Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks.

Title

Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks.

Author

Delamain, Richard, fl. 1631.

Publication

[S.l. :: s.n.],

1630.

Rights/Permissions

To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.

Subject terms

Logarithms -- Early works to 1800.

Mathematical instruments -- Early works to 1800.

Arithmetic -- Early works to 1800.

Link to this Item

https://name.umdl.umich.edu/A20103.0001.001 content_copy

Cite this Item

"Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A20103.0001.001. University of Michigan Library Digital Collections. Accessed June 5, 2025. content_copy

Contents

• illustration
• title page
• To the High and Mighty King, CHARLES King of Great Britaine.
• By a Friend upon the Mathe∣maticall Ring.
• The same to the inventer of the Logarythmes projected in Circles.
• By a Lover, and Student in the Mathe∣maticall Arts.
• In mirificam Logarythmorum pro∣jectionem Circularem.
• To the courteous and benevolent Rea∣der that affects this instrumentall practise of the Logarythmes projected circular Intituled Grammelogia, or the Mathematicall Ring.
• To the Reader.
• illustration
• Of the projecting and dividing of the Circles of the Mathematicall Ring, and of the inlargement of the Invention, either in a single projection, or in a double, and that severall wayes.
  • 1. Of the graduati∣on of the Circle of Numbers.
  • 2. Of the graduati∣on of the Circle of Sines.
  • Of the graduati∣on of the Circles of Tangents.
  • How the Ring may be framed or com∣posed.
    • subsection
    • 2. How the Projection may be formed in a Ring, so that all the graduations may be upon one side onely.
    • 3. How to fit the Circles of the Ring by a single Projection into a Ring, so that all the Cir∣cles may be upon one side onely.
  • Of the enlarging, or augmenting, of the Project∣tion of the Circles of the Mathematical Ring, to worke accuratly Trigonometrie.
  • Of severall wayes how the Circles of the Mathematicall Ring (being inlar∣ged) may be accommodated for Practicall use.
    • subsection
    • 2. A second way of accommodating this inlarged Logarythmall Projection, for practicall use, with a single Index onely.
    • 3. A third way of fitting of the Circles of the Projection for Practi∣call use.
    • 4. A fourth way, how the Logarythmall pro∣jection of my Ring inlarged, may be fitted in an instrumentall forme for practicall use in Cal∣culation.
  • Of the description of the Grammelogia or the Circles of my Mathematicall Ring on a Plaine, according to the diagramme that was given the King (for a view of that projection) and after∣wards the Ring it selfe.
    • A more particular description of each Circle, and first of the Circle of equall parts.
    • Of the Equator, and Meridian.
    • Of the Circle of time.
    • Of the Circle of Tangents.
    • Of the Circle of Sines.
    • Of the Circle of Decimalls of money.
    • Of the Circle of Equated figures, and Bodies.
    • A description of these Tabular notes.
    • Of the Circle of Numbers.
• How to performe the Golden Rule, or to finde a Proportionall Number unto another Number, as two other Numbers are in proportion amongst themselves.
  • Example 1
  • Example 2.
  • Example 3
  • Example 4
  • And so of other under roomes to other uses.
  • 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.
  • More uses upon the Golden Rule, in the division of Lines.
  • To diuide a Line into any number of equall parts.
  • To divide a Line in such sort or proportion as another Line is already d vided.
• Notions or principles touching the di∣sposing 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.
  • How to proportion a Fraction that is not Decimall, into a decimall.
  • Of the Golden Rule, or Rule of Proportion, in respect of Lines and Quantities in plaine Figures.
  • Of the Golden Rule, or Rule of Proportion in respect of Lines, and the quantity of Solids.
  • To finde what Proportion in Quantity there is betweene two or more Solids.
  • How to divide one number by another.
    • Further uses upon Division.
    • More uses upon Division.
  • To multiply one Number by another, or to finde the Product of two Numbers.
    • Vses upon Multiplication.
    • Other uses upon Multiplication.
    • Further upon Multiplication.
    • How to square a Number.
    • How to Cube a Number.
  • To finde Numbers in continuall proportion unto any two Numbers assigned.
  • To finde a meane proportion, or many betweene any two Numbers given.
  • Of the Extracting of square, and Cubicke Roots, and others.
  • To finde a meane proportionall with more facility then is formerly delivered
  • To finde a meane proportion betweene two numbers.
    • Some uses upon meane Proportionals.
  • How to extract the Square Root by the Grammelogia.
