TACTOMETRIA. SEU, TETAGMENOMETRIA. OR, The GEOMETRY of Regulars practically proposed; After a new, most artificial, exact and expeditious manner (together with the natural or vulgar, by way of mensurall Comparison) And in the Solids, not only in respect of Magnitude or Dimension, but also of Gravity or ponderosity, according to any Metal assigned. Together with several useful Observations and Experiments falling in by the way, concerning Measure and Weight. And withal, the like artificial practical Geometry of regular-like Solids (as I term them) in both the foresaid respects: And moreover, of a cylindrical Body, for liquid or Vessell-Measure (commonly called by the name of Gauging) as is for solid measure; it being therefore a most exact and expedite way of Gauging: With sundry new and exact Experiments, Observations, and Rules concerning the same. And lastly, an A-TACTOMETRIE, or an APPENDIX, for the most ready and exact discovering of the dimensionall quantity of any irregular kind of Body, whether solid or concave, which in itself will not admit of an ordinary or orderly way of measuring. And this from certain new and exact Experiments also, made for that purpose. A Work very useful and delightful for all such as are either ingenuously studious of, or necessarily exercised and employed in the practice of the Art metrical. By J. Wybard  
LONDON, Printed by ROBERT LEYBOURN, for Nathaniel Brooks, at the Angel in Cornhill, MDCL.   
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉: Velure, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. PLATO. De quo PLUTARCH lib. 8 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, prob. 2. Omnia quaecunque à primaeva rerum natura constructa sunt, numerorum videntur ratione formata: Hoc enim fuit principale in animo Conditor is exemplar. BOETIUS Arithm. lib. 1. cap. 2. Veniet tempus quo ista quae nùnc latent, in lucem dies extrahat, & longioris aevi diligentia. SENECA, nat. quaest. lib. 7. cap. 25. Est natura hominum novitatis avida. PLINIUS.  

TO All that are wellwillers to the Mathematics, in general: But more especially to the candid, impartial, intelligent, and practical Reader.  
THere is now two Lustres, or one Decade of years elapsed, and some more time, (gentle and courteous Reader,) since I happened first to hit upon this artificial kind of Mensuration (or more artificial practical Geometry for regular and regular-like Magnitudes,) which I here deliver: and which was than but in two or three Particulars thereof, here first of all laid down in the three principal Propositions contained in the first Part, and the same demonstrated practically. And so having since by degrees, very much enlarged my Conceits & inventions herein (and indeed as far I think as possibly may be, & that only by way of Mathematical exercitation & recreation from other Studies and employments) I thought good at length, thus to put them together, (with other things by the way, pertinent thereto,) and so to exhibit and expose them to a public View, examination and trial in general, and in special to thy candid and courteous censure: which I was the more emboldened and encouraged to do, considering that when at first I propounded those things which I had than conceived in this way, to several able Artists, not only in several parts of this Kingdom where I happened than to come, and with whom I had the opportunity to converse, but also in some parts beyond the Sea (where not long after I fortuned to spend some time, for the prosecuting of other Studies, which I than chiefly aimed at and intended, and have since for the part followed and embraced) most of which were Professors and Teachers of the Mathematics, and so such as are usually soon acquainted with all the new mathematical inventions that are any way made known) I found that it was to them a mere novelty, (according as I conceived it would be) and that moreover at first they somewhat doubted of the same, saying, that it were indeed an excellent way, if it would generally certain and true; and so to some others of less judgement, it seemed to be a thing so very improbable (they having made no trial thereof) as that they would suddenly and unadvisedly conclude it to be impossible, only that it might hit right now and than by chance, but not constantly.  
And than besides this, I having for my further satisfaction herein, taken the pains (both at first, and also again of late) to make a strict search and enquiry into all the chiefest Authors which have hitherto treated of practical Geometry either in Latin or English, and more especially for instrumental practice, in the way of manual or mechanical Mensuration; could not found the way here proposed, so much as barely hinted by any of them in the lest kind. And so have I now here at length (by the favour and permission of God) according to my earnest desire, brought that to a General, which at first I had conceived or apprehended only in a few Particulars, and so have completed the Invention, as that I may think there is hardly any room left for additi on thereunto.  
Here than first (friendly and ingenious Reader) shalt thou found the most artificial and exquisite quadrature of a Circle, in a practical, or organical way, as to the immediate obtaining of its Area in any measure appointed. For as Joseph Scaliger saith of the squaring of a Circle in a general way, (or of the general quadrature of a Circle) Elem. Cyclomet. 1. or Elem. Cycloperimet. Desin. 5. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. Circulum quadrare, est Circuli areae aequale Rectilineum invenire. that is, To square a Circle, is to found a rightlined Plane equal to the Area of the Circle. Here shalt thou found that rectiline Plane to be the very Square of the Diameter, (or of the Circumference) according to Quantity discrete; in which it is artificially diminished by a Line of measure, so as to be made equal with the Circle itself, whereas naturally or geometrically it is greater than the Circle. And so I may say, that more properly, precisely and nearly to square a Circle, is to found an exact Square equal to the Circle given: And the like understand for the squaring of any other regular Figure; and so thou shalt here found the like artificial quadrature of all rectiline regular Planes as of a Circle, by their lateral, diametral, and diagonal lines, for the immediate producing of their superficial contents, which I have here performed (by way of practical demonstration) in two of the first of them.  
Than next shalt thou here found the like most artificial and excellent cubing of a Globe or Sphere, for the immediate producing of its solid content in any measure assigned. And what was said before of the squaring of a Circle, both general and particular or special, the like I may say of the cubing of a Sphere; That the same generally understood; is to found a rightlined or plain Solid, equal to the solid Area of the Sphere: and so more particularly and properly, to cube a Sphere, is to found an exact Cube equal to the Sphere given; which thou shalt here found to be made artificially (in quantity arithmetical) the very Cube of the Diameter (or of the correspondent Circumference) which naturally, or in quantity geometrical, exceeds the Sphere itself; and this not only in respect of solid measure, but also of gravity or ponderosity, according to any Metal assigned: And the like understand for any other regular Solid: And so shalt thou here found the like most artificial and admirable cubature of all the five famous plain ordinate Bodies in Geometry, or rectiline regular Solids, as of a Sphere, in both the foresaid respects; and that not only as considered simply and absolutely in themselves alone, but also in relation to a Sphere, as being described either within or about the same. And what is here performed in all these regular Solids for their solid dimensions, by way of cubature; the like is done in them for their superficial dimensions, by way of quadrature; (and which therefore I conceive, may not altogether unaptly and improperly be termed the Quadrature of these Bodies, as to their superficial or external part.)  
And than shalt thou here found after the like artificial manner, the dimension both solid and superficial of all such Solids, as are not exactly ordinate or regular, but somewhat like the same, and so which I call regular-like Solids, as namely right Cylinders and Cones; and all Prisms & Pyramids constituted upon regular Bases; and both which Dimensions aforesaid, may sometimes hap in these kind of Bodies, to be artificially of the same nature with those in exact regular Bodies: viz. cubatory and quadratary; as I shall show in their Dimensions; though indeed in their solid Dimension, there is always a Quadrature, in respect of the Base. And here likewise what is understood for solid measure, must be understood for gravity or weight, according to any Metal proposed.  
And all these several dimensions afore-named, are here performed by Lines of equal parts only, according to a decumane, decimane, or decimal division, in which therefore consisteth the excellency of the performance: and the same demonstrated, not only in respect of the practical use thereof, but also the theorical grounds and reasons, according to certain Propositions laid down for that purpose. And which Lines are here generally set forth by Number, denoting or expressing their magnitudes from any Measure given or appointed.  
And now, although that learned Mathematician, Mr. Edm. Gunter (sometime Professor of Astronomy in Gresham-Colledge in London, and long since deceased) a man excellent for Instrumental inventions, hath (among other Lines) upon his Sector, certain Lines of quadrature, (as he termeth them) as for to make a Square equal to a Circle given, by the semidiameter thereof, & contrà: and so for the like quadrature of certain rectiline regular Planes, by their sides; yet those are of a much different kind and nature from ours, (as any one may plainly see) being not several Lines of measure divided into parts any way, whereby to give immediately of themselves, the side of the equal Square arithmetically (as I may so speak) or in quantity discrete, as from any certain, set, denominate Measure, according to a common way of measuring; as our Lines do, (& consequently, the superficial content of the Square, for the Area of the Figure) but only one Line, drawn twice over upon the Sector; or (if you will) two like, correspondent, or congruall Lines, drawn upon each leg or shank of the Sector, from the Centre thereof, and undevided; containing in them only certain points, in which the Sector must be opened, according to the semidiameter of the Circle given, or the sides of the other Figures to which they belong, being there expressed by the numbers of their Sides; and so give the side of the equal Quadrat, only geometrically (as I may so speak) or by Line, crossing the Sector between the two points of quadrature at the ends thereof, and so parallel to the line for the semidiameter of the Circle, or side of other Figure, to which the Sector is opened: and which line being taken of from thence with Compasses, and so applied to any certain Line of measure divided, will than indeed give the side of the equal Square in the parts of measure, as our Lines of quadrature do: but yet Mr. Gunter doth not apply his Lines of quadrature to such a use, or any way mention the same, but only to the laying down of the exact side of the Square equal to a Circle given, etc. and so to reduce the Circle into a Square, geometrically, after a mechanical manner: and which can hold but only in a small Circle, whose semidiameter (or other ordinate Plane, whose side) may not exceed a convenient extent or opening of the legs of the Sector, (or of a pair of Compasses) so as to take of the side of the equal Quadrat, in its due place upon the Sector: for indeed, the largest Sector or Compasses that are usually made, (or can be made convenient for use) will open or extend (at the utmost) but to a very short line, in comparison of those which frequently fall out to be measured; and therefore his Lines (or way) of Quadrature, cannot extend to any large Dimensions, and so serve for a general measuring as ours do, if they should be applied to the very same use that they are, namely, the superficial dimension of Circles, and other ordinate Planes, (in a quadratary way) according to any measure assigned, and thereupon can serve but to very small purpose. Nor indeed had lacquainted myself with those his Lines of Quadrature, or any other upon his Sector, when I first apprehended and conceived in my mind this artificial way of measuring which I here propose, (though I had often seen that Instrument, and had much studied his Book in other things) nor till I had in a manner perfected the same throughout all the Dimensions to be performed thereby, and so was come to the Close of this Book. Neither did any of those Artists, to whom I ever yet propounded the same (as aforesaid) mention to me thereupon in the lest kind, Mr. Gunter's Lines of quadrature (or any other upon his Sector) in reference to any measuring; although some of them I am sure (if not all) were well acquainted with Mr. Gunter's Sector, & all his other Instruments, & taught the use of them to others. But this only by the buy. Nor do his Lines of Superficies in general, upon the Sector, being also two like, congruall Lines drawn on the two leg● thereof from the Centre, and divided unequally into 100 parts, serve for the measuring of Superficial Figures, as to the immediate producing of their contents simply by themselves, in any kind of measure given; but only in a way of proportion, to found out the superficial content of one Figure, by the superficial content of another like (or unlike) Figure given or known; together with several other uses noted by him. And so his Lines of Solids in general, being two like Lines, drawn upon the Sector in like manner, as the Lines of Superficies, and divided also into 100, (or rather 1000) parts unequally, serve only for the like uses in Solids, that the Lines of Superficies do in Superficies.  
And than for his more particular Lines of Solids (as I may term them) upon the Sector, called the Lines of inscribed, and of equated bodies; having reference in particular to the five foresaid plain regular Solids, and a Sphere: the first sort of them, in respect of the inscription of the said five bodies in a Sphere; and the other sort, in respect of their equation to a Sphere, and also of one to another; are of a different nature and kind from our artificial Lines pertaining to these bodies, and a Sphere, for the immediate producing of their solid contents in any Measure appointed, according to an exact, absolute cubature, both simply in themselves, & also in respect of inscription & circumscription to a Sphere given, as aforesaid: but are the first of them, for the finding of the Sides of these five bodies, as being to be inscribed in a Sphere, by the semidiameter thereof given; and the other, for the finding of their Sides, as being equal in magnitude to a Sphere, and this by the Diameter of the Sphere given; & contrà: or as to be made equal one to another by their sides: and all this in a mere geometrical sense; and thereupon these latter Lines of his, will give of from the Sector, the side of the Cube equal to a Sphere, by the Diameter thereof; & to any of the other regular bodies, by the sides thereof given, in the same manner, that his Lines of quadrature give the side of the Square equal to a Circle, by the semidiameter thereof, and to the other regular Planes or Superficies, by their sides given; these Lines of equated bodies (as also those of inscribed bodies) being of the like kind and nature with those of quadrature, (which by the same reason, may aswell be called Lines of equated Superficies, for that they do not only equal a Square to a Circle, or other ordinate Plane given; but also equal them all one to another) and are drawn upon the Sector accordingly; & so are of no further or better use than they are.  
Than lastly, his Lines of Metals (so called) upon the Sector, (being inserted with the lines of equated bodies, because there was spare room, and much of the like kind with them, and so are contrived together on the same two lines, on each leg of the Sector, being drawn from the Centre) are not like our lines of Metals (as I may so term them) which are for the immediate discovering of the weight of a Sphere, or other regular (or regular-like) body made of any Metal, in the very same (cubical) manner, that their solid contents are obtained in any measure, by their respective artificial Lines as aforesaid; but those (together with his Lines of Solids) do serve only to found the proportion, as it were, between several Metals (as he saith) in their magnitudes and weights, and that according to the experiments of Marinus Ghetaldus, in his book entitled Archimedes promotus; that is (as he saith) In like Bodies of several Metals, and equal magnitude, by having the weight of the one, to found the weight of the rest; & contrà: together with two other uses noted by him.  
These things (courteous and judicious Reader) I thought good here to insert by the way from Mr. Gunter, to show the difference between his Lines and ours; especially those of his, which for their use, may seem to come most nearly to ours, namely his Lines of Quadrature, and of equated Bodies: & which as they cannot be so generally useful as (or not considerable for their use, in respect of) ours, according to what I shown before; so neither can they be altogether so exact in the performance of those small Dimensions which they can reach unto.  
And what I have here performed Geometrically, or by Line (or Scale) in an artificial way of measuring; I have also set forth Arithmetically, or by Number, in a way of Proportion after the most exquisite manner that may be, as from the natural Measure; according to the same division or partition of the Unity, as is of the Lines of measure, both natural and artificial; and by which therefore the artificial measure may be readily deduced from the natural, or the natural Measure be reduced to the artificial: together with a multitude of other metrical conclusions besides, in most of the geometrical Figures which I have here particularly handled, by way of practical demonstration; none of them having been done before by any man, that I do know of; except those in the Circle, being set forth by Mr. Gunter, and from him by Mr. Wingate, and perhaps some others; but yet not all of those Proportions in so ample terms, as I have here done them, which therefore I extracted again anew.  
And than moreover shalt thou here found, not only the most artificial and expeditionall way of measuring thus all regular Bodies, and such as do come very near a regular form (which therefore I call regular-like) but also of such as are of an irregular form; and first, of concave Bodies, or Vessels for Wine and Beer, (which commonly do somewhat imitate the form of a Cylinder, and may be called Cylindroidall, and so admit of a Cylinder-like dimension, being first reduced by art to a Cylinder: to which end I bestowed some considerable pains (and a little cost too) in the making of sundry experimiments, for the discovering of the true contents of the Standard-measures for Wine and Ale or Beer, pertaining to the City of London, (which are kept at the Guild-hall) as being commonly taken for the most generally received Measures for this purpose, throughout the Kingdom; and by which I have seen the Measures which have been made for some eminent Towns far remote from London, to be sized and sealed, (though here I will not contend about them) and have here accordingly fitted a gauging-Line to each of them, according to our artificial way of measuring, being also therefore Lines or Scales of equal parts in a decimal division, by which the liquid content of any Vessel will be obtained immediately in Gallon-measure, after the same manner (and with the same expedition in a manner, the irregularity of the Vessel being considered) that the solid content of any exact Cylinder is had artificially, according to any Measure appointed.  
And than last of all, do I here show (by way of Appendix) the most easy and exact way for the discovering of the solid capacities of all other irregular kinds of Bodies whatsoever, both solid and concave, which of themselves are altogether unmeasurable in the usual way of measuring (or whose Dimensions can in no wise be taken by a Line of measure) which therefore is a work of a contrary nature to all the former: and have here exemplarily illustrated the same from experiment, in a certain regular solid body, easily and exactly measurable, for a confirmation thereof; (and which way also, hathnot been set forth by any man before, that I do know of) Together with several other new mathematical experiments and observations, very useful, and worth the noting. All which (friendly Reader) I commend to thy courteous consideration & acceptance; hoping thou wilt receive not less delight (and benefit also) in the perusing and practising hereof, than I have taken delight and contentment in the study and exercise of the same, though surely with no small pains and industry, beside the expense of some time now and than, from my occasions of serious concernment, according as the same would reasonably permit: and which therefore I could do to no other end, than only to enlarge and advance (so far as here I might) the practice and exercise of this so noble and admirable an Art, being drawn thereunto by that Genius, which hath heretofore much disposed and inclined me to mathematical contemplation and exercitation in general: so that, thy friendly acceptation hereof, is all I expect for my labour. And if any after me shall hap to raise any further Conclusions from what I have here laid down, in any particular thereof; than surely will these my pains be yet thereupon so much the more to purpose. But now as I must expect this work of mine will meet with some Momaicall or Zoilan Spirit, so I shall not regard the same, or be terrified thereat, seeing that the best conceits and inventions of men that were ever yet published to the world, have been obnoxious to the obloquys and obtrectations of such malevolent and malignant spirits: and which hath been the complainr of the most learned men in all ages. And thus, courteous, ingenious, and ingenuous Reader, I friendlily bid thee farewell, resting,  
London; the first of May, 1650. Thine hearty wellwisher, J. WYBARD, DM.   

THE General Contents of this Work, consisting (almost all) in measure alone.  

PART I  
SECT. I. OF the artificial Dimension of regular and regular-like Figures, in general Page 1.  
SECT. II. The Dimension of a Circle p. 9  


The dimensional Proportions in a Circle, in the most exquisite terms, and all the variety that maybe. p. 19  
The solid dimension of a Sphere. p. 21.  
The same another way; as also another superficial dimension of a Circle, artificially. p. 28.  
The superficial Dimension of a Sphere in both those ways. ibid. and p. 29.  
The dimensional Proportions in a Sphere, both for solid and superficial measure, like as in a Circle. p. 30.  
The solid Dimension of a Cylinder and Cone. p. 32.  
The same another way artificially. p. 39    
SECT. III. The superficial dimension of a Cylinder and Cone. p. 45.  


The dimensionall Proportions in the Cylinder and Cone both for solid and superficial measure. p 54    
SECT. iv Of the differences between the natural and artificial Measure, in the dimension of Figures. p. 56.  


And of the grounds or reasons of the artificial Mensuration. p. 59     

PART II.  
SECT. I. OF the artificial Dimension of rightlined regular Planes or Superficies in general; and the same demonstrated particularly in two of the first of them. p. 71.  


As also the dimensionall Proportions of the same two Figures, expressed in all the variety that may be. p. 92 etc.  
The artificial Dimension of Triangles in general. p. 78.  
And from thence, of any Rhombus, Rhomboides or Trapezium. p. 103. 105.    
SECT. II. Of the Dimension both solid and superficial of regular-based Pyramids in general, etc. p. 106.  


And the same demonstrated particularly in the three first kinds of them; & that (for solid dimension) in three of the five plain ordinate Bodies, or rectiline regular Solids, viz. in the Tetrahedron, Octahedron, and Dodecahedron, with a brief description of all the said five Bodies. p. 108, 116, 119, 127.  
The dimensionall Proportions in the said three first kind of Pyramids, both for solid and superficial measure. between p. 136 and 137.  
Of the artificial Dimension of regular-based Prisms. p. 137.    
SECT. III. Of the Dimension of the foresaid five famous ordinate Bodies (commonly called the Platonical Bodies) both solidly and superficially, several ways; after the most exquisite manner that may be. p. 139.  
SECT. iv The same Dimensions, with a multitude of other metrical Conclusions in the said five Bodies, expressed Arithmetically by way of Proportion, in the most exquisite Terms that may be; like those of the other Figures beforegoing, here particularly handled. p. 166 etc.  
SECT. V Of the Dimension of all regular Solids aforesaid, in reference to grav●ty or weight, according to any Metal proposed; in the same artificial manner, as is for solid measure: And the same demonstrated particularly in a Sphere of a certain Metal proposed. p. 198.  


The P●oportions of gravity and magnitude together, pertaining to a spherical body of that metal, expressed in the like Terms with the other Dimensions beforegoing. p. 241.  
The like reason of Dimension, as before, for the gravities of all regular-like metalline Bodies. p. 246 and 258.  
W●●re, a demonstration, or illustration arithmetical, of the artificial Lines for the solid dimension of a Cylinder and Cone, in particular; according to Theor. 3. and consequently thereby, for all the other kinds of Dimension in general, according to Theor. 1 and 2. And withal the general ground and reason of the same Dimensions fully declared.     

PART III.  
SECT. I. Of the measuring or gauging of Vessels, in general. p. 262.  
SECT. II. Of the Quantities of the Wine and Ale-gallons, in reference to Gauging; from sundry new and exact Experiments. p. 264.  
SECT. III. The practice of Gauging, from our artificial way of measuring; as also from the natural (by way of comparison) for a confirmation of the other; as in all the precedent Dimensions. p. 288.   

Appendix.  
FOr the most ready and exact discovering of the solid capacity or dimensionall quantity of any such irregular Body whether solid or concave, as cannot of itself be measured in a plain geometrical manner. p. 301.  
The Conclusion for the more speedy absolving or expediting of all the foregoing Dimensions. p. 338   

The more particular Contents of this Work, as falling in by the way, in reference to the general: consisting most of them in measure and weight together, and being experimental.  
1. THe Proportions or Comparisons of all the principal Metals, in gravity and magnitude, expressed in general terms; according to the experiments of Marinus Ghetaldus. p. 201  
2. The same laid down particularly in our Troy-weight, in Spheres of equal magnitude. p. 237.  
3. Of the true magnitude of the Roman Foot, and the same compared with our English Foot, according to the observations and experiments of our Countryman Mr. John Greaves. p. 206, etc.  
4. The Roman Weight, and our Troy or Goldsmith's Weight compared together, from the said Mr. Greaves. And so from thence, a proportion set between them, in the lest terms. p. 209.  
5. Of the gravity of a certain serreall Sphere or Bulk, both in Avoirdupois and Troy-weight, from a new and exact experiment. p. 217  
6. Several Proportions set between the Troy and Avoirdupois-weights, compared together: And first in an Arithmetical manner, according to the Doctrine of Proportions. p. 220 and 232.  


