quadrant, howsoever the supposed Author (having the free liberty granted to see my Epistle before it was printed, and alter whatever he thought fitting, (I being unwilling to oppose his desires) did dash out whatsoever I had there to my knowledge justly attributed to M^{r}. Gunter, because he said it did belong to him primarily, yet I must say that I was especially behoulden to the labours of Mr. Gunter who is now at quiet in his Grave, and therefore not to bee wronged. And should I search the originall from the first, neyther M^{r}. Gunter nor [Page] especially hee may challenge so much unto himselfe, since the maine draught out of which theirs is extracted, was extant before they were borne, namely in that auntient Geographer Munster in his Dyalling, where it stands obuious, as also in the famous [...] Orontius, as I mentioned in my former Epistle, but I suffered him to put it out, being (it should seeme) unwilling to have his owne dismantled; As also at that time, I shewed him in our English Blagrave the like Scheme, in both their workes, long ere hee or M^{r}. Gunter committed any such thing to a publike view; howsoever, I had not the least touch of furtherance from him, or from any man breathing, either by transcripts, or verball direction, (but onely what I have formerly acknowledged out of M^{r}. Gunters Booke of the Sector, page 64. 65. 66. though in other triviall matters I doe and shall acknowledge freely) as hath bin falsely alledged by some loose tongued Instrument, that I had the making of that Quadrant by a sight of the Authors letter, whose honest relation may bee suspected, seeing hee makes so little conscience, to detract from anothers reputation whose detraction is grounded upon as great a stabilitie, as vncertainty, whose Basis is onely a bare supposition, conjoyned with an indisposed evill eye, as full of envie, as emptinesse of charitie, and this upon examination he is able to make good, and no more, which a generous or tender breast scornes to harbour, much lesse so partially to divulge: But it will no doubt make such Tale tellers so much the more odious to the sweet disposition of such noble spirits, when they shall be possessed with the manifestation of the contrary, which will no doubt shake the foundation of that Projection, by letting the world know it from the originall, that he which hath given way and gloried at the countenancing of that aspertion, and as the supposed Author of that Projection, perhaps may be challenged by others, as I mentioned before, howsoever therein very ingenious, by so assuming the Projection to himselfe (as in his owne words) but not ingenuously enough acknowledging from whence he had it: for my particular, I take God to witnesse I have without any equivocation or mentall reservation, declared in every particular the very naked truth, in the Epistle to the Reader in that Booke of the aforesaid Quadrant, from whence I had that Quadrant, and how I produced it, and it might have bin easily so composed, and published, as I have done it, by another, (without the helpe of the sight of a Letter,) that is but indifferently versed in delineations, seeing the Projection in my Author M^{r}. Gunters Booke, page 65. 66. as aforesaid lies so plaine & conspicuous [Page] (notwithstanding) in which letter to my remembrance I saw bu [...] some ordinary uses, and a checke cast upon Mr. Gunter, but no directions in it for the making of the Instrument; which uses I slighte [...] as meane, and triviall, and other things I saw not: and this I speake not to shuffle things off, but out of a true sincerity, but perhaps since the sight thereof there may bee inserted somewhat else, to make my opposites assertion good, howsoever I needed not such a helpe by the sight of a Letter, seeing Ioyners, Carpenters, and other Mechanicks about this Towne & else where, yea, Schoole Boyes in imitating the Projection aforesaid in Mr. Gunters Booke of the Sector, and following the directions therein in the 66. page of the use of the Sector aforesaid, have drawne the Projection fully and compleate by the Booke alone; and others having onely had but a simple view of my Quadrant, many yeares before I published the use thereof, have from the aforesaid Booke of the Sector page 66. made the like, having not had the least assistance from any, but the direction of the Booke onely, as upon oath they have beene examined, and doe acknowledge, and will testifie it when occasion requires: beside I know sundry Gent. and others in this kingdome, that are yet living, that have drawne the same Projection by the Booke, alone immediatly after the publishing of it 1624. as at this instant they wil be ready to confirme, and my selfe in Mr. Gunters time 1622. (besides many others) have drawne the same Projection, for our particular uses, and are yet to be shewen, w^{ch} was long before either I, or they ever heard of a new Author, of that Projection besides Mr. Gunter onely: And to make my assertion yet more absolute, I did not onely draw the Projection in Mr. Gunters time, but before his death did also shew the making of it to others; therefore (as before) I could not circumvent the said supposed Author, to have any assistance from him in making of my Quadrant, either by a verball declaration, or by the sight of a letter: or otherwise, which in its due place God permitting life, and health, shall be confirmed more at large in every title.

