MIRIFICA Logarithmorun Projectio Circularis NIL FINIS, MOTUS, CIRCULUS VLLUS HABET. GRAMMELOGIA Or, the Mathematical Ring. Extracted from the Logarythmes, and projected Circular: Now published in t● enlargement thereof unto any magnitude fit for use: showing any reasonable capacity that hath not Arithmetic, how to resolve and work, all ordinary operations of Arithmetic: And those that are most difficult with greatest facility, the extraction of Roots, the valuation of Leases, etc. the measuring of Plains and Solids, with the resolution of Plain and Spherical Triangles applied to the Practical parts of Geometry, Horo●ogographie, Geographie, Fortification, Navigation, Astronomy, etc. And that only by an ocular inspection, an● a Circular motion, Invented an● first published, by R. Delamain, Teacher, an● Student of the Mathematics. Naturae secreta tempus ●aperit. Typus proiectionis Annuli adaucti, ut in Conclusione. lybri praelo commissi, Anno 1630 promisi. To the High and Mighty King, CHARLES King of Great Britain. Dread Sovereign, HAd not your gracious favour given life unto my first birth, I should not have dared thus boldly to rush into your sacred presence, but only in humble confidence I am encouraged, that, as your gracious Majesty pleased to accept of my imperfect endeavours, the same blessed eye will not now reject that work 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 toilsome working by Sines, Tangenis, Secants, etc. My pains formerly was conversant about to bring them likewise into a shorter way for practical uses, by an instrumental Projection of these numbers in Circles, of which I composed my Mathematical Ring. Every thing hath his beginning, and curious Arts seldom come to the height at the first; It was my promise then to enlarge the invention by a way of decuplating the Circles, which I now present unto your sacred Majesty as the quintessence and excellency there of, whereby Mathematical operations which tend to ordinary and usual affairs and such of a higher nature, are performed accuratly, without tedious extraction of Roots, operating by natural Sins, Tangents, secants, or the Tables of Logarythmes themselves, and that by an inspection of the eye alone and a Circular motion. It is now again your gracious acceptance and the public good, which my life & pains are studious of: the one I hope will acknowledge, the other defend these mine honest labours which some endeavour to rob me of. I have no better thing as yet to express a loyal subjects heart, and affections by: but only my self, which with my poor endeavours in all humble submission I cast down at your sacred feet, accounting it my chiefest Happiness to be Your Majesty's most humble, true, and Loyal Subject, Rich. Delamain By a Friend upon the Mathematical Ring. Within a little Circle, or a round Of old, the double-six-signed course was found, And much it was, the whole years toil to bring Within so little compass as a Ring. Yet here enclosed not one year's work you see, But what without it scarce is done in three. The same to the inventor of the Logarythmes projected in Circles. None e'er in Circles found an End they say, Or saw Beginning where about it lay: How was there then by thee beginning found, Of that which in itself is perfect round? And yet (only to prove the saying true) Some do deny it was found out by you: So will have no beginning; If their own, The ancient Axiom still is overthrown; But if it were not thine, how dared thou say Thou wouldst augment the same another day? Which day is come, and now that's decuplate, Which single was before. Thus 'tis thy Fate, Still to increase thine error, and of one Make ten, to show thy good Invention: Look to't, if one has bred so much ado, 'tis ten to one, this will be challenged too. Well, guilty be thou first of this law's breach, And for thy fault this censure shall thee teach, That though Beginning thou to this didst lend, Yet of its copious use thou's ne'er find End. By a Lover, and Student in the Mathematical Arts. Thrice noble Nepier by his learned strain, Invented profound numbers with great pain, From whence rare projections do arise, That praise their Author who did them devise, Which now again, anew this Instrument Doth those his numbers quickly represent. And as the Orbs by motion always show Divers apparences which still are new, So infinite performances thereby Are pointed also out unto the eye, In multiplication, division, and Beside it makes you how to understand, To find out mean proportionals, that so, You may by sight of it proportions know. Besides all this, which is a thing more high, It helps to work in Trigonometry, Scorning Roots extracting, by partition, Or gradual numbers circuition: It equates regular figures, and doth show The solid bodies and the plains also; Giving their sides, and perfect symmetry, And speedily their true solidity, Their circuits, and spheres will all appear, Both circumscribed and inscribed here; Their Diameters and transmutations, In duplications and triplications. And as the Logarythmes did at first exceed, In their Inventions, all that did proceed, Or were before in any former age, So that which now this Author doth enlarge, Amongst these numbers is a work of grace, By expeditious use for future race, In weighty things much to delight his mind, Who studious is the practic way to find, If after ages seeks by Instrument to know, With greater ease makes ease excessive grow, In mirificam Logarythmorum projectionem Circularem. NEperum meritò totus mirabitur Orbis, Qui Logarythmiacos repperit arte modos: Radicem extractam dempsit, Tangensque secansque, Atque sinus, duram difficilemque viam: Ritè Mathematicam & subitò cognoscere praxin, Si quis amat, praestant Organa rara nova: Rara Logarythmi projectio, circulus isque Mobilis aspectu,, haec ardua cuncta docet. Hîc tibi cernuntur proportio recta, triumque Regula, divisi & multiplicantis opus: Hîc tibi radicum est extractio tracta figuris, Corporis hîc plani, sic solidique metron: Analysis plani tibi sphaeratique trianglî, Cernitur, hîc praxis quamque mathesis habet; Et latet hîc usus quem non natura reclusit, Jllum mirificè ast Organa adaucta docent. To the courteous and benevolent Reader that affects this instrumental practice of the Logarythmes projected circular Entitled Grammelogia, or the Mathematical Ring. AS that Honourable Lord Nepeir (the first inventor 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 Instrumental Projection as springing from that invention, I called Grammelogia, the speech of lines; for being so projected as they are, these lines graduated, do promptly teach one what to speak in proportional operations; which invention of projection of Logarythmes Circular, I promised to enlarge at the end of my first publication, with this entitled Grammelogia, or the Mathematical Ring, Printed, Anno. 1630. notwithstanding it was there sufficiently perspicuous, how it might be augmented unto any magnitude assigned, even to operate unto minutes, and seconds in trigonometry, and to find Roots, and proportional numbers in common Arithmetic unto six 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 be the parts of Circles, and so there may be a quadruplicitie of them, one of which Quadruplicities, may co●●●ine the Loga●●th●es ●f ●●●bers, another may comprehend the L●●arythmes of sins and the other two may be 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 be properly called the Circle of numbers: in the next quadruplicitie of Circles there are graduated the Logarythmes of sins, from 34. m. 22. se. unto 90. gr. noted with the letter S. containing two revolutions, that is, two perfect 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 gins 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 sins, may be therefore called the Circle of sins. Lastly, the next two quadruplicities of Circles which are noted with the letters T. T. do contain four 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 gins 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 gins 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 gins 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, Sins, 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, Sins, and Tangents, or in respect of operation (as some others lately called the Circles of my King singly projected on a Plate or plain) 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 of numbers, another for the Logarythmes of Sines, and the other two for the Logarythmes of Tangents. If the projection be of 50. Circles, than 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 facility 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 help of a thread & lead, or a single or double Index at the Centre, or peripheria: upon which I deliver several ways in this following Treatise. How it may please some men's affections I know not, my intentions and desires are free. Now since the publication of the first kind of Logarythmes Projection, Anno 1630. or the original of this enlarged, it hath pleased many about this City and Kingdom to take liking thereunto, some contenting themselves with the double projection, with a movable and fixed Circle, some with the single having an Index at the Centre: but generally the most part have been by some invited if not forced to that which carries, as they say, with it the applause 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 Centre, is a better way than that of my Ring, or that of a double Projection on a Plain: The authority of whose words to some ignorant mechanic composors of that Instrument, was a sufficient motive ever since to crown his words with a divulged rumour out of their borrowed knowledge to maintain his assertion, to put off their commodities, howsoever if the saddle (as the proverb 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 appear in it, and not our supposed Author, which assumption of fitting of it so with an Index on a Plate, had been enough if not too much, without such a divulgation of endeavouring what in him lieth, and in others, to annihilate and beat down the way which I writ upon, and to glory in the raising up of his supposed own: thereby not only possessing men with an untruth, but making me also ignorant in my choice, 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 Centre, had I first writ upon and left the other way, which is the double Projection for some other to write upon: then might he have used not indirectly that comparative aspersion of better. But before I writ of the Natural sympathy of this projection, I was not unadvised which projection to present unto the King, and to the public view of the world first, but considered intentively with myself the excellency of both ways, 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 aptness 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 help it in operation, but the motion of the Circle itself, 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 enlarged; and instantly there was presented all other numbers in the same proportion; By an Index on a Plate in a single Projection it was gross and course for the form (in respect of a Ring) and for operation there must be besides extending the feet of the Index to the members, upon every several question a new search of numbers with a new motion (which extending of the feet of the Index, was the same with Mr. Gunter's Invention of his Ruler and no new invention) Besides if the single Projection be enlarged 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 seems not a little harsh & difficult; all which the Ring or the way of the Ring on a plate, in the double Projection enlarged, doth naturally avoid, and not only caries a facility in its operation, but retains in it also a special advantage in its performance, once rectified, for the eye and the hand may work together, and what the eye finds in proportion, the pen may presently express 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 form and facility (though perhaps the succeeding times may) either as before to disannul the work, or for the difficulty that was found in an unexpert workman in the true composing and making of the work, for if the Circles on the Ring, or double Projection on a plain being not exactly composed and graduated may cause some small error in operation (which is only from an excentricke motion) the single Projection hath not only the same defect, but also a second to help it, to wit the Index, for by how much the legs of it are long and the Instrument large, by so much the more is it subject to error, 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 pass by the errors that are subject to the best kind 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 Projection is better than a mooveable and fixed Circle, which I conceive to have reference unto four generals about the Instrument. First, either in the form of the Instrument; Secondly, in the ordering of the Circles thereon; Thirdly, the expedition that is found in the practice thereof; And fourthly and lastly, the copiousness of the uses of the Instrument: and other c●uses I conceive not, why this single Projection with an Index is better than 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 be an equal extendure of magnitudes unto both Instruments, and so as a thing common unto them, and no wise different. And for the price it may be made as cheap, if not cheaper hereafter as I shall order it for these that aflect them: Now in the first place as touching the form, upon that I have spoken somewhat already, (as afore said) and may be sufficient: as for the second general 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 Book noted with the letter A, for if the Circle of numbers noted with N, N, be cut through, and the Centre of that Circular plain be fastened, so that it may move upon the Centre of the other Circle, it shall fully represent the projection of my Ring upon a Plain, to which may be placed a small single Index as in the scheme of the title page B, to help the eye for the finding of opposite numbers, these Circles of the mooveable, or fixed Circles on the plain 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 piece some other thing on the other side, therefore to avoid 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 Plain before mentioned, noted with A. But perhaps it may be objected that the Circles are easily continued on a Plain, and the Index being at the Centre the edge of it, doth accuratly cut each Circle in the proportionals, which intersections in a Ring are defective and difficult to find 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 compliments to 90. gr. but for greater expedition in working they may be continued as is seen in the great scheme A, which continuation I learned not from another (as may be 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 seems to be troubled to point out some opposite numbers, a small edge of metal may easily supply that (as many use to do) but the graduations being so ne'er the edge of the Circle, the proportionals are sufficiently given by the eye alone without such an edge. Now if in these respects the single projection with an Index, is better than that of a mooveable and fixed Circle being easily supplied as aforesaid, it is but a poor one, in common sense. But if the way with an Index on a single Projection be not better than that of a double for the former respects, than it may be in the third general to wit the Instrumental expedition: in which there needs little declaration to prove the double Projection to have a greater expedition than the single projection with an Index, seeing it appears so obvious, that what can be quicker; then having moved 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 feet that it fall upon his third number, and afterward to look for the second foot 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 than that which I have delivered in a double projection. But to pass by all the former generals as trivial and of small consequence, let us weigh seriously things more material touching both ways of these Instruments. If the said single projection on a plate with an Index at the Centre be not better in respect of its expedition in operation, then must it necessarily be (to prove the Author's assertion) in the Instruments fourth general, to wit, in its copious performance, which I hold either to be in the general, or particular use; In the general I considered the finding of proportionals, and that's 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, either 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 than one kind of proportionals, the Ring, or a mooveable and fixed Circle on a Plain, scorning as it were such lameness, or such an injurious tie from its natural property, showeth by motion infinite operations in various proportionals, even through the whole body of the practical part of Mathematical Art, which would be too copious for me to declare, or for the Readers patience to peruse, only some common uses by such motion I will deliver, somewhat to prove my assertion, that the single projection with an Index, is not better than that with a mooveable and fixed Circle, therefore. 1. First, the movable being moved about at pleasure, as 1. in the movable passeth by any multiplier in the fixed, so doth any multiplicand in the movable, point out its product in the fixed, or contrarily, as any divisor in the movable doth pass by, 1. in the fixed, so doth any dividend in the movable point out his quotient in the fixed, so as 12. (the months in a year) or 52. (the weeks in a year) or as 365. (the days in a year) in the movable in motion doth pass by 1. in the fixed, so any somme of money in the movable, doth point out its monthly, weekly, or daily expenses in the fixed. 2. Secondly, as 7. in the movable doth pass by 22. in the fixed (Archimedes Proportion between the diameter of a Circle and its Circumference,) so doth any diameter in the movable, point out its Circumference in the fixed, vel contra. 3. Thirdly, as a hundred weight of any commodity in the movable, (or any other weight or measure) passeth by its price under 100 pound in the fixed, so right against 1. in the movable, is the price of a pound weight of that commodity amongst the decimals in the fixed vel contra. 4. Fourthly, the movable being moved about at pleasure, the Interest of all sums of money according to any rate in the hundred is given; for as 100, in the movable passeth by its interest in the fixed, so every sum of money in the movable, doth point out its interest in the fixed, vel contra. 5. Fiftly, as 1. in the movable passeth by any sum of money in the fixed, so any number of years in the Circle of years, doth point out the amount of that money in the fixed, according to the term of years that th● money was forborn. 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, etc. in the movable passeth by a greater, or lesser measure, or weight in the fixed (in homogenial things) so the measures of the parts of any of these wholes in the movable will point out in the fixed the proportional parts of any other whole by way of augmentation, or diminution. 7. Seaventhly, as 1. in the movable passeth by the square of the side of any of the ten regular Plains, so doth each plain note in the movable point out right against it, its Area in the fixed; and as any kind of measure to the Pole in the movable, passeth by its quantity in the fixed, so doth any other kind of Pole point out its quantity or Area, being measured by that Pole, etc. 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 conjoined with the Circle of Numbers in operation, or the Sins with themselves or joined with the Tangents, then by motion you have the sides and Angles of infinite plain and Spherical Triangles for practical uses, either in Geometry, Astronomy, Navigation, Fortification, Horot ogographie, Geographie, etc. 1. First, as the sine of 90. passeth by 60. in the fixed amongst the numbers, so the sine compliment 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 Tropical point, in the fixed, so the sine of any degree of the Sun's distance from the Equinoctial points will point out the sine of the sun's declination answerable to that distance. 2. Secondly, as the sine of any Rumbe in the mooveable from the East or West, sailed 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 depress the pole a degree in that Latitude. 3. Thirdly, as the sine of any Latitude in the movables passeth by the sine of the Tropical point in the fixed, so the sine of the Sun's distance from the Equinoctial points in the mooveable, that passeth by the sine of 90. in the fixed, doth point out the sine of the Sun's greatest degree of the distance from the Equinoctial points that the Sun will be due East in that Latitude. 4. Fourthly, as the sine of 90. in t●e mooveable passeth by the sine Compliment of any Latitude in the fixed, so right against the sine Compliment of all Declining plains in that Latitude in the mooveable, are the sins of the degrees of the styles heights in Horologographie agreeable to these declining plains in the fixed. 5. Fiftly, as the sine Compliment of any Latitude in the mooveble passes by the Tropical point in the fixed, so the sine of the Sun's distance from either of the Equinoctionall points, will point out right against them the sins of the Sun's Amplitude. 6. Sixtly, as the Tangent compliment of any Latitude in the mooveable passeth by the sine of 90, in the fixed, so the Tangent of the Tropical point in the mooveable, doth point cut in the fixed the sine of the greatest difference of ascension for that Latitude 7. Seventhly, as the sine of the Sun's position at his setting or rising, or the sine of the hour from 6. at that instant in the mooveable passeth by the sine of 90. so the sine of the Sun's declination in the former, & the Tangent of that declination in the latter, will point out the sine of the height of the Equinoctial in the former, but the Tangent of the same in the latter. In this nature you have infinite operations performed by motion in this double Projection of a mooveable and fixed Circle; which by a single Projection with an Index cannot as before possibly be performed, therefore if in this regard the way of the Index on a single Projection be not better but is fare inferior 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 Instrumental 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 shows many pleasant, and useful propositions. But this none, or very few at all, as only the Logarithmes of numbers, the natural sins and the Tangents of the Logarythmall, etc. But the double projection with a mooveable and fixed Circle doth not only show that, but being at any position carries with it such an excellency that it assumes unto itself 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 only: a touch of which I will unfold & unveil which never yet came to a public view. 1. First, the Instrument lying upon a Table open to the eye and being at any position, mark what numbers in the mooveable and fixed are opposite one unto another, according to which proportion there is represented infinite other proportionals, in the same proportion, for one number is opposite to another through the whole Circle of numbers, sins, and Tangents, from which one might apply the proportionals in numbers to the use of things, to expenses, to proportions in Buildings, to fortifications, measurations, but too great a prolix discovery would tire the Reader in that which he may easily from it apply hereafter unto himself. 