And than the same compared, and examined by the Balance upon the weighing of several Bodies. 223 & 276.    
7. The proportion between forged or fine Iron, and cast or bullet-iron, deduced from new exact experiments p. 225.  
8. Of the weight of water, in relation to its solid measure; or the ponderall and dimensionall quantity of water compared together, from several new experiments, for discovering the solid measure of any irregular body thereby, whether solid or concave. p. 269, 270 & 287.  


The same compared together several ways from the foresaid experiments, and that in measure both unciall and pedal, and in weight both Troy and Avoirdupois: And so the nearest weight of one Inch, and of one Foot cubique or solid of water: and conversly, the magnitude, or solid measure of one Ounce, and one Pound, etc. and so, some apt terms of proportion raised for a general and ready use. p. 302 and 305.    
9 How to found out the solid content of any irregular body by the weight of water, etc. p. 308.  
10. How to found the exact quantity of water (or other-like liquid body) which is equal in magnitude or dimension to a solid body given. ibid.  
11. How to found immediately the gravity of the water (or other like liquid body) which is equal in magnitude to a solid body given, by the gravity of the solid body only: And so the manner of weighing a solid body in Water, whether it be heavier or lighter than the same. p. 310 and 311.  
12. The three last Conclusions beforeing, exemplarily illustrated from exact experiments made upon a spherical Stone-Body, or Marble-Bullet; and so confirmed accordingly. p. 313, etc.  
13. Of the difference of gravity in water; or the different gravities of several ●●rts of water, in relation to our Atactometricall practice, or Dimension of irregular Bodies. 319.  
14. The same most exactly experimented by the Balance sundry times, upon eight several Waters; and all compared together. p. 325.    

WHosoever shall think good to make use of the Lines for Gauging, or for any other Dimensions described in this Book: or of any other Instruments for mathematical practice in general; they may have the same very accurately made by Mr. Christopher Flower, dwelling in the Bulwark, near the Tower of London: besides divers others about this City.   

Courteous Reader,  
seeing that hardly ever any Book passeth the Press, free from typographical errors, notwithstanding all the care and diligence that can be used (and in which I was not here wanting) especially Books of this nature: and than that they being commonly set at the end of the Book, are seldom taken notice of, till the Book be read throughout; and so many times the mistakes of the Printer are by ignorant or malevolent Readers, put upon the Author as errors: Therefore the chief of those few faults which have here escaped, (the most of them being but merely literal) I thought good to put at the beginning; that so th●u mightest in the first place know them, and consequently amend them: assuring thee withal, that of that multitude of Numbers which are comained in this Book (the most of them denoting Lines, and Proportions of measure; and so upon which dependeth the speedy working and resolving of many excellent practical Propositions: and are therefore specially to be regarded; and in which a fault cannot be espied by bore inspection,) there is not one of them defective in any one figure thereof; such was my (more than ordinary) care therein.  

ERRATA.  
PAge 4. line 7. put (before commonly. l. 8. deal of. l. 20. r. partior. p. 7. l. 28. for hath, r. both. p. 8. l. 22, r. in respect of the Measure from which it is taken, as also of the Figure etc. l. 30. deal the. p. 32. l. 11. r. Sect. 4. p. 59 l. 14. for matter of, r. manual or. p. 75. l. 2. r. whose. p. 91. l. 24. r. shall. p. 173. the beginning of the last line save one, r. And so. p. 214. l. 2. deal the comma at Measure, and put one after Diameter. p. 316. l. 14. deal the first as. Other faults thou mayst meet with, which being only literal, and so not worth the noting here; thou mayst easily amend in the reading.    

Tactometria, Or the most Exquisite practical Dimension of all Regular, and Regular-like Figures in general.  

And first, of the Circle, Sphere, Cylinder, and Cone, in special. PART. I.  

SECT. I. Of the nature and division of the Lines of Measure in general, for the performing of all the aforesaid Dimensions.  
Forasmuch as to the due measuring of any Magnitude or Quantity continual (in practical Geometry) there is required some certain Measure first to be given or appointed: Therefore first and more generally, by Lines, we do her understand any right Line assigned for a certain Measure; such like as Euclid, Elem 10. Defin. 5. calleth 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, which is as much as certain, definite, determinate, speakable or expressible by voice, or otherwise expressible by Number: And so the most Noble, illustrious, and learned, Franc. Flus. Candalla, a most diligent and industrious restorer of Euclid's Elements interprets it Certa, a line certain, as first put or proposed and made manifest, and divided into parts certain and known: And it is also called Famosa, a measure famous, that is, (as P. Ramus, and Adr. Metius do note) first 
Ram. Geom. lib. 1. El. 8. Et Scholar Mathemat. lib. 21 Adr. Met. Geom. pract. Par. poster. seu Gaeodaes.  spoken or expressed, etc. But most of the Latin Geometers do call! such a line Rationalis, a Rational line, for that (as Ramus saith in the places here cited) it is rational to itself, as are all magnitudes equal among themselves: and Clavius saith it is called Rationalis, because 
Clavius in Def. 5, El. 10.  it is always put certain and known, whereas all other lines which are compared to this (as lines infinite in multitude, may be according to Euclid in the place forenamed) are not certain & known, though taken apart by themselves, they are, seeing that every one may be divided into any number of equal parts, and being compared to this for their Measure, they are all either symmetrall or asymmetrall, and so are said to be either 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, or 
* So put by Theon, whereas it should rather be 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ●s it is opposed privatlvely to 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉,: Sec Flussate upon the place, and also in Proem. 1●. Elem.  〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, in Def. 6, 7. explicable or inexplicable, rational or irrational; (but with this point we meddle not here,) and so we here understand this Line to be most properly called Rational, as comprehending or containing in itself the dimensionall reason of alother lines measurable thereby. And therefore here we (for brevity) will with most Latin Translatours and Commentators, and also our 
* H. Billingsley, a Citizen of London, and Lord 〈◊〉 〈◊〉. 1596▪  English Translator of, and Annotatour on Euclid's Elements, understand any right line so first set, put or proposed, by the name of the Rational Line (and this may be applied to any Measure whatsoever, and it is one of Euclid's Data in Lib. Dator. Defin. 1.) or (in respect of the ensuing work) the prime, simple, true, or natural Rational line. And this we mean when any where we say simply the Rational Line.  
Secondly, and more especially by Lines we here understand any such line augmented or diminished by a certain convenient segment or portion of the same, for the more artificial and speedy mensuration of the aforenamed Figures: And this Line we may (not unaptly) call the second supposed or artificial Rational Line, as being derived from the former, and so substituted in place thereof: like as in Numbers the Logarithmes are usually called Artificial Numbers, as being substituted instead of the natural numbers, from which they are deduced, and whose place they supply in a most excellent and admirable manner, by performing all arithmetical operations with that facility, expedition, and compendiousness (and exactness also in some cases, as I shall afterwards upon occasion show) which the natural numbers themselves cannot, for that by these, the two most tedious and troublesome parts or species of Arithmetic (to wit, Multiplication and Division) are wholly avoided and abolished, and that most difficult branch (or operation) thereof called the Extraction of Roots, is mightily abbreviated and facilitated: And so the Arithmetic performed thereby is usually called artificial Arithmetic.  
Now seeing that every continual or continued Quantity falling under Measure (in practical Geometry) is referred and reduced to the discrete, that thereby its dimensions may be made more manifest to us; so that Geometry hath perpetually need of Arithmetic for the explicating and expressing of its magnitudes in their dimensions: Surely no kind of Numeration can be so accommodate to this thing, as that which considereth the Intger of Measure (as the Unit) in a decumane, decimane, or decimal solution, for that by this the practical, instrumental, or mechanical part of measuring, or of the art metrical, commonly and improperly called by Ramus, Metius, and of some others Geodasia which properly signifies the division or partition of right lined Superficies, as Pediasimus, de mensuratione & partitione Terrae, well observes, saying Terrae mensuratio duas in parts dividitur, Geometriam, scil. & Geodaesiam: Areae namque secundum aertem mensuratio, & terrae mensuratio est et meritò Geometria vocatur; Unius verò & ejusdem areae, seu loci divisio inter diversas personas, partitio quaedam est terrae, & jure optimo Geodaesia appellatur, (and which from him Clavius noteth Geomet. pract. lib. 6.) for that the greek words, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, do (poetically) signify the same that the Latin words divido and portior, and so of which, and the word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, Terra, comes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, i e. Ter●ae divisio seu partitio) is made much more facile accurate and expeditious.  
For indeed this Mathematical solution of unity or continuity, is of all others the most absolute and certain, and the most perspicuous and rational, and by how much the more numerous it is in the parts thereof, by so much the more exact it is, and consequently the work effected by it. And what utility it hath brought to the Mathe●●atiques in general, may be sufficiently witnessed by that ●ost noble and useful part of Geometry, called Trigonomeirie, and that in the Radius of a Circle (which is the very Basis and Root of all trigonometrical operations) where (to wit) first, that greatly renowned Mathematitian Johannes Regiomontanus, having for a long time used the Sexagenary solution (as Ptolemy and others before him) did at last bethink himself of the decimal, as being much better (seeing that the Unit would perform a far greater Compendium than the Senarie) and indeed the best of all (and so put the Radius, to 10 millions of parts, and next after him, Rheticus in his great Trigonometrical Canon, to 10000 millions, and afterwards (to make that his Canon most absolute and perfect) he proceeded to 1000 milliots (as I term them) or millions of millions, whereby the art of Trigonometry & consequently other Mathematical arts, as Astronomy, Geography, etc. depending thereupon) did become, in the practice, far more facile and expeditious than before: for where the first proportional Term of the trigonometrical proposition, is the Radius or total Sine (which very frequently happens, or may be so made for the most part, as the learned Pitiscus excellently showeth amongst other his compends in working) there the proposition or question is solved only by a 
* This is to be understood in working by the Natural Numb.  conjunct or manifold 
Pitiscus, Trigon. lib. 5.  composition (or mutual implication or induction) of the second and third terms, seeing that the Unit altereth nothing in a conjunct or manifold resolution, but the Number of composition immediately becomes the number of resolution, only distinguishing between the absolute or integral, and the fractional part thereof. And moreover the benefit of this solution of unity is excellently seen in that most excellent Arithmetical operation, vulgarly called The Extraction of Roots, wherein (to wit) seeing the Roots of numbers not explicable or rational (which the Algebrists or Cossists commonly call Surd Numbers, and so their Roots furred Roots) cannot be exactly had, than those numbers are reduced into some kind of decimal parts (or parts of a great denomination, as Ramus termeth them) as C●ntesms, Millesmes, etc. and that figurate, as Quadrate, Cubique, etc. that thereby their Roots may be had more certain and nearer to the truth, 
Ramus lib. Geom. 12 de Quadrato & 24 de Cubo.  than they can by the natural or vulgar extraction, as Ramus showeth in the aforesaid Books, where he (the first) shown this 
See Wingate Arithm. 1 Book, ●1 Chap. 3 Sect.  kind of Extraction which to some seems to have been the very foundation of decimal Arithmetic, although Ramus hath no where else in his Mathematical Works made any other use of this kind of numbering, or made any mention of the same: But indeed that of Regiomontanus in the Radius of a Circle seems to me to have given the first light thereof to the World; so that the trigonometrical Numbers which now we use, may be termed decimal, as they are derived from that Radius: For all the Sins to a Quadrant, and the Tangents to an Octant or semiquadrant, are decimal parts or fractions of the Radius, but indeed the least Secant is greater than the Radius: And so we will here make use of this kind of Numeration, as being the fittest for our purpose, as we said before: And indeed for that this present work of ours cannot conveniently be performed by any other.  
Now the Geometrical Figures which we have here first named, and so for which we have first extracted these kind of metrical lines (or linear numbers) are those four which Archimedes himself more especially treated of, and which are as it were, the beginning and ground of all the rest, namely, the Circle, Sphere, Cylinder and Cone, and accordingly the like Lines may be extracted for all ordinate Planes and Solids whatsoever, as we afterwards show; for that to these four may be aptly referred all other regular and regular-like Figures; As to the Circle may be aptly referred all ordinate Planes, to the Sphere, all exactly ordinate Solids; to the Cylinder, all ordinate-based Prismes; and to the Cone, all ordinate-based Pyramids: And though a Cylinder and Cone, and the like, cannot properly be reckoned among Regular Figures, according to the strict acception of an ordinate or regular Figure in Geometry, yet in respect of the regularity of their Bases, and also the regularity and uniformity of their other, and more special superficial part besides (whether the same consist of one entire plane only, as in the Cylinder and Cone, or of several planes equal and like, as in all right or erect regular-based Pyramids and Prismes) the same may in a sort be termed regular (especially the Cylinder and Cone) and which therefore, for distinction sake, I call regular like solid Figures.  
But now the quantities of the artificial metrical Lines, first extracted for the four Figures first before named, according to any prime Rational Line, and that to a Decumillenary division of the same, are numerally thus.  

I. A Line for the most excellent superficial mensuration (or diametral Quadration) of a Circle, is 1. 1284 ●eré.  
II. A Line for the most exquisite Solid dimension (or diametral Cubation) of a Globe or Sphere, is 1. 2407.  
III. A Line for the Square Solidation (as I may term it) or Rectangular Parallelepipedation of a Cylinder, by its Diameter and Axis or Altitude (as to an exact quadrate Base) is 1. 0838.  
iv A Line for the like dimension of a Cone, is 1. 5632 ferè.   
And so divers other the like Lines, for the dimension both of these, and also of other Figures hath Superficial and Solid, I shall afterwards show in their several places.  
Every such second supposed, or artificial Rational Line, must be divided decimally, as is the first, true, or natural Rational Line, from which it is taken, according as the length thereof can conveniently bear: But if the first Rational Line cannot well admit of so many parts as are here set down, than they may be abbreviated or contracted; as to Millesmes, Centesmes, or to prime or simple Decimals (or Tenths) only; though indeed our least common Measure in use, (viz. an Inch or Pollicar) may be distinctly divided in●o 100 (or 1000) parts, if it be rightly handled according to the more artificial (or diagonal) way, as hereafter we shall have occasion to show. And so we have here set forth the Rational line in parts of a large denomination, that so it might serve for more exactness in use, because the more parts, or the greater divisionall denomination the Integer of measure (or the unite) is of, the more exact will be the work performed by it, as I noted before, and as afterwards I shall make plainly to appear.  
Furthermore, every such artificial or supposed rational line, may be said to be two fold, to wit, general or universal, and special or particular: General, in respect of the number by which it is indicated and explicated, for that the same linear number, doth serve alike to all prime Rationalllines: And particular or special in respect of the Figure (Superficial or Solid) to which it is appropriated and applied, because that every such particular Figure as is mentioned in this Book, doth peculiarly and properly claim to itself (and that several ways) such a line for its more artificial and expeditionall mensuration.  
Now the use of the four artificial Lines before-going, for the measuring of the four first Figures aforenamed, we will deliver in the three practical or problematical Propositions following in the next Section; and withal the magnitudes and uses of all the other artificial metrical Lines pertaining to the said four Figures.   

SECT. II.  

PROP. 1.  
If the Diameter of a Circle be taken by its proper Line of Measure (according to any Rational Line) I say than, that the Quadrat of the Diameter shall be the Area of the Circle (according to the same Rational line) which I prove thus demonstratively.   geometrical diagram 
Now that there may be a comparing of our new or artificial mensuration with the common or natural, and thereby a confirmation of the same: I draw the right line C D, for the second, compound, supposed, artificial (or quadratarie) Rational Line, to the length of A B, the primary, simple, true, or natural Rational line, and moreover. 13 ferè of the same (for so much is the additament or additional segment in centesimall parts) and than I divide C D likewise into 100 equal parts (or first into 10 parts, and than only one of those parts into 10 parts, which will be sufficient, as in the Line A B) which done; I measure the line A B 2.00 (for the Diameter of the Circle) by the line C D (the Line or Scale of quadrature) and find it to be thereof 1.77 (for the line A B, simply upon the line C D, falleth about the middle between .88 and .89, and so the double of A B in one entire line at length being measured by the Line C D will fall about the middle between 1.76 and 1.78, which is about 1.77) which squared, gives 3.1329 for the Area of the Circle, agreeing with the former area exactly in the integral part: But now the fraction-part of that area, viz. 1/7 being converted into decu-millesmes or square centesmes, is .1428, which exceeds our measure by .0099, viz. 99 square centesmes, which difference is not considerable in common practice, as I shall afterwards plainly show: but yet our measure wanteth of the true area, found by the more new & true terms of Cyclometry, or Circular Tetragonisme (which here we use in all Cyclometricall operations) not so much by 12 square centesmes, viz. but 87 such centesmes as I shall straightway show: For those of Archimedes, of the Diameter to the Circumference 7 to 22 subtripla-sesquiseptima, 
Archim. de dimen. Circ. prop. 2, 3.  and consequently of the Square of the Diameter to the Circle 14 to 11 supertriundecima, or supertripartiens-undecimas (though sufficient enough for any ordinary mechanical use, as he only meant them, and that especially in smaller Circles, yet) are in strictness of Art too large, and so give the area somewhat greater than indeed it is, and the more, the greater the Circle is, and hereupon hath arisen that difficult and curious question so much controverted among Artists about the Quadrature of a Circle; and in which many learned men have bestowed great pains, as to the finding out of the nearest proportion between the Diameter and Circumference, etc. among which that excellent Artist 
* Lib. de Circ. & Ad scrip. Here also is seen the excellent use and benefit of decimal Numeration in the Quadrature of a Circle. Snel. lib. Cyclomr prop. 31. See Lansberg. Cyclome●. lib 1. Porism● 3. to 29 places, agreeing with Ceulen.  Ludolph van Ceulen (alias Cullen & Collen) hath hitherto generally carried away the greatest commendations, having set forth the same in decimal terms to 36 places, which he willed to be engraven upon his Tombstone, as a Testimonial and Memorial of those his painfully sustained and finished Labours (as W. Snellius noteth, who afterwards produced the very same number) and which I have seen upon the same in the great Church in Leyden, called St. Peter's Church, there being drawn thereupon a large Circle, and upon the Diameter-line an unit with 35 cyphers for the number of the Diameter, and round about the Circumference, the number for the same to 36 places; of which so many as are needful here to deliver, as being sufficient for ordinary use, are 3.14159, answering to the Diameter 1.00000, or more briefly 3.1416, the Diameter being 1.0000; which according to the foregoing Archimedean terms of 7 and 22, will be 3.1428, etc. But those of that famous Mathematician * Metius the elder, sometime Geometrician to the Estates of the confederated or United Belgic Provinces, of 113 and 355 do agreed with the Ceulenian terms to the first seven places, viz. 3141592, answering to the Diameter 1000000, for that he demonstrateth the proportion of the Periphery to its Diameter, to be less than 3 17/120 (that is 377/12) but greater than 3 15/106 (that is 33/106) whose 
* Metius in lib. advers. Quadratur. Circ. Simonis a Quercu. Et Adr. Met. in Lib. Geomet. pract. part. prior. Cap. 10. prop. 3. & part. post. Cap. 4. prop. 1.  intermedian 
The mean Arithmetical is in the lest Terms ● 1801/12720 which is Decimally 3.1415880503, etc. and the mean Geometrical (or mean proportional) is decimally 3. 1415880493, etc. But his mean 3 16/1●3 is decimally 3.141592920, etc. which exceedeth both the other, & that of Cullen is 3.14159265, etc.  proportion is (saith he) 3 16/113 (or 355/113) and which is a little larger than that of Ludolph Van Cullen, yet so, as that the difference is less than 1/10●000. And this proportion doth give the Area of the foregoing Circle 3 16/113, which is by decimal conversion of the fraction-part into square Centesms, 3.1416, agreeing exactly with that which is produced by the Ceulenian proportion, and this is the true Area of the Circle: but the Area 3 1/7 produced by the vulgar or Archimedean proportions, being decimally to square Centesmes 3.1428 exceedeth this Area by 12 Square Centesmes (and so near the old and new Cyclometry do here agreed) & so the area of the circle found▪ by our way 3.1329. wanteth of the true area but 87 square centesmes, as I said before: and this defect (not considerable in common practice, as I noted before) happeneth, in regard that the double of the first line measured by the second, falleth not precisely on 1.77, but is somewhat more, though indeed upon the said line itself, it is very hardly discernible, being but very little: Wherhfore if both the Rational Lines were divided into more parts, as 1000, 10000, etc. than the work would prove still more and more exact (but indeed here we could not draw a line actually capable of a greater decimal division, or denomination of parts than 100, according to the plain, vulgar, simple, or natural division of a Line, which here chiefly for plainness we have used:) For first, if the prime rational Line (A B) be divided into 1000 parts; than the second (C D) will be of the same. 1.128, and the first compared with the second (being also divided into the like number of parts) will be .886 or near thereabouts, which falleth about the middle between .88 and .89 of the said second line being 100, for these two converted into Millesmes are but .880 & 890, and so the double of the first Line measured by the second will be 1.772, or near thereabouts, which is somewhat more than 1.77 upon the second Line centesimally divided, for 1.77 converted into Millesmes, is but 1.770, which shows that 1.77 was somewhat too little for to produce the true content of the Circle: Now 1.772 being squared, gives 3.139984 for the area of the Circle (which is in square Centesmes, 3.1400 ferè) and the area produced by the truest terms of Cyclometry, both Ceulenian and Metian, is to Milli-millesms, or square Millesms, 3.141593 ferè, (whereas the vulgar area 3 1/7 is 3.142857) which our area is wanting of by .001609 ferè, viz. 1609 square Millesms, or but 16 square centesms, and agrees with the same in square tenths, viz. 3.14) which comes much nearer the matter than the centesimall operation, and indeed as near as need be desired: but yet this inconsiderable difference (by way of defect in ours) happens also for the reason aforesaid, in that the double of the first line measured by the second is not just 1.772, but somewhat more (as the first line simply applied to the second, is not precisely 0.886 but somewhat more) and therefore if the first be divided into 10000 parts; the second will be thereof 1.1282 ferè, according to the ancient or Archimedean Cyclometry, but 1.1284 ferè according to the later or Ceulenian and Metian Cyclometry: and so the first line compared with the second (being also 10000) will be 0.8862 (or somewhat more) and thereupon the double of the first, being measured by the second, will be 1.7724, which shows that 1.772 was somewhat too little to produce the true area, for this in decumillesms is but 1.7720, but our number is 1.7724 (and indeed somewhat more) which squared yields 3.14140176 for the Circular area, agreeing yet much nearer with the true area, which is now 3.14159265 (to which comes very near that of 3 16/113 being by decimal conversion of the fractional part into square decumillesms 3.14159292, beginning but now to exceed the true area produced by the Ceulenian Cyclometricall terms) our measure wanting thereof now but 19089 square decumillesms, which by contraction of the parts is hardly 2 square centesms.  
And if we proceed one operation further, namely, to a centu-millenary solution of the rational Lines (where A B being 100000, CD will be thereof 1.12838 ferè) our measure will be found to agreed with the true natural measure, exactly to square centesms, and to want thereof hardly 14 square millesms: for AB 2.00000, being measured by CD, made 100000, will be found thereby 1.77245, (which shows that the artificial or quadratary number in the precedent operation, was not exactly 1.7724, but would fall about the middle between that and 1.7725 upon the artificial line of measure, and so gave the area of the Circle short of what it should be) which being squared affords 3.1415790025, for the area of the Circle, which wanteth still of the true area (being now correspondently 3.1415926536 ferè) 136511 square centumillesms, which by contraction of the parts is but 1365 square decumillesms, and about 14 square millesms, and not one square centesm.  
And if we go on yet further to a milli-millenary solution of the Rational Lines (where AB being 1000000, CD will be of the same 1.128379) the natural and artificial Measure will be found to agreed exactly to square millesms: For AB 2.000000 entirely taken, being measured by CD (put 1000000) will be found 1.772454 ferè, whose Quadrat is 3.141593182116 ferè for the superficies of the Circle, which now exceedeth the true measure (being here correspondently 3.141592653590 ferè) by 528526 square milli-millesms, which by abbreviation or contraction, is 5285 square centumillesms, or but about 53 square decumillesms, and not one square millesm.  
By all which it is already sufficiently evident, that the more parts the Unit or the Integer of measure given, is divided into, the more exact will be the work performed by it, as I noted before, and as may be also further seen in the following dimensions.  
But indeed if the Rational Lines be divided but into 100 parts they will be sufficient for any ordinary use: For the greatest difference arising here between the true natural measure and ours, being that in the first operation (by centesms) where the superficial content of the Circle found by our measure 3.1329. falleth short of the true content (found by the truest terms of Cyclometry) 3.1416, by 87 square centesm●, is in vulgar terms but ●/115 of the prime rational Line A B (as the Integer of the measure given) squared, which is but as one part of a square unit divided into 115 parts; and surely this difference is of no moment in common practice. And whereas in the other operations, the differences happening between the true content and outs, in parts of a greater denomination, may seem to such as do not well apprehended this matter, to be very great; yet being reduced to vulgar Arithmetical terms (which are better understood by them) will appear to be still lesser and lesser, and as nothing: As in the fifth or last operation of this first Example or Demonstration, being under a solution of the Rational Lines to a million of parts, where the area of the Circle found by its proper Line of diametral quadration, exceeds the true area by 528526 square milli-millesms, that is, 528526 parts of the prime Rational Line being 1000000 parts squared, and so resolved into 1000000,000000 parts: which difference, though it may seem great by reason of the multitude of the decimal fraction-parts, yet considering the greatness of their denomination, and being reduced into common arithmetical terms (whose numbering part is 1) they will appear to be as nothing, being hardly 1/1●92●54 of the entire prime Rational Line (as the Intiger or Unity of Measure) squared, which is but as one part of a square Unit containing 1892054 parts.  
And if the Line of Measure first given, be so short as that it cannot be distinctly divided into 100 parts, according to its own self simply. than may 10 parts reasonably suffice: As suppose here a Rational Line to be of the former line A B, .1, and divided equally into 10 parts, viz. A b. (which is A B 0. 10) than the second Line, or Line of quadration, viz. C d. will be thereof 1.1 (which is as A B 0.11) answering analogically to A B 1.10, and which wanteth of the true additional parts .03 ferè, because the total additional centesimall segment of A B is. 13 ferè, as was noted before) And let the Diameter of a Circle, be A b, 1.5 (which is as A B 0.15) than the periphery will be (according to the same division) A b 4.7 (which is as A B 0.47) whose moiety A b 2.35 (or A B 0.235) together with the semidiameter A b 0.75. produceth the Circle itself in square Integers and parts of A b, 1.76. Now the diameter A b 1.5 being measured by C d (being also of a denary or simple decimal division) becomes but 1.3, whose square is 1.69 for the superficies of the Circle; which wanteth of the true content 7 square primes or tenths only, viz. 7 parts, of the prime Rational Line, or Line of Measure first given, A b 10 parts (as the Integer or Unit of measure) squared, and so resolved into 100 parts, which in vulgar Arithmetical esteem are hardly 1/14 of the same live squared, the said line itself being than near 4 parts. Or the diameter being taken in centesimal parts only of the Line A B, viz. 0.15 the content of the Circle will be only in square centesms, .00176, exceeding the Square of the quadratarie parts CD 0.13, viz. 0.0169 by 7 square seconds, or centesms only, which in common account make but 1/1429 ferè of A B squared (the line itself being than near 38 parts) which is much less than 1/14 for the difference between the two Measures: so that you may here see, how much 100 parts in the Line of Measure are better than 10; and therefore they that understand the more artificial way of dividing lines (commonly called the diagonal way) may better divide the lest Line of Measure, into 100 parts: For this latter assumed Line A b being but. 1 of the first Line A B, and but about 1/4 of an Inch or Pollicar may be distinctly divided into 100 parts, being taken according to the power thereof, so as that any number of centesms may be taken exactly therefrom.   geometrical diagram 
And here we now gather these profitable and delectable proportional conclusions in the Circle, and that to a milli-millenary solution of the unite.  
1 The proportion of the Diameter to the Circumference 3.141593 ferè Side of the Square squall to the Circle (which is the most near, precise, and proper squaring of a Circle.) as 1.000000 to .886227 ferè Side of the inscribed square .707107 ferè, √ q 1/2   geometrical diagram 