  • How to extract the Cubicke Root.
  • Vses upon the square Root.
    • Pro. 1. There are two square formes, the one is 12. every way, and the other 16. every way, if of those two were made one, how many should it be every way in the side.
    • Pro. 4. How to encampe horse or foot, according to any proportion assigned.
    • In mentall reservation of a number, to finde that number.
  • Further uses upon the Grammelogia in the resolution of Questions, touching Interest, Purchases, valuation of Leases, and such like.
    • Pro. 1. To finde what a summe of money comes to, at the end of any number of yeares, accounting 8. li. for 100. pr Annum.
    • Pro. 2. To finde what a summe of money which is due any number of yeares to come, is worth in ready mony, allowing 8. li. for 100. li.
    • Pro. 3. A yearely Rent of a Lease, or a Pension to be sold for any number of yeares, to finde the worth thereof in ready money: Or the Rent for any number of yeares being unpaid, to finde what it amounts unto, acounting 8. li. for 100. li. per Annum.
  • If the Rent were behind unpaid 12 yeares.
    • Pro. 4. A summe of money borrowed, and a Lease ingaged for that money, to finde how long the Lease ought to be kept.
    • Pro. 5. A Lease to beginne for yeares to come, and then to continue for any nuber of yeares, to finde the worth thereof in present, accounting Interest Compound at 8. li. for 00. li. per Annum.
  • Conclusion.
• privilege (monopoly)
• Vpon his Ring.
• By a Friend.
• To the Reader.
• Of the Examination of the Graduation of the Circles of the Ring, which may serve as an in∣ducement and furtherance to the Learner, to fit and acquaint him how with prompt∣nesse to conceive of opposite numbers in the answering of Questions following.
  • A Type of the Ringe and Scheme of this Logarithmall projection, the use followeth.
  • In Astronomie.
    • Pro. 1. The sunnes place or distance from the Aequinoctiall points knowne, to find his Declination.
    • Pro. 2. To finde the Sunnes amplitude, or distance of rising or setting from the East or West knowing the Sunnes place and Latitude.
  • In Astronomie. But if the Declination of the Sunne, or a Starre be knowne; the Amplitude may be found thus.
    • Pro. 3. In any Latitude to finde what higth the Sunne most have to be due East or West knowing the Sunnes place.
  • In Astronomy.
    • If the declination of the Sunne or the Starre bee knowne, you may finde the Altitude thereof at the point of East in any Lati∣tude by this Rule.
  • In Astronomie.
    • Pro. 4. Knowing the height of the Sunne at the point of East or West, in any Latitude to finde the houre of the day.
  • In Astronomie.
    • Pro. 5. To finde the time of the Sunne Rising or Setting in any Latitude, the Sunnes declination knowne.
  • In Dyalling.
    • Pro. 1. To finde the distance of the houres, in horizontall plaines for an oblique Sphaere.
    • Pro. 2. To finde the distance of the houres for a direct south Dyall in an oblique Sphaere.
  • In Dyaling.
    • Pro. 3. Knowing the Declination of a verticall plaine, and the latitude of the place, to finde what Angle the Axis makes with the plaine, common∣ly called the height of the stile.
    • Pro. 4. Knowing the Latitude of the place, and the Declination of a verticall plaine to finde the number of degrees betwene the Meridian of the place, and the Meridian of the plaine, which may be called the diffe∣rence of Meridians.
    • Pro. 5. To finde the houre distances in Dyalling in a Declining plaine by knowing the former, viz. the stiles height, and the difference of Meridians.
  • Of Plaine Triangles. Praecognita.
    • Axiome 1. The Axiome for the resolution of this proposi∣tion is thus.
  • How to dispose the three Termes knowne in any of the former Triangles fit for operation.
    • The construction by the Ring is thus.
  • Vpon Plaine Triangles in Dementions. How to measure an Inaccessible height scituated upon a Hill, the Practice being not upon a Plaine.
• Ʋpon his Ring.
• By a Friend.
• illustration
• addition
  • How to operate, in the finding of Proportionalls by my Logarythmall Projection of Circles inlarged, eyther by a mooveable and fixed Circle, or by a single Projection, with an Index at the Peripheria, or Center.
  • Or without considering the absolute revolutions, you may operate by the difference of Circles noted on the single Index thus.
  • An Example of the operation upon the Projection of the Circles of my Ring inlarged, according to the conclu∣sion of my first Booke, in a Scheme or Instrument where the Circles of Numbers, Sines, and Tangents are decuplated, the diame∣ter being but 18. Inches onely.
• to the reader
• To the Reader
  • The Answer upon my Quadrant.
  • Ʋpon my Ring.
  • In the behalfe of vulgar Teachers & others.
  • section
• part