## Ʋpon my Ring.

IF I circumvented not the said Author, nor that in a loving manner he opened his intent unto me by assisting of me in the making of my Horizontall Quadrant; Then to make his assertion good, it must necessarily be in the other Instrument, which I produced and published, to wit, that of my Mathematicall Ring (that is, how I might compose [Page] or make the same) but concerning this latter Projection of the Ring or any thing to that effect, in a loving manner then, or at any time, or otherwise that he discovered his intent unto me, it can hardly bee collected (certaine circumstances seriously weighed, and considered) The whole ground of which being from as weake a principle, to open the way unto me for the making of my Ring, as the sight of a Letter to shew me the making of my Quadrant; for about Alhalontide 1630. (as our Authors reporteth) was the time he was circumvented, and then his intent in a loving manner (as before) he opened unto me, which particularly I will dismantle in the very naked truth: for, wee being walking together some few weekes before Christmas, upon Fishstreet hill, we discoursed upon sundry things Mathematicall, both Theoricall and Practicall, and of the excellent inventions and helpes that in these dayes were produced, amongst which I was not a little taken with that of the Logarythmes, commending greatly the ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, in the lines of proportion, extracted from these Logarythmes for ordinary Practicall uses; He replyed unto me (in these very words) What will you say to an Invention that I have, which in a lesse extent of the Compasses shall worke truer then that of Mr. Gunters Ruler, I asked him then of what forme it was, he answered with some pause (which no doubt argued his suspition of mee that I might conceive it) that it was Archingwise, but now hee sayes that hee told mee then, it was Circular (but were I put to my oath to avoid the guilt of Conscience, I would conclude in the former.) At which immediatly I answered, I had the like my selfe, and so we discoursed not a word more touching that subject: all which sundry times ingenuously according to the very truth the said supposed Author hath acknowledged before divers persons, who doe and will testifie the same: Then after my comming home I sent him a sight of my Proiection drawne in Pastboard: Now admit I had not the Invention of my Ring before I discoursed with the supposed Author thereof, it was not so facil for mee or any other (to an eye not partiall) to raise and compose so complete, and absolute an Instrument from so small a principle, or glimpse of light, but was knowne unto me, (as I have produced it) long before that time, and being now published as it is, the composure of it seemes most facil, (as all inventions doe, once knowne) that I have much wondred with my selfe, that Mr. Gunter or some other produced it not so to the world as now it is, seeing it was so easie, and caries with it such an excellency, above that which is in the Lineary forme, for in a [Page] Circle it is naturall, & perfect, in a line defective & imperfect: But he, or they, perhaps saw it not, though their sight I confesse (no wise to disparage their worth) might penitrate further in other things; but gifts may not be attributed to naturall ingenuitie, but to God the giver of them, who disposeth where his goodnesse pleaseth. And who knowes to the contrary but many private men in this kingdome, or elsewhere, might have this Invention long before my selfe, or he that now challengeth it, seeing the gifts of God in his donations, to severall persons, are oftentimes in one and the same thing, and if God bestowes his Talents on us, we ought to communicate them to others, which not to doe, is to hide them in a Napkin, and who so concealeth them are not free from some reprehension: our Author may answer for himselfe, who reporteth now that hee had the Invention for more then twelve yeares past, but put it not to use, which in a sense was injurious both to God and man, if hee saw the copious, and compendious use of it then, as now hee doth; to mee and to many others it hath seemed strange, that he should hide such an excellencie, so long in obscuritie; But it is supposed, that hee had then but a glimpse of its performance (untill I writ upon it, by opening the Cabenet, and shewing its treasure to the world) and so regarded it not then as a Iewell, but since. But when I had a sight of it, which was in February 1629. (as I specified in my Epistle) I could not conceale it longer, envying my selfe, that others did not tast of that which I found to carry with it so delightfull & pleasant a taste, knowing that as the Logarythmes (as a Iewell) that did excell all Inventions that ever went before it, so this kind of Projection, as the daughter springing from so noble a Mother, was the rarest Instrumentall Gemme, that ever Art in all preceeding