Contrarily.  2 The proportion of the Circumference to the Diameter, .318310 ferè Side of the Square equal to the Circle, (which is the second most proper quadrature of a Circle.) as 110 .282095 side of the inscribed Square . 2250●9  
3 The proportion of the Quadrat. diametral or circumscribing to the Circle itself, as 1 to .785398 Circumferential .079577  

Contrariwise.  4 The proportion of a Circle to the Quadrat diametral or circumscribing as 1. to 1.273240 ferè. circumferential 12.566371 ferè.  
And here you may observe the excellency of these proportions, as also of those of the like kind in the Sphere, and all other the Figures following; in that the antecedent or first Term is always an Unit, accompanied (or supposed to be accompanied) with so many Ciphers, as are places in the second term, which may here signify either a milli-millenary composition of the Unity, or the like resolution of the same, according as the Terms are taken; by means whereof the solution of the question or proposition is mightily facilitated and expedited, according to what I said in the beginning: and therefore in the extraction both of these in the Circle, and also of those in all the other Figures following, I have used all possible industry and exactness.   

PROP. 2. Of the solid dimension of a Sphere.  
If the Axis or Diameter of a Sphere, be taken by its proper Line of solid Measure; I say, that the Cube of the Diameter shall be the solidity of the Sphere, according to the prime Rational Line: which I thus demonstrate practically.  
ADmit the former Rational Line A B (divided as before) and let the Diameter of a Sphere be of the same Line entirely taken 2.00, so the solidity thereof (according to the commonly received Archimedean proportion of the Cube of the Diameter to the Sphere, 21 to 11, super-decupartient-undecimall) will be 4 4/21: which appears also by the Circumference of the greatest Circle of the Sphere, 6 2/7 (in respect of the subtriple sesquiseptimall proportion of the Diameter to it) for this being enfolded with the diameter, gives the total convex superficies of the Sphere, 1● 4/7, whose 1/3, viz. 4 4/21 being enfolded in like manner with the Semidiameter (A B 1.00) or the 1/6 viz. 2 2/11 (answering to the square Plane or Base of the inscribed Cube) enfolded with the whole diameter, produceth the total spherical solidity 4 4/21 as before.   geometrical diagram 
Again, suppose the Diameter of a Sphere to be of the lesser, or simple decimal Rational Line A b, 1.5 (which is in parts only of the greater or centesimall Line A B 0.15) so the greatest periphery will be of the same, 4.7 (in parts only of A B 0.47) as formerly in the second practical demonstration upon the Circle) which too enfolded together, do produce the superficies of the Sphere in square or superficial Integers and prime decimal parts of A b, 7.05 (in square centesimall parts only of AB 0.0705) whose 1/3 viz. 2.35 (or 0.0235) being enfolded with half the Diameter A b 0.75 (or A B 0.075) or the 1/6, viz. 1.175 (or, 0.01175) with the whole Diameter, A b 1.5 (or A B 0.15) do produce the total Sphere in solid Integers and prime parts of A b, 1.762 (in solid centesimall parts only of A B 0.001762) Now A b 1.5 for the Diameter of the Sphere, upon E f the proper correspondent Line of Cubation (being A b, 1.2 (viz. A B 0.12) which in Centesms make but 1.20, wanting indeed of the true additional parts .04, for that the total segment of A B in centesms, to be added, is .24, as before hath been showed) divided also into 10 parts, is 1.2, whose Cube is 1.728, for the solid content of the Sphere wanting of the true content, only 34 cube-primes, or simple decimal parts, viz. 34 of 1000, which in a common arithmetical account, are only about 1/29 of the Line A b (as the Integer of measure) cubed, and which difference will be in the centesimall operrtion (by parts only of the Line A B) but 34 cube-centesms, viz. 34 of 1000000, or more vulgarly 1 of 29412 ferè: For that the Diameter of the Sphere being put A B 0.15, the same will be upon the correspondent Line of Cubation E F 0.12, whose Cube, 0.001728, is for the solidity of the Sphere, which wanteth of the true solidity, 0.001762, only as aforesaid.  
And here we may observe by the way, how this Sphere and the second Circle before supposed, do agreed, to wit, in that they both having the same diametral number, have also the very same dimensionall or are all number, the one superficial, the other solid; viz. in Integers and parts of the Line A b, 1.76, etc. in parts only of the Line A B .0176, in the Circle, and .00176 etc. in the Sphere: which will further appear, if their common Diameter be taken according to a more ample division (or partial denomination) of the Rational Line, and so consequently, the superficial content of the one, and the solid content of the other, how far soever they be extended decimally, (and than also indeed they will be found somewhat greater.) And this may also plainly appear by the more usual or common arithmetical expression of their Diameter (and so of their other parts of dimension) viz. A b 1 1/2, (or A B 3/20, in the lest proportional terms to A B 0.15) and so their periphery, in the reason of the common (or Archimedean) Cycloperimetricall terms (7 and 22) A b 4 5/●; or rather of the Metian terms, 113 and 355 (before noted) 4 161/226 (in proper parts answering to A B 3/2● is A B 1065/2260) whose 1/2 A b 2 161/452 (or A B 1065/4520) together with the semidiameter A b 3/4 (or A B 3/40) do produce the area of the Circle, 1 1387/18●8; which converted into square primes, or simple decimals, (according to the division of the Line A b) is 1.76: or the area in parts only, 3195/180800 converted into square seconds or centesms (according to the partition of the Line A B) is 0.0176, as before. Than the Diameter, and so the Circumference of the Sphere being the same with those of the Circle, these two conjunctly, produce the Spherical superficies according to A b, 7 31/452, (in parts only according to A B, 3195/●4520) whose 1/6 viz. 1 483/2172 (or 3195/27●2●●) together with the diameter 1 ●/2 (or 3/20) or the 1/3 viz. 2 483/1356 (or 3195/135600) with the Semidiameter 3/4 (or 3/40) do produce the Spherical solidity, 1 4161/3414, which in the decimal expression to millesms, or cubical prime decimal parts (according to the division of the Line A b) is 1.767 (and so much it is really; exceeding the solidity formerly cast up, by 5 cube-primes:) And the solidity in parts only, 9585/5424000 is in milli-millesms, or cube-centesms (according to the division of the Line A B) 0.001767. So as that the superficiality of the Circle, and the solidity of the Sphere being put in vulgar fractional terms (either proper in respect of parts only, or improper in respect of Integers and parts) viz. the Circle 3195/1●●8 00, and the Sphere 9585/5424 000, they will be found to be proportional among themselves, viz. as the Denominatours are each to other, so also are the Numerators; for as 1808 -- 00 to 5424 -- 000, so 3195 to 9585, and so convertibly.   geometrical diagram 
And here note, that what we have now done in the Diameter of the Circle and 
Another superficial (or quadrate) dimension of a Circle, and solid (or Cubick) dimension of a Sphere: viz. By their Circumfe: rences: and the Artificial Lines of Measure for the same.  Sphere, to make the Square of the one, equal to its Circle, and the Cube of the other equal to its Sphere: The like may be done also for their Circumferences: so the Line for squaring the Circumference of a Circle, will be of the prime Rational Line (under a decumillesimal partition) 3.5449. and for Cubing the greatest or true Circumference of a Sphere, will be 3.8978: and which perhaps may be of more general use than those for the diameters, because that commonly the Circumference of a Circle and Sphere (in materiate things, where the Centre is not apparent) must first be had, before their diameter can exactly, (especially in a Sphere) for than that must be had by proportional argumentation; though indeed the diameter of a Sphere may be taken at first as well (if not better) by a pair of Callaper Compasses, where the same may be had ready upon occasion, and than the Line for the diameter may better be used.  
And other Lines also of the like nature may be fitted to the Diameter 
The superficial dimension of a Sphere, only by squaring its Diana. & Circumference. And the artificial Lines of Measure for performing the same.  and Circumference of a Sphere, as to Superficial Measure: As namely to make their particular Quadrats, equal to the convex spherical superficies: So the Line of measure for this purpose, will be for the diameter, of the Rational Line (deficiently) 
* ●q. Diana, ad periph. 1 viz. .1 --. 68-169. etc.  0.564 etc. (viz. as A B 10000, etc.— .4358, etc. from the segment of diminution) and for the (greatest or true) Circumference (redundantly) 
* ●q. Periph. ad Diana. 1. as in page 19 and see page 30. & 31. Numb. 2.  1.772 etc. And hence we may here raise this practical proposition. viz. If the Diameter or Circumference of a Sphere, be taken by their proper and peculiar artificial Lines for superficial measure; That than their several Quadrats, shall be equal to the Superficial Area of the Sphere (or to the Spherical) according to the prime or natural Rational Line.  
ANd so supposing the former Sphere, whose diameter being first put (of the prime Rational Line) exactly 21, the spherical convex superfice was found (according to the true or greatest Periphery 65.97 etc.) 1385.4424. Now the said Diameter being taken by its proper Line of Quadrature (in a centesimall partition) will be found 37.22, whose Square is, 1385.3284, for the spherical superfice; agreeing almost exactly (in the very parts of measure) with the other.  
And so again likewise the greatest Periphery of a Sphere, being first put (of the Rational Line) exactly 66, the true Diameter will be found (of the same) 21.008 etc. and thereby, the spherical superficies, 1386.5579 ferè. Now the said Periphery being measured by its proper Line of Quadiature (under a centesimall solution) will be found, 37.24 ferè, which squared, gives 1386.8176 ferè, for the convex superfice, or the Surface of the Sphere; exceeding the other, only about 1/4 of a square integer (or unit) as of the Measure first appointed.  
And here we now gather these useful and excellent proportional Corollaries in the Sphere, viz.  
1 The proportion of the Axis or Diameter to the Periphery of the greatest Circle, & contra: is the same with that of the Diana. of a Circle to its Circumference, & contra. Side of the Cube equal to the Sphere (which is the nearest, precisest, & properest cubing of a Globe or Sphere.) as 1. to .806040 Side of the Quadrat equal to the convex spherical superfice, (which may not altogether unaptly, be termed the squaring of a Sphere; but most truly the squaring of the Spherical: and that the nearest and properest also.) (ferè. 1.772454  
2. The proportion of the greatest or true Circumference to the Side of the Cube equal to the Sphere (which is the next most proper cubing of a Sphere.) as 1. to .256556 Side of the Quadrat, equal to the convex superficies; (which may be termed he next most proper squaring of the Spherical.) .564189  
3. The proportion of the Cube of the Axis or Diameter to the Sphere itself as 1. to .523599 Greatest Periphery .016887 f.  
Contrariwise.  
4 The proportion of the total Sphere to the Cube of the Axis or Diameter. as 1. to 1.909859 Greatest Periphery. 59.217626  
5 The proportion of the Quadrat of the Diameter. To the total superficies as 1. to 3.14159, etc. being the same with that of the Diana. to the Circumf. Periphery 0.318310 ferè, being the same with that of the Circumference to the Diana.  
Conversly.  
6 The proportion of the total superficie of a Sphere to the quadrat of the Diameter Is the same with that of the Peripheriall Quadrat to the total superficies, (or the Circumference to the Diameter.) Periphery Is the same with that of the diametral Quadrat to the total superficies (or the Diana. to the Circums.)  
As for the proportion of the Diameter and greatest Periphery to the side of the inscribed Cube; I shall deliver the same afterwards, among the proportions in the five plain Regular bodies inscribed in a Sphere, Part 2. Sect. 3.   