2. Secondly, mark what number in the fixed, (in the Circle of numbers) is against any number of years in the mooveable, which suppose a Legacy to be paid for so many years to come or a sum of money forborn 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 Legacy or sum of money in the fixed. 3. Thirdly, in Horologographie, you have the distance of the hours in a poler, and Meridional plain, without operation, for mark 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, etc. of the styles hight) so right against the Tangent of the equal hours in the mooveable you have the hour distances in the Circle of Numbers in the fixed. 4. Fourthly, the hours of a horizontal or vertical dial, for some one Latitude or other is shown, for mark what degree amongst the sins in the fixed (which represents the Latitude) is against the sine of 90. in the mooveable, so the Tangent of the equal hours in the mooveable, doth point out the hour distances (for that Latitude) amongst the Tangents in the fixed, and these hours serve for a vertical Dial in the compliment 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 movable, such are the sides of those figures whose quantities are equal 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 equal solidities the one unto another. 7. Seventhly, you have infinite obliqne angled plain Triangles represented, and such who have equal Altitudes but different Basis the sides of several parallellograms, equal unto one and the same square or the quantity of a Triangle given in Acres, the perpendicular and Base is also given: For first the sins of the Angles on the mooveable will point out their sides of the Triangle amongst the numbers in the fixed, vel contra. And secondly, mark 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, than the equal 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 equal unto the square made of the Triangles perpendicular. And thirdly, the quantity 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 half perpendicular according to the Area given. 8. Eightly, mark 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 spheres Circumscribing diameters, but if the said number in the fixed be supposed to be the semediameter of a sphere, the numbers in the fixed (in the Circle of numbers) against the solids circumscribed notes in the mooveable shows the sides of these regular bodies that will circumscribe that sphere. 9 Ninthly, mark 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 sphere, the side of a plain figure, or that of a solid body: so the numbers in the fixed in the Circle of numbers against the potential notes in the mooveable shall represent the diameter, Axis, or side of its homogenial figure, or solid, according to the proportion of these potential notes in the mooveable. 10. Tenthly, mark what numbers in the Circle of numbers in the fixed are against the notes of the regular figures, such shall be their Areas, and the numbers in the fixed against the regular Bodies convexities, such is their superficial convexity: 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 several 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 Nobility or Gentry with the attribute of juggling, against the simple modesty of the Author,) I have in some measure supported their honours in that particular in the Epistle at the end of this Book: and that we may say something more upon the excellency of this Instrument without multiplying of tautologized and needless prefixed gradual numbers, or Circuitions, if not Circumlocutions, in the naked truth of this Instrumental projection, according to its natural property. The Roots of all square and cubic numbers without partition are given; and that by an inspection of the eye only. Thus I might have extended myself more copiously in the excellent use of this my mooveable and fixed Circle, and even from the Instrumental position by an inspection of the eye only without motion, compile a large Book of its ample performance, but in that which I have delivered I have only but scatteringly glanced upon things, as making way for many occasions, and as a motive to a further inquiry: It's an ancient proverb amongst us, good wine needs no Bush, but the wine must not be fast locked up then, that none can come by it, if so it wants both bush and key, and to some such needless expressions might be avoided, the Instruments own excellency will to the more learned easily present itself that which I have published: concerning it, I glory not in, but only desire to satisfy those who would see the difference of both ways, with, and without a mooveable Circle, & to let others know the truth of things which are conceited, and carried away with opinion only, that the way of the Index on a single projection is better than the way of a mooveable and fixed Circle, which both in regard of expedition as also copiousness of the Instrumental use, by motion or without motion comes short of the other. What means the Author's divulgation then, that the way of the Index is better than the way of a mooveable and fixed Circle, I know not, whose known skill in the whole Systeme of Mathematical learning will easily free him from the suspicion, that the way can be made, or the subject unvailed for him. But I have now a little more made bold to unveil the subject 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 enlargement, which hitherto lay in obscurity, and as a general benefit to those that affect the way of this Instrumental practice. It were good that the divulgers would prove their aspersions, touching the word better, that others might participate substantially of their better way by the Instrumental performance, either 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 beat down the use of the other, which savours of too high a conceit of the one, and too great a detraction from the other: Too great & too lose an aspersion hath been 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 world's rumour as it hath since the last publication of this Logarythmall projection Circular; howsoever, here is my comfort, the guiltlessness and innocency of my cause, which may teach me, and others carefulness hereafter, how and what we publish to the world, seeing there are such carpers, and maligners even of the most useful and best things, yea, such busy bodies who mar that which others make, who scorn to have a second, knowing all things and admiring nothing but themselves, such who have stings like Bees, and Arrows always ready to shoot against these whom they dislike, such who while they will needs have many callings neglect their own; sharp witty cryticks, Diogenes like, snarling at others, and not looking home unto themselves, but by all means endeavouring to take away the mantle of peace, and rend the seamelesse coat of love and amity. If things be not done well by others than they triumph and send forth their invectives, if well, they profess it nothing, and cannot pass without their censure. To speak ill of a man upon knowledge shows want of Charity; but to raise a scandal upon a bare supposition, & to act it in Print, argueth little humanity, less Christianity: but enough of this if not too much, I am sure some have casted too much already, perhaps others hereafter may help to bare a share, for my own part I desire no favour but the truth and equity of my cause, and the due weighing of things with their real circumstances. Veritas non quaerit Angulos. I desire no shifting, or pretences, but if I have done others wrong let me suffer; If I have been wronged by others let me have truth, and right done me, that's all I require. Who am An ever well wisher to the truth and thee. R. D. To the Reader. SInce my first publication of the uses of my Mathematical Ring or the Logarythmes projected Circular, I have been oftentimes invited by sundry persons for the way of the projecting, and dividing of the Circles of my Ring upon a Plain, so that it might be made in Pasteboard to avoid the charge of the Instrument in metal, for such which have not abilities to buy, and for others, who would first see the practice on it, before they would be at the cost of the Instrument in metal: for whose sake, and use, desiring to satisfy the affectionate, and for a public benefit, (rather than aiming at mine own 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 be used on a Plain as it is there described, noted with the letter A. and the other comprehending that Projection enlarged, noted with the letter B: that so such may make use of them more readily, to avoid the labour of dividing the Circles; which schemes being pasted on a Pasteboard are ready for use. And yet further to satisfy those that are desirous, I have delivered also in the first place ensuing, how those Circles are projected & divided, that so they may be made according to any magnitude. In the second place how several ways they may be framed in a Ring: In the third place I show the enlarging of the Instrumental Invention in these Circles to as great a magnitude for use as may be desired. In the fourth place I deliver several ways how these Circles enlarged may be accommodated for Practical use. In the fift place, I make a description of the Grammelogia, or Instrument in the particular Circle of my Mathematical Ring, projected on a mooveable and fixed plain, to wit, of the former scheme A. And in the sixth and last place, I will declare the excellent uses of both these Instruments, in the Practical parts of Arithmetic, Geometry, Astronomy, Horolographie, Navigation, &c, Detailed depiction of the logarithmic ring (circular slide rule) Of the projecting and dividing of the Circles of the Mathematical Ring, and of the enlargement of the Invention, either in a single projection, or in a double, and that several ways. FOr the first, according to any semediameter describe several Circles concentrical, as here are represented by the figure A, the outmost of which may be noted with the letter E, serving for the Circle of equal parts, and be divided into 100 1000 or 10000 equal 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 be set the letter S, which may represent the Circle of Sines, and the inner Circle may be noted with the letter N, which may represent the Circle of Numbers: These Circles of Numbers, Sins and Tangents, may be divided out of the Table of the Logarythmes of Numbers, 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, mark what number there is against 2. in the Table of Logarythmes, which is 3010. and account it from E, in the Circle of equal parts, (if the Circle be divided into 10000 equal parts, but if the Circle be divided but into 1000 equal parts, then account only 301. parts) which is against 2. in the Table of Logarythmes (as aforesaid) and will be at α, lay a Ruler upon it and the Centre, and intersect the Circle of Numbers N, in 2. then mark what number is against 3. in the Table of Logarythmes, which is 4771. and account it from E in the said Circle of equal parts, viz. at β, and lay the Ruler upon the Centre, and to the said 4771. or at β, and intersect the Circle of Numbers N, in 3. Again 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 proceed until 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 capital divisions of that Circle. Then may the spaces between 1, and 2. between 2, and 3, etc. be subdivided by the consequent numbers in the Table of Logarythmes thus. Mark what number is against 11. in the Table of Logarythmes, which is 413. account this number from E. in the Circle of equal parts, viz, at δ, and laying a Ruler upon the Centre A, and the said δ, or 413. note the Circle of numbers N, in ε, then mark what number is against 12. in the Table of Logarythmes, which is 791. account it from E. in the Circle of equal parts, viz. at ζ. and lay a Ruler upon it and the Centre, cut the Circle N, in n. and so go on until you have divided the whole space between 1, and 2. which will be when you come to 20. against which in the Table of Logarythmes is 13010, etc. 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 between 2, and 3. in the Circle of Numbers N. from the Logarythmes which are against the Numbers between 20. and 30. and so of the rest of the spaces between 3. and 4. between 4, and 5, etc. until you come to 1, which represents 100 then if you will subdivide the space between N, and ε, mark 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 Centre and cut the Circle N, in ι. than mark what number is against 102. which is 86. account it also from E, in the Circle of equal parts; and lay a Ruler upon it and the Centre, and cut the Circle of Numbers N, in λ, and so go on, until 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 only the several Revolutions. In like manner may be divided the space between ε, and n, in the Circle of Numbers N out of the Logarythme between 100, and 200. and so of the rest: But if the Instrument be large than it is requisite to divide the space between N, and ι, between ι, and λ, etc. which will be from the Logarythmes which are between 1000 and a 10000 Therefore in a small Instrument you may begin to divide it at the Logarythme of 101. until you come to 1000 but in Instruments more large begin from the Logarythme of 1001, until you come unto 10000 and if the divisions fall large at the beginning, they may be divided successively from 10001. until 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 Sins, 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 enlarged; therefore I have endeavoured to be very plain in the opening of the original of the work, that so the ensuing more compendiously may be conceived. 2. Of the graduation of the Circle of Sines. 2. To divide the Circle of Sines, you ought to have reference to the Table of Sines, and the work for the graduation of it is in a manner nothing different from the former, only here may be noted, whereas all the Logarythmes of Numbers are comprehended under one absolute revolution of a Circle, the Sins (which are sufficient for practise) will comprehend or require two revolutions; The first of which gins 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 gins again at the said 5. gr. 44. m. 22. s. and ends at 90. gr. and is divided thus: In the Table of Sines; mark what number is against 1. gr. under the title Sines, which is 8. 2418. (the Index 8. being rejected) of which account only but 2418. from E, viz. at a, and lay a Ruler upon the Centre, and upon the said a, or the Number 2418. and intersect the Circle of Sins upon the inside in 1, which represents 1, gr. then mark what number is against 2. gr. under the title Sines, which is 5428. account this from E, in the Circle of equal parts, viz. at b, and lay a Ruler upon it and cut the Circle of Sins upon the inside, as before in 2, which stands for 2, gr, and so proceed until 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 minutes, 5ths. of minute's 10ths of minutes, etc. according to the greatness or smallness of the Instrument: The inner side of the Circle of Sines being thus divided, mark 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 equal parts, viz. at e and lay a Ruler thereto and the Centre, and intersect the Circle of Sines S. on the outside in 6. gr. then mark what number is against 7. is 858, account from E, in the Circle of equal parts, viz. at f, and lay a Ruler thereto and intersect the Circle of Sines S, in 7. gr. and so proceed until 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. Look in the Table for 1. gr. and mark 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 Centre, & intersect the lower Circle of Tangents T, on the outside in 1, which is opposite to 1. gr. in the Sins, 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 Centre, & intersect the Circle T, in the outside which shall represent 2. gr. in that Circle, and so go on until you come to the line of Conjunction, which will be at 5. gr. 42. m. 40. s. then mark what number is against 6. gr. under the title Tangents which is 216. account this Number from E, in the Circle of equal parts, viz. at e, and lay a Ruler upon it and the Centre, and cut the Circle of Tangents T, on the outside of it in 6. gr. then look what number is against 7. gr. under the title Tangent which is 891. account it also from E, in the Circle of equal 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 go 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 mark 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 equal parts, viz. at g, and lay the Ruler upon the Centre, and intersect the Circle of Tangents T, on the inner side in 46. gr. then look 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 equal parts, viz. at h, and lay a Ruler upon it, and the Centre, and cut the Circle of Tangents T, on the inner side in 47. gr. and so go on in dividing the inner side of this Circle until you come to the line of Conjunction, which will be at 84. gr. 17. m. 21. s. then seek 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 equal parts, from E. viz. at 1. and lay a Ruler upon it, and the Centre, and cut the lower Circle T, upon the inner side in 85. gr. and so proceed with the rest of the graduations until 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 only the compl●ments of the former, so the compliment of the Tangent of 40. gr. is 50. gr. the middle space between which is 45. again, the compliment of the Tangent of 35. is 55. the middle of which is at 45. Now the distance between 50. and 45. is the same between 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 be so inserted upon the edges of Circles of mettle turned in the form of a Ring, so that one Circle may move between two fixed, by help of two stays, 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 Sins noted thus, 90, 90. 80, 80. 70, 70, etc. unto 6, 6. and each degree subdivided, and then over the former division and figures, 90, 90. 80, 80. 70, 70. etc. you have the other degrees, viz. 5, 4, 3, 2, 1. each of those are divided and subdivided by several points. Secondly, (if the Ring be great) near the outward edge of the side of the fixed, against the Numbers, are the usual divisions of a Circle, and the points of the Compass: serving for observations in Astronomy, or Geometry, and the sights belonging to the Ring may be placed on the mooveable Circle. Thirdly, opposite to these Sins 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. etc. unto 45. 45. which numbers or divisions serve also for their compliments to 90. as before: so 40, gr. stands for 50. gr. 30. gr. for 6●, gr. 20. for 70. etc. and each degree here 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, etc. 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 Sins, in every thing answerable to the first description of Sins on the other side. Fiftly, on the edge of the Ring is graduated a part of the Equator numbered thus, 10, 20, 30. unto 100, and thereunto is adjoined the degrees of the Meridian enlarged and numbered 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 concern Latitude, Longitude, Roumbe, and distance in Nautical 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 hours and parts with an Axis, to show both the hour, and Azimuth, and within this Circle may be hanged a Box and Needle, with a socket for a staff to slide into it, and this accommodated with screw pins, to fasten it to the Ring, and Staff, 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 form in a Ring, so that all the graduations may be upon one side only. THis may be done by a double projection, if the mooveable Circle be so fitted that it move upon a Plate, and be of sufficient breadth to contain all the Circles of the single projection, and that one of the fixed Circles retain the same breadth with the mooveable: as for the innermost fixed Circle, that may be but as an edge only, then may there be a small channel 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 moved to and fro at pleasure, in the Channel, which Index may serve to show the opposition of Numbers; then upon the other side of the Ring may be placed what the fancy 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 only. THis is done by having a narrow Circle turned, to move about the outward edge of a broad Circle, so that being framed together, it may to the eye seem but as one Circle only, then may all the Circles of the Projection be described and graduated on the broad Circle, and to the narrow mooveable Circles may be placed two like edges of Mettle as the parrs of a double Index, which may extend unto the largeness of the broad Circle, which may move somewhat strait in a Channel made in the narrow Circle, for then these edges being placed unto any two proportionals in the Projection, if you shall also move that Circle which carries these two edges; as the one edge passeth by any Number in that projection, the other will show the fourth proportional, in the same projection, and this way doth avoid the moving of an Index at the Centre; for that is supplied by the motion of the Peripheria of the Circle, and so according to this direction you may retain the form of a Ring, and have all the Circles to the eye upon one of the sides of the Ring only; which form is more Gentile, and Mathematical, then if the projection were placed on a Plate or Plaine; Many other forms might be delivered, 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 work accuratly trigonometry. THis augmentation of the projection is very facile, being but only the doubling, tripling, quadrupling, quintupling, decupling, centupling of Circles in the single projection, and so to conceive such decuplation, centuplation, etc. only 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 only one continued Circle, and the way how, we will somewhat open by this general Rule, Divide 10000 (which is the Radix of numbers) by that number which you intent to have the projection enlarged upon, according to a Ratio, or proportion assigned, and the Quotient shall show 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 be Quadruple to that which is single, that is, four times greater, then, having described four 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 four Circles) must be divided into 25000. equal parts, therefore let it first be divided into 25. equal parts, each of those parts divided into halves, and every one of them divided into 5. parts, so the whole Circle shall be divided into 250. equal parts, then may each of those divisions be divided again into halves, and every one of these halves may be divided also into 5. other equal parts, and then