Ages did afford: which in its production heretofore lay obscure, but now being published how cleare doth it seeme to bee in its composure, and if it were so easie at the first to produce (as some account it) why did not some one all this while since the publishing of it (being now more then two yeares past) inlarge that Invention, or deliver others in that nature of their owne, being easier to adde unto an Invention, now being divulged, then to deliver its originall? But hitherto it hath seemed difficult, many having attempted and endeavoured, but have fayled in their ends, howsoever I know not what some private mens industry hath produced by way of augmentation to that invention of my Ring, or the Logarythmes Projected Circular: At the Conclusion of my Booke I gave a sufficient and perspicuous direction, and invitation thereto, that so by the former [Page] labours, and the other there intended, the way might be made faire, and the Ice broken that they might wade the easier: and so produce at last something for a publike use by Augmentation to this Invention, at which I shall much rejoyce, rather then any wise envie: But I have as yet all this while demurred in my pretended purpose, and being often times by sundry men importuned, according as I have promised, to inlarge that Invention, seeing none as yet hath done any thing therein, I have now at last therefore for their benefit, & others, to a publike view delivered what is mine owne to that end: and that by sundry wayes, and how easy doth the inlargement of that Invention now also seeme to be, being now produced. With the like facility did I compose long agoe my Helicall line of Roots which affords five places accuratly, by an inspection of the eye onely, which being not as yet divulged to a common view may to some seeme difficult for its composure: The secrets and intentions of the mind, how closely are they lodged in the breast, and who can search the heart, that from a word barely delivered the whole may be conceived; therefore, it was not easy for me to know the Authors intent in his project by a bare word, had not God long before opened the way unto me, how this projection, or Invention might be composed; what meanes then these words; that in a loving confidence the Author discovered his intent unto me? is it not to give a faire flourish upon a untrue subject, to delude, if not to possesse Men with a falsitie, to detract from another his good name; besides the barrennesse of the word we will a little consider the straightnesse of time, in the original, for the producing of the Booke, and Invention, as is challenged by the Author to be about Alholantide, 1630. & that then his loving intent to me he opened, and so was circumvented; I will put my cause into the hands of any indisterent Iudge to censure of it, that having had no other direction, or light from the said Author then formerly is specified, (for more hee dares not avouch with a cleare conscience) how could I from it so easily forme my Projection: Mr. Gunters Ruler, as some thinke, was a furtherence to me, but it was rather a hinderance, for his line of Numbers was as impertinent for me to follow, as such a double composure in the projection was superfluous, therefore such a bare verball (as before) or Instrumentall dictate was not used, or could be sufficient to compose so high a worke: but it was from a long intentive precogitation of many yeares with my selfe, how the Logarythmes in the Tables might be so compacted, that all Numbers in these Tables should bee proportionall one unto another, and according to a divers, and variable proportion assigned; which to [Page] effect I found at the first very difficult, and could not conceive how otherwise it might bee done, but either by fitting of Tables to all proportions, which so to doe would not onely be too great for operation, but also breede confusion; or it must be from some graduated insertion of these numbers Instrumentally, so that by motion numbers might be mooved one unto another; and for expedition of which I found no figure more apt then a Circle, and on that my Meditations fixed, and there I rested, and so (as my few houres could permit me) I made severall projections, that the senses might see the effect of it in a perfect beauty, which the intellect saw before but in obscurity: about which how often was I afterwards interrupted in my desires to looke further into the mysteries that lay open in that new Invention, having not scarce an houre in a day, and sometimes not two houres in a weeke of serious privacie (by reason of my calling) to sport my selfe in operation thereon (for it seemed so to me then as a recreation, as all new Inventions at the first doe to any that invents them) It was many moneths therefore before I made tryall of it, in the generall uses that it might bee put unto, in matters of Arithmeticke, for common operations, and to the measuring of