PROP. 3. Of the solid dimension of a Cylinder and Cone.  
If the (basiall) Diameter, and the Axis of a right Cylinder, be taken by their proper Line of Measure, and the Quadrat of the Diameter be augmented by the Axis; I say, that the resulting rectangle, regular-based Prisma or Parallelepipedon, shall be solidly equal to the Cylinder. And the like in a Cone. Both which I do here practically confirm.   geometrical diagram  geometrical diagram 
But indeed (which is the most absolute, complete, and compendious way) the very same artificial dimension which is used in a Cylinder; will hold good in a Cone, and so may be as properly and fitly applied thereunto, (according to the last Proposition) whereby the solid content thereof shall be immediately produced, as that of a Cylinder (even as if it were a Cylinder) notwithstanding the continual diminution of its body between the Base and the top or point. And the artificial Line of measure for this purpose; I noted at the beginning, to be of the Rational Line in general, 1.5632 ferè, and which for the present purpose will here be (according to the foregoing practical demonstrations) of the centesimall Line, A B 1.56, which is here the Line I K, (divided as the former Lines) By which the basiall Diameter of the fore-supposed Cone, (naturally A B 0.25) being taken, the same I find to be thereof, 0.16, whose Quadrat, 0.0256, is for the Base of the Cone: And than the Axis (naturally A B 0.75) being also measured by the same Line, becomes thereof, 0.48 ferè, which wholly enfolded with the whole artificial Base, produceth the solidity of this accute-angled Cone, 0.012288 ferè, which exceedeth the true solidity before-noted, viz. 0.012175, by 113 solid centesms only, and which in vulgar terms, make but 1/8850 ferè, viz. as one part of a solid Integer or unit, divided into 8850 parts.   geometrical diagram 
And so a Cone being of equal Altitude and Diameter (in the Base, and so of equal base) with the foresaid Cylinder, the sub-triple thereof shall be the total Oxygoniall Cone, viz. 153.938040 (or 153 106/113) which by the first base 38 1/2, is 154 exactly; and which agreeth exactly with the Circle last handled whose Diameter was put 14) as appeareth by the most usual dimension of a Cone, by infolding the Base with a trient of the Altitude, which here being (A B) 4.00, produceth the conical solidity as before, (or a trient of the base with the Axis produceth the same.)   geometrical diagram 
And here observe, that as in the Circle and Sphere, we shown, how that the superficiality of the one, and both the superficiality and solidity of the other, were obtained artificially, not only by the bore Quadration and Cubation of their Diameters, but also of their Circumferences, by the like artificial Lines of measure accommodated to them: So likewise in the Cylinder and Cone, may the solidity be had, not only by the induction, implication, or involution of the Quadrat of their Basiall Diameter into their Axis (as we have already showed in both of them) but also of the Quadrat of their Basiall circumference into their Axis, these being (both of them,) 
The solid dimension of a Cylinder and Cone by their Basiall Peripheries, and their Axes together. And the artificial Lines of measure for the same.  taken by one and the same Line of measure: So the Line for squaring the circumference (of the Base) of a Cylinder, (for its Base) will be of the Rational Line, 2.32489: and for squaring the Basiall circumference of a Cone, 3.3531; which two Lines being divided as the former, and the Basial circumferences (as also the Axes) of these two Bodies taken thereby, may sometimes prove to be of more satisfaction than those for the basiall Diameters, to wit, where the basiall centres of these Bodies are not apparent, (especially if the Bases of the Cylinder or Cone to be measured be very large) according as I noted before in the Circle and Sphere. But indeed, where the Diameters may be first exactly taken, either in these two Figures, or in the other two before going, it will be much easier and readier in practice, than by the Circumferences, in regard both, that the artificial Lines of measure serving thereunto, are much shorter, and also that the diametral numbers being much lesser or smaller than the circumferential, the Arithmetical operations following thereupon, in casting up the superficial and solid contents, will be sooner expedited, unless the same be performed Geometrically (as I may term it) or Instrumentally, viz. by Scale and Compasses, or the like (as I shall in the very close of this Book by way of conclusion declare) for than the latter may be performed thereby, even as soon as the former.  
And now you may observe here by the way, how that although the base of a Cylinder and Cone be a Circle, and its Area be here had also, by the only quadration of the Diameter or Circumference, as before was done in the Circle, whereby the same artificial Lines of measure that are there used, might also perhaps seem to serve here, (and which indeed in some sort may, as I shall strait way show) yet for the obtaining of the solidities of these Bodies, by the implication or induction of the Quadrat of their Basial Diameter or Circumference into their Axis (either of these two basiall Lines, with the Axis, in each of these two Bodies, being taken by one and the same proper, distinct, artificial Line of measure, as before hath been sufficiently showed) the Lines there appropriated to a Circle cannot here hold for their Bases; for the Diameters or Circumferences here of being taken by those Lines, will be either greater or lesser (in the number or parts of measure; according to quantity discrete) than by their own proper Lines; (and so their Quadrats for the Bases, and consequently the solid contents will be greater or lesser than they aught to be; what artificial Line of measure soever, the Axis should be measured by.) For so in a Cylinder and Cone, the Diameter of the Base being taken by the artificial Line belonging to the Diameter of a Circle, for its quadrate dimension thereby (being simply and absolutely considered in itself alone as a Circle) will be found in the Cylinder to be somewhat less, and in the Cone much greater, (in the number or parts of measure, as in a Quantity discrete) than being taken by its proper artificial Line for the quadrate dimension of the Base, as in reference to the solid dimension of the Cylinder and Cone, performed wholly by the said Line of Measure, from the Basiall Diameter and the Axis together.  
And than, both in a Cylinder and Cone, the circumference of the Base being taken by the artificial Line, belonging to the circumference of a Circle simply, for its quadrate dimension thereby; will be found less (as falling under the notion or nature of a Quantity discrete, as aforesaid) than by its proper, respective, artificial Line, for the squaring of the base thereby, as in relation to the solid dimension of the Cylinder and Cone, wholly performed by the said Line, from the Basiall Circumference, and the Axis together.  
As in the Cylinder and Cone last handled; where the Diameter of the Base being put naturally (A B) 7.00, the same was there found to be artificially (G H) 6.46 ferè, and (I K) 4.48 ferè, and which will be found by the Line (of Quadrature) pertaining to the Diameter of a Circle, simply (C D) to be 6.20, viz. somewhat less than that of G H, and much greater than that of I K.  
And so there again, the Circumference of the Base being naturally (A B) 22 ferè; the same will be found to be artificially, by the Line for squaring the (Basiall) Circumference of a Cylinder (as in reference to its solid dimension by the Circumference and Axis together) 9.46; and by the Line for squaring the basiall Circumference of a Cone (as in relation to its solid dimension in the like manner) 6.56, and which by the Line for the circumferential quadration of a Circle, simply, will be found 6.20 (as the Diameter before, by its proper Line of quadrature (C D) which is less than either of the other.  
Therefore, if the basiall Diameter or Circumference of a Cylinder and Cone, be taken by the Lines of (diametral and circumferential) quadration, properly, peculiarly, and simply pertaining to a Circle, and so its Quadrat be made the Base of the Cylinder or Cone; than must the Axis be taken by the Prime Rational Line: (And so the dimension will be mixed.) For that here the Area of the Base, will fall immediately, in the true, natural, measure, (as under the dimensionall reason of the prime, true, or natural Rational Line) according as hath been demonstrated before in the dimension of a Circle.  
As here the Basiall diameter of the Cylinder and Cone, naturally (A B) 7.00 taken by the Line C D is 6.20 (or 6.2, as I shown even now) whose Quadrat. 38.4400 (or 38.44 only) for the Base, (which was found before, to be most truly and naturally, 38.4845) being drawn into the Axis, put before, naturally, (A B) 12.00, will give the cylindrical solidity (in the dimensionall reason of A B) 461.280000, (or 461.28 only) and so the conical solidity, 153.76; which differ (by way of defect) from the true, natural solidities (produced wholly by the Line A B) viz. of the Cylinder, 461.814120, about 1/2 of a cubique Integer or Unit, as of A B cubed: and so of the Cone, viz. 153.938040, about 1/6 of a cube-integer only, as of the same Line Cubed; And the solidity of the Cylinder, produced artificially by its proper Line of measure (G H) viz. 461.968812 ferè, differeth therefrom (by way of excess) only about 1/6 of a cube-integer; and the solidity of the Cone produced in like manner, by its proper artificial Line of measure (I K) viz. 154.140672 ferè, only about 1/5 of a cubique or solid Integer of the natural measure (as of A B considered cubically) as before hath been showed,  
All which several solid dimensions (both natural and artificial, and mixed of both) do so nearly agreed one with another, as that their differences are altogether inconsiderable. And the same will hap in the solid dimensions of these two Bodies, by the other Line of Quadrature pertaining simply to a Circle, as to the quadration of their Basiall Peripheries, for their basiall Area's, and withal by the prime or natural Rational Line, as to the dimension of their Axes or Altitudes.  
And now, as for the Axis (or Altitude) of a right or erect Cone, you may here observe, that though the same cannot immediately be taken Instrumentally; or by a Line of Measure (as it is within the body of the Cone) as that of a Cylinder (being parallel to, and so agreeable with the side,) by reason of the inequality of its Body between the Base and the Cusp, or vertical point; but being obtained purely Geometrically, must be had by mediation of the Side (as being the side of a rectiline rectangle Triangle, subtending the right angle, and therefore potentially equal, or equally potent to the two containing, comprehending, or including sides. by E. 1. p. 47.) and of the basiall Ray, (being one of the sides about the right angle (and most commonly the lesser,) and so the side of the Cone, and the radial line of its base, being first taken Instrumentally or Mechanically, the Axis will be had Trigonometrically, according to the reason of the forecited prop. of Euclid.) Yet may the same be obtained most readily (out of the Cone) by a Line of measure, if from a Plane constituted in the top, or vertical point of the Cone, parallel to the Base, you let down a Perpendicular-line to the Plane on which the Base is, (and which may fall precisely upon the Basiall periphery) for that (being measured) shall be equal to the true Axis of the Cone. And thus also may the altitude of any obliqne, scalene, or inclined Cone be taken; and also of an obliqne, scalene, or inclined Cylinder; if from the top of the Cone to the Plane in which the Base is set, or from the superior base of the Cylinder, to the Plane of the inferior base, be let fall a perpendicular-line; for that (being measured) shall be the altitude of the obliqne, or inclined Cylinder or Cone; and so being enfolded with the whole base, if a Cylinder (whether measured by the natural or the artificial Line of measure) shall make the solidity of the same: or being enfolded with a trient of the base, if a Cone, (or the base with a trient of that) and measured by the natural Line; or else the whole perpendicular of altitude with the whole base, if measured by the artificial Line proper to a Cone, shall produce the solidity of the obliqne Cone; seeing that such a Cylinder and Cone, is equal to a right Cylinder and Cone, having the same base and altitude 
See also E. 11. p. 31, & E. 1. p. 35, 36, 37, 38.  with it, (according to the reason of E. 12. p. 11, & p. 14.)  
And after the same manner will the true altitude of any Pyramid, whether right or obliqne, be had, as of a right or obliqne Cone, and the altitude of an obliqne or inclined Prism, as of an obliqne or inclined Cylinder; the dimension of which bodies, especially the Pyramidal (both solidly and superficially) according to our new artificial way (together with the natural or vulgar, by way of dimensionall comparison) I shall show next after the dimension of rightlined ordinate Planes or Superficies, seeing that upon any of them may be erected or constituted a Pyramid or Prism; as a Cone or Cylinder upon a Circle.    

SECT. III. Of the Superficial Dimension of a Cylinder and Cone.  
ANd as we have here shown the most artificial and expeditious dimension of the Cylinder and Cone, in respect of their Stereometry, or solid measure; so we shall likewise demonstrate their dimensions in respect of Planometry, or superficial Measure. And the artificial Lines for the performing hereof, I find, for the Cylinder, (in respect of its Diameter & Side together) to be of the prime Rational Line, the same with that which was formerly found to be for the obtaining of the Superficies of a Sphere, by the only quadration of the Diameter, viz. 0.564, etc. And for the Cone (in relation to its basiall diameter, & its side conjunctly) 0.79788 (which according to the parts of diminution, is as A B 1.00000— .20212) and in relation to its basiall periphery and side together, it will be 1.4142, &c, (viz. √ q 2) which several Lines being exactly set of from the prime Rational Line, and than divided as the same, and so the basiall diameter, and the side of a Cylinder, and of a Cone; and also the basiall Circumference, and the side of a Cone together, be taken by their peculiar, respective, distinct artificial Lines of Measure, and so multiplied together, their several products shall be the superficial contents of the Cylinder and Cone, according to the prime Rational Line: where, by the superficies of a Cylinder and Cone, must be understood without their Bases; for so the Superficies of these two Solids are generally taken by Artists, and are called by the Latin Geometers, especially Ramus, (in one word) Cylindraceum, and Conicum, that is as much as to say in English, the Cylindraceall or cylindrical, and the conical; and so the Superficies of a Sphere is called by them Sphaericum, viz. the Spherical. But now for a brief dilucidation of these superficial dimensions in the Cylinder and Cone, I shall lay down an example in each of these three artificial metrical Lines (though here I do not draw them, but only express them by number) whereby the verity of these our dimensions also, may plainly appear, to those that shall please to make trial thereof. And first we may hereupon raise this practical Proposition, viz.  If the (basiall) Diameter, and the Side of a right Cylinder; And the basiall Diameter or Periphery, and the Side of a right Cone, be taken by their proper and peculiar, distinct artificial Lines of Measure, and the same be severally enfolded together: That the resulting rectangle Parallelogram, shall be equal to the cylindrical and conical Superficies, (according to the reason of the prime Rational Line.) 
THerefore, suppose here first the Cylinder last handled, whose Diameter was put (of the Prime Rational Line) exactly 7, and so the Circumference (according to the most common Cycloperimetricall terms) was exactly 22, and the altitude (which is the same with the side) was put exactly 12, and these two enfolded together, do produce the Superficies of the Cylinder (without the two Bases) exactly, 264: whereby this Cylinder becomes absolute in all its dimensions; But by the other Cycloperimetricall terms, the Circumference was but 21.99 etc. (or 21 112/113) whereby, the Superficies becomes now (to square centesms of the Rational Line) but 263.8938, (or more vulgarly 263 101/113) Now if the Diameter of the Cylinder naturally 7, be measured by its proper artificial Line of superficial measure (being as A B 0.56, made 100) the same will be found, 12.41 ferè, and the Side of the Cylinder naturally 12, being measured by the same Line, will be found, 21.27 ferè, which two multiplied together, produce 263.9607 ferè, for the superficies of the Cylinder, agreeing with the true superficies, exactly in Integers (or Units) of measure, and differing therefrom in the fraction-parts (by way of excess) but as 1/14 of a square integer or unit. And if the first or natural Line of measure be made 1000, and so the second or artificial Line (being thereof 0.564) be also made 1000, than the said Diameter taken thereby, will be 12.407, and the Side will be 21.269, which two multiplied together, will produce the Cylindrical Superfice, 263.884483, differing from the true one, viz. 263.893783 (now by way of defect) only as 1/108 ferè, of a square integer or unit of the appointed measure. Which excellent compend in the superficiary dimension of a Cylinder by its Diameter and Side only, will further and fullier appear, if it be compared with the most natural or vulgar way, in respect of the Side and Diameter only given: where, by the Diameter, the Circumference being proportionally obtained, must than be multiplied into the side, to make the Superficies (for that the Superficies of a right Cylinder is most naturally, a Plane made of the Circumference and the side thereof) and the obtaining of this by the Diameter and Side; Mr. Oughtred in his Book entitled The Circles of Proportion, Part 1. Chap. 7. Sect. 10. delivers in this proportional manner.  
As 7 to 22, Or 1 to 3.1416.  
So the Diameter and side multiplied together,  
To the Superficies (viz. without the two bases.) And so in this our Cylinder, the Diameter being put 7, and the side 12, I say,  
As 7 to 22, or rather, 1 to 3.14159 etc. (or 113 to 355) So 7 into 12, viz. 84,  
To 264, the Superficies,  
Or rather 263.89378 &c (or 263 101/113) as first by the Circumference and side together: which two several operations you see do consist of two several Multiplications, and one Division, whereas ours consisteth of one Multiplication only, to wit, of the Diameter and Side together.  
And from this Analogical operation used by Master Oughtred, for the discovering of the superficies of a right Cylinder, by its side and (basiall) Diameter, you may here note, how that it is demonstrated by Archimedes, Lib. 1. de Sph. & Cylind. prop. 13. that the Superficies of a right Cylinder (without the Bases) is equal to a Circle, whose Semidiameter is a mean proportional Line between the side of the Cylinder, and the Diameter of its Base.  
And so in this our Cylinder, the side being 12. and the Diameter 7, the mean Proportional between them, will be 9 3/19, or rather by an immediate decimal extraction of the parts, 9.16515 &c (whereas the other is decimally by conversion of the parts, but 9.15789 etc.) which being put as the Semidiameter of a Circle, and so the Square thereof 84, the Area of the Circle will be found 263.89378 etc. agreeing exactly with the Superficies of the Cylinder. Or the Area is by the Metian Cyclometry, 263 101/113, for the Superficies of the Cylinder, as before.) For seeing here, that the Semidiameter of the Circle equal to the Superficies of a right Cylinder is the mean proportional between the Side and (basiall) Dia metre of the Cylinder; and that the Analogy holds the same from the Quadrat of a Circles Semidiameter to the Circle itself, as from the Semidiameter to the Semicircumference, or the 
* Quae est ratio totius ad totum, ●adem est ratio dimidii ad dimidium: Et sie alic●rus partis ad aliquam partim consimilim sin iogneminem.  Diameter to the Circumference, (only the one Analogy is Lineal, and the other Superficial.) Therefore the Analogy of the side and (basial) Diameter of a right Cylinder enfolded together, to the Superficies thereof (without the two Bases) will hold the same, as of the Diameter of a Circle to its Circumference.  
Again, supposing the basiall Diameter and the side of a right, erect, or Isoskelan Cone, to be the same with those of the foregoing Cylinder, viz. 7 and 12; the superficies will be half the superficies of the Cylinder: for that the superficies of a right or Isoskelan Cone is a rectangle Plane made of the basiall semiperiphery and the side (or the basiall periphery and the semi-side) and so the true conical superfice or surface, will be 131.9469, (or 131 107/113, agreeing decimally with the former, or by the most common account, 132 exactly) And the finding of thetrue, genuine superficies of a right Cone by the side and basiall Diameter, the foresaid Mr. Oughtred also showeth in the 8 Sect. of th● same Chapter before named, by this Analogy, viz.  
As 7 to 22, Or 1 to 3.1416:  
So the Semidiameter of the Base multiplied with the side,  
To the Superficies (viz. without the Base) And so the side of the Cone being here put 12, and the basiall Diameter 7, I say,  
As 7 to 22 (or 113 to 355) or rather 1 to 3.14159 etc.  
So 3 1/2. (or 3.5) with 12, viz. 42. to 132 exactly:  
Or more truly, 131 109/113, or 131.9469, just as before, for the Conical. So that in both these ways also, are two several Multiplications and one Division, besides the bipartion or mediation of the basial diameter: or the same is performed by half the side with the whole Diameter, (for both come to one pass.)  
And from hence you may observe, that Archimedes, lib. 1. de Sph. & Cyliud. prop. 14. (from whence the foresaid Analogy is deduced) demonstrateth the superficies of a right or Isoskelan Cone (without the Base) to be equal to that Circle, whose semidiameter is the mean proportional line between the side of the Cone, and the Semidiameter of its Base. And so in this our Cone, the Side being 12, and the basiall Semidiameter 3 1/2 (or 3.5) the mean proportional between them is 6 6/13, or rather by immediate decimal production of the parts, 6.48074 etc. (whereas the other is by decimal reduction, but 6.461538 etc.) which being put for the semidiameter of a Circle, and so the Quadrat thereof 42, the Area will be found 131.9469 etc. agreeing exactly with the foregoing Superficies of the Cone: (or the Circular Area, is from the Metiau Cyclometry, 131 107/113, which is decimally, 131.9469 etc. agreeing exactly with the true conical Superficies, found out decimally at first.  
And here you may by the way take notice 
Archi●●. lib. 1. de Sph. & Cyl. pro. 15  of the next Proposition of Archimedes concerning the superficiary dimension of a Cone, laid down Analogically thus, viz. That the superficies of every Isoskelan Cone, hath that rationality to the Base, as the side of the Cone hath to the Semidiameter of the Base. For so here,  
As 12 (the side)  
To 3 1/2, or 3.5 (there Radius of the Base)  
So 131 107/113, or 131.9469 (the true superficies)  
To 38 219/45●● or 38.4845 the Base; (And so was the same found to be before, in the solid dimension of a Cone, and of a Cylinder,) And contrariwise will the reason of the Base be to the superficies of the Cone, as of the Semidiameter of the Base to the Side of the Cone: For so,  
As 3 1/2 or 3.5, ro 12;  
So 38 219/452 or 38.4845, to 131 107/113, or 131.9469. Now the Base being added to the foregoing Superficies, there will arise the total Superficies of the Cone, 170 195/452, or 170.4314 etc. which by the most common account, will be 170 1/2, the Conical being just 132, and the Base 38 1/2.  
And the very same Analogy with the former, holdeth in the right, erect, or rectangle Cylinder, from the Superficies to the double of the Base, (or both the Bases jointly) and contrarily; because that the Superficies of the Cylinder is double to the Superficies of the Cone, having the same Side, and Diameter in the Base, as I noted before: And so here,  
As 12 (the Side)  
To 3 1/2 or 3.5 (the basiall ray)  
So 263 101/113 or 263.89378 etc. (the Superficies)  
To 76 219/226, or 76.9690 etc. (the double of the Base, or the aggregate of the two Bases.) And so contrarily, will the Analogy hold from the double of the Base, or the two bases conjunctly (76 219/226, or 76.9690 etc.) to the Cylindrical Superficies (263 101/113 or 263.89378 etc.) as from the Radius of the Base (3 1/2 or 3.5) to the side of the Cylinder (12.) And lastly, if to the foresaid cylindrical Superficies 263 101/113, or 263.8938 ferè, you add the aggregate of the two Bases, 76 219/226, or 76.9690, you will have the total external superficies of the Cylinder, 340 195/116, or 340.8628 ferè, which by the most vulgar account, would be 341 exactly; the Cylindraceal (or Cylindrical Superfice) being just 264, and the two Bases together. 77  
But now for a more speedy dimension of the Superficies of the foregoing Cone, and first by the Side and basial diameter only, together: I say, that if the side of the Cone (naturally 12) be measured by its proper Line of Measure for this purpose (which is in centesimal parts only of the prime Rational Line, 0.80 ferè) under a centesimal solution; the same will be found thereby, 15.04; and the basial Diameter (naturally 7) measured by the same Line, will be found 8.77, which two being multiplied together, do produce 131.9008, for the Conical Superfice, which wants of the true content, only 1/22 ferè of a square or superficial Integer or Unit of the Measure first assigned. And so again, if the basial Periphery of this Cone, naturally, 22 ferè: be taken by its proper, respective, artificial Line of Measure for this purpose, (being of the prime Rational Line, in a centesimal solution, 1.41) centesimally divided; the same will be found thereby, 15.56 ferè: and the Side of the Cone being also taken by the same Line, will be found 8.48; which two multiplied together, do yield 131.9488 ferè, for the Superficies of the Cone, which agreeth somewhat more nearly with the true Conical, viz. 131.9469, than that which was produced in this kind by the basiall Diameter and the Side together, viz. 131.9008. for this latter (by the basial periphery and the Side) differeth from the true one (by way of excess or redundancy) but 19 square or superficial centesms ferè, viz. 19 of 10000, which in a vulgar Arithmetical expression, are hardly 1/526 of the prime Rational Line (as the Integer or Unit of measure) squared: so that both these ways for the finding of the superficies of a Cone, consist also but of one Multiplication only, as that for the Cylinder.  
But for finding the superficies of a Cylinder in this manner, by an artificial Line of measure, peculiarly fitted to the Circumference and Side together, the same is not to be done, seeing it is performed immediately by the natural Line of measure, or the Prime Rational Line itself, with the same expedition as that in the Cone by an Artificial Line; for that the Plane made of the Circumference and Side of the Cylinder, taken by the prime or natural Line of measure first appointed (according as I said at first) doth constitute the true Cylindraceall, or the cylindrical superfice. And so from the premises it appears, that as a Cylinder in regard of solid dimension, is the triple of a Cone having the same Base and Axis, or Altitude: So in respect of superficiary dimension, it is the double of a Cone having the same Base and Side or Longitude.  
And as for the Bases of a Cylinder and Cone, if they be required in their superficiary dimensions, the same may most readily be obtained by either of the Lines of Quadrature pertaining to a Circle, according as the Diameter or Circumference thereof shall hap to be taken: For so the basiall Diameter or Circumference of the foregoing Cylinder and Cone (naturally 7.00, and 22 ferè, viz. 21.99) being measured by the said Lines of diametral and circumferential quadration (under a centesimall solution) will be found each of them severally; to be 6.20 (as I noted before in their solid dimensions) which squared, yields 38.4400 for the Base; and the true base was formerly found, 38.4845; from which ours differs, (by way of defect) not so much as will make in vulgar terms, 1/22 of a square Integer or Unit.  
And thus having sufficiently declared and demonstrated the dimension both Solid and Superficial of a Cylinder and Cone (both theorically, and) practically, according to the Instrumental part of Geometry; and that as well naturally (by way of metrical comparison) as artificially, for the confirmation and verification of our artificial way of measuring: I shall next lay down the same Dimensions numerally, in Terms analogical (by way of comparison) from the natural Measure to the artificial; and that under an ample or numerous solution of the unity (according as I did in the Circle and Sphere) by means whereof, the artificial Measure may readily be deduced from the natural; or the natural measure be reduced to the artificial: And first for the Cylinder,  
The true or natural Measure is to the artificial, in respect of the Axis, or Altitude, conjunctly with the (basial) Diameter as 1. to .92264 ferè Solid measure. Circumfer. .430127. Side (which in a right, erect, or rectangle Cylinder, is equal to the Axis or Altitude) with the (basial) Diameter, as 1. to 1.772454 ferè, (being the same with that which was noted before in the Sphere, for the proportion of the Diameter to the side of the Quadrat equal to the Spherical.) Superficiail Measure.  
Than for the cone;  
The natural Measure is to the artificial, in respect of the Axis (or Altitude,) conjunctly with the basiall Diameter as 1. to .639839 Solid dimen. Periphery .298377. Side (or Longitud) Diameter 1.253314 Superficiaty Dimension Periphery .707107 [ferè √ q 1/2.  
The last of which proportions, is the same with that of the Diameter of a Circle to the side of its inscribed Quadrat, noted before in the dimension of a Circle.   