the whole Circle shall be divided into 2500. parts, and if the divisions be great enough, let every one of them be divided again into 10. other equal parts, so the whole Circle shall be divided into 25000. equal parts; if they cannot be divided into so many parts, yet we may conceive them so to be divided; and so of other divisions for this projection, which Circle of equal parts may be noted at each capital division thus, 1, 2, 3, 4, 5, 6, etc. unto 25. How to divide the projection enlarged. Now to graduate these quadruplicities of Circles from the said Circle of equal parts, you ought to keep the same method that was used in the dividing of the single Circles in the single projection, only 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 cyphers, 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 seek the Logarythme of the said 17. which is 123044. (the Index 1. being rejected) account from E, in the Circle of equal parts in the scheme B, only 23044. and laying a Ruler upon it and the Centre, intertersect the lowermost Circle of the former, which belongs to the projection of numbers in 17. but if 18. were to be inserted into the projection, seek for it in the Table, so right against it is 125527. account beyond the line of conjunction only the 25527. and lay a Ruler upon it and the Centre, 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 equal parts, the other part of the number, viz. 875. and lay a Ruler thereto, and to the Centre, and intersect the former Circle in 19 and so proceed for the dividing of the other numbers until 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 it 52, etc. and coming 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 proceed until you come to the first Radix, viz. 100000. or to the line of Conjunction at the point E. this being fully conceived, and having a scheme here already graduated, it shall be easy from this to divide any other projection which is to be augmented in a different manner: But if the projection enlarged be of a Decuple, or Centuple proportion, than the outmost Circle is to be divided only into 10000 equal parts according to the Circle of equal parts E. in the single projection, and then the Logarythmes in the Table will divide them without any consideration (the Index being only rejected) and as the whole Circle of Numbers in its quadruplicities of Circles may be 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 projection in this kind. Thus for the augmenting of the Circles of the projection of my Mathematical Ring, and of the dividing of them. Of several ways how the Circles of the Mathematical Ring (being enlarged) may be accommodated for Practical use. FIrst the Circles being projected singly upon a Plain, an Index with two feet or a flat Compasses may be placed at the Centre to open at any two terms assigned: and so to move it about as occasion requires, but in stead of it a semicircle may be fastened 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 moved Circular, as the Radius of the semicircle passeth by any number in the projection, so the edge of the Index in motion shall show the proportionals in that projection; or a paper edge, and a thread, with a bead may be sufficient, seeing the Circle of equal parts will show the equal distances of the proportionals, etc. But the way of operation upon this single projection with an Index, etc. to a learner will be somewhat troublesome for the several observations that necessarily depend upon that way, and cannot be avoided, But a double projection makes it very facile. 2. A second way of accommodating this enlarged Logarythmall Projection, for practical use, with a single Index only. THis is done by having a double Projection enlarged on a Plain, the one to be fixed and the other to be movable, (agreeable to the scheme A, in the title page) for being so fitted it shall operate with the same facility and expedition, as though it were a single Projection, the proportional numbers being always by opposition, & in like differences of Circles, one caution considered; and the Index at the Centre, is only but to help the eye in finding them. 3. A third way of fitting of the Circles of the Projection for Practical use. THis is performed by help of a double projection, the one to be fixed, and the other to be mooveable: But so that the Circles of the mooveable being described, every other Circle may be cut out, that is, that there may be a vacuity between each Circle, then let the edge of the mooveable Circle, be divided and graduated answerable unto the graduation and divisions of the projection on the fixed, but so that the whole mooveable Circle being placed at the Centre, the divisions, graduations, and figures on the said fixed Circle, may be seen conspicuous through these Channels, and vacuities which are cut out in the mooveable, so this Projection shall also show, or give the proportionals by opposition, as the former. 4. A fourth way, how the Logarythmall projection of my Ring enlarged, may be fitted in an instrumental form for practical use in Calculation. THis may be done according to my great Cylinder which I have long proposed (in which all the Circles are of equal greatness,) and it may be made of any magnitude or capacity, but for a study (he that will be at the charge) it may be of a yard diameter and of such an indifferent length that it may contain 100 or more Circles fixed parallel one to the other on the Cylinder, having a space between each of them, so that there may be as many mooveable Circles, as there are fixed ones, and these of the mooveable linked, or fastened together, so that they may all move together by the fixed once in these spaces, whose edges both of the fixed, and mooveable being graduated by help of a single Index will show the proportionals 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 Mathematical Ring on a Plain, according to the diagramme that was given the King (for a view of that projection) and afterwards the Ring itself. THe parts of the Instrument are two Circles the one mooveable and the other fixed, the mooveable Circle is that unto which is fastened a small pin to move it by, the other Circle may be conceived to be fixed; upon the mooveable and fixed there are described thirteen distinct Circles, notwithstanding, on the mooveable ●nd fixed there are 24. several Circles considerable, 〈◊〉 which there are four in the mooveable, answerable to four 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 Calendar. E. Represents the Circle of equal 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, years, and months. A more particular description of each Circle, and first of the Circle of equal parts. FIrst, the Circle of equal 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 do stand for themselves, or such numbers unto which a cipher, or Ciphers may be conceived to be added, or taken away: so 10. stands for 1. or 10. 100 1000 etc. and so of the rest, and here note that the space between any two figures, must be conceived to comprehend the difference in denomination between them, that is, so much as one number is greater than another, so many divisions must be contained between these numbers, so if the 20. stands for 20. the 30. then shall stand for 30. and because the 30. doth exceed the 20. in its denomination by ten, there must therefore be conceived to be 10. divisions between the said 20. and 30. but if the 20. had stood for 200. and the 30. for 300. then the distance between 20. and 30. must be conceived to contain 100 divisions, and so of the rest: which Circle of equal parts, in its divisions represents such numbers as one hath occasion to use, and is of special and singular use if it were particularly applied, (though some one said upon the description of it, it was scarce of any use, but only that by help thereof the given distances of number may be multiplied, or divided as need requires) had I the way made for me, and the subject unvayled to help my fight, I should see its use better than now I do, & would not conceit myself to be so sharp sighted, but be thankful to any one that would unveil it for me yet ( 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 myself 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 equal parts doth not only represent itself, but also a part of the Equator, or Equinoctial Circle, containing 100 equal degrees, each degree being divided actually into 10. parts, so that each part doth contain 6. minutes, and is numbered 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 enlarged, or the unequal degrees of Latitude, according to wright's projection, figured & noted thus, 5. 10. 15. 20. 25, etc. unto 70. gr. each degree being divided into 10. parts, so that each part containeth 6. minutes 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, etc. unto 30. and each of the spaces is subdivided into 12. equal parts and is called the Circle of time, or years, the spaces noted with figures being years, and the subdivisions, months, and serves for to operate such questions as concerns interest and valuation of Leases, but the outmost Circle, which is noted with letters upon the inner side, is the Calendar of time, divided into months & days, and is adjoined to the degrees of a Circle, noted with the Signs that so readily knowing the day of the month, the Sun's place may be found, and so contrarily; which Circle of degrees serves also to take the Sun's high, and for the observation of Angleses. Of the Circle of Tangents. FOurthly, the next Circles to that of equal 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 near a Quadrant or 90. gr. and is graduated and numbered on both sides, having its beginning at the line of conjunction E N, at 34. m. 22. s. the numbers that are minutes are noted with a touch over them thus, 35● that so they may be 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 equal 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 than 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 minute, and from 10. gr. unto 80▪ gr. every degree is divided into 30. divisions, each division being 2. minutes then from 80. degrees 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 numbered 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 numbered thus, 6. 7. 8. 9 10. etc. unto 90. every degree in this Circle of fines is divided into parts according to the greatness of degree, and the Circle of sins on the movable is the same with this on the fixed. Of the Circle of decimals of money. Sixthly, next unto the Circle of sins, 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. etc. unto 23. and each space subdivided into 4. parts: which divisions so numbered 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. etc. into 19 parts & the spaces between each figure is divided into 12. parts, (or as many parts as each space will contain) which whole Circle so divided (is supposed to represent a pound of money,) is the decimal of shillings, or of a pound of money: the like may be done for the Decimals of weight. Of the Circle of Equated figures, and Bodies. SEventhly, the Circle which is a like situated unto this on the movable that is noted with the letter E. is the Circle of equated figure● and Bodies, serving excellently to facilitate and expedite such operations which concern regular figures, and the five platonical Bodies, with other occurrents, a touch upon which was shown 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 which contains 100 notes, 50. of them serving for Superficies, and the other 50. for Solids. Tabula notarum corporum solidorum & figurarum regularium. 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 Notae Solidorum. 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 Pentagon. Hex. stands for Hexagon Hep. stands for Heptagon No. stands for Nonogon De. stands for Decagon these letters joined with the former, as e. signify equal. in signify inscribed. c. signify circumscribed. 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 Sphere. these adjoined, s. signify solidity. e. signify equal. x. signify convexity in. signify inscribed. cir. signify circumscribed. These notes, P. 16. ½ feet to a Pole. P. 18. feet to a Pole. P. 20. feet to a Pole. P. 21. feet to a Pole. Ac. represent Acres: F.B. represent Foot of Board. F.T. represent Foot of Timber. f. t. represent Foot of Timber equated. Cir. D. represent Circles Diameter. Cir. c. represent Circles Circumference. Cyl. D/ s represent cylindrical. solidite. Cyl. e/ ● Cyl. d/ f represent cylindrical foot measure. these notes represent metals, viz. ☉ stands for Gold. ☽ stands for Silver. ☿ stands for Quiksi. ♀ stands for Copper. ♃ stands for Tin. ♄ stands for Lead. ♂ stands for Iron. 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 unequal 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 cipher or Ciphers 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. etc. but if 1. stands for 1. by augmentation, than the figure of 2. shall represent two, the figure of 3. shall stand for three, etc. and the space between each figure shall be the parts of 1. Secondly, if the figure of 1. stands for 10, than the figure of 2. stands for 20. the figure of 3. for 30. etc. hence it followeth that between the figure of 1. and the figure of 2. or between 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, numbering till you come to 30. or any other number, which divisions noted thus is only to help the eye in numbering. 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 between 200. and 300. and 100 more between 300. and 400. and so of others. Now seeing that ten ten make an hundred, there must be 10. divisions between the said 200. and 300. and every one of those divisions do represent 10. so the said note shall be half of the said 100 therefore at that note you may read 250. and the middle between 300. and 400. viz. at read 350. etc. Every one of the divisions which stand for 10. is divided again into 10. other divisions, the middle of which hath its division a little higher than the rest, to help 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 speak 243. here the denominations are Hundreds, Ten, and Unites: therefore the figure of 2, shall stand for 200. the four great divisions next the ●. shall be 40. and the next three small divisions shall represent 3. (which is within seven small divisions of ) and so of others. Thus for the description and numbering upon the Circle of numbers on the fixed. The numbers & divisions on the movable Circle, are the very same with that on the fixed; for if you move 1. in the movable to 1. in the fixed, there is represented to every number or division his opposite, not only 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 always fixed, and the other to be moved; and if any number in the movable be moved, all other numbers move with it: so if you move 25. in the movable in the Circle of numbers, unto 30. in the fixed on the Circle of numbers, right against 26. in the movable, is 31. and 2. tenths in the fixed, and right against 27 in the movable, is 32. and 4. tenths in the fixed, right against 30. in the movable is 36. in the fixed, against 46. in the movable, in 55. and 2. tenths in the fixed. Again, if 108. in the movable be brought to 15. in the fixed, right against 16. in the fixed is 115. in the movable, and right against 12. in the fixed, is 86. and 4. tenths in the movable. Thus what denomination you give unto the numbers in the Circle of numbers in the movable, you are successively to keep 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 perform the Golden Rule, or to find a Proportional 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 general, as well in Mathematical Calculations, as in Arithmetical Computations, and therefore may not unfitly be so called, and the Instrumental 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 work, because of expedition and facility: and the way of operation is thus: Seek the first number in the movable, Constructio. and bring it to the second number in the fixed, so right against the third number in the movable, is the answer in the fixed. Example 1 If the Interest of 100 li. be 8. li. in the year, what is the Interest of 65. li. for the same time. Bring 100 in the movable to 8. in the fixed, Constructio. so right against 65. in the movable is 5. 2. in the fixed, and so much is the Interest of 65. li. for a year at 8. li. for 100 li. per annum. The Instrument not removed, ☜ you may at one instant right against any sum of money in the movable, see the Interest thereof in the fixed: the reason of this is from the Definition of Logarithmes. Proportionales Logarithmi aequales habent differentias. Definitio. Necess● est igitur proportionales Logarithmos in proport●one Lincari distantias aequales habere. Example 2. If a Troop of 50. Horse have for their pay 140. li. how much shall suffice to pay a Troop of 64. Horse. Constructio. BRing 50. to 140 then right against 64. in the movable is, 179. 2. in the fixed, the monthly pay of the said 64. horse. And there immediately may you see the monthly charge of any number of Horse, for, the number of Horse given in the movable, right against it, is their pay in the fixed. Example 3 It is said that the proportion between the circumference of a Circle to his Diameter is ●. 22. Bring therefore 7. in the movable to 22. in the fixed, then immediately 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 movable, is the Circumference in the fixed; or right against any Circumference in the fixed, is his Diameter in the movable: Thus for the simple Rule. Example 4 Further uses of the Golden Rule in ordinary service in proportionating of things. Let FLX. represent the Perimeter of a Pentagonal Fort, and let the distance between the points of the Bastines, FL. be 926. foot, or KL the square side of a Building 470. foot, and the other dimensions, both of the Fort, and the Building according to the here under inscribed Tables. The distance between the points of the Bulwark. 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 Flank D E. 100 The line of defence D L. 700. The face of the B●stine E F. 264. The capital li●e A F. 224. The distance from the Centre to the Bastine A C. 564. From the Cottine to the Centre C I 456. The breadth of the Bulwark. G E. 310. The greatest square side of the Building K L. 470. foo● Q. A court within the middle of the Building. The distance between the middle of the Court and any out angle, as K A. 236. The least inner square of the Court E F. 200. Between any out corner of the Building, as RX. 180. 0. 0. 0. etc. a stone Gallery in breadth 36. And so of other under rooms to other uses. NOw admit another like Fort, or another like Building is to be erected, whose greatest distance between the aforesaid points of the Bastines, can be but 750. foot, or the greatest side of the piece of ground where the Building is to be made, is but 400. foot, what shall the several measures of this new Structure be, so that the Fort to the Fort, or the Building to the Building, in all parts be proportional? 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 several known measures in the movable, you have the several 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 movable 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 movable is 285. 9 170. 2. 153.1. 8 in the fixed. These numbers found out by the ordinary way of Arithmetic may trouble a nimble Arithmetician a whole hour or more, and therein subject to much error, but others 6. or 8. hours at the least, if not more; but by this Grammelogia, they are found out in less time than half a quarter of an hour: for so quick 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 down 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 return is gotten 50. li. how much shall A. B. C. D. and E. have, according to their several 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 several metals, allotted to make a Statue, Vessel, Bell, etc. A Gold, B Silver, C Coper, D Latin, and E Tin; now when the Metals were melt and cast, there was left a piece which weighed 50. li. how much Gold, Silver, Coper, Latin, and Tin doth it contain, that so the worth of that piece may be known. Or if there were 5. Companies, or 5. Captains, 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 keep them from mutiny, the General 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 lays in 84 tuns, B. 72. C. 48. D. 54. and E. 42. tuns; in the voyage by reason of tempest, for safeguard of their lives and Ship, there was cast over board 50. tuns of the loading, how much shall A bear of the loss, 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 yearly 84. li. B. 72. li. C. 48 li. D. 54. li. and E. 42. li. how much shall each one pay, etc. THus I might infinitely dilate myself upon one subject, tending to admirable uses, I only in this glance by things, making but way to the occasions: The resolution of which, and all othets of this kind, is drawn from this ensuing Axiom. Axiom. There is such proportion between any whole, and his parts, as between the like whole, either greater or lesser, and his parts: or between the parts and the parts, as between the whole and the whole. So in the first example, Declaratio. Add 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 proportional 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 movable unto 50. in the fixed, Constructio. so right against any particular part in the movable is his part proportional in the fixed, as there apparently is seen, and from thence they are taken and placed in a Table, as here under appears. 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 find a Line that shall keep any proportion assigned unto another line given. Constructio. Measure the line A by a scale of equal parts, then bring 3. unto 5. so against the measure of the line A in the movable, 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. etc. Propositio. 2 To divide a Line into any number of equal parts. Declaratio. Let it be required to divide the Line A into 23. parts: first, by a seal of equal parts measure the Line A, Constructio. which admit to be 51. parts, bring then 23. in the movable unto 51. in the fixed. So right against 1. 5. 10. 15. 20. in the movable, 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. than may those parts be easily subdivided. Propositio. 3 To divide a Line in such sort or proportion as another Line is already d vided. Declaratio. Let the Line B. C. be 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 contain 37. parts, bring 51. unto 37. so against the parts of R in the movable, 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 find a line in continual proportion unto two given lines, or a proportional 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 work 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 always accounted or sought out upon the movable Circle. Example. 