Plaines and Solids, but especially how it might be applyed to the doctrine of Plaine, and Sphericall Triangles, in Astronomicall Calculations, Nauticall practices, and Horologicall conclusions, all which in every particular I practised on it at such times as convenience would permit mee; And having for many moneths thus satiated my selfe, and fed my fancie upon its Theoricall contemplation, and its Practicall operation, though oftentimes I found many Rubs and impediments in the Practise, in applying this instrumentall Invention at the first so generally and copiously as I did; And it cannot bee denyed of any, that the wayes of new Inventions lies not so obvious, or so easie, to bee discovered with such celerity, as a long premeditation might produce; I was desirous therefore to make the world participate thereof also; Therefore at severall times, having but a little time at anytime, by little and little did I compose and produce a method pleasing to mine owne fancie; doing one weeke such a peece, another weeke another, and so going on untill I had runne through the whole parts of Arithmeticke, in the Goalden Rule direct, and indirect, in division and in multiplication, applying all these to manifold uses in combination of Numbers, to common affayres, in fortification, to mensuration, and fractionall operation, then I applyed the Instrument to the finding of numbers in continuall proportion, in finding of meane Proportionalls, and the [Page] extracting of Rootes, I laboured further to make the Instrument more complete that it might worke all usefull proportions touching Interest, or valuation of Leases: and last of all I applyed the Instrument in the Circles of Sines, and Tangents, to some uses of Astronomy, in the finding of the Declination of the Sunne, and its ascention through all Latitudes; These things I drew up in a Booke at severall times aforesaid, as a part of the practises that I made upon it: And it was a task sufficient for a man that had cōmand of his houres, by allotting them soly to that end, in as long a time as was betweene Alhalantide & Christmas, in the opening of the way of that new Invention, & by applying of it so, (to use for a publike view) rather then by one that was commanded by them, especially at that time and all the Tearmes before: for how few are the houres that a man of any imployment gets to set upon a new method for a new Invention at such a time, I leave to any indifferent eye to judge; and it would have beene so much the more harsher, and difficult to have it so suddainly produced, had it beene then at Alhalantide but onely conceated, and agitated on, but it was premeditated on long before, as is aforesaid, and intended for a new yeares gift for the King, which accordingly I gave him on new yeares day, though a fortnight, or three weekes before Christmas, his Ma^{tie} received from mee a Scheme of the Projection in Pastboard, with a manuscript of the Booke which is now published, agreeable to it in every tittle, (the Epistle excepted) a Copy of which was then at the Presse: and was Printed foure dayes before new yeares day, 1630. Now that all these Practices, and many Transcripts that were drawne, with the doubts and hinderances that did arise in the fitting of these things, could bee made, so ordered, and produced in so little a time, as the scattering of a few houres as betweene Alhalantide and a moneth before Christmas, I leave to the Iudicious Reader to censure: I broke the Ice, and made the way facil for another that came behinde mee; yet if hee tooke a yeare and more to meditate and write upon my endeavours after my publication (thereby not to signifie that hee wanted time, but takes liberty enough if not too much, to the losse of time) as long a time, halfe of it, or a quarter of that time, may by a Charitable boone be granted mee. Which consideration of time, will easily also cleare mee from the imputation of the Authors loose assertion, that hee was Circumvented by mee, nor that in a loving confidence he opened his intent unto me. Yet in the last place I will hasten [Page] to vindicate his untrue declaration before the Courts of justice; if restitution be not speedily made, where true witnesses shall bee produced, and that which is now but in agitation I will bring to action, and prove that before Alholantide or very neare that time, my Invention was produced to a publike view; therefore it was very injurious in the Author to possesse the world with so an untrue aspertion, upon a bare supposition, in that I should have the Invention frō him, and that in a louing manner he opened his intent unto mee, and was Circumvented, as before, whose assuming disposition hath not onely bin busie to take from me my good name, & labours by this his loose aspertion, but hath also endevoured (by too rigid and generall a censure sparing none in some sence) to bring all in a common disrepute in their Callings, therefore,