SECT. iv Showing briefly, the theorical reason of the differences happening between the natural and artificial Measure, in the superficial and solid contents of Figures. And moreover, some observations concerning the grounds and reasons of the Artificial Mensuration in general.  
ANd now again, as for the differences happening between the superficial and solid Contents of Figures, found by the natural or vulgar way of measuring, and our artificial way; we have formerly showed, how small and inconsiderable they generally are▪ and also the practical, instrumental (or Geometrical) reason thereof; viz. that the several Lines of dimension in the several Figures (either naturally belonging to them, or artificially and commonly abscribed to them) as namely, the Diameters and Circumferences of the Circle and Sphere; and so the Diameters. Circumferences and Axes, or Altitudes, and Sides, of the Cylinder and Cone, before going, (and so of all the other Figures following respectively) taken by their proper, respective artificial Lines of Measure, are seldom or never exact and precise indeed in the parts of measure, but either deficient or redundant in the same, and so give the are all contents of those figures either a little lesser or greater, than indeed they are (though for the most partlesse, especially in the two first decimal partitions of the Lines of measure, viz. into Centesms, or only into prime or simple Decimal parts, or Tenths) as appeared formerly (and will also afterwards) by a continual decimal augmentation (or subdecuplation) of the parts of those Lines, whereby the Contents were had still nearer to the truth; Which reason ariseth from (and so dependeth upon) the more true, natural, theorical (or the Arithmetical) reason of these differences, lying in the extraction of the Square and Cube-Roots: For that the Roots of numbers not exactly Square and Cubical, cannot be exactly had, but are always defective, so as that they being inverted, or drawn again into themselves, do not tender the numbers precisely, out of which they were extracted; but the further that the extraction is extended or continued decimally, by the adjection or apposition of more Figures, (or of Ciphers where there is need) the nearer still to the truth will the Root be had; as I noted at the 
* Where I might also have particularly expressed one thing more (to the young practitioner) amongst other particulars concerning the decimal solution of Unity, (and which I may here, not altogether un-opportunely do, though it be there included and understood in the General,) viz. That thereby, all the tedious and troublesome operations of Arithmetic in Fractions, by the vulgar way, are wholly avoided: for that here all fractional numbers, whether coming alone by themselves, or together with integral numbers, are wholly and universally wrought as integrals, without any manner of preparatory operation, (as Reduction or transmutation of terms, one way or other,) which in the working of vulgar Fractions is necessarily required.  beginning, where I took occasion to speak of the excellency of decimal Arithmetic, as in reference to the work in hand: So that what error (though inconsiderable) may at any time arise in our work, doth proceed primarily and principally from the extraction of the Square and Cube-Roots; our artificial Lines of measure giving immediately the Square and Cube-Roots of the Figures to which they are appropriated and applied, (or the sides of their equal Squares and Cubes, as nearly, almost as may be) according as their dimension is superficial or solid, as being naturally (as it were) procreated or derived from them; except it be in the Cylinder and Cone, and the other regular-like Solids following; as regular-based Pyramids and Prisms: but else in all truly ordinate Superficies and Solids; as the Circle and Sphere before-going, and so the equiterminall and equiangular Superficies and Solids following, it holdeth so: And so likewise in the Cylinder and Cone, and other regular-like Solids, both for solid and superficiary dimension, where there is a congruity between their Lines of dimension, by which their solid and superficial Conetnts are obtained, as the Diameters, Peripheries, Sides, and other dimensionall Lines of their Bases with their Axes or Altitudes for solid dimension, and with the sides of their Bodies for superficial dimension. And therefore if there were no defection in the aforesaid Radical extractions, there would be none in our work; for that all our artificial Lines of measure (or Lines of artificial measure) being thus Radically produced, are themselves in the nature of Roots quadrate and cubique. And so also, all the Numbers or Terms of Quadrature and Cubature, &c, by which the sides of the equal Squares and Cubes of Figures, (in the natural Mensuration) and also some other particular lines, or numbers of dimension, in the artificial Measure, deduced from the natural) are proportionally obtained several ways (according to unity) as from their several lines of dimension before named (and which also are immediately given by our artificial Lines of measure) are produced by Radical extraction, from the respective figures, to whose dimension (superficial or solid) they serve, as being their Roots, or the sides of their equal Squares and Cubes, according to their foresaid several dimensionall Lines, put unitly. But indeed, the greatest error that can commonly arise here, (in respect of the difference of our measure from the true measure, whether superficial or solid,) will be of no moment, as I have showed before (and shall also show after) in several examples. And if our measure agreed with the true measure, but exactly to integers or units (as it almost always doth, and much nearer also; as even to small parts of an Integer or unit, (superficial or solid) of the appointed Measure) it will be sufficient in any matter of mechanical Mensurations; for which, this our artificial way of Measuring, was chiefly devised and intended.  
But now further, as to the ground and reason, briefly, of this kind of dimension (or of these artificial metrical Lines,) the same may be understood 
Concerning the grounds and reasons of the Artificial measure in general.  to be twofold, to wit, general or universal; and special or particular; that consisting in Unity alone; this in the solution of Unity: For the general reason is by itself absolute, simple, and certain, without relation or limitation to any the proper, compounding, denominate (or other) parts of the Rational Line, (as being the Integer or Unit of measure) but considereth the same generally (and absolutely in itself) as some one entire or whole thing, (of which I shall (God willing) speak more fully afterwards, in the close of the second part of this Book, as being a place convenient.) But the particular reason (which I shall chiefly insist and proceed upon) is limited and confined to the certain, set, or commonly known parts of every particular rational Line (they being considered discreetly or Arithmetically (taken as a common, known Measure, and that, according to several places & customs, (or such like other parts, as the Geometer or Artificer shall in his mind think fit privately to impose on the same for his use) and so cometh by them to that of the General. (Every particular, figurate Quantity or Magnitude, measurable in this artificial manner, being here considered in those parts, according to their powers Quadrate or Cubick (or the Parts considered in their said powers Arithmetically, in every particular Figurate Magnitude, according as the dimension is superficial or solid.) For every common or customary Line of measure is usually divided into certain denominate parts, of which it doth primarily and properly consist: As our Foot is vulgarly said to be divided into 12 parts immediately, called Inches, which do compose or constitute the same: Or (as in divers Countries beyond the Sea) a foot may be understood with us to be composed first (and that most nearly) of 4 Palms, or Hand-breadths; a Palm (being composed) of 4 Digits, or Finger-bredths; and a Digit of 4 Grains or corns of Barley; according to the Latin Distich,  
Quatuor ex granis, Digitus componitur unus:  
Est quatèr in Palmo digitus: quatèr in Pede palmus.  
And this according to the description of 
Vitru●. Lib. 3. Architect, cap. 1.  the ancient Roman Foot by Vitruvius and others: So that thus the Foot contains 16 Digits, answering to 12 inches with us, for that a 
* See Circles of Proportion, Part 1 Chap. 9 Sect. 4.  Palm (namely Palmus minor) is said to be 3 of our Inches, and so 4 Palms 12 Inches: But we in England usually taking no notice of the Palm and Digit in measuring, but only of the Foot and Inch (besides the Yard and el etc.) divide the Foot immediately into 12 parts, called by the Latins unciae & Policies or Pollicaria, and so also was the Roman Foot anciently divided (and still is) and so is the Foot in many other places. So that every such greater Measure is commonly composed of the next lesser, being some certain times reiterated or repeated: As our greatest common Geometrical Line of measure, viz. a Perch or Pole. (for Land-measure) is commonly composed of 
* Statute measure A●. 33. Edw. 1. other (customary) measures there be, as the Pole or Perch of 18 feet, usual for Wood-land measure, etc.  16 1/2 Feet, a Foot being composed of 12 Inches (as I said before) and an Inch of three barley-corns in length, picked out of the middle of the Ear: but a Barley-corn being the lest of all Measures (or rather no measure at all, being but the very beginning of measure, as an Unit is usually counted not number itself, but only the beginning of Number) cannot be composed of any other.  
And although Arithmetic in general, naturally taketh no notice of these lesser Measures as the proper composing parts of the greater Measure given; but immediately considers every particular Measure (as an Unit) according to a simple or natural Arithmetical division into parts (or a division into parts simply or arithmetically denominate) as halves, quarters, and the like; or more especially (and more tightly) as decimal Arithmetic; into Tenths, Centesms, etc. and so taketh the Contents of Superficies and Solids, to such parts thereof, Square and Cubick (or otherwise Superficial and Solid) in general; as Square and Cubick parts of a Foot, and of an Inch; square parts of a Perch or Pole; and so also, square parts of an Acre, etc. Yet this being understood only by Artists, and so not sufficient to satisfy the vulgar: these simple or natural Arithmetical parts (or parts merely divisionall) must at last be reduced to the proper compounding, denominate, or commonly known parts of the Measure given or appointed; (or the Geometrical or mensurall parts of the said Measure (as I may term them) they being by themselves alone, put as measures certain, and entire, and so compounded of, (or dividuall into) other the like kind of parts, or inferior Measures, (according as I noted even now) and so are to be considered as Quantities continuate:) As the parts or fractions of a Foot into Inches; of a Perch or Pole into Feet, (if they be required) and so likewise of an Acre into Perches, etc.  
And now again, the special, particular, or partial reason aforesaid, of this artificial dimension (or of the Artificial Lines of Measure) or the Reason of the Parts, as I may term it, from what I have declared before:) may be considered in a twofold respect: viz. either more generally; as relating to the denominate, compounding or Geometrical parts of every Measure first given or appointed (as the prime or natural Rational Line) only for the discovering or producing of the artificial Lines of measure, as considered in the general Reason, (or the Reason of the Whole;, as I may call it, from what I have said before concerning the same) such as are all the artificial Lines beforegoing, and also the other following, being expressed by Quantity discrete, or number, which show their Magnitudes (or quantities in measure) from the entire, prime, or natural Line of Measure in general, from which they are to be taken; And which therefore I may call (not unaptly, I conceive) their Indicant or Exponent Numbers. Or more particularly, precisely and properly; as relating merely to the foresaid parts of every particular Measure given (considered discreetly or Arithmetically) for the constituting of the said artificial Lines of Measure so, as to give the superficial and solid contents of 
Here note another the like artificial dimension of Figures (as the former) not mentioned before; which is only according to the compounding, denominate (or Geometrical) parts of Measures given or appointed.  Figures, quadrately and cubickly, etc. (or the sides of their equal Squares and Cubes, etc.) from their several lines of dimension belonging to them, and by which severally they may be thus artificially measured; as the forementioned artificial Lines do) only, (or for the most part) according to those parts, (being considered merely in the nature of such parts of the prime or natural Line of measure appointed; which otherwise taken apart by themselves alone, may be put as measures entire (or Integers of measure) as I noted even now; and than are considerable in our general Reason of Measure, or Reason of the Whole, before named.) And which Lines therefore being most properly considered and laid down from the foresaid parts of the natural Measures appointed, (as before, the Lines are from the whole entire Measures themselves) will altar continually in every particular Figure, (in respect of quantity discrete, according to their Arithmetical Indices or Exponents, which express their magnitudes in the forenamed parts of Measure) not only in regard of the different dimensionall lines thereof, by which it may be propounded to be artificially measured, as aforesaid, and so to which they are respectively fitted: (as the artificial Lines  do, according to their like Indices or Exponents before declared; especially for the most part; though it doth hap otherwise sometimes, as that one and the same artificial Line, is found either wholly throughout, or sufficiently in part, (that is, in respect of the fractional or decimal part of the natural Line of measure, from which it is taken, showed by the Arithmetical Index or Exponent, how far soever the same be continued or extended decimally) to serve unto several Dimensions; as I show afterwards, in Part 2. Sect. 3.) but also, in respect of each one and the same particular dimensionall line thereof, by itself alone, according as the said parts of the Measure proposed, do (arithmetically) altar: whereas the arithmetical Exponents of the first mentioned Lines, relating generally to any whole Measure appointed (as I noted at the beginning) do continued the same (without alteration) in every particular, distinct Dimension of one and the same Figure, according to its several lines of Dimension aforesaid. And so these two severally mentioned (or supposed) sorts of artificial Lines, will hereupon differ in every several dimension of one and the same Figure, in respect of quantity discrete or Arithmetical, (according to their Indices or Exponents of measure) though they do not, in respect of quantity continuate or Geometrical; or of Measure itself in general, as I shall show by and by: And than also they differ herein; That as the former artificial Lines being immediately of the whole entire natural Lines (either redundantly or deficiently, according as their Arithmetical Exponents do show) considered without respect of parts, compounding or Geometrical, (but only Arithmetical, or merely divisionall, as is always necessarily required, for the exactness of measure) and so giving the Areall contents of Figures accordingly; are themselves to be considered (in the artificial Dimension) as whole entire Lines of measure in like manner: These latter mentioned (or supposed) Lines must (for the contrary respects aforesaid) be considered (in the like Dimension) as Lines of measure containing (or compounded, as it were, of) certain parts, answering (Arithmetically) to the primary or compounding parts of the natural Lines, from which they are (most properly) derived, and so which they do artificially represent, and consequently, according to which (chief) they give the superficial and solid contents of Figures (as aforesaid.) And which Lines (for the reasons before alleged, as also for distinctiorsake) we may well call, the particular or second artificial Lines (or the Lines of parts) as the other may be called the prime or integral artificial Lines (or Lines of the Whole) And so the Measure arising there from may accordingly be called, the one, the prime or integral Measure (or Measure of the Whole) as having its denomination simply and absolutely from the whole entire natural Line of Measure appointed: And the other the particular or partial Measure (or Measure of the parts) as being denominated chief from the parts of the said Line of Measure.) And therefore, as those first Lines do artificially represent the respective natural Lines from which they are taken, considered simply and entirely in themselves, as the Integer (or Unity) of Measure severally: So these second Lines, do accordingly represent the said natural Lines, as they are composed of certain denominate, or mensurall parts; (viz. of some inferior or lesser Measure, considered as a part of some greater Measure, and so some certain times iterated or ennumerated, for the making up of the same, according as I lately showed,) And so these second Lines are really none other than the first, divided into the like number of parts, as are the composing, constituting, (or Geometrical) parts of the respective natural Lines from which they are deduced; and which parts of these (supposed) second artificial Lines, we may conveniently call (by way of distinction from other parts) their prime or Geometrical parts: and these being than divided severally into some certain number of parts, as is requisite for the exactness of Measure, as aforesaid (according as the correspondent parts of the natural Lines are) especially decimal; we may call the same, their second or Arithmetical parts, (or the particles of measure, as being indeed only the parts of the other Parts.) So that these second Lines are never exactly decimal, as the prime Lines perpetually are, unless it hap, that the prime or compounding parts of the natural Measure appointed be in a decimal number, for than as the natural Line, so likewise the correspondent particular, or secondary artificial Line will be exactly decimal; their said prime parts being divided decimally; but yet however, they do generally perform the Dimensions wrought by them, after a decimal manner (though secondarily, viz. after a reduction of the Measure taken by them, into their prime parts, etc. As first (for example) in the natural Mensuration; the most common, mechanical way of measuring by the foot with us, is, as the same is vulgarly divided into 12 Inches, which are, as its composing, denominate, or Geometrical parts (for so every measure is commonly said to be divided into the parts, of which it is properly composed) and each Inch divided into some certain parts (as is necessary for exectnesse in measuring) which may best be decimal: and so the lines measured hereby, do fall out most frequently, in Feet, Inches, and parts of Inches together: Now if the sides or other dimensionall lines of any Superficies or Solid propounded to be measured, be thus found (in this mixed measure (as they will for the most part) than must the same be first of all reduced into Inches, etc. before the content of the Figure, superficial or solid, can be conveniently cast up, seeing that the Measure thus taken is mixed (as it were) of several parts or kinds of measure, and so is of different denominations; and therefore must be reduced into one kind of measure (or one mensurall denomination:) And than if the parts of the Inch be decimal, the work will be afterwards performed in a decimal manner, in Inch-measure only: But if the said lines thus measured, be found wholly in Feet (without Inches, etc.) than will the contents be had immediately in whole Foot-measure, which otherwise must be had in the like measure by Reduction from Inch-measure. And so seeing that the second artificial Lines do represent the natural Lines, only as being composed of certain denominate, or mensurall parts, and so are to be considered themselves accordingly, as aforesaid: Therefore, if the side or other dimensionall Line of any Figure measured thereby, for the obtaining of its content superficial or solid, by way of Quadrature, Cubature, etc. and so artificially representing the Side of the equal Square or Cube, etc. be found mixedly, in Integers and prime parts, etc. of the same Lines; (as for the most part they will) than must the Measure so taken, be first reduced wholly into their said prime parts, etc. for the casting up of the content as aforesaid, (according to what was showed before in the natural Mensuration, after the most plain or vulgar way by a common natural Line of parts) And than those artificial prime, or mensurall parts, being divided Decimally; the dimension of the Figure proposed, will be performed in a decimal manner, according to those parts only: But if the side, or other line of the Figure to be measured, do fall out in Integers only of the said artificial Lines, (viz. in whole Lines without any parts;) than will the Areall content be obtained immediately, and exactly, in Integers of the natural Measure appointed (according to the reason of the prime artificial Lines, or Lines of the whole Measure) which otherwise can be had in the integral measure (or the measure denominated only from the whole natural Line) only, by way of reduction from the primary, compounding, denominate, o●metricall parts of the same, in which it is first found, as before was showed.  
And now therefore it appears from hence, that the artificial Dimension performed by these second or particular artificial Lines, or Lines of parts, so called; or the artificial Lines, as they are considered merely in the particular or special reason of the artificial Dimension (or Reason of the Parts) taken in the latter respect; is not considerable in comparison of the artificial Dimension performed by the prime or integral (or more general) artificial Lines, or Lines of the Whole; or the artificial Lines, as they are considered in the general reason of the said artificial Dimension (or Reason of the Whole) they giving the sides of the equal Squares, and Cubes, etc. of Figures (and so their superficial and solid contents accordingly) immediately in In●egers (and decimal parts) of the prime Rational Line, of the natural Measure appointed; which the particular Lines do give (for the most part) mixedly in Integers and compounding parts and particles together (or in Integers and parts primary, and secondary) and so we must come at last to the Areall content in the former Measure, by way of reduction from those Parts, as aforesaid. But yet however; for the variety of operation and Art, in this kind of Mensuration, I thought it would not be amiss to manifest thus much concerning this latter way; that so the ingenious Reader that shall please to exercise himself (practically or experimentally) in this artificial way of measuring, may (by comparing the effects or results of the general and particular, or special reason thereof together, both in the extraction or production of the artificial Lines themselves, & also in the said two several ways of working by them) receive a more full satisfaction, and the thing itself be accordingly confirmed.  
And now seeing that the aforesaid parts of Measures given, are various (as in quantity Geometrical, so usually in quantity Arithmetical) according as the Measures themselves are in magnitude various: therefore I shall first and principally prosecute our said special or partial Reason of the artificial dimension, and that chief in relation to the first, or more general acception or consideration thereof, (namely, for the producing of the artificial Lines, as they are considered in the general Reason, by the parts of the Measure given (or the prime artificial Lines by the second, as we have differenced or distinguished them first of all, in Quantity discrete, or in the Number of their measure from the natural Line, according as the same is understood, either without or with the foresaid kind of parts.)  
And this I shall accordingly lay down in three special Theorematicall Propositions (contained in the second part of this Work) answering to the three principal problematical or practical Propositions beforegoing (laid down in the Figures particularly handled in this first Part, namely the CIRCLE, SPHERE, and CYLINDER, together with the CONE) and which will generally serve for all other ordinate or regular, and regular-like Figures whatsoever, for the like occasion (according to what I noted at the beginning) And the last of these only I shall demonstrate or illustrate by Number, as being sufficient for all, though indeed there needeth no manner of demonstration or illustration of any of them, they being all so very plain and perspicuous.    

 printer's device of Robert Leybourn, featuring an old man standing by a olive-tree with falling branches (McKerrow 310) 
NOLI ALTUM SAPERE    

PART II. Containing the most artificial and expeditious practical Dimension, of all rightlined ordinate or regular, and regular-like Figures in general.  

SECT. I. Proposing the foresaid Dimension in all rightlined regular Planes or Superficies in general: And demonstrating the same particularly in two of the first of them.  

THEOREM I Exhibiting particularly, the fore mentioned Lines, for the Quadrature of a Circle; from our particular or special grouna and reason before declared. And consequently, the Lines for the like dimension of all rectiline or angular ordinate Planes in general.  
IF the Diameter of a Circle equal to the Quadrat from the Parts aforesaid, of the Rational Line, be found; The same shall be the respective Line of Quardature, according to the Parts: And the reason between that, and the correspondent or congruall Tetragonismall Line, according to the whole Measure will be such as the reason between the Parts, and the Whole; which is as the reason of their Squares. And the like for the periphery.   geometrical diagram  geometrical diagram 
As a, d, b, or a, d, c, 90º— 10,0000000 Rad. to a b, or a, c, 12— 1,0791812.  
So a b d, or a c d, 60º— 9,9375306. S, A. to a d, 10.3923 etc.— 1,0167118.   geometrical diagram 
Or again, by the same reason;  
As b, a, d, or c a d, 30°— 9,6989700. S, A to b, d, or c d, 6— 0,7781512  
So a b d, or a c d 60°— 9,9375306. S, A to a d, 10.3923— 1,0167118. as before. And other wales also may the same be found out; either by Sines alone, or by Sines and Tangents together, according to Axiom. 1 Planor Pitisc. and Consect. 1 and 2. And Prop. 1. and 2. Planor. in Trigon. Brit. and more particularly cap. 4. before cited, probl. 1. 2. 3. But this way of working is the most plain and vulgar.  
Now the Diameter or Perpendicular of the Trigon, a d, 10.3923 being drawn into the semi-base (or semi-side) b d, or c d, 6; there resulteth the Area of the Trigon (according to the reason of E 1. p. 41.) 62.3538. Or the same Trigonal Area will also be produced, by comparing each side of the Trigon severally, with the semi-aggregate of all the sides, and than infolding the said semi-aggregate and the difference of each side thereto, continually together; for so the Root quadrate of the total Result, shall be the Area of the Trigon: As;  

See upon this kind of Triangular dimension (or Geodesie, as Ramus terms it) Ram. lib. 12 Geomet. theor. 9 but more espcially the reason thereof demonstrated by him, in fine lib. ult-Schol. mathemat.    geometrical diagram 