1 As if 30. li. do rend 45. Acres of Land yearly, how much doth the yearly 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 movable be brought to his answer in the fixed: that is, bring 45. to 30. so, right against the thing demanded in the movable: that is, against 84. shall be the answer in the fixed, viz. 56. and so many pounds will rend yearly the said 84. Acres. Secondly, note further, that those three numbers as 45. Acres, 30. li. and 84. Acres, are distinguished by numeral 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 alway such proportion to the third number, as the second is to the first: Vel contra. From which by a more general name, it is called The Rule of proportion, for that it proportionateth things unto any proportion assigned; so is the said 56. a proportional 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 four numbers, 45. 30. 84. 56. are proportional 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 be greater than the second number. Contrariwise, if the third number be less than the first number, the fourth number is less than the second number. But in Reciprocal proportion this fourth number is inverted; so if the third number be greater than the first, the fourth number is less than the second. Example 1 So if 45. men in 30. days, will do a service, in how many days shall 270. men do it. Here the denominations alike are 45. Men and 270. Men, the answer to 45. men is 30. days, the answer to 270. men is required. Constructio. If you move 45. in the movable to 30. in the fixed, right against 270. in the movable is 180. days in the fixed: which answer is absurd, seeing there is more men allotted to do the work, there must most necessarily be less time. Therefore in all Questions of Reciprocal proportion, let the demand be sought out upon the movable, viz. 270. and brought to the first numbers answer in the fixed, viz. 30. so right against the first number in the movable, viz. 45. is the answer in the fixed, viz. 5. and in so many days will 270. men do that service, if 45. men do it in 30. days. Quest. 2 Again, if 3840. soldiers are victualled for 10. months, how many men may it serve that the said provision may last 12. months. Constructio. In this and all others (as before) bring the third number 12. in the movable, to the other numbers answer in the fixed, viz. 3840. so against the first number 10. months in the movable, is 3200 men in the fixed, and so many men will the same provision serve for 12. months. From which direction, those ensuing questions, and the like, may be resolved. Ques. 3 If I lend 140. li. for 7. months, if I should borrow of him 200. li. how long might I keep it; facit 4. months and 9 10. Quest. 4 According to the Statute, if Wheat beat 50. s. the quarter, the penny loaf should weigh 6. ounces and a half, 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 contain 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 do the same, etc. facit 35. ounces. How to proportion a Fraction that is not decimal, into a decimal. So if 8. & 12. 40. were to be used, 12. 40. must be changed into a decimal, thus: bring 40. in the movable to 10. in the fixed, so right against 12. in the movable 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. Again, let 63. 84. be a fraction which is to be used, this cannot be found out upon the Grammelogia: change it therefore into a decimal. Bring therefore 84. (the Denominator) to 100 in the fixed, so 63. (the Numerator) in the movable, gives 75. in the fixed; so 63. 84. is now changed into a decimal 75. 100 the same in value with 63. 84. and so of any other Fraction that is not decimal. This for Lineary Proportion. Of the Golden Rule, or Rule of Proportion, in respect of Lines and Quantities in plain Figures. Pro. 1 IF the demand be of the quantity, As if ●he Diameter 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 known to the other line, that is 7. to 18. so right against 38. and 5. 10. in the movable is 99 in the fixed, which looked out in the movable, 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 plain Figures. Pro. 2 If a piece of Land of 20. Pole square bedworth 30. li. what is a piece of Land of the same goodness worth, which is 35. Pole square every way. Constructio. Bring 20 to 35. so right against 30. in the movable you have 52. and 5. 10. in the fixed; and right against this 52. and 5. 10. in the movable you have 91. and 8. 10. in the fixed, the worth of that land. Pro. 3 If a piece 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 piece of ground which is 40. paces square Constructio. Bring 50. to 40. so right against 1600. in the movable is 1280. in the fixed, Pro. 4 Our English land measure is 16. foot and a half to the Pole, the Irish Pole hath 21. foot, how many English Acres doth 30. Irish Acres make. Constructio. Bring 16. and 5. 10. to 21. then right against 30. in the movable is 38. and 2. 10. in the fixed, and right against this 38. and 2. 10. in the movable is 48. and 6 10. in the fixed, and so many English Acres is contained in 30. Irish Acres, etc. ☞ Our usual measures in England to the Pole are 16. foot and a half 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 known in the movable, 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 be supported with Cubes of Silver, or other rich Material, 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 mettle be worth that is but one inch more in the side, viz. 5. inches. Bring 4. to 5. so right against 12. in the movable, Constructio. is 15. in the fixed, and right against this 15. in the movable is 18. and 75. 100 in the fixed, and right against this 18. and 75. 100 in the movable is 23. li. and 4. 10. in the fixed, and so much will the second Cube cost: this might be applied to the weight, worth, or quantity of other Solids. Pro. 2 A Pe●ce of 5. Inches boar or Diameter, requires for her charge 16. pound of Powder, what quantity of Powder will serve another Piece of 4. Inches in the boar. Bring 5. to 4. so right against 16. in the movable is 12. and 8. 10. in the fixed, Constructio. and right against 12. and 8. 10. in the movable is 10. and 24. 100 in the fixed; and right against this 10. and 24. 100 in the movable is 8. and 2. 10. in the fixed: the answer of Powder according to Cubick proportion, but Canoniers do somewhat qualify this proprotion. To find what Proportion in Quantity there is between 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 between the Solids, or how often doth the greater contain the lesser. Bring 10. to 4. so right against 100 in the movable is 40. in the fixed, Constructio. against this 40. in the movable is 16. in the fixed; and right against this 16. in the movable, is 6. and 4. 10. in the fixed; so the proportion between the Solids are as 100 to 6. and 4. 10. But how often the greater doth contain 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 movable, is the quotient in the fixed. Declaratio. So if it were demanded, how many days there is in 216. hours, because a day natural contains 24. hours, that therefore is the Divisor. Move then 24. to 1. and right against the said 216. in the movable is 9 in the fixed, and so many days is 216. hours. ☞ 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, than the Quotient shall have one place more. Example 2 So if it were further required, how many days there were in 360. hours, 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 movable, is the answer in the fixed, so right against 360. in the movable is 15. in the fixed, and so many days are there in 360. hours. This note serves only 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 years there is in 14600. days, there being 365. days in the year: this therefore is the Divisor. Bring then 365. to 1. so right against 14600. in the movable is 4. in the fixed, but by the former note ☞ it must be 40. and so many years is there in 14600. days, the Instrument not moved, right against any number of days, as 5000. 10000 20000. etc. in the movable, is the years in the fixed. With the same expedition and facility may you divide by fractional numbers. Further uses upon Division. Example 4 IN a year are 52. weeks or 365. days. If I would know the weekly expenses of any yearly sum of money. Bring 52. to 1. then right against any sum of money in the movable, you have the weekly expenses in the fixed: Constructio. But if you move 365. to 1. then right against any sum of money in the movable, you have the daily expenses in the fixed. So if the expenses yearly were 1000 li. or the charge of a certain Company of Soldiers: 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 year, 50000. li. or 100000. li. a year: for right against the charge or expense in the movable, is the answer in the fixed. More uses upon Division. Example 5 It is said that Land is bought after the rate of 14. year's purchase: if 14. be therefore brought to 1. right against any sum of money in the movable, you have the Annual 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, do but speak the sum 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 find the Scale to divide the Meridian line in a Sea Chartley, according to any breadth, & to a Latitude assigned. Example 6 Let the breadth 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. wright's projection, are 12. & 24 100 Bring this 12. & 24. 100 to 1. so right against the breadth of the Card in the movable, you have the Inches, or parts of Inches in the fixed to make your scale by to divide the Meridional line. So if the breadth or the Card were 33. Inches, Declaratio. right against it in the fixed is 2. Inches, 7. 10. the largeness of a degree of the Equator: if the breadth were 24. & 5. 10. right against it is 2. Inches: if 20. & 8. 10. then the breadth of a degree is 1. & 7. 10. if 14. & 7. 10. then 1. & 2. 10. if 8. & 5. 10. then 7. 10. etc. To multiply one Number by another, or to find the Product of two Numbers. Constructio. MOve 1. to the multiplier, then right against the Multicand in the movable 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 be one figure or place less 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 bear 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, than the first figure of that Product towards the left hand is ten or ten; if the Product have three figures, than the first figure of the Product towards the left hand is hundreds, etc. Example 2 To Multiply 18. by 5. Bring 1. to 5. then right against 18. in the movable 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 be multiplied by 4. move 1. to 4. so right against 35. in the moved is 140. by the last note ☞. Example 3 To multiply fractional, 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 movable 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 movable is 4. 10. in the fixed. Uses upon Multiplication. Example 3 12. Monetht make a year, bring 1. unto it, so right against any monthly expenses in the movable you have the yearly expenses in the fixed according to the note ☞: So if the monthly expenses were 75. li. right against it in the fixed is 9 which by the former note ☞ makes 900. if ● 50. li. for a month, right against it in the fixed is 1800. li. the yearly charges or expenses. Other uses upon Multiplication. Example 4 60. Minutes make an hour, bring 1. to 60. so right against any number of hours in the movable is the minutes of those hours in the fixed. Further upon Multiplication. Example 5 Admit lands be sold at 14. year's Purchase, bring 1. to 14. so against any Rent in the movable 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 movable 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 movable is 38. and 4. 10. in the fixed, and right against 38. and 4. 10. in the movable is 228. in the fixed, the Cube of 6. and 2. 10. Again to Cube 6. bring 1. to 6. for right against 6. in the movable is 36. in the fixed; and right against this 36. in the movable is 216. in the fixed, the Cube of 6. etc. To find Numbers in continual proportion unto any two Numbers assigned. Constructio. BRing the first number to the second, then right against the second upon the movable, is the third number in the fixed, and against this third number in the movable, is the fourth number in the fixed, etc. Declaratio. So if the numbers to be continued in proportion be 2. to 4. move 2. to 4. so right against 4. in the movable is 8. in the fixed, and 8. in the movable gives 16. in the fixed, and those numbers, 2. 4. 8. 16 etc. are said to be in continual Proportion. Example 2 Again, it I would continue a Proportion, as 2. to 3. move 2. to 3. then 3. in the movable shall point out 4. & 5. 10. in the fixed, and 4. & 5. 10. in the movable shall give 6 & 7. 10. i● in the fixed, and so on (if need were) to find others: and those numbers are said to be in continual proportion one unto another. ☞ The increase or interest of Mon●y from this ground is easily found, seeing the increase of the Money must be in continual Proportion to the Principal, as 100 li. is to his Interest. Example 3 As if the Proportion were to be continued to 40. li. as 100 to 108. Constructio. Move 100 to 108. then against 40. in the movable is 43. li. 2. 10. in the fixed: the first year's Interest and its Principal, & against this 43. li. 2. 10. in the movable, is 46. li 8. 10. in the fixed, which is the seconds years Principal and Interest: in like manner may you proceed to other years. ☞ The Instrument being at this stay, the eye may denote out at one instant the Interest of any sum of Money: for right against your number in the movable, is both Principal 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 years end, and so all other sums of money do offer themselves at one instant to the eye in their resolutions. To find a mean proportion, or many between any two Numbers given. MArk what number of equal parts in the fixed is against each of the given numbers, (which equal parts represent the Logarythmes of these numbers, if the Logaryt●mall Index be put unto them, which is a unity less than the places of any given number) and add these Logarythmes, or equal parts together, then take the half of that sum, which sought out in the former Circle of equal parts, right against it in the Circle of numbers is the mean proportional required, or the half difference of these two Logarythmall numbers, or equal parts being added to the lesser Logarythme will give the same, or sub. etc. But if many mean proportionals be required divide the differential Logarythme of the two numbers, or number of equal parts, by a unity more than the number of mean proportionals, which quotient being by succession added to the Logarythme or equal parts belonging to the lesser number, doth show the several mean proportionals required. So if between the Cubic numbers 27. and 64. two mean proportionals were required, the third part of the difference of equal parts between 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 mean proportional unto which 556. again add successively the said 125. which makes 681. against these equal parts in the Circle of numbers is 48. the other mean proportional. According to the same manner between 243. and 1024. four mean proportionals might be found, to wit, 324. 432. 576. and 768. Or proportions may be found to a term assigned, between two numbers, either by augmentation from a greater, or diminution from a lesser. Of the Extracting of square, and Cubic Roots, and others. THe construction of this depends upon the latter, because the square Root of any number, is nothing but a mean proportion between a unity and the given number, the Cubic Root is the first of two mean proportionals, between the unity and the Cube proposed▪ the Biquadrat Root, is the first of three mean proportionals between the unity and the given number, etc. But because the former direction specified in finding of mean proportions, adheres unto the way and nature of Logarythmes, it being more facile by them, than so to apply it Instrumentally, therefore we will somewhat compendiate that labour by the Instrument alone, and avoid the search of the Logarythmes of numbers, and their partitions, and as an ease for Radical extractions. To find a mean proportional with more facility than is formerly delivered Constructio. Mark what number of equal parts in the Circle of equal 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 movable to these parts in the Circle of equal parts noted with Q between Q A A Q if these two numbers have like places or exceed one another even places. exceed one another by odd places. so the like number of equal parts in the fixed in the Circle Q which is against the greater number in the Circle of equal parts E, being sought out between Q A, in the fixed shall right against it point out the mean 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, etc. For the extraction of square and Cubic Roots more compendiously, that's done by an inspection of the eye only, 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 Roots, the Root doth contain in places always the just half of the places of the number given if it hath even places, but if it have odd places, than the Root hath as many places as the greater half comes to. Now as every two places in square numbers, affords one place for its Root: so Cubic numbers, affords for every third place, or ternary one for its Root: but if the number have any places above, the ternary or ternaries of places, than the Root shall be one place more than the number of ternaries. ☞ Note further, that in seeking of mean proportionals it may be doubtful what denominations to give unto it when it is found on the Instrument, which may be discovered in this manner, find the places that the two extreme numbers given would make if they were multiplied together, which the Rule in Multiplication, Pag. 14. will show you; then having the number of places for the product, the former Rule which doth allude to the places for the square Root, will tell you what denomination to give the mean proportion sought for. To find a mean proportion between 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, until 1 in the fixed be equally distant between them, which the divisions in the pricked Circle A B will help you; so right against 1. in the fixed, is the mean proportion in the movable. Pro. 2 If the two numbers exceed one another by one, or three places, move the numbers to and fro, until 1. in the fixed be equal distance between them; so right against B in the movable is the mean Proportion. Some uses upon mean Proportionals. Pro. 1 To find how much is taken in the 100 li. in Loane of money. If 40. li. be lent for two years, Declaratio. and at the end thereof were received 48. li. and 4. 10. what was taken in the 100 Find a mean proportion between 40. and 48. 4. 10. which will be 44 according to the last rule; so right against 48. 4. 10. in the movable, is 110. in the fixed, which is the Principal 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 several services, the one containing 500 Soldiers, the other 320. Soldiers, there is a third place, neither so strong as the latter, nor so weak as the former, therefore a mean number of Soldiers is thought convenient for the defence thereof: what number shall it be? Find a mean proportion between 500 and 320. facit 400. and this is a mean proportional number between 320. and 500 and the number of men required. Pro. 3 Pro. 3. To find 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; find a mean proportion between this and the form. 8. Acres, which is 6. & 67. 100 and this stands against 1. in the fixed, which represents 10. his scale, but 8. in the movable 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 seek that number to be extracted in the movable, if it have 1. 3. 7. or 9 places, etc. bring the number towards the left side of 1. in the fixed: but if the number have 2. 4. 6. or 8. places, etc. bring it twards the right side of the fixed 1. and move your number to and fro, until 1. in the movable be as fare distant from 1. in the fixed, as your given number is from 1. in the fixed: (the equal parts in the Circle A B will help you in this) so the number in the movable 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, etc. How to extract the Cubic Root. ☞ Upon the movable there are those letters A. B. C. the distance between A. B. is divided into 10. equal parts, and each part subdivided: the distance between 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 always between A and B in the movable for the Extraction of Cubic Roots, and move the movable to and fro, until that the given number and 1. in the fixed be of like number of parts distant from A in the movable. So if the given Number have 1. 4. 7. or 10. 2. 5. 8. or 11. 3. 6. 9 or 12. Places, etc. the Cubic 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, etc. Uses upon the square Root. Pro. 1. There are two square forms, 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. find a number in continual proportion to 16. as 12. to 16. facit 21. & 3. 10. add this to 12. facit 33. and 3. 10. Then by 1. Pro. pag. 17. find a mean proportion between that 33. and 3. 10. and 12. facit 20. the side of the Square required. Pro. 2 Otherwise we might apply the Pro. thus, A B is the breadth 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 be as before 20. Pro. 3 A and C are two Towns, Allies West of the Meridian of C 16. miles, and C lies North of the Parallel of A 12. miles, the distances of the two Towns would be as before 20. miles, etc. Pro. 4. How to encamp horse or foot, according to any proportion assigned. Pro. 4 240. men or horse are to be embattled, Declaratio. that the Flank to the Front shall be in proportion, as 3 to 5, how many shall be in the Front, and how many in the Flank. Bring 3. to 5. so against 24. in the movable 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 Flank. In mental reservation of a number, to find that number. Pro. 5 Let the number be broken into two parts, and to the product of the parts add the square of the half 〈◊〉 difference of the parts, the Root, Quadrate of the Agragate is half th● number conceived, etc. 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 movable, there is charactered 1. 2. 3. 4. 5. 6. 7. 8. 9 10. 11. 12. 13. 14. 15. 16. 17. etc. all of equal distances, those serve for the number of years as occasion requires. Pro. 1. To find what a sum of money comes to, at the end of any number of years, accounting 8. li. for 100 p●r Annum. Pro. 1 Declaratio. So if 20. li. were forborn 12. years, how much doth it come to allowing Interest compound at 8. li. for 100 li. Constructio. Bring 1. in the movable to 20. in the fixed, so right against 12. years in the movable you have 50. li. 4. 