## In the behalfe of vulgar Teachers & others.

BY way of advertisement to you, and my selfe, & to whom it may concerne, vulgar Teachers of Mathematicks about this Towne and kingdome, as you are styled, which had bin faire (if no worse) seeing you are not professors, or publike Readers, but common Teachers, it behooves you, and me, that we indeavour to avoyd the disreputation of a scandalous attribute in our profession of so noble a science, that our doctrines be not onely Practicall, but also Rationall, and Theoricall, that we may not bee ranked with Iuglers, and teachers of Tricks, as we are lately glanced upon publikely, but a charitable breast howsoever (I perswade my selfe) hath a better opinion of us, that according to the Talent that God hath given unto us, out of the Riches of his bountie, no doubt in our callings we use them rightly, and doe not (by deceiving) derogate from the end, which is, to glorifie God in these gifts in a true and sincere use of them; which otherwise would not onely be a blemish and staine to our name but also a dishonour to our noble profession, the large testimony of which (no doubt) every one of you in his particular can produce, (notwithstanding such an uncharitable censure) can amply testifie in your callings your sufficiency, and therein remoove all scruples to confirme your integritie; and as the imputation hath lighted heavy upon us, so hath it not rested there, but rebounded likewise (if the words be truly scanned) unto such Nobility, and Gentrie, to which we have beene servants to or in present are, in so honourable and laudable a service, which aspertions [Page] are as highly backed with arguments, as he was to forwords that divulged them: whose judicious and censorious eyes, hath beene two much busied to shew the wayes of the Iugling of others, and prove delinquents in the same things themselves, whose words are these, That it is a preposterous course of vulgar Teachers, to begin to teach with Instrument, which was not onely to despight Art, to betray willing and industrious wits to ignorance, and idlenesse, but was also losse of precious time, making their schollers doers of tricks, as it were Iuglers. Which words are neither cautelous, nor subterfugious, but are as downe right in their plainnesse, as they are touching, and pernitious, by two much derogating from many, and glancing upon many noble personages, with too grosse, if not too base an attribute, in tearming them doers of tricks, as it were to Iuggle: because they perhaps make use of a necessitie in the furnishing of themselves with such knowledge by Practicall Instrumentall operation, when their more weighty negotiations will not permit them for Theoricall figurative demonstration; those that are guilty of the aspertion, and are touched therewith may answer for themselves, and studie to be more Theoricall, then Practicall: for the Theory, is as the Mother that produceth the daughter, the very sinewes and life of Practise, the excellencie and highest degree of true Mathematicall knowledge: but for those that would make but a step as it were into that kind of Learning, whose onely desire is expedition, & facilitie, both which by the generall consent of all are best effected with Instrument, rather then with tedious regular demonstrations, it was ill to checke them so grosly, not onely in what they have Practised, but abridging them also of their liberties with what they may Practise, w^{ch} aspertion may not easily be slighted off by any glosse or Apologie, without an Ingenuous confession, or some mentall reservation: To which vilification, howsoever, in the behalfe of my selfe, & others, I answer; That Instrumentall operation is not only the Compendiating, and facilitating of Art, but even the glory of it, whole demonstration both of the making, and operation is soly in the science, and to an Artist or disputant proper to be knowne, and so to all, who would truly know the cause of the Mathematicall operations in their originall; But, for none to know the use of a Mathematicall Instrumen, except he knowes the cause of its operation, is somewhat too strict, which would keepe many from affecting the Art, which of themselves are ready enough every where, to conceive more harshly of the difficultie, and impossibilitie of attayning any skill therein, then it deserves, because they see nothing but obscure propositions, and [Page] perplex and intricate demonstrations before their eyes, whose unsavoury tartnes, to an unexperienced palate like bitter pills is sweetned over, and made pleasant with an Instrumentall compendious facilitie, and made to goe downe the more readily, and yet to retaine the same vertue, and working; And me thinkes in this queasy age, all helpes may bee used to procure a stomacke, all bates and invitations to the declining studie of so noble a Science, rather then by rigid Method and generall Lawes to scarre men away. All are not of like disposition, neither all (as was sayd before) propose the same end, some resolve to wade, others to put a finger in onely, or wet a hand: now thus to tye them to an obscure and Theoricall forme of teaching, is to crop their hope, even in the very bud, and tends to the frustrating of the profitable uses, which they now know, and put to service, and to the hindering of them in their further search, in the Theoricall part, which otherwise they would apply themselves unto: being catched now by the sweet of this Instrumentall bate; which debarring would not onely injure the studeous but also cause the Mechanicke workemen of these Instruments, to goe with thinner clothes, and leaner cheekes.