The artificial Dimension of Triangles in general.  
And here our artificial Mensuration, may be applied to any other kind of Triangle, either regular-like, or altogether irregular, (as I may term them) viz. Isoskelan or Skalene, for the finding of their areal contents; but it will be in a way somewhat different from that of the regular or Isopleural and Isogonial Triangle; (being indeed according to the vulgar or natural way of measuring,) viz. in that, as there the Area is had immediately by the only squaring of some one dimensional line of the Triangle, as the Side or Perpendicular, etc. and so the same is the Quadrat thereof in Quantity arithmetical or measure numeral: here it will be the Rectangle made of the whole Base and Perpendicular together (as it is naturally and really the half of that Rectangle, by E. 1. p. 41. before cited) which two lines, if they hap at any time to be equal, than will the Area be produced under the form of a perfect Quadrat, from either of them, as that of an Isopleuron, or regular Trigon; seeing that the Rectangle made of them both, is no other than the exact Quadrat of either of them singly: And so therefore, after this latter way may an Isopleuron also be artificially measured, the way being general. And this resembles the artificial dimension both solid and superficial of regular▪ like Solids, as the Cone, and all regular-based Pyramids, etc. and so differeth from that particular and peculiar dimension of the ordinate or regular Triangle (and of all other regular Planes) as the dimension of regular-like Solids doth from that of Solids exactly regular: And so, as the solid content of any obliqne, Scalene, or inclined Cone or Pyramid, etc. is obtained both naturally and artificially, as that of a right, upright, or Isoskelan Cone or Pyramid &c. (as was showed before in the dimension of a Cone, and shall be afterwards in the dimension of Pyramids) So is here the superficial content of any obliqne Scalenal or irregular Triangle obtained (as well artificially as naturally) like that of an upright or Isoskelan Triangle by any side thereof, put as the base, and the perpendicular lot fall from the opposite angle thereto, whether that side need to be continued out or produced, or not; seeing that every Skalene Triangle is equal to an Isoskelan, having the same base and perpendicular of altitude, (or they being constituted upon the same base, or equal bases, and in the same Parallels, as Euclid, and his interpreters do speak) by E 1. p. 37. and 38. as every obliqne or Skalene Cone and Pyramid, etc. is equal to a right, or Isoskelan Cone and Pyramid, upon the same (or equal) base, and of the same altitude, by the reason of E. 12, p. 11. and 14. beforecited in the Cone, etc. and p. 5. and 6. afterwards in the Pyramids, etc. And now the Line of measure, for the performing of this latter or general artificial dimension of Triangles, will be the same with that, which was noted formerly for the superficial dimension of a Cone, by its side and basial Periphery together, viz. of the Rational Line in general, 1.4142, etc. √ q 2. And here the proportion of the natural Measure to the artificial, in the two forenamed lines of a Triangle together, will be the same with that in the foresaid Conical dimension; viz. 1. to .7071 etc. √ q 1/2 But indeed seeing that the Area of any Triangle may be obtained as readily in a manner by the natural Line of measure, or common way of measuring, as by this latter kind of artificial Line, or general artificial way of Triangular dimension: (according to what I noted even now about the same.) Therefore I shall press this point not further, than what is only for variety of Art and operation in this kind.   geometrical diagram 
SUppose next the side of the ordinate Pentagon, A B C D E, to be the same with that of the ordinate Trigon before going, (as from the Rational Line A B,) 12.00; than the true Area thereof, will be found, 247.7487 etc. and not 240, as Ramus makes it, showing the Geodesie (as he terms it) of ordinate Polygons in general, and particularly of an ordinate Pentagon, whose side is 12; where he makes the Ray of the inscribed Circle, (which in 
Ram. Geom. lib. 19 el. 1. p. riff in Epitome Rami. And Oront. Fin. lib. Geom. pract. cap. 24.  this pentagonal Figure, is F G) to be exactly 8, (and consequently the Ray of the circumscribing Circle F C or F D, just 10) and thereby the Area, but 240; and which from him P. riff also hath in his Epitome of Ramus: And so Orontius Fineus also measures this Pentagon; and thus they take it one from another (as Ramus from Fineus, and riff from Ramus) without any further examination; And hence they give the faid Pentagon, absolute in all its dimensional Numbers. But indeed, the side of an ordinate Pentagon, being 12, the Ray of the inscribed Circle, will be truly, 8.26 ferè; and the Ray of the circumscribing Circle (which here we need not) will be 10.21 ferè, and so the Pentagonal area will be exactly (to square centesms) 247.7487. All which we shall here demonstrate Trigonometrically. For seeing that an ordinate Pentagon is a Triangulate, consisting of, (or resolvable into) five equal and like Isoskelan or equicrural Trigons, meeting vertically in the centre of the Pentagon, (or of its circumscribing Circle) and so whose Bases are the sides of the Pentagon, and whose shanks are Rays of the said Circle; such as is the Triangle C F D; the vertical (or Centrical) angle, F, will answer to a Fifth of the said circumscribing Circle's circumference, as being measured thereby; which the side of the Pentagon C D (as the base of the Isoskelan Trigon) subtending, the same is consequently the Pempta-chord (as it were) or Hypodia-pempte of the foresaid Circle, as the Hypotenuse, Inscript, or Chord Pentagonal thereof; the said circumscribing Circle being Pentachordal. Which Isoskeles being biparted or bisected into two Orthogonials, viz. C G F, and D G F, by its perpendicular F G (which is a Ray of the inscribed Circle, as aforesaid) whose Hypotenusals (specially so called) F C, and F D, as the two equal sides of the Isoskeles, are two Rays of the Pentagon's circumscribing Circle, as aforesaid; the semi-base of the Isoskeles, C G or D G, as the lesser containing side of the right angle in the two foresaid orthogonial Trigons; will be 6 (as being the semi-side of the Pentagon) and he Angle C F G or D F G, will answer to a Tenth of the foresaid ambient Circle's circumference, as being half the vertical angle of the Isoskeles C F D; and so consequently, the basial or greater acuteangle F C G, or F D G, will be the compliment of the other two angles to a Semicircle (according to E. 1. p. 32) or the Compliment of that other acute angle to a Quadrant: and hereby the perpendicular of the Isoskeles F G, as the perpendicular let fall from the centre of the Pentagon to its side, (or the Ray of the inscribed Circle) being the thing next to be inquired, will be found. And therefore in the rectangle Triangle, C G F, or D G F, seeing all the angies are given, together with the lesser containing side C G or D G; the greater and common containing side F G required, will be had these two several ways following: and first according to Axiom. 2 Planor. Pitisc. and Prop. 3. Trigon. Brit. etc. before-cited, most easily and readily by Logarithmo-trigonometrical supputation, or artificial Trigonometry, thus;  
As C F G, or D F G, 36º— 9,7692187. S A to C G, or D G, 6.— 0,7781513.  
So F C G, or F D G, 54º— 9,9079576. S A to F G, 8.26 ferè, viz. 8.25829 etc.— 0,9168902.   geometrical diagram 
Or secondly; seeing that in a recti-line rectangle Triangle, any side may be put for the Radius of a Circle, by Axiom. 1 Planor. Pitisc. and Prop. 1 Planor. Trigonom. Brit. Therefore in the foresaid Rectangle C G F, or D G F, the lesser containing side C G or D G given, being put as Radius; the greater (and common) containing side, F G sought for, will (by the foresaid Axiom, and Prop. 2. Planor. Trigon. Brit.) be the Tangent of its opposite angle F C G, or F D G; Whereupon it followeth accordingly, in this Trigonometrical Reason;  
As C G, or D G,— 10,0000000. R. to F G— 10,1387390, T A, 54ᵒ.  
So C G, or D G, 6— 0,7781513 to F G, 8.258 &c,— 0,9168903 as before.  
Which being enfolded with the semiperimeter of the Pentagon, A B C G, 30, there will result the Area of the Pentagon, 247.7487, as I said at first. Or the said F G, as the perpendicular of the foresaid Isoskeles C F D, being enfolded with the semi-side of the Pentagon, C G or D G, as the semi-base of the Isoskeles, produceth the Area of the Isoskeles, C F D, 49.54974, etc. which augmented by the number of the composing Isoskelan Trigons, produceth the total Pentagon, 247.7487 etc. as before; and which is most readily and accuratly obtained by Logarithmical numeration, only by a simple composition of Numbers, thus;  
F G, 8.258 etc.— 0,9168903 A  
C G, or D G, 6— 0,7781513 A  
Isoskeles C F D, 49.5497 etc. 1,6950416 aggreg. A  
5 A 0,6989700 aggreg. A  
Pentagon, 247.748745 etc.— 2,3940116. aggreg.   geometrical diagram 
Or, according to the first operation, more briefly thus;  
F G, 8.258 etc.— 0,9168903 A  
A B C G, 30— 1,4771213 A  
Pentagon, 247.7487 etc.— 2,3940116, as before.   geometrical diagram 
Now as to the Radius of the circumscribing Circle, which Ramus, & the others beforenamed, do make exactly 10, (though here we make no use thereof, yet for the truth's sake) we will demonstrate the same to be 10.21 ferè, thus: And first in the rectangle Triangle C, G, F, or D G Faforesaid; the lesser acute angle C F G or D F G, and the lesser side about the right angle, viz. C G or D G (as the semi-side of the Pentagon) subtending the said acute angle, as the right Sine thereof, being first of all given: the Hypotenuse F C, or F D, being the Ray of the ambient Circle, sought, will be found thus,  
As C F G, or D F G, 36º— 9,7692187. S A. to C G, or D G, 6— 0,7781513.  
So C G F, or D G F, 90º— 10,0000000. R. to F C, or F D, 10.21 ferè, viz. 10.2078 etc.— 1,0089326.  
Or again secondly in the same Triangles, the greater acute angle F C G, or F D G (being half the Pentagonal angle) and the greater and common side about the right angies, viz. F G, subrending the two said equal acute angles, ●eing only first given; the Hypotenusal side, F C, or F D, inquired, will be had thus;  
As F C G, or F D G, 54º— 9,9079576 S A to F G, 8.25829 &c— 0,9168902  
So F G C, or F G D, 90º— 10,0000000 R. to F C, or F D, 10.2078 etc. 1,0089326, as before.  
Or again, thirdly, in either of the said rectangle Triangles, where the foresaid greater acute angle being only known and the lesser side about the right angle, viz. C G, or D G, and the same put as Radius; the side subtending the right angle, viz. F C, or F D inquired, will be the Secant of the said greater acute angle, F C G, or F D G, (as the greater and common side about the right angle, and the common Subtense of the said two acute angles, viz. F G, was said before in the like case, to be the common Tangent of the same angles) whereupon it followeth in this Trigonometrical Analogy;  
As C G, or D G— 10,0000000, R. to C F or D F— 10,2307813, Sec. 54ᵒ.  
So C G or D G, 6— 0,7781513. to C F or D F, 10.2078 etc. 1,0089326, as before.  
Or fourthly, in either of the said Rectangles, the lesser acute angle C F G, or D F G, being only given, and the greater containing side of the right angle, viz. F G, and the same put for Radius; the Hypotenusal F C, or F D, will be the Secant of the said lesser acute angle, (as the lesser containing side C G or D G will consequently be the Tangent of the same acute Angle.) Therefore it followeth;  
As F G— 10,0000000 to F C or F D— 10,0920424. Sec. 36ᵒ.  
So F G, 8.258 ●c.— 0,9168902 to F C or F D 10.2078 etc.— 1,0089326, as before,  
Or fifthly and lastly, in the Isoskeles C F D, where all the angles and the Base C D (as the side of the Pentagon) being known; the side of the Isoskeles F C or F D sought for, will be obtained by the most common Trigonometrical operation, thus;  
As C F D, 72º— 9,9782063. S A. to C D, 12— 1,0791812.  
So C D F, or D C F, 54º— 9,9079576, S A to C F or D F, 10.2078 etc.— 1,0089325, as before.  
Now the Radius of the circumscribing Circle, viz. F C or F D (to which A F is equal) being these five several ways found out, to be 10.21 ferè; the same with the Radius of the inscribed Circle, viz. F G found before, 8.26 ferè, will make up the total Diameter or Perpendicular of the Pentagon, viz. A G (which is the altitude thereof) to be 18.47 ferè; which according to the account of Ramus, and the other Auth●rs before named, would be exactly 18. For that here you may observe by the way, how that in all such ordinate Planes, as have their angles and sides in an uneven or unequal number (and so the angles and the sides severally, are not exactly opposite one to another, that is, angle to angle, and side to side, but contrarily, (as the Trigon, Pentagon, Heptagon, Hennea-gon, and the like; the Diameter is composed of the semidiameters of the Circle circumscribing and inscribed: And in all such ordinate Planes, as have their angles and Sides in an even or equal number, (and so the angles and sides severally, are directly opposite one to another, that is, angle to angle, and side to side) as the Tetragon, Hexa-gon, Octa-gon, Deca-gon, and the like; the Diameter is no other than that of the circumscribing Circle. And so by these Diameters may the areal contents of these Figures be obtained artificially as by their sides, according as I noted at first: For so the artificial Line of measure for squaring of an ordinare Pentagon by its said Diameter or Perpendicular, will be found (by the reason of the foregoing Theorem, etc.) to be of the prime Rational Line in general, 1.1732 ferè. And from 
The artificial Line in general for the quadrate dimension of an ordinate Pentagon by its Diameter or Perpendicular of altitude.  the premises it appears, that if the Diameter of a Circle be measured by any of the artificial Lines pertaining to the Diameters of the latter kind of ordinate Planes here mentioned, the Quadrat thereof shall be equal to the respective inscribed Figure; where, by the Diameters, are meant their angular Diameters, or longer Diagonals passing between two extremely opposite angles, through the Centre of the Figure; where as there is to be understood another Diameter, passing between two opposite sides, through the Centre, (and to which is equal in the Hexagon the line passing between the ends of the two opposite sides, and so subtending the Angle of the Figure, and which is more peculiarly and specially called the Diagonall-line, or Diagonie of the same Figure) which is no other, than the Diameter of the inscribed Circle; And so by this other diametral line, may the said Figures be also artificially measured as before, by Lines of Measure appropriated thereunto; And therefore, if the Diameter of a Circle be taken thereby, the Quadrat thereof shall be equal to the respective circumscribed Figure: And so also may these ordinate Polygonal or Polypleural Planes in general, be artificially measured by their shorter Diagonal-lines, which subtend their angles singly, (and are equal, neither to the diameters of their Circles circumscribing, nor inscribed, nor composed of both) & are more peculiarly called their Diagonials or diagonies; and which is in the Hexagon, the Side of its inscribed Trigon; in the Octagon, the side of the inscribed Tetragon; & in the Decagon, the side of the inscribed Pentagon, etc. And which sort of line is drawn and handled (amongst others) in the next Pentagonal figure, set forth for the dimensional proportions in a Pentagon. And so the artificial Line for the quadrate dimension of an ordinate Pentagon by its said diagonie, or angular Subtense, will be of the prime Rational Line, (in the general Reason of Measure) 1.2336 ferè.   geometrical diagram 
And thus having demonstrated the dimensions of the three first ordinate, Planes, geometrically, in every respect: we she next of all deliver the same purely arithmetically, by way of proportion, as we have extracted them in all the variety thereof that may be; some of which may be of singular use, not only for the more easy and speedy superficial dimension of these Figures simply, according to the natural Measure; but also for the solid dimension of the Pyramids raised or constituted upon them (and so consequently of the soresaid five plain, or rectiline regular Solids, after the natural and vulgar way of measuring them, which is the most difficult) as to the speedy discovering of the semidiameters of the Circles circumscribed to their Bases; and so thereby, and by the side of the Pyramid together, the Axis or Altitude of the same; as I shall show in the next Section; like as I did before, for finding the Axis of a Cone. And first therefore for Linear Proportions (as I may term them) or Proportions of Linear dimension, in the ordinate or regular Trigon.  
1 The Side (a, e or a, ●) is to the Ray of the Circle, circumscribing (a. o.) as 1. to .57735 √ q 1/3. inscribed (o. u.) .288675. And so to the diameter or Perpendicular (a. u.) as 1. to .8660254 √ q 3/4 or. 75. And viceversâ, the Perpendicular (a, u) is to the Side (a, e, or a, i) 1.1547.   geometrical diagram 
2 The Radius of the circumscribing Circle (a, oh) is to the Side (a, e, or a i,) as 1. to 1.73205 √ q 3. Ray of the inscribed Circle (oh, u,) 0.5 dupla. And consequently to the Diameter or Perpendicular, (a, u) as 1. to 1.5 subs●squi ●●tera.  
3 The Radius of the inscribed Circle (oh, u) is to the Side (a, e, or a, i) as 1. to 3.4641 Ray of the circumscribing Circle, (a, oh,) 2. subdupla. And so to the Perpendicular or diameter (a, u) 3. sub●ripla.  
Whereby it appears, that the Radius of the circumscribing or containing Circle, is 2/3 of the Trigonal Perpendicular or diameter, and the Radius of the inscribed or contained Circle is 1/3 of the same, and which is farther manifest by E. 14 p. 18.  
Secondly, for superficial Proportions in the ordinate Trigon; or Proportions of superficial dimension:  
4 The Quadrat of the Side is to the Trigon itself, as 1. to .4330127 Diameter .57735. √ q 1/3.  
Which latter, is the same with that of the side to the Ray of the circumscribing Circle.  
And contrarily;  
5 The total Trigon is to the Quadrat of its Side as 1. to 2.309401 Diameter, 1.73205. √ q. 3  
And so consequently.  
6 The Side, is to the side of the Quadrat equal to the Trigon, as 1. to .658037. Diameter, .759836 ferè.  
Next for the like proportions in the ordinate Pentagon; and first in respect of Linear dimension;  
1. The side of the Pentagon (a. b.) is to the Ray of the Circle circumscribing (a. g) as 1. to .85065. inscribed (g. h.) .68819. And so to the Diameter or Perpendicular of the Pentagon (a. h.) as 1. to. 1.53884. diagonal, or the Subtense of the pentagonal angle (b. e.) 
* Flussat. El. 16. p. 2.  cutting the Diameter by extreme and mean proportion) as 1. to 1.618034 ferè.  
4. The Radius of the ambient Circle (a. g.) is to the Side of the Pentagon, (a. b.) as 1. to 1.17557 Ray of the inscribed Circle, (g. h.) .809017 (And so to the Pentagonal Diameter (a. h.) as 1. to 1.809017.) Pentagonal Diagonie, or angular Hypotenuse (b. e.) as 1. to 1.902113.   geometrical diagram 
3. The Radius of the inscribed Circle (g. h.) is to the Side of the Pentagon, (a. b.) as 1. to 1.453085 Ray of the ambient Circle, (a. g.) 1.236068 (ferè. (And so to the Diameter or Perpendicular (a. h.) as 1. to 2.236068) Diagonal or Hypothenusal (b. e.) as 1. to 2.351141  
4. The Diameter of the Pentagon (a. b.) is to the Side (a. b.) as 1. to. 649839. Ray of the Circle (of which two circular Rays it is composed.) circums. (a. g.) as 1. to .552786 inscrib. (g. h.) .44721 Pentagonal Diagony (b. e.) as 1 to 1.051462.  
And to its greater portion or segment (f. h.) (being parted by the said diagonal or hypothenusal, according to extreme and mean reason, as aforesaid) as 1. to .618034, and to its lesser portion● (a. f.) as 1. to .381966: which two segments are Algebraically or Cossically, the irrational Apotomies, √ 1 1/4 (or 1.25)— 1/2 (or. 5) for the greater segment; and 1 1/2 (or 1.5)— √ 1 1/4 (or 1.25) for the lesser segment; according to a Cossicall invention of the said two segments, agreeing exactly with the trigonometrical.   geometrical diagram 
5. The diagonal or Hypotenusall (b, e) is to the Side, (a, b) as 1. to. 618034. Ray of the Circle circumscrib. (a. g.) as 1. to .525731 inscribed, (g. h.) .425325 And so to the Diameter, (a, h) as 1. to .951056. And to the greater segment thereof (f, h) (as being divided by the diagonal, according to extreme and mean reason, as aforesaid) as 1. to .587785 And to the lesser segment of that division or partition, (a, f) as 1. to .363271.  
Which two diametral segments, are Cossically the irrational Apotomies √ 1.130635 etc.— 0.475528, for the greater segment; and 1.426584 etc.— √ 1.130635 etc. for the lesser segment, according to a Cosficall computation of these two segments, which we found to agreed exactly with the trigonometrical, as the former.  
Which Pentagonal proportions before-going, (those of the Diagony being secluded) would be according to the Dimension of the foregoing ordinate Pentagon, by Ramus etc. exactly rational in the lest terms, thus;  
1. The Side of the Pentagon to the Ray of the Circle circumscribing, as 6 to 5, sesquiquinta, (which is decimally irrational, 1. to .833333 infinitely, & defective.) inscribed, as 3 to 2 sesquialtera, (which is decimally irrational, 1. to .66666 etc. infinitely, and defective.) And so to the Diameter, as 2 to 8, subsesquialtera (which is decimally irrational 1. to 1.5, (or integrally, rational, 10 to 15, defective.)  
2. The Radius of the circumscribing Circle to the Radius of the inscribed Circle, as 5 to 4, sesquiquarta, (which is decimally irrational. 1. to .8 (or integrally, rational, 10 to 8. deficient. And consequently to the Pentagonal Diameter, as 5 to 9, sub-superquadriquinta, or subsuperquadrapartiens-quintas, (which is decimally irrational, 1. to 1.8 (or integrally, rational, 10 to 18 deficient.)  
And so the Radius of the inscribed Circle, to the said Diameter, as 4 to 9, subdupla-sesquiquarta, (which is decimally irrational, 1. to 2.25, (or integrally rational, 100 to 225) and excessive or redundant.)  
By which the reciprocal proportions may be had by inversion of the Terms: And here the proportion of the Pentagonal Diameter to the side, willbe the same with that of the side to the Ray of the inscribed Circle, before noted, viz. 3 to 2 sesquialtera.  
Than secondly in relation to superficiary Dimension, the Pentagonal proportions will be exactly, as followeth.  
1. The Quadrat Lateral is to the Pentagon itself, as 1. to 1.720477 Diametral 0.726543 ferè Diagonial 0.657164 ferè  
And contrariwise  
2. The Pentagon is to the Quadrat Lateral as 1. to .581234 Diametral 1.37638 Diagonial 1.52169  
And so,  
3. The Side is to the Side of the Pentagonal Quadrat, as 1.10 1.31167. Diameter .85237. Diagonial .810656.  
Which three last Proportions are the most precise and proper Tetragonismal terms of an ordinate Pentagon; but chief the first of them.  
And which superficial Proportions (secluding those sore the Diagony) would be, according to the former Linear Proportions, deduced from the foregoing Pentagonal dimension of Ramus, etc. in these Terms;  
1. The Quadrat Lateral to the Pentagon itself, as 1. to 1.666666 infinitely, & defective; being in vulgar terms, as 1. to 1 ●/●, or 5/1 sub-super-bitertia, or sub-super-bipartienstertias. diametral. .740740, infinitely, & excessive; being more vulgarly, 1 to 20/27, superseptu-partiens-vigesimas; which is very near sesqui-tertia in the lest terms, it being superpartient 7/20.  
Contrarily;  
2. The Pentagon to the Quadrat Lateral as 1. to .6 (or integrally, 10 to 6) excessive; being more vulgarly, 1 to 3/5, superbitertia, viz. superpartient 2/3. diametral 1.35, (or integrally, 100 to 135,) defective; being in vulgar terms, as 1 to 17/2●, sub-super-septupartiens-vigesimas.  
And so,  
3. The Pentagonal. Side to the Side of the Pentagonal Quadrat, as 1. to 1.29099, desicient. Diana. .86066, redundant.  
But the former or Linear Proportions are erroneous; Therefore, the latter or superficial.  
Than as for the Proportions of dimension in the Tetragon; there be only these two considerable; viz.  
1. The Side is to the Diagony (or circumscribing Circle's Diameter) as 1. to 1.4142, etc. √. 2. (which is the general Linear or Scalar Number, for Diagonial Quadrature, noted before)   geometrical diagram 
And Viceversâ.  
2. The Diagony is to the Side, as the diameter of a Circle to the side of its inscribed Quadrat, noted formerly in the dimension of a Circle; viz. 1. to .7071, etc. √ 1/2:  
And so is the side to the semi-diagonie, (or the circumscribing Circle's semi-diameter.)  
And here, as the side of the Tetragon, is the Tetrachord, or the Chord tetragonall of the circumscribing, comprehending, or containing Circle, (as subtending a Quadrant of its periphery) So is the Diagony, the like Chord of the Circle described out of some angular point of the Tetragon, according to the side thereof, as the Radius; viz. the Circle described about the diagoniall Quadrat; the said Diagony being than the Side of the Tetragon inscribed; And so those Circles are Tetrachordall.  
And here we may take in by the way, another kind of tetragonall Plane, very variable in respect of its angles, but regular-like, (as I may term it) being equilateral, though notequi-angular, called Rhombus, being the only obliquangle equilateral 
Concerning the dimension of a Rhombus.  Parallelogram, and which therefore is a Quadrat, as it were, dislocated in its Terms, or compressed in the angles, as Ramus speaketh lib. Geomet. 14. el. 7, 8, and so is a Triangulate, consisting as it were, of two equal and like isoskelan Triangles, meeting upon one common base; And if the Triangles hap to be equilateral, equiangular, or exactly regular; than will the Rhombus be artificially in the nature of an exact Quadrat, as that kind of Triangle is, and so all other regular Planes artificially are: and so will admit of the like artificial, or quadrate dimension, by its side, (or by either of the Diagonall-lines which may be drawn in it, and so by which it may be resolved into 4 equal and like rectangle Scalenons; the lesser or obtuse angle-diagonall, which is the common base of the said two Isopleurons, being equal to the side of the Rhombus, as being a side of its composing ordinate Triangle) according to an exact Quadrature. And the artificial Line for this particular Rhomball Dimension, will be (according to the reason of the foregoing Theorem, etc.) of the rational Line in general. 1.0745, etc.   geometrical diagram 
And by the artificial Line for the dimension of Triangles in general, may the Area of any Rhombus be obtained, its two diagonal-lines being measured thereby; for so the rectangle Parallelogram resulting therefrom, shall be equal to the Rhombus; as the Rectangle of those two lines, is naturally and really double to the Rhomb; or the Rhomb is half the Rectangle made of those two lines, being measured by the natural Line of measure; viz. the Rectangle from one whole diagonal enfolded with half the other. And thus also may be artificially obtained the Area of any Rhomboides or Trapezium.     