10. in the fixed. And so much will 20. li. amount to being forborn 12. years. ☞ The Instrument not removed, you may at one instant see the amount of the said 20. li. for any number of years of parts of a year; for right against the time in the movable, you have the answer of the money in the fixed. Pro. 2 Pro. 2. To find what a sum of money which is due any number of years to come, is worth in ready money, allowing 8. li. for 100 li. Declaratio. So if the said 20. li. were due 12. years hence, what is it worth in Present. Constructio. This is only the converse of the former: bring therefore 12. years in the movable to 20. in the fixed, so right against 1. in the movable 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. years hence. Pro. 3 Pro. 3. A yearly Rent of a Lease, or a Pension to be sold for any number of years, to find the worth thereof in ready money: Or the Rent for any number of years being unpaid, to find what it amounts unto, accounting 8. li. for 100 li. per Annum. Declaratio. Let a Lease or Pension of 20. li. per Annum be sold for ready money, which is in being 12. years, how much is it worth? Bring 8. to 100 then right against 20. li. in the movable is 250. li. in the fixed; Constructio. unto this 250. in the fixed bring 12. years; so right against 1. in the movable is 99 li. 3. 10. in the fixed, which taken out of the said 250. there remains 150. li. 7. 10. the worth required. If the Rent were behind unpaid 12 years. THen bring 1. to the said 250. so right against 12. years in the movable 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 forborn 12. years at 8. li. for 100 li. per Annum. Pro. 4. A sum of money borrowed, and a Lease engaged for that money, to find how long the Lease ought to be kept. Pro. 4 Let 300. li. be borrowed upon a Lease in being 20. years, Declaratio. of 50. li. a year, 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 movable is 625. in the fixed: Constructio. from this 265. li. subtract the money 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. years 5 10. in the movable, 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 Sum and time to find the Rent. Pro. 5 Pro. 5. A Lease to begin for years to come, and then to continue for any number of years, to find the worth thereof in present, accounting Interest Compound at 8. li. for 100L. li. per Annum. Let a Lease of 40. li. per Annum begin 7. years hence, Declaratio. and then to continue 10. years after; if it were to be sold, what is it worth in ready money? By the third Pro. find the worth thereof in the present for the 10. years, facit, Constructio. 268. li. 4. 10. then by the second Pro. find what that 268. li. 4. 10. is worth in present if it were to be received 7. years hence, facit, 156. li. 6 10. and so much is the said Lease of 40. li. per Annum worth, which is to begin 7. years hence, and then to continue unto 10. years. Thus I might have gone further in those matters, but I intended not to be large in this Tract, only showing what weighty, and difficult matters in this kind by the Grammelogia, or Mathematical Ring, may be easily and speedily resolved. Conclusion. IF there be composed three Circles of equal thickness, A. B. C. so that the inner edge of D and the outward edge of ● be● answerably graduated with Logarithmall signs, 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 again the Logarithmall signs oppositly to the former graduations, it shall be fitted for the resolution of Plain and Spherical Triangles. Example. So if you move the Sign of 90. Degrees unto the Tropical point in the fixed, you have the Declination of any Degree of the Ecliptic only by an ocular inspection, for right against the Sun's longitude in the movable amongst the Signs, is the Sun's declination in the fixed. Again, in the 〈◊〉 of Tangents, if you bring the compliment of any Latitude in the movable 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 Sun's ascentionall difference: and in plain Triangles the operations are performed with like facility. Hence from the form, 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 yard's Diameter, or there about, and the line Decupled, it would work trigonometry unto seconds, and give proportional number● unto six places only by an ocular inspection, which would compendiate Astronomical calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability so health and time. FINIS. This Instrument is made in Silver, or Brass for the Pocket, or at any other bigness, over against Saint Clement's Church without Temple Bar, by Elias Allen. CHARLES by the grace of God, King of Great Britain, France, and Ireland, Defender of the Faith, etc. To all our loving Subjects whom it may concern greeting: Whereas Richard Delamain, Teacher of the Mathematics, hath presented unto Us an Instrument called Grammelogia, or The Mathematical Ring, together with a Book so entitled, expressing the use thereof, being his own Invention; We of our Gracious and Princely favour have granted unto the said Richard Delamain and his Assigns, Privilege, Licence, and Authority, for the sole Making, Printing, and Selling of the said Instrument and Book: straightly forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument or Book within any our Dominions, during the space of ten year's next ensuing the date hereof, upon pain of Our high displeasure. Given under our hand and Signet at our Palace of Westminster, the fourth day of january, in the sixth year of our Reign. Upon his Ring. AS in a secret Circle wrapped lies nature's mysteries, Which time brings forth, & industry makes plain to all men's eyes, So that what erst was hard to find, at length is wrought with ease, Rules of Proportion, the Roots Extraction, and these With ocular inspection now: Also Triangles Plain And Spherical; for Questions soon an Answer to obtain; That it makes Arts laugh his quickness to see, then use this Ring, The Circle where, hid Arts lie now themselves discovering. If after times shall seek more ease than in this easiness By Instrument, he makes Arts lame, with eases great excess. By a Friend. THe Egyptian Sages, who were wont to sing In the Hiroglyphicks, their Philosophy; Portrayed the year in semblance of a Ring, Cause so the year round in itself doth lie. Thy Mathematic Ring is more profound, A world of Art lies in that little Round. FINIS. To the Reader. NOw by way of advertisement to the Reader, in this Circular projection of Logarithmes, you may make use of the Projection of the Circles of the Ring upon a Plain, having the feet of a pair of Compasses (but so that they be flat) to move on the Centre of that Plain, and those feet to open and shut as a pair of Compasses (which some call a Sector abusively) now if the feet be opened to any two terms or numbers in that Projection, 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 show his proportional 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 express 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 writ, showing some uses of those Circles amongst themselves, and conjoined with others, in the resolution of such Questions which are ordinarily practised in Astronomy, Horolographie, in plain Triangles applied to Dimensions, Navigation, Fortification, etc. as a preparative ground for a more ample work, 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 promptness to conceive of opposite numbers in the answering of Questions following. FIrst, to examine the Circle of Numbers, bring any number in the movable to half of that number in the fixed: so any number or part in the fixed shall give his double in the movable, and so may you try of the thirds, fourth's, &c. of numbers, vel contra. 2. Bring 2. in the movable unto 3. in the fixed, so against 3. in the movable is 4. and 5. tenths in the fixed, against 4. and 5. tenths in the movable is 6. and 75. 100 in the fixed, and so may you go on in trying the divisions of the Circle of Numbers in continual 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 movable is 4. and 5. tenths in the fixed, and against 4. in the movable is 6. in the fixed, but against 4. in the fixed is 2. and 66. 100ths in the movable; against 5. in the movable is 7. & 5. tenths in the fixed, but against 5. in the fixed is 3. and 33. 100ths in the movable, and so may you proceed in examining farther, according to the Table B. in which M. at that head of the Table signifieth movable, and F. at the head of the Table signifieth fixed, and so against the numbers under M. or F. is one another's answer. 4. Bring 3. in the movable unto 2. in the fixed, so right against the sine of 90. in the movable is 41. gr. 44. m. in the fixed; ●gainst 60. gr. is 41. gr. 02. m. against 75. gr. in the movable is 40. gr. 05. m. in the fixed, and so you may examine further, as in the Table C. 5. To examine the sins amongst themselves in continual proportion, as 6. gr. to 7. gr. bring 6. gr. in the movable to 7. gr. in the fixed, so right against 7. gr. in the movable is 8. gr. 10. m. in the fixed, and right against this 8. gr. 10. m. in the movable is 9 gr. 32. m. in the fixed, and so may you go on in examining other sins on the Instrument in continual proportion, according to the Table D. 6. The Instrument being at this stay against 10 gr. in the movable as 11. gr. 41. in the fixed, but against 10. gr. in the fixed is 8. gr. 34. m. in the movable; against 15. gr. in the movable is 17. gr. 34. m. in the fixed, but against 15. gr. in the fixed is 12. gr. 50. m. in the movable, and so may you examine other sins, 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 movable is the Tangent of 7. gr. & the sine of 7. gr. 3. m. in the fixed; against the Tangent of 7. gr. in the movable 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 movable 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 Ring and Scheme of this Logarithmall projection, the use followeth. These Instruments are made in silver or Brass by john Allen near the Savoy in the strand The movable Compass The f●● ed Projection on a plain The movable Projection In a Ring M. movable F. fixed In Astronomy. Pro. 1. The sun's place or distance from the Equinoctial points known, to find his Declination. Constructio. BRing the sine of 90. in the movable unto the sine of the Tropical point, viz. 23. gr. and a half in the fixed, so right against the sine of the degree of the Sun's nearest distance from ♈ or ♎ in the movable is the sine of the Sun's declination of that degree in the fixed. Declaratio. So if the Sun's place where in the beginning of ♊. ♌. ♐. or ♒. which is 60. gr. of distance from the Equinoctial points; right against this 60. gr. in the movable, 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 movable gives 3. gr. 58. m. in the fixed, if the Sun have 3. gr. of distance this in the movable gives 1. gr. 12. m. in the fixed. Otherwise you may turn the 3. gr. or such which are less) into minutes by allowing 60. minutes to a degree which 3. gr. makes 180 m. minutes, this sought out in the movable amongst the Numbers gives 72. minutes in the fixed, which is 1. gr. 12. m. of Declination as before: But if you make a degree to contain 100 minutes or parts, than the 3. gr. will be 300. minutes or parts, so right against this in the movable amongst the Numbers is 100L. 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 part of the Ecclipticke, as for 1. minute for right against 60. seconds in the movable (which is answerable to 1. minute) is 24 seconds in the fixed, the declination belonging to 1. minute, etc. Pro. 2. To find the Sun's amplitude, or distance of rising or setting from the East or West knowing the Sun's place and Latitude. Amplitude. in the fixed. Declaratio. So the Latitude being 51. gr. 30. m. the Compliment is 38. gr. 30. m. Bring this to the sine of 23. gr. 30. m. in the fixed: now if the Sun have 90. gr. of Longitude: right 'gainst this 90. in the movable 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 movable (or any other) is the Sun's 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 near the Equinoctial point, the Amplitude of it may be had on the Numbers, as in the former Example. In Astronomy. But if the Declination of the Sun, or a Star be known; the Amplitude may be found thus. Constructio. BRing the Sine of the Compliment of the Latitude in the movable unto the sine of 90. in the fixed, so right against the sine of the Sun or Stars declination in the movable, is the sine of the Sun or Stars Amplitude in the fixed. The Instrument not removed you may for any Declination have the Amplitude of it: for right against the Declination in the movable is the Amplitude in the fixed, etc. and there may you see what Declination such Stars have, which never rise, or set in that Latitude. This Proposition may be inverted and applied to practice in Navigation to find the Latitude, by knowing the Sun's place and Amplitude, for if you bring the degree of the Sun's Amplitude amongst the Sins in the movable, unto the degree of the Sun's place in the fixed; right against 23. gr. 30. m. in the movable, is the degree of the Compliment of the Latitude in the fixed. Pro. 3. In any Latitude to find what high the Sun most have to be due East or West knowing the Sun's place. Declaratio. So if the Latitude were 51. gr. 30. m. bring the Sine 51. gr. 30. m. in the movable, unto the Sine of 23. gr. 30. m. in the fixed: and if the Sun's place were from ♈ or ♎ 90. gr. 80. 70. 60. 50. 40. 30. 20. or 10. gr. G. M. Right against this 90 In the movable is 30. 38 in the fixed, the Sun's 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 between 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 movable softly along, that as the sine of any Latitude in the movable passeth by the sine of the Tropical point in the fixed, so any degree in the movable that passeth by the sine of 90. in the fixed, doth show 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 show his degree of Longitude in the movable, or any degree of Longitude in the movable will show at what Altitude the Sun 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 movable would give the same degree of Altitude in the fixed to be due East or West; so if the Sun be in Longitude 20. gr from ♈ or ♎ than the Sun's A●titude would be also 20. degrees when it comes due East or West: if the Sun's place be 50. from ♈ or ♎ than the Altitude would be 50. to make due East, etc. But if the Latitude were less than 23. degrees, 30. m. as admit 15. degrees bring the sine of this 15. in the movable 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 movable: so when the Sun cometh 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 days, until he come to the 59 degree, 32. m. of ☊, and then every day after the Sun will cross the Vertical Circle of East or West, until the S●nne pass the other Equinoctial point of ♎. If the declination of the Sun or the Star be known, you may find the Altitude thereof at the point of East in any Latitude by this Rule. Bring the sine of the Latitude in the movable unto the sine of 90. in the fixed: so right against the sine of any Declination in the movable is the sine of the Sun's Altitude in the fixed. Hence you may conceive, that if the Zenith be between the Tropics, than the Declination that is equal to the Latitude, shows the greatest Altitude to be East: and any greater Declination shows you that the Sun will not be East: but if the Declination be less than the Latitude, then right against it in the movable is the Sun's Altitude in the fixed. In Astronomy. Pro. 4. Knowing the height of the Sun at the point of East or West, in any Latitude to find the hour of the day. Declaratio. Let the Latitude be 51. gr. 30. m. and the Sun's Altitude bring the Tangent of 45. gr. in the movable unto the Sine of 38. gr. ●ed, so right against the Tangent of 14. gr. 45. m. in the movable 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 Sun was due East that day 37. minutes after 6. in the morning, or due West 37. minutes before 6. in the afternoon. The Instrument not removed, you have the time of the Sun's being East or West for any other Altitude: for right against the Sun's Altitude in the movable is the degree of time that the Sun comes due East or West in the fixed. But if you move the movable softly along, as the Tangent of 45. in the movable passeth by the Tangent of any Latitude in the fixed, so the degrees of Declination in the movable passeth by their degrees of time in the fixed, amongst the Sins at what hour any such degrees are due East or West, until the Tangent of 45. in the movable be opposite unto 45. gr. in the fixed: for then as the Tangent of any Latitude less than 45. in the movable passeth by the Tangent of 45. in the fixed, so any degree of Declination in the movable will show his degree of time amongst the Sins in the fixed, that the Sun is due East or West. This Proposition of finding the hour of the Sun's being East or West, may serve to great use both on Sea and Land, in rectifying of Glasses, Watches, or such like, to keep and Regulate the account of Time. In Astronomy. Pro. 5. To find the time of the Sun Rising or Setting in any Latitude, the Sun's declination known. Constructio. BRing the Tangent of 45. gr. in the movable 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 Compliment of Latitude in the movable unto the sine of 90. in the fixed.) So against the degree of Declination amongst the Tangents in the movable, is the degree of the Ascentionall difference among the Sins on the fixed. G. G. M. So as 45. passeth by the Latitude of 25 The Tropical point, viz. 23. gr. 30. m. will pass 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 dialing. Pro. 1. To find the distance of the hours, in horizontal plains for an obliqne Sphere. ☞ Note, that the hours in a right Sphere are equal, the one unto another, viz. 15. gr. for one hour, 30. gr. for two hours, 45. gr. for three hours, etc. hence here after they are called equal hours, etc. Declaratio. So if the Latitude were 51. gr. 30. m bring the Tangent of 45 in the movable unto the sine of 51. gr. 30. m. in the fixed, so right against 15. gr. in the movable amongst the Tangents is 11. gr. 50. m. in the fixed amongst the Tangent, and so much is the distance of the hour 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, half hours, quarters, etc. For right against those equal hours, viz. 15 Are the unequal hour distances, viz. 11. gr. 50. m. For the hour 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 try for what Latitude ordinary pocket dials are made, knowing the distance between the hour of 11. and 12. for if you bring the Tangent of 15. gr. in the movable unto the Tangents of the distance in the fixed, right against the Tangent of 45. gr. in the movable is the sine of the Latitude in the fixed. Pro. 2. To find the distance of the hours for a direct south Dial in an obliqne Sphere. Constructio. This differeth 〈◊〉 little from the former, only, here bring the Tangent of 45. gr. in the movable unto the fine Compliment of the Latitude in the fixed, so right against the Tangent of the degrees of the equal hours, are the Tangents of the degrees of the hour distances in the obliqne sphere. Declaratio. So if the Latitude were as before 51. gr. 30. m. bring the Tangent of 45. gr. in the movable, 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 hour 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 movable softly along as the Tangent of 45. gr. passeth by the sine Compliment of any Latitude, so the degrees of the equal hours, will show the degrees of the unequal hour distances, in comparison of one Latitude to another. This may be otherwise represented in this fundamental Diagram, if EZ, and EH be divided out of the Table of natural 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 show the hour distances in a horizontal plain, and the hour distances in E Z for a vertical plain, and this kind of projection and motion may serve to other excellent purposes, etc. In Dyaling. Pro. 3. Knowing the Declination of a vertical plain, and the latitude of the place, to find what Angle the Axis makes with the plain, commonly called the height of the style. Constructio. BRing the Sine of 90. in the movable unto the sine Compliment of the latitude in the fixed, so right against the sine Compliment of the Declination, in the movable, is the degree of the styles height amongst the sins in the fixed. Declaratio. So if the Latitude were 51. gr. 30. m. the compliment of it is 38. gr. 30. m. which seek amongst the sins in the fixed, and bring 90. gr. to it, now if the Declination of the plain were 10. 20. 30 40. 50. 60. 70. 80. etc. The Compliment of those are. 80 So right against these in the movable are 37. 49 in the fixed the styles height 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 vertical plain to find the number of degrees betw●ene the Meridian of the place, and the Meridian of the plain, which may be called the difference of Meridian's. Constructio. Bring the Tangent of 45. gr. in the movable 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 Plain be 38. gr. 3. m. then the difference of meridians is 45. gr. But if the Declination be less than 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 movable. So if the Declination were 35 Right against these in the fixed are 41. 49 In the movable, 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 along by the Tangentiall degrees of Declination in the fixed, until 45. gr. in the movable 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 Meridian's amongst the Tangents in the movable. Lastly, if the Declination be above 45. gr. than the Declination most be accounted amongst ●he degrees of the movable, and if you move the movable 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 Meridian's in the movable passeth by the sine of the Latitude in the fixed, etc. Pro. 5. To find the hour distances in dialing in a Declining plain by knowing the former, viz. the styles height, and the difference of Meridian's. Declaratio. LET the Declination of a Plain be S. W. 50. gr. according to the former Instruction, the styles height would be 23. gr. 35. m. and the difference of Meridian's 56 gr. 43. m. Now before the hour distances can be known which are always unequal, there may be made a Table of equal hours thus. First, place down 56. gr. 43. m. the difference of Meridian's against 12. as in the Table, then by adding 15. gr. unto this difference of Meridian's, viz. to this 56. gr. 43. m. it makes 71. gr. 43. m. for the distance between the hour of 11. and the Meridian of the plane: unto this 71. gr. 43. m. add another 15. gr. makes 86. gr. 43. m. for the hour distance between 10. and the Meridian of the Plain: But for the hour of 1. 2. 3. that 15. gr. must be taken from the said 56. gr. 43. m. so the hour distance of 1. is 41. gr. 43. m. the hour 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 hour 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 hour distance of 4. will be 3. gr. 17. m. now unto this add 15. gr. makes 18 gr. 17. m. for the hour of 5. then add 15. gr. to that, and so prosecute the Table which may be called the Table of equal hours from the Meridian of the plain. Now to find the true hour distances, bring the Tangent of 45. gr. in the movable unto the sine of the styles height in the fixed, viz. 23. gr. 35. so right against the Tangent of any equal hour in the movable under 45. gr. is the Tangent of the true hour distance in the fixed A. But if the equal hour distance be above 68 gr. 12. m. (which is right against 45. gr. in the movable) then right against the equal hour distance in the fixed, is the true hour distance in the movable B. Lastly, if the equal hour distances be between 45. gr. & 68 gr. 12. m. then move the movable softly along, and as the Tangent of any equal hour in the movable passeth by the sine of the stiles ●e●ght in the fixed, so right against the Tangent of 45. gr. in the movable is the Tangent of the true hour distance in fixed. So the equal hour distances being A 3. 