Neither doth the use of Instrument to a man ignorant of the cause of its operation, any wayes oppose or dispite Art, seeing that the end of producing and inventing of Instruments, is their Practicall use; Besides, its impossible to shew the use of an Instrument but in teaching there must needes be laid downe some grounds or prolegomena [...], as, what is meant by such, or such names; what are such, and such tearmes, and therefore the beginning of a mans knowledge even in the use of an Instrument, is first founded on doctrinal preceps, and these precepts may be conceived all along in its use: and are so farre from being excluded, that they doe necessarily concomitate & are contained therein: the practicke being better understood by the doctrinall part, and this later explained by the Instrumentall, making precepts obvious unto sense, and the Theory going along with the Instrument, better in forming and inlightning the understanding, &c. vis vnita fortior, so as if that in Phylosophy bee true, Nihil est intellectu quod non prius fuit in sensu, and things the more they be objected to the sense, are more fully represented to the understanding, then it must needes follow that a doctrinall proposition laid first open to the eye, and sense, and well perceived enters more easily the dore being opened, then if the intellect by the strength of its active sense, should eliciate, or screw out the meaning by a long exeogitated operations; Its not therefore requisite unto all Capacities, to have Instrument severed from science, [Page] in many things: is though the use of one could be without the other lesse or more.

Neither doth the practise on them betray willing & industrious wits to ignorance and idlenesse, that assertion therefore is very ridiculous, for he that is industrious, and willing to spend his houres, for the attayning of any science, according to doctrinall method, busieth himselfe not with Instruments, but applies himselfe to such Authors, or such compendious abstracts, as are taken from them, which doe not onely open the essentiall parts of the subject in the Theory, but also layes downe such documents and principles which may (in a higher nature and way) induce him to practise them, then possibly by Instrumentall operation for exactnesse can bee attayned unto; for such is the excellency of Art by Theoricall doctrine, that all things tending to practise may bee done by the science onely, without the helpe of Instrumentall operation, certaine propositions being granted, which originally and principally are proper unto Instrumentall observation, being the Basis or foundation of the whole; Therefore science, as before, hath a principall dependance on Instrument, but is it in their observation, rather then in their operation, and the Inventions that are daily produced in that kind, are onely to compendiate and facilitate practicall things, which the learned in those Arts (having the science) scarce use at all, if at any time they use them, it is in small and triviall things to satisfie the sense, and not the intellectuall part; which Theoricall way doth not onely augment the desires of the Teachers in the accommodating things, and remove of difficulties, by making them conspicuous to the learner; But also by so proceeding causeth the Teacher to adde many ends both pleasant and usefull, by the way, for the Practicers incouragement. To begin with Instrument is unprofitable for the Teacher, though advantagious to the learner, (if his ends be but to know some uses upon an Instrument, as it is with many) for it is easier for a man to learne more usefull Practise upon some Instrument in one houres Instruction, then to know the cause in 20. houres of some Instrumentall operation: and yet there are many Instruments almost as facil in their demonstration as in their operation; Which kind of beginning to teach, or usuall proceeding therein is not as vulgar Teachers use, but is as indirect in the Method as false in the assertion (if they may possibly avoid it) for that were to teach against their owne profit, and the dignitie of learning; Yet that any Teachers of Mathematicks should be so nice as to deny the use of Instrument, to such Gent. or others, who perhaps desire the Theory to [Page] contemplate on hereafter: but the practicke for the [...]r private ends, for the present, were not onely to frustrate their desires, but injure their occasions; who might rather laugh at the teachers pride, then contemne his Art.

Lastly, that the practise on a Mathematicall Instrument should be losse of precious time, (to any one that knowes how to use them in their practise though not in the Theory) is not onely ridiculous but also untrue and absurd, for to what end did the manifold and laborious calculated Tables of the learned in their Subtenses, Sines, Tangents, and Secants serve, were they not to avoyd the great toyle of Algebreticall worke in Radicall extractions, which otherwise in Trigonometrie (the practicall end of Theory) must necessarily be used, but they were Invented & produced to avoid the losse of time, to the more learned (who know the causes of their operation in the Genises) and as a facilitie to open the way unto the unlearned, though thousands working Trigonometrie by them knowes not scarce the cause of such operation, which in a manner is as meere Mechanicall, as if it were Instrumentall (though more accurate in its performance) But to come nearer unto these present times, how hath the Invention of Logarythmes, taken away the labour and losse of time, that was used in former calculation, for that which cannot be done by the common Tables of Sines, Tangents & Secants in 20. hours, is now done by the helpe of the Logarythmall Tables in one houre, & there are thousands in the world that also worke by them for their private use, by reason of their great quicknesse, in hard and laborious matters, who know not the cause also why these Numbers should so expedite such a difficultie, with so great a facilitie, & celeritie: yet they use them though the performance be also mechanicall, that they may not likewise lose time: Now Instruments though they bee extracted from Tables, yet according to their capacitie in practise they exceede these Tables, in all common serviceable uses, that I dare maintain that what the Logarythmes by the Tables, for giving of proportionalls, either in Numbers, Sines, or Tangents, will doe in 20. houres, my Ring or the Circles so projected from it, will doe in one houre. Which Instrumentall Invention, I have also produced to helpe the studeous to avoyd the losse of pretious time, which to omit and keepe them from it (in some occasions) were also misspending of time: Therefore in respect of private mens ends, to begin with Instrument is not to Iuggle, nor to doe tricks, neyther to oppose or dispite Art, or to betray willing and industrious wits to ignorance and idlenesse, nor losse of pretious time.