SECT. II. Setting forth the Dimension, both solid and superficial, of regular-based Pyramids in general, and their Compounds: And demonstrating the same particularly in the three first kinds of them.  
ANd now having sufficiently showed our artificial Dimension in the three first rectiline or angular ordinate Planes in particular; namely the Trigon, Tetragon and Pentagon, simply in themselves, (but chief the Trigon and Pentagon, as being in them, only requisite) and so consequently the like Dimension of all ordinate polygonall or polypleurall Planes whatsoever, by the same metrical reason: We shall next proceed to the like kind of dimension in them, as in order and relation to all Pyramidal Bodies, both prime or simple, and compound, (as I may so speak) or Pyramids, and Pyramidates, as being their Bases: or the dimension of these Solids, being founded, (as it were) constituted, or erected upon such Planes; and so denominated from them accordingly, as I have said before. And this I shall particularly show in the three first sorts of Pyramids, constituted or raised upon the three first Planes before named; and more especially for that, as those three Planes do concur superficially, to the composition, (or to the superficial composition) of the five famous ordinate Bodies, or rectiline regular Solids, as I said before; namely, the Trigon, to the Tetrahedron, Octahedron, and Eicosahedron; the Tetragon to the Hexahedron, and the Pentagon to the Dodecahedron: so the three kinds of Pyramids erected or constituted upon them, do concur in like manner solidly, to the composition, (or to the solid composition) of the said five Bodies, as I shall show particularly in each of them.  
And seeing that of all the kinds of Pyramids (which may be as infinite in number, as the Figures, for their Bases, upon which they are raised and constituted, and so from which they take their special denomination, as whether the same be trigonal, tetragonall, pentagonal, or howsoever polygonall, and so the respective Pyramids be denominated accordingly) there is but one kind exactly ordinate or regular, and so is specially and peculiarly, (for the excellency thereof) called Tetraedron, and by Euclid, simply by the name of Pyramid, in E 13. p. 13; it consisting of four equal ordinate Trigons compact together by solid angles (by E 11. d. 26) which therefore are in number subtriple the plain, superficial, or Trigonal angles constituting the same, (so that to the constitution of one solid angle, do here concur three superficial angles; and therefore this solid angle is contained under two plain right angles precisely, and so is 2/3 of a solid right angle, as its composing or basiall angle is 2/3 of a plain right angle) and so the angular lines, or sharp edges, called the sides of the Pyramid, (made by the connexion of the sides of the ordinate trigonal Planes) are in number, subduple the trigonal sides constituting the same) And so any one of the said trigonal Planes may be put for the Base of this Pyramid, and thereupon the solid angle opposite thereunto, made by the inclination, connexion, or concursion of the other three like Planes, (according to their vertical angles) shall be the top or vertical point of the same; between which, and the Centre of the Base, shall be adjudged the perpendicular altitude, (or the Axis) thereof: We shall first therefore show our artificial dimension (both solidly & superficially) of the first kind of Pyramid, in the first of the foresaid five regular Bodies, namely the Tetrahedron; and that by the Side of the Base, (which is the general side of this Body) in answer to the foregoing quadrate dimension of an ordinate Trigon by its side) and the Axis together; which we shall compare with the natural or vulgar dimension, according as we have done in all the Figures beforegoing; whereby the same may withal be understood by such as are yet to learn.   geometrical diagram 
And though a Cone do somewhat resemble a Pyramid, and a Cylinder, a Prism; yet a Cone cannot properly be called a Pyramid (as some do call it) nor a Cylinder a Prism; they being not plain Solids, rising from a rectiline or angular Base; but various, or variable gibbous Solids, rising from a curviline, obliqueline, or circular Base; but yet a Cone may be understood to comprehend in its self, any kind of Pyramid; and so a Cylinder to comprehend any kind of Prism; their Bases being the Circles circumscribed to the Base of the Pyramid or Prism, and so their whole Bodies circumscribed to the whole Pyramid or Prism, viz. their concave superficies just touching the angular lines, or the sides of the respective Pyramid or Prism inscribed. And contrarily, may any Pyramid be understood to comprehend or include a Cone, & any Prism a Cylinder; the rectiline Planes of the Pyramid and Prism, just touching the convex Superficies of the inscribed Cone and Cylinder, (and so the angular or rectiline base of the Pyramid and Prism, completely circumscribed to the circular base of the Cone and Cylinder,) according to the nature of Mathematical Inscription and Circumscription, as you may see it fully set forth in E. 4. d. 1, 2, 3, 4, 5, 6, for Superficies; And E, 11. d. 31 and 32, for Solids.  
But now to the main thing in hand, to wit, the solid dimension of the foregoing Pyramid, according to our artificial and compendious way (as before in the Cone) and here, by the side of the Base, & the Axis, or line of altitude; the artificial Line of Measure 
The artificial Line for the solid dimension of a trigonal Pyramid by the side of its base and its Axis or Altitude together.  for which purpose, I found (according to the reason of the 3d. Theorem, etc. following) to be of the prime or natural Rational Line in general, 1.9064, ferè; which being duly set of therefrom; the side of the aforesaid Tetrahedrum's Base (put naturally 12.00) will be found thereby, 6.29, (in a centesimall partition) whose Quadrat is 39.5641, for the artificial base: And the Axis or Altitude of the Tetrahedrum, found geometrically before, 9.7979, etc. (which according to a centenary solution of the unit, or of the Rational Line, is 9.80 ferè.) will be found by the foresaid Line, 5.14, for the artificial Axis; which being wholly enfolded with the whole Base, will produce the rectangle regular-based Prism, or Parallelipipedum, 203.359474, sor the artificial solidity of the Tetrahedrum; which differeth from the true, natural, solidity, viz. 203.646753, (by way of defect) not so much as 1/3 of the prime Rational Line cubed, or 1/3 of a cube-unit. And by a further solution of the two Lines of Measure, (viz. the natural and artificial Rational Line) the difference will be found much less, according to what I have said, and also plainly demonstrated in all the precedent Dimensions.  
Now for the superficiary dimension of this kind of Pyramid; if it be exactly 
Concerning the superficial dimension of atrigonal Pyramid.  ordinate, as the Tetrahedron, than one of the Planes being had, the whole Superficies is easily had, by the quadruplication of that Plane: As the Plane of the foregoing Tetrahedrum, being found by the natural Dimension, 62.3538; the whole superficies will be 249.4152: And which may be artificially obtained by either of the Lines for the quadrate dimension of an ordinate Trigon; but most readily and properly, by that for the side. As the side of this Tetrahedrum being 12, will be found by the Line of Lateral quadration of a Trigon, 7.90 ferè (as before the side of the Trigon simply) and so the Square thereof, 62.4100 ferè, for the Plane of the Tetrahedrum; and the quadruple of this, is 249.6400 ferè, for the total superficies thereof; which exceedeth the former, or true superficies, not so much as 1/4 of a square-unit, or Integer.   geometrical diagram 
Now for a trial of this dimension artificially, and that by the foresaid perpendicular-line, and the Side of the base of the Pyramid together, whereby the rectangle Plane made of them, shall agreed with the rectangle Plane made naturally of the semiperimeter of the Base, and the said hedrall perpendicular, for the true Pyramidal superfice, viz. without the proper base: The artificial Line of Measure specially serving hereunto, I found (according to the reason of the 3d. Theorem, etc.) to be of the Rational Line in general, deficiently, 0.8165 ferè, (which is Apotomally in Number, 
The artificial Line for the superficial Dimension of a trigonal Pyramid, by the Side of its Base & the perpendicular-line of its lateral trigonal Plane, (or the altitude of the same) together.  according to our general reason of Measure, 1— .1835 ferè) which being set of from the said Line, and the side of the foresaid Pyramids base measured thereby (in a centesimall partition) will be found, 14.70 ferè; and the Perpendicular being also measured by the same Line, will be found, 12.73. ferè; which two being multiplied together, there will result the rectangle Parallelogram, 187.1310 ferè, for the Pyramidal superfice,  (viz. without the Base) which exceedeth the like superficies formerly found most truly, 187.0614, not fully so much, as will make in vulgar terms, 1/14 of a square-unit.  
And thus much for the Dimension both solid and superficial of the first kind of Pyramid in general, whether ordinate or inordinate, so as the Base be ordinate: And so in special, of the first of the five ordinate plain Bodies, namely, the Tetrahedron; according to a Pyramidal dimension only.  
As for the first kind of Pyramidate, called in general Prisma, which I have mentioned before in the first Part; I shall afterwards speak a little of the like dimension thereof, as is of a Pyramid itself.   geometrical diagram 
But now the whole difficulty of this Dimension of the Icosahedrum, (and ●o of the other two soresaid Bodies) consists in the investigation of the Axis, or line of altitude, of the compounding Pyramid (as it did before in the Tetrahedron) if the same be inquired geometrically (as Geometricians speak) to wit, by first having the side of the ordinate Solid only, and so coming at length, after many tedious and troublesome operations, both arithmetical and geometrical, to the said perpendicular-line of altitude, (which is no other than the Radius of the Sphere inscribed within the plain Solid) which way therefore I shall here pretermit, as being needless for me to demonstrate; the same being showed by divers practical Authors, especially Ramus and Clavius in Latin; and more abundantly in English, by our Countryman, Mr. Digges long since, in his learned Discourse of geometrical Solids, annexed to his Pantometria, as a part thereof; to which Authors I therefore refer the diligent practizer for a full satisfaction in this point; my intention in this place, being only to bring in our new or artificial way of Pyramidal Dimension, in the aforesaid ordinate Bodies, for the more easy and speedy obtaining of their solidities in a Pyramidal way, the Hexahedron being excepted. And therefore I shall here only show, how, instrumentally or mechanically to get the said Pyramidal Axis or altitude, by getting first the altitude of the whole body, (which 〈◊〉 no other than the total Dimetient of the inscribed Sphere) being according to what I shown before, for getting the altitudes of Cones and Pyramids in general. Therefore, if from the superior Plane of the Icosahedrum, being produced or extended, that is, from the inferior or interior superfice of any Plane placed upon the uppermost Plane of the Icosahedrum, a perpendicular be let down to the inferior or opposite Plane or Base thereof, in like manner produced, that is, to the superior or exterior superfice of a Plane placed under the Icosahedrum (and so, upon which the same lieth) the said perpendicular-line (being accurately measured) shall give the altitude of the whole Icosahedrum, whose half shall be the Axis or Altitude of the composing Pyramid sought for.  
So that the great difficulty in the dimension of the Icosahedrum, arising by the foresaid geometrical investigation of this Pyramidal altitude, is by this means quite taken away, and the thing made very easy. Ane therefore whensoever you would measure an Icosahedron (or a Dodecahedron, for the same reason holds in it for the altitude) it is best to use this way; for so the solidity of the same will than be obtained with little labour, especially according to our foregoing artificial pyramidal Mensuration; For the said Axiscr altitude of the Icosahedron's composing Pyramid, being taken by the foresaid artificial Line for the solid dimension of a Trigonal Pyramid, and the same be tiplied into the Quadrat of the side, taken by the same Line; the product shall be the solid content of the said compounding Pyramid; whose vigecuple willbe the solidity of the total Icosahedrum. But how, readily to obtain the Axis of the compounding Pyramid, both of this, and also the other ordinate bodies (where need is) I shall afterwards show among the dimensional Proportions in these Bodies; by first having the side of the Body, and which is very easily taken, without any trouble at all.  
And so the total superficies of this Solid, will readily be had by the Line for the Lateral quadrature of an ordinate Trigon (according as I shown before in the superficies of a Tetrahedron) for the side thereof being measured thereby, its Quadrat willbe the area of one of the bases or Planes, whose vigecuple willbe the total Icosahedral superficies.   geometrical diagram  geometrical diagram 
And if the solidity of an Octahedrum be immediately requiree; than the total Axis or Diagony thereof (which is double to the Axis of the foresaid Pyramid, and so no other than the Diagony of its base, or the Diameter of the Circle circumscribing the same, and also the Axis or Diameter of the Sphere circumscribing the Octahedron) being enfolded with a trient of the compounding Pyramids base, (which is no other than the Octahed●on's Lateral Quadrat, as was showed before) or the whole base wi●h a trient of the said Axis or Diagonial; there will immediately result the solidity of the Octahedrum: For so the true Axis or Diagonial of the foresaid Octahedrum, 16.97, etc. sufficiently produced by Radical extraction, or otherwise, viz 16.97056274, etc. being enfolded with a trient of the aforesaid pyramidal base, viz. 48. (or the whole base 144 with a trient of the axis, viz. 5.65685424, etc.) there, wilimmediatly result the true total Octahedral solidity, 814.587012 as before. And so the total axis or Diagony of the Octahedrum, found by the former artificial Line, (in a centesimal solution) 11.77 ferè, being enfolded with the foresaid total common base of the two compounding Pyramids, produced by the same Line, 69.2224; there will also immediately result the solidity of the Octahedrum, 814.747648 ferè, which comes much nearer the true solidity, than the former dimension; this differing there from (now by way of excess) not so much as 1/6 of a cube-unit, or integer, of the appointed measure; which Octahedral solidity is exactly quadruple to the foregoing Tetrahedral solidity, these two Bodies having here one and the same side in measure. And here therefore it appears in brief, that if the total Axis or Diagonial, and the Side, of an Octahedrum, be taken by the foresaid artificial Line of measure for a Tetragonal Pyramid, and the Quadrat of the Side be augmented by the Axis; the resulting rectangle regular-based oblong Prism, or Parallelepipedum, shall completely contain the solidity of the Octahedrum.  
As for the Diagonial, Axis, or Altitude of the Octahedron, the same may be also obtained instrumentally or mechanically, according to what I shown in the Tetrahedron, and Pyramids in general; and also for the altitude of the Icosahedron, and Dodecahedron; the altitude of this Body being considered according to a perpendicular-line comprehended between its two opposite angles, for as much as it is composed of two equal and like quadrangular Pyramids joined together in their bases, as I said before.  
As for the superficiary dimension of this kind of Pyramid which now we have in hand, the 
The superficial dimension of a tetragonal Pyramid; And the artificial Line for performing the same by the Side of the base, and the perpendicular line of the trigonal Plane together.  same may be most readily performed in the same artificial manner as that of a trigonal Pyramid, to wit, by the side of the Base, and the perpendicular-line of its triangular Plane together. And the artificial Line of Measure for this purpose, I found (according to the reason of the same Theorem) to be of the Rational Line in general, 0.7071, etc. √ 1/2, which is Apotomally in Number, as 1— .2929 ferè) which Line being set of there from, and divided in a due manner, and than the two forenamed Lines of the Pyramid commensurable by the same, be accordingly measured thereby, the product arising by their mutual multiplication, shall be the superficies of the Pyramid, (to wit, without the base) agreeing with that which is produced naturally, by the semiperimeter of the base, and the foresaid trigonal perpendicular multiplied together, according as I fully demonstrated before in the superficiary dimension of the trigonal Pyramid. As in the foregoing tetragonal Pyramid, the side of the base being 12, the perimeter thereof will be 48, and so the semiperimeter 24: & the side of the Pyramid, or of its trigonal Plane, being the same with the side of the base, the perpendicular of the said Plane, will be 10.39, etc. as was showed in the superficiary dimension of the Tetrahedron, and before that, in the dimension of the Trigon alone; which two multiplied together, (the said perpendicular being further produced, as formerly) there will arise 249.4153, for the superficies of this Pyramid, without the base; agreeing with the total superficies of the foregoing Tetrahedrum, the Planes of that & this, being all one. Now the side of the base of this Pyramid (naturally 12.00) being measured by its proper Line, for superficial measure, will be found artificially, the same that the Diagony or Diameter of its base is naturally, viz. 16.97, etc. and the foresaid perpendicular will be found by the same Line, (in a centesimal partition) 14.70 ferè; which two multiplied together, do produce 249.4590 ferè, for the Pyramidal superfice aforesaid; exceeding the true superfice, only so much as 1/23 of a square-unit or integer, it being decimally, .0437, or 437 of 10000  
And thus may the superficies of an Octahedrum be obtained, it being only double the superficies of its compounding Pyramid: For so, the superficies of the foregoing Octahedrum, will be by this latter or artificial Measure, 498.9180 ferè. And therefore if the side of an Octahedrum, and its hedral perpendicular, be taken by this artificial Line, and one of them be doubled; the Rectangle Plane made thereof, will be the total superficies of the Octahedrum: And so the hedral perpendicular of the foresaid Octahedrum being taken by this Line, and doubled, will be 29.39. which multiplied by the side of the Octahedrum, found by the same Line, 16.97; the Plane produced therefrom, will be, 498.7483, for the superficies of the Octahedrum; which comes a little nearer the true superficies, than the former; that exceeding the same only, .0874 ferè, which in vulgar terms, is not fully 1/11; and this wanting thereof, but .0823, which in vulgar account, is hardly 1/12; the true superficies being 498.8306, (double to the superficies of the foregoing Tetrahedrum:) so that the side and hedral perpendicular of an Octahedron being taken by the prime Rational Line, and doubled; the Plane arising from their mutual multiplication, shall be the true superficies of the Octahedrum; being no other than that of the basial semiperimeter, and the trigonal perpendicular of its tetragonal compounding Pyramid, doubled; As before, the double product of 24 and 10.39, etc. (or 12, and 20.78, etc.) being now the single product of 24, and 20.78, etc. Or the side being quadrupled, and the said perpendicular taken single, shall together produce the same; being no other than that of the whole basial perimeter of the foresaid Pyramid, and the perpendicular of its triangular Plane together, for the double superficies of the Pyramid; which is the supersicies of the Octahedrum: As here the product of 48, and 10.39, etc. Or the superficies of this Solid, may be had again artificially by the Lines of quadrature pertaining to the Trigon, as I shown before for the superficies of a Tetrahedrum; for so, one of the Planes or bases being had; the Octuple thereof shall be the total Octahedral superficies.   geometrical diagram  geometrical diagram 
Or again, seeing that the rectangle Solid (Prisma or Parallelepipedum) contained under the Perpendicular from the Centre of any regular plain Body, to any of its bases or Planes, and a trient of the total superficies, comprehends the solidity of the whole body (according to what I shown formerly in the dimension of a Sphere, for the producing of its solid content by the semidiameter, and a trient of the Superficies, the plain Solid arising therefrom, being equal to the spherical Solid: or which is all one, the plain Solid made of the whole Diameter, and a sextant of the spherical superficies,) in as much as that which is contained under the said Perpendicular, (which is here no other than the Axis of the compounding Pyramid, and the semi-axis of the inscribed Sphere) and a trient of one of the bases or Planes (which is the base of the compounding Pyramid, as aforesaid) comprehends the solidity of one of the compounding Pyramids; and so consequently that which is contained under the said Perpendicular, and a trient of all the bases or Planes together (as being the bases of all the compounding Pyramids together) must needs comprehend the solidity of all the Pyramids together, and so of the whole ordinate Body; (as the rectangle Plane contained under the perpendicular from the Centre of any rectiline regular Plane, or superficial Figure, to any of its sides, and the semiperimeter of the same, comprehends the Area of the whole Figure (as I shown formerly in the Pentagon, and which answereth to that of the dimension of a Circle, for the producing of its Area by the semidiameter, and semiperiphery, the rectangle Plane or Parallelogram resulting therefrom (by their mutual implication) being equal to the Circular Plane: or which is all as one, the rectangle Plane made of the whole diameter & a quadrant of the Periphery, or of the whole Periphery & a quadrant of the diameter) in as much, as that which is contained under the said perpendicular, (which is no other than the perpendicular of altitude of the Figure's compounding Trigon, and the Radius of the inscribed Circle) and half the side, (as being, half the base of the said Trigon) comprehends the supercies, or area, of one of the compounding Trigons; and so, that which is contained under the said perpendicular and half the perimeter of the Figure (as being half of the bases of all the compounding Trigons together) must needs comprehend the superficial Content of all the said Trigons together, and consequently of the whole regular Figure itself.) Therefore the Base of the Dodecahedrum being 61.9371, etc. the total superficies thereof, will be 743.2462 etc. whose subtriple, 247.7487, etc. being augmented by the foresaid perpendicular (or axis of the compounding Pyramid) 6.68, etc. there will result the total Dodecahedral solidity, 1655.2339, etc. exactly as before: which may be plainly seen by the subsequent Logarithmical operations, whereby these Pyramidal and Dodecahedral dimensions, are most readily and accurately performed.  
Therefore first,  
Side of the Dodecahedron, or of its compounding Pyramids Base, 6.  
 