17 The true hour 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 hour distances from the Meridian of the Plain, they may be placed against the hours as in the Table, and protracted thus, draw the hour of 12. C. M. and on C. describe a semicircle: now seeing in the Table that the Meridian of the Plain is from the hour of 12. in its true distance 31. gr. 20. m. protract it in the Circular Ark from D. to S. (because of West declination) otherwise contrary, and draw C. S for the Substiler; from this S. protract all the hours' distances, as S. R. 4. gr. 45. m. for the hour of 3. S. Q. 11. gr. 22. m. for the hour of 2. etc. then may we draw the hour lines CN. CO. CP. CQ. CR. CT. CV. CW. CX. CΩ. and upon the said CS. place the style A. B. C. perpendicular to the plain, so it shall be fitted for the casting of shadows upon the said hour lines, Of Plain Triangles. Praecognita. Theorem. 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 equal to two right angles, that is twice 90. or 180. gr. by the thirtenth of the first of Eue. therefore knowing one of those Angleses, the other is also known▪ 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. Theorem 2. The three Angles of any plain Triangle either Right, or Obliqne by the 32. of the first of Euc. are equal to two right Angles, therefore one acute Angle known in a right angled Triangle, and two Angles known in an Obliqne Angled Triangle, the third Angle is likewise known: 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 Obliqne Triangle A H I. the Angle at A and I put together being 135. the Angle at H must be 45. gr. etc. 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. Obliqne Angled Two Angles, and a side, or two sides and one Angle opposite to either of those sides Axiom 1. The Axiom for the resolution of this proposition is thus. The sides in all such Triangles, bear proportion the one to the other, as the sins of their opposite Angles do. Or the sins of the Angles are directly proportional to their opposite sides, by the seven & twentieth of the first Book of Regiomontanus, the thirtenth Chap. of the 1. of Copernicus, and by the second Axiom of the third book 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: Again in the obliqne 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, etc. ☞ 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 terms, is primarily, and principally to be considered. How to dispose the three Terms known in any of the former Triangles fit for operation. FIrst, if a side be required in a right, or obliqne Triangle, & a side be known, and the Angles opposite to those sides be also known: the Angle opposite to the side known 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 known the third place. Secondly, if an Angle be required, and an Angle known with the sides opposite to those Angleses, than the side opposite to the Angle known 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 known shall have the third place: the disposing of the Terms thus considered, the Angles have reference to the Circle of Sins upon the Ring, and the sides to the Circle of Numbers. The construction by the Ring is thus. Bring 60. gr. amongst the Sins in the movable to the Sine Compliment of it in the fixed, viz. 30. gr. so right against AB. 25. in the Circle of Numbers in the movable is 14. and 43. 100 in the fixed. But if the side AB. had been any other number, you might instantly have the said BC. according to the same proportion. ☞ Here note generally, that if the two first Terms be upon the Circle of Sines, the other two terms are then upon the Circle of Numbers, vel contra. * ☞ Note further, that in all trigonometry, the terms or parts given either Angleses or sides, are noted with a small stroke of the Pen thus— and the Terms or parts required either in the Angleses or in the sides, thus O. so the sides AB. AD. A ●●. with the Angles at A. are given the sides AC. CB. A●. and ●●. and the Angles at E. and G. are demanded; now the mutual proportion of these parts one unto another, is according to the former Axiom; by which infinite propositions may be resolved of admirable consequence, lying under the habit of some one of those whose excellent use in ordinary Practical things, I will illustrate in several kinds by sundry propositions; first, In Dimensions; secondly, in Fortification; thirdly, in Navigation; fourthly, in dialing, as followeth. Upon Plain Triangles in Dimensions. How to measure an Inaccessible height situated upon a Hill, the Practice being not upon a Plain. THe example upon this I will take from mine own observations which I made upon one of the goodliest Hills in this Kingdom, which belongs to Sir Richard Newport in Shropshire, called the Wreaken, not fare from Shrewsbury, which Hill I found to be near 6. miles and a half 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 sunk from the Top of the Hill at D to be levill with the spring C, viz. at R: here it may be demanded how deep this Well may be sunk, viz. D R, how fare it is from the spring C, to the botume of the Well R, and how fare it is from the said spring C, to the Top of the Hill D. Constructio. First, with conveniency I made choice 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. Chains, and at B I caused another Stay or Rest to be erected of the same height that the former at A was: Then looking 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 known 20. Chains, 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 known, 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 movable be brought to the sine of 4. gr. 44. m. in the fixed, right against 20. in the Circle of Numbers in the movable is 1. Chain, and 65. 100 of a Chain in the fixed, which according to Page the ninth is 108 Foot, and 9 tenth. this shows that the station at B was so many Foot higher, than that at A. By the same Rule A X. would be found to be 19 Chains, and 93. hundreths of a Chain, or 1315. Foot, and 5. tenths, those two dimensions 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 down, 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 Theorem 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 several Angles, Add 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 obliqne Triangle A B D, and A B C are known in each of them two Angleses, and so consequently by the aforesaid second Theorem, Pag. 64. The other Angles are also known (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 Axiom seeing that in the Triangles A B D, all the Angles are known, and one side to wit A B. 20. the side A D may be also known. Constructio. Bring the sine of the Angle A D B 6. gr. 16. m. in the movable unto the sine of the Angle A B D. 1●2. gr. 31. m. (that is to the Sine of the Compliment of that Angle, viz. 17. gr. 29. m.) in the fixed, so right against 20. in the movable amongst the Numbers is 54. Chains and 93. 100 the. in the fixed, which in feet is 3625. and 6. tenths. and so fare 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 Theorem Page 64. all the other Angles are known, viz. A D H 74. gr. 4. m. A H D 90. and seeing by the last work that A D was found to be 54. Chains, and 93 100ths. the other part● of the Triangle by the afore said Axiom will be likewise known. 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. Chains, and 07. 100ths. of a Chain, which by Page the ninth makes 995. foot the high of the hill. In like manner may you find the other side of the Triangle viz. A H, that is from the eye under the top of the hill, viz. 52. Chains, and 82 hunderths, or 3486. foot, and 4. tenths. Fourthly, in the Obliqne Triangle A B C. all the Angles are known (as afore) with the side A B, and therefore the side B C would be found to be 32. Chains, or 2113. foot. Fifthly, in the rect angled Triangle B Y C the Angle C B Y was formerly known● to be 16. gr. 45. m. the Compliment of which is the Angle B C Y. 73. gr. 15. m. and the side C B was formerly found to be 32. Chains. therefore by afore said first Axiom of plain Triangles, the other sides of the Triangle, viz. B Y. will be found to be 30. Chains, and 65. hunderths. or. 2023. foot, and 3 tenths, and the side C Y. to be 9 Chains, and 23 hunderths, or 608. foot, and 9 tenths. Sixthly, seeing B X. is equal unto Y Q. unto the said C Y. 608. foot, and 9 tinths, add. B X. formerly found, viz. 108. foot, and 9 tinths makes Q C. 717. foot, and 8. tenths. but this Q C. is equal to H R. which taken from the high of the whole hill D H, viz. 995. foot leaves D R. 277. foot, and 2. tenths, the depth of the well. Seventhly, for as much as B Y was found to be 2023. foot, and 3. tenths. which is equal to X Q and A X being as before 1315. foot, 5. tenths, those two put together makes A Q 3338. foot, and 8. tenths, which taken from A H, which was formerly found to be 3486. foot, 4. tenths leaves Q H. 147. foot. 6. tenths, which is equal to C R. the distance between the spring C, and the bottom of the well R. Eighthly, and lastly knowing D R. 277. foot, and 2. tenths, and C R 147. foot, and 6. tenths C D is found to be 313. and 8. tenths, according to the first Proposition, Page 19ths. the example being the same with the second Proposition: thus for the Instrumental way: such as desire to examine the work by numbers may proceed 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. ABDELLA A B. 1320. A D. 3625. & 6. 10ths. B A D. 11. gr. 11. m. 43. s. ABDELLA. 162. gr. 31. m. 45. s. AND 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. Upon his Ring. AS in a secret Circle wrapped lies nature's mysteries, Which time brings forth, and industry makes plain to all men's eyes, So that what erst was hard to find, at length is wrought with ease, Rules of Proportion, the Roots Extraction, and the●e With ocular inspection now: Also Triangles Plain And Spherical; for Questions soon an Answer to obtain; That it makes Art laugh his quickness to see, then use this Ring, The Circle where, hid Arts lie now themselves discovering. If after times shall seek more ease than in this easiness By Instrument, he makes Art lame, with eases great excess. By a Friend. THe Egyptian Sages, who were wont to sing In the Hiroglyphicks, their Philosophy; Portrayed the year in semblance of a Ring, Cause so the year round in itself doth lie. Thy Mathematic Ring is more profound, A world of Art lies in little Round. FINIS. DELAMAIN IN circularem Logdrythmorum projectionem adauctam (ut in fine libri, Anno. 1630. praelo commissi promissum erat) viam demonstantem qua continua circulorum decuplatione aut alio modo, adhuc magis augeatur, ita ut operationes Trigonometricae fiant ad minuta, numerique proportionales, & radices dentur ad quinque aut sex locos idque motu circulari, & tantum oculari inspectione. How to operate, in the finding of proportionals by my Logarythmall Projection of Circles enlarged, either by a mooveable and fixed Circle, or by a single Projection, with an Index at the Peripheria, or Centre. THe way of operation is drawn from the nature of Proportional Logarythmes, that as they keep equal differences, so in a lineary or Circular Instrumental projection of these Logarythmes, Proportional numbers shall always have equal distances, which as a fundamental ground may serve to the more learned both for a full demonstration, and direction in operation. But to make things more obvious, & to remove such scruples as may arise in working by this Projection; the numbering ●f the Circles especially is to be considered, that is either by augmentation or diminution, in continuation or discontinuation, and th●●●ath relation to the line of Conjunction E. T. which showeth the breaches of the Circle, or the uniting or continuing of the parts: which multiplicity 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 conjunction, 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 mark the several revolutions or Circles between them ascending or descending; for then the fourth Proportional 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 between the first and second numbers. So if the line of conjunction on the mooveable, be on the right or left side on the line of Conjunction on the fixed, and the first and second numbers be between them, and also the third, or all these three numbers be not between them, the fourth number or proportional is had without any consideration, but only by the same number of Circles, as was between the first and second numbers. But if the third number be on the otherside of the line of Conjunction, and that the proportionals did augment, or diminish, the same number of Circles or revolutions accounted ascending or descending from the third number, will likewise show the fourth proportional 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 mark the difference of Circles, either ascending or descending between them; then whensoever in operation the said first and second number, and also the third number are neither of them between the lines of Conjunctions, or all of them are between them, than the fourth proportional is had in the fixed right against the third number in the mooveable, by the same difference of Circles ascending or descending, as was between the first and second numbers. But if the line of conjunction on the movable, be on the right side of the line of conjunction on the fixed, and the first number be in a lower Circle than the second and be not between the lines of Conjunctions, but the third, be between the lines of conjunctions & not the third than the fourth proportional is in one Circle more less than the difference of Circles between the first and second numbers. But if the first number be in a higher Circle than the second, than the difference more will be less, and that which is less more. Contrarily, if the line of Conjunction on the mooveable be 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 enlarged, with a double Index to move on the Peripheria, or Centre of a Circle, if in stead of moving one number to another, you extend the feet of the Index, as followeth. An Example of the operation upon the Projection of the Circles of my Ring enlarged, according to the conclusion of my first Book, in a Scheme or Instrument where the Circles of Numbers, Sins, and Tangents are decuplated, the diameter being but 18. Inches only. SVppose the Sun being in the Tropic, 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 compliment of the Latitude, is unto the Sine of the Sun's declination proposed, so is the Sine of 90. gr. unto the sine of the Sun's Amplitude required; the Diagramme and Demonstration of which, see pag 57 and the Instrumental operation may be ●hus. Constructio. Place one of the edges of the Index unto the sine of 23. gr. 31. m. (the Compliment of the Latitude proposed) and extend the other edge unto the Sine of 23. gr. 30. m. (the Sun's greatest declination) then move the edge of the first Index unto the Sine of 90. gr. so the edge of the second Index shall give the fourth proportional sought for. But because the answer in this Proposition falls near the sine of 90. gr. (where the degrees are small, and the graduations close) the answer is not so exactly discerned in the minutes as if it were larger. To supply which, or such like as may fall out in Practice, I have continued the Sins of the Projection unto two several 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 Projection) the other beginning at 86. gr. 6. m. 48. s. and ends at 90. gr. (being the last of a ternary of decuplated revolutions, or the thousand part of tha● Projection) and may be thus used. Lay the edge of the Index upon the former 23. gr. 31. m. and mark what equal parts in the Circle of equal 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 equal parts 0069. Then take ten times the distance of these two numbers in the Circle of equal parts between the feet of the Index by decuplation, that is, place one foot of the Index upon 99 and extend the other foot unto 69. so the edge of the first foot being placed on the sine of 90. gr. the edge of the second foot 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 Tropic aforesaid. Again, admit that in the Latitude of 51. gr. 32. m. the Sun's place were required by knowledge of his Meridional Altitude, which suppose was found to be 61. gr. 57 m ⅙. Now the common Axiom according to the Diagram of Pag. 56. grounded on Mathematical doctrine to operate which is: As the sine of the Sun's greatest declination, viz. 23. gr. 30. m. is unto the sine of 90. gr. so is the sine of the Sun's Declination given viz. 23. gr. 29. m ⅙ unto the sine of the answer. Therefore Instrumentally open the Index unto the first two terms proposed, and place the edge of the first foot upon the third number, so the edge of the second foot shall give the fourth proportional required. But for as much also as the answer in this doth fall amongst the small divisions near the sine of 90. gr. in the decuplated Projection, 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 Projection, thus; Lay one of the edges of the Index upon the sine of 23 gr. 30. m. and mark the number of equal parts which it cuts in the Circle of equal 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 equal parts between the feet of the Index, that is, place one of the edges of the Index upon 70. m the Circle of equal parts by Centuplation, and extend the other edge unto 64 (for the Decuplating, and Centuplating of the equal parts, do 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 Sun's 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 public view my Projection of Logarythmes enlarged by way of decuplating, Centuplating, etc. of the Circles, as I promised at the Conclusion of the first publishing of this Invention, to the world, and have in some measure shown the Accurate working of Trigonometry by it, near the Sine of 90. gr. where difficulty seems to be; for being thus enlarged the greatness of a degree between the sine of 89. gr. and the sine of 90 gr. is more than 4. Inches, and if I should have enlarged the Circle of Tangents according to the Sins, the capacity of a minute at 45. gr. would be more than 8. Inches which enlargement amongst the degrees which falls near the Sine of 90. gr. doth operate as true (according to the former Diameter but if 18. Inches) as if Mr. Gunters excellent Lines were extended or projected unto 4000 foot. And if I should frame a Ring as is specified in the conclusion of my first publication of this Invention, being of two yard's diameter, and apply it to Astronomical Calculations, no doubt I might show a way (or others may easily) to compendiate many operations therein; and sufficiently clear my intentions then delivered touching the Prostaphereses of the motions; though some one in contempt of my good endeavours divulged after it came to the world's view, it could not be done, nor possibly a minute expressed at the sine of 90 gr. (as I have now produced it) thereby endeavouring to annihilate my labours and to spread an unsavoury rumour, which might seem to argue not only his ignorance of my intentions, but also of the manner of extending and enlarging 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 maturity, that my jealous opposite may be no more mistaken with a suspected untimely birth. I confess these single Circles before, were something untimely, in regard of these which are of fuller growth, and yet may have further application, without waiting his time to perfect them. Courteous Reader HItherto the world hath been abused as well as my self, with a false Rumour (raised by some rude ignorant tongue) that my Invention both of Ring and Quadrant was got, or borrowed, or stolen (even as they please to miscall it) from another Man, by the sight of a Letter, by some private Conference by— I know not what means (nor they neither) but only by their malicious fancy: which how true, how just it is, that the world and my self may have our right, and they the shame, by this following just defence and future event, I hope thou mayest fully be informed. I did not intent 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; I desire all may now judge who is in fault; and let this ensuing discourse be my Plea. To the Reader HOw undeservedly oft are the single and sincere endeavours of some men by the malevolent disposition of envious detractors backbited, (which sometimes rebounds back aversly upon them) not only by bare assertions, but also by injurious, and contumelious aspersions; In which kind I have not a little lately suffered: for having for a general end (more than aiming at mine own particular) published the making & use of my horizontal Quadrant, with my new invention of the Projection of Logarythmes Circular by a former book of the use of it, entitled with this, Grammelogia, or the Mathematical Ring, since that time I have been deeply glanced upon, and scandalised 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 disturb the quiet and Peace which formerly I enjoyed, but did also disorder and slack my intentions, in the publishing of the enlarging of the Invention of my Ring, as I promised to do in the Conclusion of the aforesaid Book of the use of that Ring, which would serve as a help for such as affect Mathematical practices for the working of trigonometry unto minutes, and to give accuratly Roots and Proportional numbers, unto 5. or 6. places, as is specified before in the description of the projection, and had ere now been published with the excellent use thereof, (not out of any Mercenary respect nor interlased with untruths, delusions, and bombast stuff, 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 Centre, 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 himself, but ingenuously acknowledged it being so made up, and contrived, to be the same projection with my Ring, But the manifestation of it, so first to the world, properly and solely belongeth unto him; which Circles so projected on a Plate with an Index, was not also unknown unto me, as by good testimony I can produce, before any of these things came to the world's view, though publicly I writ not first upon them: the Learned in those Arts, and those that understand the projection, know it to be 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 turn a Garment in and out; A motion must perform th● operation to give proportionals 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 been prevented by some others, whos● callings might have invited them to spend their hours better, tha● to snatch with greediness that out of another's hands which wa● not their own: for these Circles of my Ring projected on a Plate o● Plaine, so fitted with an Index