BVt I would be loath while I seeme to vindicate the losse of time, cause others to lose it, by my too prolix defending of it; the vindicating of the truth indeed was my cheefest ayme, which I hope will be my spokeseman & defend it selfe, meane while not to hould my Reader too long, I wish with all my heart that such occasions were taken away, then might he spend so much time in reading better things, and I in propagating, not in defending my right; detraction, Calumnie and defamation come close to a man, and therefore my Reader may excuse me if I be so tender of it, since it may be his owne case (how innocent soever) perhaps the very next of all. There is (and I am not the first nor it m [...]y be the last truly sensible of it) a generation in the world true salamanders whose delight is to live in the fire of contention, endeavouring to loade others with Calumnie and detraction, and to raise unto themselves a proud Babell out of others ruins; a vice odious both to God and man, thereby forgetting the rules of equitie which ought to be observed, so to deale with others as one would be dealt withall, thereby blemishing the good name of another, who are not ignorant (or at least should not be) how pretious the name of a man is: is it not above Riches, above pleasure, aboue Gold it selfe? Calumnie indeede is the fire this Gold is cast into sometimes, onely heeres the difference, this purifies the one, and that strives to pollute the other; nay, what speake I of those grosse things, is it not dearer then life it selfe, being the very life of the life here, and eternitie of a [Page] mans life hereafter? If he that steales but a small matter from his neighbour is obnoxious to punishment, sometimes unto death, is not he much more worthy of the same, and greater punishment that layeth violent hands upon that which is above life it selfe? If reports were true, private admonitions may win men, if false why are they then divulged? and grant the reports be true yet, is every report to be spoken? may not a man breake the ninth Commandement that divulges these reports though true, which tend to the infamy of another? and how much more is he guilty of this breach, that is not onely a blazer of such reports, but a compiler? Is not Satan stiled from this word of Calumniation? doth he not Calumniate God to man, and man to God, and one man to another? and shall any man tread in his steps? is it not the precept of the Psalmist, or rather of God by the Psalmist that every one should set a watch before the dore of his lips, and keepe in that unruly member, that is, a world of wickednesse? what is the stroake of the hand to the stroake of the tongue? the one wounds the body, the other wounds the soule; the one may be easily cured, the other hardly; the one strikes them that are present, the other those that are absent; the one striks but one at a time, the other strikes many at the same time, at least three at once, with the selfe same blow; viz, him whom he doth traduce, him who heares him, and himselfe, and many times the last most of all; and ought wee not to take great care how we order this unruly tongues what Gall, what Poyson, what Wormewood, what Wildbeast is to bee compared with the tongue, which lancinates, teares, and wounds the soule, by the virulencie of its words, and reports: Looke into the heavie judgement of God against such, 1 Cor. 6. that [Page] revilers shall not escape punishment. Such was the account of the preciousnesse of a mans goodname in ancient times, that there hath beene Lawes utterly to banish such from the society of men, who by their Calumny and detraction separate all good societie amongst men, making foes of friends, and breaking the sweet bands of peace and charity, causing nothing but brawles and dissention, and as neare as they can dissolving this goodly frame into the ancient Chaos of confusion. Againe, some of the Antients herein are so violent that they call it Grave malum, Turbulentum Daemona Pestem pestiferam, and professe plainly, Qui detractioni student diabolo serviunt, so that as Coales are to burning Coales (saith the wiseman) and wood to fire, so is a contentious man to kindle strife The words of a Tale bearer are wounds, saith he, they desturbe a mans Peace and goe downe into the inwards of the belly, and a whisperer separateth deare friends. I wonder what neede these Lamiae gad so much abroad having worke enough at home, Te prius corrigas quam alterum corrigas, If such would turne their eyes inward they should finde matter enough to mende in themselves first, and afterwards they might reprehend others, sed ad delicta nostra Talpae, ad aliena serpentes sumus. It was once Constantines speech. That hee would throw his mantle over his brothers nakednesse rather then lay it open; What shall wee thinke meanewhile of those who would faine discover nakednesse, where they can finde none? what should bee the plot, except it bee to feede the bosome Wolfe of envie and malice which at last will gnaw out their owne bowels? doe they envie because God gives to one man one gift, to another man another? would they have all, and others none? If the whole body [Page] were an eye where were the hearing, if an eare where were the smelling? so, were all contracted into one, what would become of the rest? No, rather let every one be thankefull unto God for that hee hath, then envious for that hee hath not; God hath store enough for all, let us not envie one another; if one pull downe the house as fast as another builds, what will become of the building? rather let al [...] build and then a goodly structure will be sooner finished,

Nature it selfe peacheth us to make much of union, without which all falls to ruine; See we how the dayes part equal [...] stakes with the nights, and if one have a greater share now, it allowes the other it afterwards.