at the Centre as aforesaid, by clear testimoney in its particulars I shall prove in his due place to be another's, and not the supposed Authors, whose conscience may checke● him and tell him herein, that he never saw it, as he now challengeth it to be his invention, until it was so fitted to his hand, and that he made all his practice on it after the publishing of my Book upon my Ring, and not before; so it was easy for him, or some other to write some uses of it in Latin after Christmas, 1630. & not the Summer before, as is falsely alleged by some one who hath made himself 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 stuff, or scrambling pieces, if not confused fragments of his own, or some others, to a public view, in obscure and various phrases, a thing supposed to be forged by sundry heads, rather than by one alone, seeing there is such roving from the Text; amongst whom to blow some smoke thereto, there was some gross one, seeing the matter is so common; for to a finer element perhaps his capacity could not assent, or ascend. A blind Guide and a Parrot's speech are not much different, the one walks he knows not whether, and the other speaks he knows not what; and such are all precepts in Arts, which lead and make men speak without Demonstration; which doth not only protract the studious, and frustrate the affectionate, but makes an ingenious spirit (who ever is more Rational th●n practical) to contemn such Circumlocutions, and laugh in private, if not in public, at the learned style of some Authors, who making themselves by their obscure kind of writing seemingly famous, stick not to calumniate others to make them infamous. Those that are but initiated unto knowledge, for their failings and defects of order, are worthy of some blame, but for others who would so are above all, and not only pick holes in the coats of the living, but also vilify the dead, 'tis a shame; making too great haste at the beginning to glory in that, which no doubt will prove shame in the ending. It's a common thing, that, one man having laboured, planted and sowed, with great pains, another reaps his Harvest with no industry; yet in this there was some honesty shown, not to take the Crop but the Glean, holding it easier to follow a beaten path then hazard a discovery, but the way was not made plain, and the veil remooved to help his sight: God that gave me the former invention without the advice of any, hath also reserved for me the manifestation of the latter, without the help 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 himself as his own, and say I had it from him: I have hitherto borne the injury of the infamy with great grief of heart, (and God that is the discerner of the spirits knows mine integrity and innocence herein.) The window hath been as yet close, and darkness possesseth the place, I will now withdraw the Curtain that the Sunshining light may appear, to expel those 〈◊〉 mists that have been 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 slackness, 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 reap unto themselves again, and with what measure they meat, the same shall be meated unto them, I will therefore in the first place answer for my Quadrant, than afterwards for my Ring, and lastly for my self, and others. The Answer upon my Quadrant. I Have not only suffered by way of divulging of the said project in of Logarythmes Circular, 1630. but also by the late publishing of my horizontal Quadrant, 1631. by a scandalous aspersion couched under the relation of the Author, in the words of his Translator (but rather his Transcriber, if not in the most of that Book as is suspected, his Compactor) which are as followeth, Which whilst I went about to do, 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 own cannot stand, well then, this preoccupating, preventing, and circumventing, and discovering in loving confidence his intention to me, must be either about that of my horizontal 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 publicly upon the same Projection, as he hath now done in his book named the Circles of projection, which tract is so fare from being answerable throughout, to that which it promiseth in the beginning, that it seems rather to pussle the studeous then in any wise to further them) of which I say no more, but advise the studeous Reader only so fare to trust as is agreeable to the Text, and a true Doctrinal Method, which being therein omitted, it doth not only 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 speak Parrot 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 neither 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 Projection long before ever I heard of the Author's name, (as in its due place I can prove) but this aspersion no doubt was from an inveterate hate of some, who endeavoureth not only 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 posterity; But I doubt not in a public Audience I shall clear myself 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 themselves, let them laugh on, as they have begun while I hasten the Issue of it. The extendure of God's hand in his donations is manifold, and where his spirit pleaseth to breath, there a door is opened; now whether the gift to the creatures be divine or humane, we should bless God as the first and principal Author, and giver of all, no wise d●spise man, as the second agent, or receiver: It's very fantastical in some therefore, who think such and such things are not worthy the general vote and allowance, if they proceed not from such, and such an one, envying they should be produced by any other; which if they be so divulged, as much as lies in them, they will hasten to possess 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 fancy, choose rather to abandon the Lore of sound reason, then to be divorced from their prejudicated affections. I will in the first place therefore make way for my Quadrant, that I did not circumvent the said Author for the attaining thereof, (as before:) But whereas he taxed me with going about to preoccupate, or prevent him, it seems somewhat soaring like, to abridge any man's free affections, in constraining them to wait on any others concealed intentions, which they never knew (in which every man hath a freedom to himself) It is (no doubt) somewhat too malapert, & too rigid, to tie the liberties of others in their actions, who desiring the good of others in not concealing things, if they shall by their industrious labours, show the excellent use of such or such a thing, to a public view, for a general 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 conceit which some entertain of their own worthiness: the said Author having had the Projection 30, years before by him (as he giveth out) wherefore did he not publish it then, or give way to it as he hath now done, how could it be so long concealed, and others never so much as hear of such a thing, was it that he would not have the same communicated to others, or that he would not be known in his name by a profitable action, or that some others might challenge an Interest in it beside himself, or that the uses of the Projection lay hidden, and obscure; or that they were not at that time so plentifully made manifest, until Mr. Gunter's ingenuity opened the mystery of it, and applied that projection so methodically, and copiously, to Horolographie in his Book of dialing in the use of the Sector, and accommodated that Projection long before in a Dial for an honourable personage, as the first that ever was made in that kind, and the same form with these that are now made, and therefore to give every man his due, and not to injure the dead, it is properly ●●en Mr. Gunter's Dial, for that composure, and not another's, (notwithstanding the inverting of things and detractions from him) But it should seem he would publish it then, either when the way was made fair for him, or when he might catch at some one, as lately he hath done in an unfit and uncharitable way: And as Mr. Gunter was copious in applying that projection to that particular of Dy●lling, so might he no doubt have been in the Astronomical uses of it also, though he delivered so few as but six observations only to a public view upon that Projection, which before the publishing and also after, both in the uses and the Projection itself I often intentively looked upon, and extracted from them many useful performances more, even in Mr. Gunter's time; and since his death have published my horizontal Quadrant, extracted from that dyagramme: in which I have abundantly supplied the emptiness and obscurity of that Projection, fitting it for a Pocket Instrument, or according to any magnitude, as a help and benefit to those that are studious of Mathematical practices; which labours, of Mr. Gunters, and my own, if they were not to unveil the subject and to make way for another to help his sight, which writes afterwards, I refer it to an eye not partial, to be judge, it being rare and wonderful for one man to see all at once, and there are fare more excellent uses yet upon that Projection, which may be also known if some one will open the veil a little more; if there be any that knows 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 original as before specified I did extract and compose from the fundamental dyagramme of Mr. Gunter's Book of the sector, page 66. as I have specified in the Epistle to the Reader, in the book 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 Mr. 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 be wronged. And should I search the original from the first, neither Mr. Gunter nor especially he may challenge so much unto himself, since the main draught out of which theirs is extracted, was extant before they were borne, namely in that ancient Geographer Munster in his dialing, where it stands obvious, as also in the famous 〈◊〉 Orontius, as I mentioned in my former Epistle, but I suffered him to put it out, being (it should seem) unwilling to have his own dismantled; As also at that time, I shown him in our English Blagrave the like Scheme, in both their works, long ere he or Mr. Gunter committed any such thing to a public view; howsoever, I had not the least touch of furtherance from him, or from any man breathing, either by transcripts, or verbal direction, (but only what I have formerly acknowledged out of Mr. Gunter's Book of the Sector, page 64. 65. 66. though in other trivial matters I do and shall acknowledge freely) as hath been falsely alleged by some lose tongued Instrument, that I had the making of that Quadrant by a sight of the Author's letter, whose honest relation may be suspected, seeing he makes so little conscience, to detract from another's reputation whose detraction is grounded upon as great a stability, as uncertainty, whose Basis is only a bare supposition, conjoined with an indisposed evil eye, as full of envy, as emptiness of charity, and this upon examination he is able to make good, and no more, which a generous or tender breast scorns to harbour, much less 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 original, that he which hath given way and gloried at the countenancing of that aspersion, 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 himself (as in his own words) but not ingenuously enough acknowledging from whence he had it: for my particular, I take God to witness I have without any equivocation or mental reservation, declared in every particular the very naked truth, in the Epistle to the Reader in that Book of the aforesaid Quadrant, from whence I had that Quadrant, and how I produced it, and it might have been easily so composed, and published, as I have done it, by another, (without the help of the sight of a Letter,) that is but indifferently versed in delineations, seeing the Projection in my Author Mr. Gunter's Book, page 65. 66. as aforesaid lies so plain & conspicuous (notwithstanding) in which letter to my remembrance I saw bu● some ordinary uses, and a check cast upon Mr. Gunter, but no directions in it for the making of the Instrument; which uses I slighte● as mean, and trivial, and other things I saw not: and this I speak not to shuffle things off, but out of a true sincerity, but perhaps since the sight thereof there may be inserted somewhat else, to make my opposites assertion good, howsoever I needed not such a help by the sight of a Letter, seeing joiners, Carpenters, and other Mechanics about this Town & else where, yea, School Boys in imitating the Projection aforesaid in Mr. Gunter's Book of the Sector, and following the directions therein in the 66. page of the use of the Sector aforesaid, have drawn the Projection fully and complete by the Book alone; and others having only had but a simple view of my Quadrant, many years before I published the use thereof, have from the aforesaid Book of the Sector page 66. made the like, having not had the least assistance from any, but the direction of the Book only, as upon oath they have been examined, and do acknowledge, and will testify it when occasion requires: beside I know sundry Gent. and others in this kingdom, that are yet living, that have drawn the same Projection by the Book, alone immediately after the publishing of it 1624. as at this instant they will be ready to confirm, and myself in Mr. Gunter's time 1622. (besides many others) have drawn the same Projection, for our particular uses, and are yet to be shown, which was long before either I, or they ever heard of a new Author, of that Projection besides Mr. Gunter only: And to make my assertion yet more absolute, I did not only draw the Projection in Mr. Gunter's time, but before his death did also show 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 verbal 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. Upon 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 horizontal 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 Mathematical Ring (that is, how I might compose 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 be collected (certain circumstances seriously weighed, and considered) The whole ground of which being from as weak a principle, to open the way unto me for the making of my Ring, as the sight of a Letter to show 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, we being walking together some few weeks before Christmas, upon Fish-street hill, we discoursed upon sundry things Mathematical, both Theorical and Practical, and of the excellent inventions and helps that in these days were produced, amongst which I was not a little taken with that of the Logarythmes, commending greatly the ingenuity of Mr. Gunter in the Projection, and inventing of his Ruler, in the lines of proportion, extracted from these Logarythmes for ordinary Practical uses; He replied unto me (in these very words) What will you say to an Invention that I have, which in a less extent of the Compasses shall work truer than that of Mr. Gunter's Ruler, I asked him then of what form it was, he answered with some pause (which no doubt argued his suspicion of me that I might conceive it) that it was Archingwise, but now he says that he told me 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 immediately I answered, I had the like myself, 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 diverse persons, who do and will testify the same: Then after my coming home I sent him a sight of my Projection drawn in Pasteboard: Now admit I had not the Invention of my Ring before I discoursed with the supposed Author thereof, it was not so facile for me or any other (to an eye not partial) to raise and compose so complete, and absolute an Instrument from so small a principle, or glimpse of light, but was known unto me, (as I have produced it) long before that time, and being now published as it is, the composure of it seems most facile, (as all inventions do, once known) that I have much wondered with myself, that Mr. Gunter or some other produced it not so to the world as now it is, seeing it was so easy, and caries with it such an excellency, above that which is in the Lineary form, for in a Circle it is natural, & perfect, in a line defective & imperfect: But he, or they, perhaps saw it not, though their sight I confess (no wise to disparage their worth) might penetrate further in other things; but gifts may not be attributed to natural ingenuity, but to God the giver of them, who disposeth where his goodness pleaseth. And who knows to the contrary but many private men in this kingdom, or elsewhere, might have this Invention long before myself, or he that now challengeth it, seeing the gifts of God in his donations, to several persons, are oftentimes in one and the same thing, and if God bestows his Talents on us, we ought to communicate them to others, which not to do, is to hide them in a Napkin, and who so concealeth them are not free from some reprehension: our Author may answer for himself, who reporteth now that he had the Invention for more than twelve years past, but put it not to use, which in a sense was injurious both to God and man, if he saw the copious, and compendious use of it then, as now he doth; to me and to many others it hath seemed strange, that he should hide such an excellency, so long in obscurity; But it is supposed, that he had then but a glimpse of its performance (until I writ upon it, by opening the Cabenet, and showing its treasure to the world) and so regarded it not then as a jewel, but since. But when I had a sight of it, which was in February 1629. (as I specified in my Epistle) I could not conceal it longer, envying myself, that others did not taste of that which I found to carry with it so delightful & pleasant a taste, knowing that as the Logarythmes (as a jewel) that did excel all Inventions that ever went before it, so this kind of Projection, as the daughter springing from so noble a Mother, was the rarest Instrumental Gem, that ever Art in all preceding Ages did afford: which in its production heretofore lay obscure, but now being published how clear doth it seem to be in its composure, and if it were so easy 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 than two years passed) enlarge that Invention, or deliver others in that nature of their own, being easier to add unto an Invention, now being divulged, then to deliver its original? But hitherto it hath seemed difficult, many having attempted and endeavoured, but have failed in their ends, howsoever I know not what some private men's industry hath produced by way of augmentation to that invention of my Ring, or the Logarythmes Projected Circular: At the Conclusion of my Book I gave a sufficient and perspicuous direction, and invitation thereto, that so by the former labours, and the other there intended, the way might be made fair, and the Ice broken that they might wade the easier: and so produce at last something for a public use by Augmentation to this Invention, at which I shall much rejoice, rather than any wise envy: 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 enlarge that Invention, seeing none as yet hath done any thing therein, I have now at last therefore for their benefit, & others, to a public view delivered what is mine own to that end: and that by sundry ways, and how easy doth the enlargement of that Invention now also seem to be, being now produced. With the like facility did I compose long ago my Helicall line of Roots which affords five places accuratly, by an inspection of the eye only, which being not as yet divulged to a common view may to some seem 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 Author's 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 means then these words; that in a loving confidence the Author discovered his intent unto me? is it not to give a fair flourish upon a untrue subject, to delude, if not to possess Men with a falsity, to detract from another his good name; besides the barrenness of the word we will a little consider the straightness of time, in the original, for the producing of the Book, 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 judge to censure of it, that having had no other direction, or light from the said Author then formerly is specified, (for more he dares not avouch with a clear conscience) how could I from it so easily form my Projection: Mr. Gunter's Ruler, as some think, was a furtherence to me, but it was rather a hindrance, 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 verbal (as before) or Instrumental dictate was not used, or could be sufficient to compose so high a work: but it was from a long intentive precogitation of many years with myself, how the Logarythmes in the Tables might be so compacted, that all Numbers in these Tables should be proportional one unto another, and according to a diverse, and variable proportion assigned; which to effect I found at the first very difficult, and could not conceive how otherwise it might be done, but either by fitting of Tables to all proportions, which so to do would not only be too great for operation, but also breed confusion; or it must be from some graduated insertion of these numbers Instrumentally, so that by motion numbers might be moved one unto another; and for expedition of which I found no figure more apt than a Circle, and on that my Meditations fixed, and there I rested, and so (as my few hours could permit me) I made several 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 look further into the mysteries that lay open in that new Invention, having not scarce an hour in a day, and sometimes not two hours in a week of serious privacy (by reason of my calling) to sport myself in operation thereon (for it seemed so to me then as a recreation, as all new Inventions at the first do to any that invents them) It was many months therefore before I made trial of it, in the general uses that it might be put unto, in matters of Arithmetic, for common operations, and to the measuring of Plains and Solids, but especially how it might be applied to the doctrine of Plain, and Spherical Triangles, in Astronomical Calculations, Nautical practices, and horological conclusions, all which in every particular I practised on it at such times as convenience would permit me; And having for many months thus satiated myself, and fed my fancy upon its Theorical contemplation, and its Practical operation, though oftentimes I found many Rubs and impediments in the Practice, in applying this instrumental Invention at the first so generally and copiously as I did; And it cannot be denied of any, that the ways of new Inventions lies not so obvious, or so easy, to be discovered with such celerity, as a long premeditation might produce; I was desirous therefore to make the world participate thereof also; Therefore at several times, having but a little time at anytime, by little and little did I compose and produce a method pleasing to mine own fancy; doing one week such a piece, another week another, and so going on until I had run through the whole parts of Arithmetic, in the Goalden Rule direct, and indirect, in