The sweet combination of Elements and inanimat bodies may teach us concord; Fire and water, earth and ayre how soever opposite in themselves in both qualities, yet in mixtion and composition sweetly conjoyne together, one not destu [...]bing but qualifying the excesse of the other, as if they without eyther sense or reason meant to teach men both: But above all, Piety and Religion enjoynes this, Are wee not all Christians, nay are wee not all Protestants (or at least should be) all professing one and the same faith, the same Baptisme? have we not all one Father, one Redeemer, are wee not all members of one and the same holy Church? whence then are these discords amongst brethren, these detractions amongst friends? Ile speake [Page] no more, least I seeme to teach those who should teach others. And this I speake not with any solace to my selfe, but with greefe of heart in the behalfe of such which are guilty of such breakings out, and make my prayers to God for them, that they may see the foulenesse of the offence, and be incited & stirred up unto repentance, and make satisfaction for the same, by calling in their sinister and untrue reports, And as they render their owne reputation, so to bee carefull of the reputation of others, and as they love their owne peace so to bee studious and tender of the peace of others; If they shall doe this there will bee some satisfaction made (though full recompensation for Calumnie can never bee made) and comfort in their acknowledgement, and repentance. If it cannot bee by this effected, I will follow the precept of my Saviour, and pray to God for t [...]em, and walke as well as I can through good reports, and bad reports, and comfort my selfe with the companie of my Saviour, of the Apostles, and of all Gods Saints, who were not exempted from the lash of the tongue, commending my cause to God (and to the equall judgement and censure of a Reader not partiall) to whose mercy likewise I commend them, and so desiring with S^{t}. Paul that we may all keepe the unity of the Spirit in the band of peace, this ever shall be my wish

SInce my first publication of the uses of my Mathematicall Ring, or the Logarythmes projected Circular I have beene oftentimes invited by sundry persons for to deliver the way of the projecting, and dividing of the Circles of my Ring upon a Plaine, so that it might bee made in Pastboard to avoyde the charge of the Instrument in metal, for such which have not abilities to buy, and for others, who would first see the practise on it, before they would be at the cost of the Instrument in metal: for whose sake, and use desiring to satisfie the affectionat, and for a publike benefit, (rather then mine owne particular) profit, I have caused two Plates of metal to be cut and ingraved, the one containing the Circles of the Projection of my Ring, noted with the letter A. and the other comprehending the Projection inlarged, noted with the letter B. that so they may make use of them more readily, to avoid the labour of dividing the Circles; which schemes being pasted on a Pastboard it is ready for use; And yet further to satisfie those that are desirous, I have delivered also in the first place ensuing how those Circles are divided, that so they may bee made according to any magnitude. In the second place how severall wayes they may be framed into a Ring: In the third place I shew the inlarging of the Instrumentall Invention in these Circles to as great a magnitude for use as may be desired. In the fourth place I deliver severall wayes how these Circles inlarged may bee accommodated for Practicall use. In the fift place, I make a description of the Grammelogia, or Instrument in the particular Circles of my Mathematicall Ring, projected on a mooveable and fixed plaine. And in the sixt and last place, I will declare the admirable, and excellent uses of both these Instruments, in the Practicall parts of Arithmeticke, Geometrie, Astronomy, Horolographie, Navigation. &c.