division and in multiplication, applying all these to manifold uses in combination of Numbers, to common affairs, in fortification, to mensuration, and fractional operation, than I applied the Instrument to the finding of numbers in continual proportion, in finding of mean proportionals, and the extracting of Roots, I laboured further to make the Instrument more complete that it might work all useful proportions touching Interest, or valuation of Leases: and last of all I applied the Instrument in the Circles of Sines, and Tangents, to some uses of Astronomy, in the finding of the Declination of the Sun, and its ascension through all Latitudes; These things I drew up in a Book at several times aforesaid, as a part of the practices that I made upon it: And it was a task sufficient for a man that had command of his hours, by allotting them solely to that end, in as long a time as was between Alhalantide & Christmas, in the opening of the way of that new Invention, & by applying of it so, (to use for a public view) rather then by one that was commanded by them, especially at that time and all the Terms before: for how few are the hours that a man of any employment 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 been so much the more harsher, and difficult to have it so suddenly produced, had it been then at Alhalantide but only conceited, and agitated on, but it was premeditated on long before, as is aforesaid, and intended for a new year's gift for the King, which accordingly I gave him on new year's day, though a fortnight, or three weeks before Christmas, his Ma.tie received from me a Scheme of the Projection in Pasteboard, with a manuscript of the Book which is now published, agreeable to it in every tittle, (the Epistle excepted) a Copy of which was then at the Press: and was Printed four days before new year's day, 1630. Now that all these Practices, and many Transcripts that were drawn, with the doubts and hindrances that did arise in the fitting of these things, could be made, so ordered, and produced in so little a time, as the scattering of a few hours as between Alhalantide and a month before Christmas, I leave to the judicious Reader to censure: I broke the Ice, and made the way facile for another that came behind me; yet if he took a year and more to meditate and write upon my endeavours after my publication (thereby not to signify that he wanted time, but takes liberty enough if not too much, to the loss of time) as long a time, half of it, or a quarter of that time, may by a Charitable boon be granted me. Which consideration of time, will easily also clear me from the imputation of the Author's lose assertion, that he was Circumvented by me, nor that in a loving confidence he opened his intent unto me. Yet in the last place I will hasten to vindicate his untrue declaration before the Courts of justice; if restitution be not speedily made, where true witnesses shall be produced, and that which is now but in agitation I will bring to action, and prove that before Alholantide or very near that time, my Invention was produced to a public view; therefore it was very injurious in the Author to possess the world with so an untrue aspersion, upon a bare supposition, in that I should have the Invention from him, and that in a loving manner he opened his intent unto me, and was Circumvented, as before, whose assuming disposition hath not only been busy to take from me my good name, & labours by this his lose aspersion, but hath also endeavoured (by too rigid and general a censure sparing none in some sense) to bring all in a common disrepute in their Callings, therefore, In the behalf of vulgar Teachers & others. BY way of advertisement to you, and myself, & to whom it may concern, vulgar Teachers of Mathematics about this Town and kingdom, as you are styled, which had been fair (if no worse) seeing you are not professors, or public Readers, but common Teachers, it behoves you, and me, that we endeavour to avoid the disreputation of a scandalous attribute in our profession of so noble a science, that our doctrines be not only Practical, but also Rational, and Theorical, that we may not be ranked with jugglers, and teachers of Tricks, as we are lately glanced upon publicly, but a charitable breast howsoever (I persuade myself) hath a better opinion of us, that according to the Talon that God hath given unto us, out of the Riches of his bounty, no doubt in our callings we use them rightly, and do not (by deceiving) derogate from the end, which is, to glorify God in these gifts in a true and sincere use of them; which otherwise would not only be a blemish and stain 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 testify in your callings your sufficiency, and therein remove all scruples to confirm your integrity; 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 Gentry, to which we have been servants to or in present are, in so honourable and laudable a service, which aspersions are as highly backed with arguments, as he was to forwords that divulged them: whose judicious and censorious eyes, hath been two much busied to show the ways of the juggling 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 only to despite Art, to betray willing and industrious wits to ignorance, and idleness, but was also loss of precious time, making their scholar's doers of tricks, as it were jugglers. Which words are neither cautelous, nor subterfugious, but are as down right in their plainness, as they are touching, and pernicious, by two much derogating from many, and glancing upon many noble personages, with too gross, if not too base an attribute, in terming them doers of tricks, as it were to juggle: because they perhaps make use of a necessity in the furnishing of themselves with such knowledge by Practical Instrumental operation, when their more weighty negotiations will not permit them for Theorical figurative demonstration; those that are guilty of the aspersion, and are touched therewith may answer for themselves, and study to be more Theorical, then Practical: for the Theory, is as the Mother that produceth the daughter, the very sinews and life of Practice, the excellency and highest degree of true Mathematical knowledge: but for those that would make but a step as it were into that kind of Learning, whose only desire is expedition, & facility, both which by the general consent of all are best effected with Instrument, rather than with tedious regular demonstrations, it was ill to check them so grossly, not only in what they have Practised, but abridging them also of their liberties with what they may Practise, which aspersion may not easily be slighted off by any gloss or Apology, without an Ingenuous confession, or some mental reservation: To which vilification, howsoever, in the behalf of myself, & others, I answer; That Instrumental 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 solely in the science, and to an Artist or disputant proper to be known, and so to all, who would truly know the cause of the Mathematical operations in their original; But, for none to know the use of a Mathematical Instrumen, except he knows the cause of its operation, is somewhat too strict, which would keep many from affecting the Art, which of themselves are ready enough every where, to conceive more harshly of the difficulty, and impossibility of attaining any skill therein, than it deserves, because they see nothing but obscure propositions, and perplex and intricate demonstrations before their eyes, whose unsavoury tartness, to an unexperienced palate like bitter pills is sweetened over, and made pleasant with an Instrumental compendious facility, and made to go down the more readily, and yet to retain the same virtue, and working; And me thinks in this queasy age, all helps may be used to procure a stomach, all bats and invitations to the declining study of so noble a Science, rather than by rigid Method and general Laws to scar men away. All are not of like disposition, neither all (as was said before) propose the same end, some resolve to wade, others to put a finger in only, or wet a hand: now thus to tie them to an obscure and Theorical form 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 Theorical part, which otherwise they would apply themselves unto: being catched now by the sweet of this Instrumental bate; which debarring would not only injure the studeous but also cause the Mechanic workmen of these Instruments, to go with thinner clothes, and leaner cheeks. Neither doth the use of Instrument to a man ignorant of the cause of its operation, any ways oppose or despite Art, seeing that the end of producing and inventing of Instruments, is their Practical use; Besides, its impossible to show the use of an Instrument but in teaching there must needs be laid down some grounds or prolegomena●, as, what is meant by such, or such names; what are such, and such terms, and therefore the beginning of a man's knowledge even in the use of an Instrument, is first founded on doctrinal precepts, and these precepts may be conceived all along in its use: and are so fare from being excluded, that they do necessarily concomitate & are contained therein: the practice being better understood by the doctrinal part, and this later explained by the Instrumental, making precepts obvious unto sense, and the Theory going along with the Instrument, better in forming and enlightening the understanding, etc. vis unita fortior, so as if that in Philosophy be 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, than it must needs follow that a doctrinal proposition laid first open to the eye, and sense, and well perceived enters more easily the door 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; It's not therefore requisite unto all Capacities, to have Instrument severed from science, in many things: is though the use of one could be without the other less or more. Neither doth the practice on them betray willing & industrious wits to ignorance and idleness, that assertion therefore is very ridiculous, for he that is industrious, and willing to spend his hours, for the attaining of any science, according to doctrinal method, busieth himself not with Instruments, but applies himself to such Authors, or such compendious abstracts, as are taken from them, which do not only open the essential parts of the subject in the Theory, but also lays down such documents and principles which may (in a higher nature and way) induce him to practise them, then possibly by Instrumental operation for exactness can be attained unto; for such is the excellency of Art by Theorical doctrine, that all things tending to practice may be done by the science only, without the help of Instrumental operation, certain propositions being granted, which originally and principally are proper unto Instrumental observation, being the Basis or foundation of the whole; Therefore science, as before, hath a principal dependence on Instrument, but is it in their observation, rather than in their operation, and the Inventions that are daily produced in that kind, are only to compendiate and facilitate practical 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 trivial things to satisfy the sense, and not the intellectual part; which Theorical way doth not only 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 add many ends both pleasant and useful, by the way, for the Practisers encouragement. To begin with Instrument is unprofitable for the Teacher, though advantageous 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 learn more useful Practice upon some Instrument in one hours' Instruction, then to know the cause in 20. hours of some Instrumental operation: and yet there are many Instruments almost as facile in their demonstration as in their operation; Which kind of beginning to teach, or usual 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 own profit, and the dignity of learning; Yet that any Teachers of Mathematics should be so nice as to deny the use of Instrument, to such Gent. or others, who perhaps desire the Theory to contemplate on hereafter: but the practice for the●r private ends, for the present, were not only to frustrate their desires, but injure their occasions; who might rather laugh at the teacher's pride, then contemn his Art. Lastly, that the practice on a Mathematical Instrument should be loss of precious time, (to any one that knows how to use them in their practice though not in the Theory) is not only ridiculous but also untrue and absurd, for to what end did the manifold and laborious calculated Tables of the learned in their Subtenses, Sins, Tangents, and Secants serve, were they not to avoid the great toil of Algebreticall work in Radical extractions, which otherwise in trigonometry (the practical end of Theory) must necessarily be used, but they were Invented & produced to avoid the loss of time, to the more learned (who know the causes of their operation in the Genises) and as a facility to open the way unto the unlearned, though thousands working trigonometry by them knows not scarce the cause of such operation, which in a manner is as mere Mechanical, as if it were Instrumental (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 loss 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 help of the Logarythmall Tables in one hour, & there are thousands in the world that also work by them for their private use, by reason of their great quickness, in hard and laborious matters, who know not the cause also why these Numbers should so expedite such a difficulty, with so great a facility, & celerity: yet they use them though the performance be also mechanical, that they may not likewise lose time: Now Instruments though they be extracted from Tables, yet according to their capacity in practice they exceed these Tables, in all common serviceable uses, that I dare maintain that what the Logarythmes by the Tables, for giving of proportionals, either in Numbers, Sins, or Tangents, will do in 20. hours, my Ring or the Circles so projected from it, will do in one hour. Which Instrumental Invention, I have also produced to help the studeous to avoid the loss of precious time, which to omit and keep them from it (in some occasions) were also misspending of time: Therefore in respect of private men's ends, to begin with Instrument is not to juggle, nor to do tricks, neither to oppose or despite Art, or to betray willing and industrious wits to ignorance and idleness, nor loss of precious time. But I would be loath while I seem to vindicate the loss of time, cause others to lose it, by my too prolix defending of it; the vindicating of the truth indeed was my chiefest aim, which I hope will be my spokeseman & defend itself, mean while not to hold 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, Calumny 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 own 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 salamander's whose delight is to live in the fire of contention, endeavouring to load others with Calumny and detraction, and to raise unto themselves a proud Babel out of others ruins; a vice odious both to God and man, thereby forgetting the rules of equity which ought to be observed, so to deal with others as one would be dealt withal, thereby blemishing the good name of another, who are not ignorant (or at least should not be) how precious the name of a man is: is it not above Riches, above pleasure, above Gold itself? Calumny indeed is the fire this Gold is cast into sometimes, only here's the difference, this purifies the one, and that strives to pollute the other; nay, what speak I of those gross things, is it not dearer than life itself, being the very life of the life here, and eternity of a man's life hereafter? If he that steals 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 itself? 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 break the ninth Commandment 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 only a blazer of such reports, but a compiler? Is not Satan styled 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 door of his lips, and keep in that unruly member, that is, a world of wickedness? what is the stroke of the hand to the stroke of the tongue? the one wounds the body, the other wounds the soul; the one may be easily cured, the other hardly; the one strikes them that are present, the other those that are absent; the one strikes but one at a time, the other strikes many at the same time, at least three at once, with the self same blow; viz, him whom he doth traduce, him who hears him, and himself, and many times the last most of all; and ought we not to take great care how we order this unruly tongues what Gall, what Poison, what Wormwood, what Wildbeast is to be compared with the tongue, which lancinates, tears, and wounds the soul, by the virulency of its words, and reports: Look into the heavy judgement of God against such, 1 Cor. 6. that revilers shall not escape punishment. Such was the account of the preciousness of a man's goodname in ancient times, that there hath been Laws utterly to banish such from the society of men, who by their Calumny and detraction separate all good society amongst men, making foes of friends, and breaking the sweet bands of peace and charity, causing nothing but brawls and dissension, and as near as they can dissolving this goodly frame into the ancient Chaos of confusion. Again, some of the Ancients herein are so violent that they call it Grave malum, Turbulentum Daemona Pestem pestiferam, and profess plainly, Qui detractioni student diabolo serviunt, so that as Coals are to burning Coals (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 man's Peace and go down into the inwards of the belly, and a whisperer separateth dear friends. I wonder what need these Lamiaes gad so much abroad having work enough at home, Te prius corrigas quam alterum corrigas, If such would turn their eyes inward they should find matter enough to mend in themselves first, and afterwards they might reprehend others, sed ad delicta nostra Talpae, ad aliena serpentes sumus. It was once Constantine's speech. That he would throw his mantle over his brother's nakedness rather than lay it open; What shall we think meanwhile of those who would fain discover nakedness, where they can find none? what should be the plot, except it be to feed the bosom Wolf of envy and malice which at last will gnaw out their own bowels? do they envy because God gives to one man one gift, to another man another? would they have all, and others none? If the whole body were an eye where were the hearing, if an ear where were the smelling? so, were all contracted into one, what would become of the rest? No, rather let every one be thankful unto God for that he hath, then envious for that he hath not; God hath store enough for all, let us not envy one another; if one pull down 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, Quod Sapimus conjungat Amor, quod vivimus uno Conspiret study; Nil dissociabile firmum. Nature itself peacheth us to make much of union, without which all falls to ruin; See we how the day's part equals stakes with the nights, and if one have a greater share now, it allows the other it afterwards. The Sun not all doth claim, but Moon and Stars have shares, And greater lights need not envy the lesser theirs: The Land not all doth keep but gives the Sea its bounds, Hills have their tops, and let the valleys have their rounds. The sweet combination of Elements and inanimat bodies may teach us concord; Fire and water, earth and air how soever opposite in themselves in both qualities, yet in mixtion and composition sweetly conjoin together, one not destu●bing but qualifying the excess of the other, as if they without either sense or reason meant to teach men both: But above all, Piety and Religion enjoins this, Are we not all Christians, nay are we not all Protestants (or at least should be) all professing one and the same faith, the same Baptism? have we not all one Father, one Redeemer, are we not all members of one and the same holy Church? whence then are these discords amongst brethren, these detractions amongst friends? I'll speak no more, lest I seem to teach those who should teach others. And this I speak not with any solace to myself, but with grief of heart in the behalf of such which are guilty of such break out, and make my prayers to God for them, that they may see the foulness 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 own reputation, so to be careful of the reputation of others, and as they love their own peace so to be studious and tender of the peace of others; If they shall do this there will be some satisfaction made (though full recompensation for Calumny can never be made) and comfort in their acknowledgement, and repentance. If it cannot be by this effected, I will follow the precept of my Saviour, and pray to God for t●em, and walk as well as I can through good reports, and bad reports, and comfort myself with the company of my Saviour, of the Apostles, and of all God's Saints, who were not exempted from the lash of the tongue, commending my cause to God (and to the equal judgement and censure of a Reader not partial) to whose mercy likewise I commend them, and so desiring with St. Paul that we may all keep the unity of the Spirit in the band of peace, this ever shall be my wish Hoc habeat concordia signum quos una fides, jungat & unus Amor. SInce my first publication of the uses of my Mathematical Ring, or the Logarythmes projected Circular I have been oftentimes invited by sundry persons for to deliver the way of the projecting, and dividing of the Circles of my Ring upon a Plain, so that it might be made in Pasteboard to avoid the charge of the Instrument in metal, for such which have not abilities to buy, and for others, who would first see the practice on it, before they would be at the cost of the Instrument in metal: for whose sake, and use desiring to satisfy the affectionate, and for a public benefit, (rather than mine own 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 enlarged, 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 Pasteboard it is ready for use; And yet further to satisfy those that are desirous, I have delivered also in the first place ensuing how those Circles are divided, that so they may be made according to any magnitude. In the second place how several ways they may be framed into a Ring: In the third place I show the enlarging of the Instrumental Invention in these Circles to as great a magnitude for use as may be desired. In the fourth place I deliver several ways how these Circles enlarged may be accommodated for Practical use. In the fift place, I make a description of the Grammelogia, or Instrument in the particular Circles of my Mathematical Ring, projected on a mooveable and fixed plain. And in the sixth and last place, I will declare the admirable, and excellent uses of both these Instruments, in the Practical parts of Arithmetic, Geometry, Astronomy, Horolographie, Navigation. etc.