Again, 2ly.  
 
Or again, 3ly.  
 
Which our Measure found by the foresaid artificial Line of Pyramidal consolidation, falleth short of indeed, about 3 integers, or units, and an half: But than if the said Pyramidal Line of Measure be increased in its parts, by a subdecuple solution of the former; the side of the aforesaid Pyramids base (naturally 6.000) will be found thereby, 4.985, ferè, whose Quadrat is 24.850225 ferè, for the artificial base: and the Axis of the said Pyramid (naturally 6.681) will be found thereby, 5.551 ferè; which two by a conjunct composition, will produce 137.943598975 ferè, for the solidity of the Pyramid (which now exceeds the true solidity, being 137.936158730, hardly so much as 1/134 of a solid integer or unit) whose duodecuple, 1655.323187700, is for the solidity of the Dodecahedron; which now differeth from the true solidity, being correspondently 1655.233904760 (by way of excess) hardly 1/1● of a solid integer or unit.  
As for the superficies of a Do●ecahedron, the same may be readily obtained by any of the three Lines of quadrature pertaining to an ordinate Pentagon, but that for the side is the most fit and proper (though all of them will produce the same thing) for so the side of the Dodecahedron being taken thereby, the Square thereof sh●l be the basial or hedral area, whose duodecuple will be the total Dodecahedral superficies. But a fare better way for the dimension both solid and superficial of this, and the other plain regular Bodies, I shall show in the next Section, which will be wholly taken up about the said five Bodies.  
As for the superficial Dimension of a pentagonal Pyramid seeing it is but the 
Concerning the superficial dimension of a Pentagon all Pyramid; And withal, the artificial Line for performing the same, by the side of the base, and the perpendicular of the trigonal Plane together.  same with that, which I have fully showed in the two preceding Pyramids, both naturally and artificially, in the most ready manner that may be; to wit, artificially, by the side of the Base, and the perpendicular-line of the triangular or lateral Plane together: Therefore I shall not need to insist upon the same in this last Pyramid here particularly handled, by way of exemplary illustration; But shall only give the artificial Line of measure for the performance thereof, as I found it to be from the natural Line of measure, or prime Rational Line in general, deficiently, 0.63245, etc. (that is Apotomally in number, from our general reason of Measure, as 1— .36754, etc. the said artificial Line being √ 2/5 of the natural Line, in its power quadratick.  
And thus may the Dimension both solid and superficial of this and all other Pyramids, be performed artificially by the other dimensional lines of their Bases (specified before in the dimension of those basial Figures simply, and by which we said, they might be also artificially, or quadratically measured;) as, the Diameters or Perpendiculars, and Diagonals, together with their Axes or Altitudes, for solid measure; and with the perpendicular-lines, or altitudes of their triangular Planes, for superficial measure: And which, though it be needles, that by the side of the Base, (with the Axis and trigonal Perpendicular) being most ready, and also most proper, according to what I noted in the beginning of this Section, in the basial Figures: Yet having for variety of Art in this kind, not spared the pains of extracting or eradicating the artificial Lines serving thereunto; I thought it might not be amiss, to set them down here also; as they are from the natural Line of measure, or prime Rational Line in general, in a decumillenary solution; those for solid dimension, (as also the other beforegoing) being all of them, thereof redundantly; and those for superficial dimension, all of them deficiently.  
The artificial Line of Measure, is for the Pyramid, trigonal, in respect of the basial Diameter, conjunctly with the Axis, or Altitude of the Body, 1.7320 √ q 3 Solid measure. Tetragonal, 1.8172 √ c 6 Pentagonal, 1.6043. trigonal, Perpendic line, or altitude of the trigonal or lateral Plane. 0.7598 √ qq 1/3 Superfie measure Tetragonal, 0.8409 √ qq 1/2 Pentagonal, 0.7846 ferè.  
And for the Pentagonal Pyramid, in regard of the Basial Diagony with the Axis, or Altitude, 1.6589 ferè. Solid measure. trigonal perpendic. 0.8045 ferè. Superficial measure.  
All which Pyramidal Dimensions beforegoing, in a Linear, instrumental, or geometrical way, I shall next briefly express in an Arithmetical way, in the proportional terms following, from an ample solution of the unit, (according as I did before in the Cone and Cylinder,) by which, the artificial Measure may be readily produced from the natural; or the natural Measure be reduced to the artificial. Therefore,  
The natural Measure is to the artificially in the Pyramid Trigonal, or Tetrahedral in respect of the Axis, or Altitude of the Body, conjunctly with the Side of the Base, as 1. to .524587 Solid dimension. Diana. .577424 Tetragonal, or Pentahedral Side .693373 √ c 1/2 Diana. .550321 Pentagonal, or Hexahedral, Side .830824 Diana. .623323 Diagon. .602815 Trigonal, Altitude, or perpendicular of the trigonal or lateral Plane conjunctly with the Side 1.224745 √ q 1 1/2 Superficial dimension. Diana. 1.316074 √ bq3 Tetragonal, Side 1.414213 √ q2 Diana. 1.189207 √ qq2 Pentagonal, Side 1.581139 √ q 2 1/2 Diana. 1.274597 Diagon. 1.243014 
Place this between Page 136 and 137.   
And what hath been here delivered concerning the artificial Dimension of Pyramids upon regular Bases; the like is to be understood for the dimension (both solid and superficial) of Prisms upon the like bases, by artificial Lines of measure peculiarly appropriated and applied to them for that purpose, in regard either of any of the basiallines aforenamed, with the Axis or Altitude for solid measure, or with the same 
Concerning the dimension of Prisms.  for superficial measure, (seeing the Axis or Altitude and the side, is all one in right or upright Prisms, as in right Cylinders, which I shown formerly.) But these I shall here pass by, leaving them to the industry of the ingenious Practitioner, that shall please to exercise himself therein; and the rather for that the artificial Lines of measure pertaining to the Pyramids, will also serve (if need be) for the dimension of the corresponding Prisms, (as I shown formerly between the Cone and Cylinder seeing that a Prism is only triple its correspondent Pyramid, according to E, 12. p. 7. as a Cylinder is triple its correspondent Cone, according to E, 12. p. 10, as I noted formerly; And as the proportion of the Prism to the Pyramid is triple, for solidity; so it is double for superficiety; the similitude, and so the reason between a Pyramid and Prism, being the same (in both Dimensions) with that between a Cone and Cylinder: And so the Lines of artificial Dimension, pertaining to the trigonal or tetrahedral Pyramid, will serve for the trigonal or pentahedral Prism; And the Lines for the tetragonal or pentahedral Pyramid, will serve for the tetragonal, or hexahedral Prism: And the Lines for the Pentagonal, or hexahedral Pyramid, will serve for the pentagonal, or heptahedral Prism; and so forward: For as the Pyramid gins à Quaternario, So the Prism, à Quinario.  
And by the same artificial manner of measuring (as by the natural) may be obtained the solid content of any obliqne or inclined Pyramid or Prism upon a regular base, as readily as of a right or upright one, according 
Concerning the dimension of obliqne or inclined Pyramids & Prisms.  as I shown formerly for obliqne Cones and Cylinders; the true altitudes of these Bodies (not their Axes) being considered; seeing that every such obliqne kind of Body, is equal in solidity to the right or erect body, having the same (or being of equal) base and altitude; by the reason of E, 12. p. 5 and 6, and p. 11 and 14, and also E. 11, p. 30 and 31, as I noted formerly.   

SECT. III. Exhibiting a more special and peculiar artificial way of measuring both solidly and superficially, the four plain ordinate Bodies, or rectiline regular Solids before handled, than was before; & this, after the most exquisite manner that may be. Together with the like artificial dimension of the other like regular Body, several ways.  
HAving now showed our artificial Dimension several ways, in Pyramids, and their Compounds, or Pyramidates; and consequently of the five forenamed ordinate plain Bodies, or rectiline regular Solids, in a Pyramidal way: (the Hexahedron excepted.) We shall next come to show the dimension of the said Bodies in a fare better way; and indeed, in the most excellent artificial, and compendious manner that can possibly be found out: and that for superficial measure, with the same speed, ease and exactness, and so in the very same manner, as that of the Circle, Trigon, God 〈…〉, etc. And for solid measure, as that of a Cube, by the natural or vulgar way of measuring the same (which weshal also here show by the like artificial way with the rest) And this only by squaring and cubing any one of their dimensional lines (as in the Sphere) by artificial Lines of measure convenient for the purpose: Which may indeed, I almost despaired of, in regard of the great difficulty which I found to be in the solid dimension of these Bodies, by the usual or natural way, according to their Pyramidal compositions, (the Hexahedron excepted) especially the the two last and greatest of them, namely the Icosahedron and Dodecahedron. But yet considering the excellency of them in their compositions, constitutions, and structures above any other solid Figures; from whence they are called by Pappus, and other of the Greeks, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, i, e. ordinata benè ordinata; and commonly the Pythagorean, and Platonical Bodies, as being first invented (as is generally supposed) by Pythagoras, and afterwards set forth briefly by 
* Plato in Timaeo, de Anima Mundi, seu Natura.  Plato in the composition and fabric of the World, as in the Heaven, and the four Elements; and so are also called therefrom, the Cosmical or mundane Bodies; and so that in the knowledge and understanding of the natures, properties and affections thereof, lieth as great a difficulty, nicety, and curiosity of Geometry, as may be; in so much as that Proclus makes the singular and admiable end of the Mathematics, to be in the knowledge of these five Bodies, in respect of their constitution, adscription, and comparation, collation, or application among themselves: And so thereupon our Countryman Billingsley, in his learned Annotations upon Euclids Elements in English, after the 25th. defin. of the 11th. book, speaking of the dignity and excellency of the said 5 Bodies; saith, that they are as it were, the end, and perfection of all Geometry, and for whose sakes was written, whatsoever was written in Geometry: Therefore for the avoiding and removing of all difficulties in their Dimensions, and so the facilitating of the same, as to the obtaining of their solid & superficial Capacities (especially solid) as easily and exactly, as of any other Figure whatsoever, and that several ways, both naturally and artificially; I resolved (by God's assistance) to prosecute my metrical conceits and invention herein, as far as in any other Figure.  
And seeing that the sides of these Bodies may more readily and accurately be taken by a Line of measure, than any other their lines of dimension ascribed to them, (and indeed very accurately, without any trouble) as being their only natural lines of dimension, and so only apparent of of themselves in them; (according to what I formerly said for rectiline Planes or Superficies in general: Therefore I shall here briefly demonstrate their dimensions both solid and superficial (by way of example) artificially by their sides only; and withal, shall by the way, give artificial Lines for the like dimension of them, by some other of their dimensional lines ascribed to them. And first for solid dimension, the Lines for the Sides of these Bodies, (or for the cubick dimension of them by their Sides) I found, by the reason of the 2d. Theorem, etc. following, to be of the prime Rational Line in general, (in a decumillenary solution,) as followeth.  
The artificial Line for solid Measure, or Line of Cubature is for the side of the Tetrahedron, 2.0396 redundant Octahedron, 1.2849 Icosahedron, 0.7710 deficient Dodecahedr. 0.5072  
Which Lines being duly set of and divided, as the former, and so the Sides of these bodies measured thereby; the Cubes thereof, shall be the solid contents of the same, according to the dimensional reason of the prime Rational Line: which I shall briefly illustrate in three of these Bodies, by the foregoing Exammples laid down in their Pyramidal dimensions, viz. the Tetrahedron, Octahedron and Dodecahedron; to which, I shall here add the like for the Icosahedron.   geometrical diagram  geometrical diagram  geometrical diagram 
iv The side of an Icosahedron being put naturally, 12.00, the solidity thereof will be found by the natural Pyramidal Dimension, upon the very point of 3770: For the Axis of the compounding Pyramid, will be found (by the proportion of the side of an Icosahedrum to the Axis of its inscribed Sphere, or altitude of its own body, hereafter declared) to be 9.069, etc. which with a trient of the base (being the same with that of the foregoing Tetrahedrum and Octahedrum, viz. 62.3538, etc.) viz. 20.7846, etc. will produce the solidity of the compounding Pyramid, 188.498398. (which by the proper artificial Pyramidal Line, will be 188.325116,) whose vigecuple, 3769.967960, is the solidity of the Icosahedrum: All which you may see most accuratly produced by the several Logarithmetical operations or artificial Numerations following, according to those formerly in the Dodecahedron. And therefore first.   geometrical diagram 
 
Again, 2ly.  
 
Or again, 3ly.  
 
Or 4ly. (to come towards our present way of measuring here proposed) the same will be most readily produced in a cubical manner, without any trouble of Calculation, or arithmetical and geometrical operation; and that first by comparing the Cube of the Icosahedrum's side with the Icosahedrum itself, according to the most exquisite terms of proportion, noted in the next Section; and this only by one simple composition of Numbers, from the artificial Numeration, thus;  
 
Or 5ly. and lastly; the same from thence, by finding the content of the Cube equal to the Icosahedrum according to the most exquisite Proportion of the side of the Icosahedrum to the side of that Cube or Hexahedrum, noted immediately after the former Proportion; (to which doth answer exactly, our present artificial Mensuration, o● organical, or mechanical Cubation) As,  
 
Which several Logarithmical operations, do agreed well with that Algebraical or Cossical computation of Mr. Diggs in his forementioned discourse of geometrical Solids. Probl. 14. where having Cossically cast up the solid content of this very Icosahedrum, he saith at last, that the same being reduced into rational numbers, will fall between 3769 and 3770.  
And with these several artificial Numerations, will be found to agreed very nearly, our present artificial Mensuration: For the side of this Icosahedrum being measured by its proper Cubatorie Line, (taken in a centenary solution) will be found thereby, 15.56 (as before, the side of its equal Cube, by the artificial Numeration) which cubed, yields, 3767.287616, for the solidity of the Icosahedrum, which indeed wanteth of the true solidity, between two and three integers of the appointed measure; but one example is not to be regarded: But however therefore, proceeding here in a more ample or numerous solution of the Line● of measure both natural and artificial, equally; the solidity of this Icosahedrum will be thus artificially produced very near the true content. As if the prime or natural Line be made 1000 parts, and so also the second, artificial, or Cubatorie Line; than the side of this Icosahedrum measured thereby, will be found 15.564 ferè, whose Cube is 3770.193726, etc. for the solidity of the Icosahedrum, which exceedeth the true Content, not so much as 1/4 of the prime or natural Line, cubed; or 1/4 of a cube-integer, or unit.  
Now for the like superficiary Dimension 
Concerning the superficial dimension of the 4 regular Solids before-going, by way of exact Quadrature.  of these Bodies, the artificial Lines of quadrature for this purpose, I found (according to the reason of the first and 2d. Theorems etc.) to be of the prime Rational Line in general (under the former solution) as followeth.  
The artificial Line for superficial measure, or Line of Quadrature, is for the Side of the Tetrahedron, 0.7598 √ qq. 2/● Octahedron, 0.5373 ferè Icosahedron, 0.3398 Deficient Dodecahedr. 0.2201 ferè  
By which Lines duly set of and divided, the sides of these Bodies being measured; their Quadrats shal● be the total superficies; which I shall also briefly illustrate by help of the examples beforegoing in the Dimension of the Trigon and Pentagon, which I made use of formerly in the superficial Dimension of the Tetrahedron and Octahedron in a Pyramidal way.  
Therefore first; the Side of the foregoing Tetrahedrum being naturally 12.00, the true total Supersicies thereof was formerly found, 249.4153, (which by the Line of quadrature pertaining to the side of an ordinate Trigon, was found, 249.6400 ferè.) Now the Line of quadrature peculiarly appropriated to the side of a Tetrahedrum (not having respect to the base or Plane simply, as being an ordinate Trigon) being centesimally divided; the side of this Tetrahedrum measured thereby, will be found, 15.79, whose Quadrat is 249.3241, for the total superficies of the Tetrahedrum, which wanteth of the true superficies in vulgarterms, only 1/11 ferè, of a square integer or unit, of the measure appointed.  
II. The side of the foregoing Octahedrum being the same with that of the Tetrahedrum, viz. 12.00; the superficial Content thereof will be double to that of the Tetrahedrum, (as I shown before) viz. 498 8306. Now the side of this Octahedrum being measured by its proper Line of quadration (in a centesimal solution) will be 22.33, whose Square is 498.6289, for the Octahedro●'s total Superficies, wanting of the true content, only 1/5 of a square-integer. Or the said Line being made 1000, it will give the Octahedron's side, 22.335 ferè, which squared, gives the Octahedron's superficies, 498.852225 ferè, which exceeds the true Content, viz. 498.830633 ferè, hardly 1/46 (in vulgar terms) of a square-unit.  
III. The side of the former Icosahedrum being the same with the side of the Tetrahedrum and Octahedrum; the superficies thereof will be quintuple the superficies of the Tetrahedrum, and so double-sesquialter the superficies of the Octahedrum, (according to what I have formerly spoken in the superficial Composition of these Bodies) viz. 1247.0766 ferè: Now the side of this Icosahedrum, being taken by its proper tetragonismal Line, under a centesimal partition only, will be found 35.31, whose Quadrat is 1246.7961, for the superficies of the Icosahedrum; which wants of the true superficies, scarcely 1/3 of a square-integer. But yet the side of the Icosahedrum being measured by its said Line under a millesimal solution, will be found upon the point of 35.314, which quadrately, is 1247.078596 ferè, for the Icosahedrum's superficies, which now exceeds the true superficies, (being 1247.076581) in vulgar terms, hardly 1/4●6 of a square integer or unit.  
iv The Side of the foregoing Dodecahedrum, being, 6, the true total superficies thereof, was formerly found 743.2462, etc. Now the Line peculiarly appertaining to the side of a Dodecahedrum, for the quadrate dimension of its Superficies (not as relating to the Base or Plane thereof simply, as being an ordinate Pentagon) being first laid down under a centesimal partition, and the side of this Dodecahedrum measured thereby, will be found 27.26, whose Quadrat is 743.1076, for the total Superficies of the Dodecahedrum, which wanteth of the true superficies, only about 1/7 of a square-integer or unit.   geometrical diagram 
II. Than for the superficial dimension of this Body by its said Axis or Diagoniall; I found the artificial Line for that purpose, to be the very same with that which was formerly found for the superficial dimension of a tetragonal Pyramid, by the side of its 
The superficial dimension of the Hexahedron artificially by its Axis or Diagony, according to quadration.  base, and the perpendicular of its trigonal plane together, viz. 0.7071, √ 1/2: By which the said Axis or Diagony being taken; the Square thereof shall be the total Superficies of the Hexahedrum: And therefore if the Diameter of a Sphere be taken by this Line, the Quadrat thereof shall be the inscribed Hexahedron's superficies.  
III. As for the solid dimension of this Body artificially, by the Diagony of the base; the Line of measure serving thereunto, I found to be the very same with that which was formerly noted for the dimension of a 
The solid dimension of the Hexahedron by the Diagony or Diameter of its Base, according to Cubature, artificially.  Tetragon by its Diagony, and also for the superficial dimension of a Cone by its whose basial periphery and side together, viz. 1.4142, √ 2: By which the basial Diagony being taken, it will be found to agreed with the side taken by the prime Rational Line; and so being cubed, must needs produce the same Hexahedral solidity.  
iv Than for the superficial dimension of this Solid, artificially, by its said basial Diagony or 
The superficial dimension of the Hexhaedron by its basial or hedral Diagony, according to quadrature artificially.  Diameter; the Line of quadrate dimension for this purpose, I found (according to the reason of the 1 & 2d. Theorems, etc.) to be of the Rational Line in general, (defectively) 0.57735, etc. which is Apot●mally in Number, 1— .42265, etc. the said artificial Line being √ 1/3 of the natural Line, taken in its quadratique power or capacity. By which Line (first duly divided) the basiall or hedral Diagony being taken, its Quadrat shall be the total superficies of the Hexahedrum.  
V And so may the superficies of this Body, be wholly obtained in the like manner, by its Side (which may be 
The superficial dimension of the Hexahedron artificially by its Side, according to one exact quadrature.  termed the most precise and proper squaring of a Cube, as to its superficiety, or superficial part; as also the two former ways by the Diagonial of the Base & of the total Hexahedron itself) and the Line of quadrature convenient for this purpose, I found (by the reason of the foresaid Theorems) to be of the prime rational Line in general (deficiently) 0 4082, etc. the Apotomal segment, or parts of diminution, being .5917, etc. the Line itself being √ 1/● of the foresaid Rational Line, in its power or capacity tetragonical. By which (first duly divided) the Side of the Hexahedrum being measured; its Quadrat shall be the total Hexahedral superficies; Which way, as also that by the Diagony of the base, as they are more artificial and excellent in themselves, so also more ready, for getting the superficies of an Hexahedron, than the vulgar or natural way, by squaring the side taken by the natural Line of measure, and than sextuplating that square, as being the Base. And here it appears, that if the Diameaer of a Sphere be taken by this last artificial Line, the Quadrat thereof shall be the total convexe superficies of the circumscribed Hexahedron.  
And this ordinate Body here last handled, may also be conceived or considered together with the other 4 before-going, under a Pyramidal composition & resolution (as all plain Solids generally are; even Pyramids themselves; as I showed before in the first and most simple kind of all, to wit, the trigonal, and that in the Tetrahedrum itself) according to its Bases or Planes, which are to be understood, as the bases of so many Pyramids equal and alike, by which they are only externally eminent or apparent, their whole bodies beside, being internally latent, and so do meet in their vertical angles, in the centre of the hexahedral Body; (or of its ambient Sphere,) This Body being composed superficially or externally of, or contained internally under, six equal Tetragons, (according to E, 11. d. 25) compact together by solid (right) angel's, which are in number, subtriple the plain, superficial, or hedral (right) angel's constituting or including the same, as in the Tetrahedrum and Dodecahedrum. And hereupon this regular Solid will also admit of a Pyramidal dimension, for the obtaining of its solid Area, like as the other; which will be most easily performed, seeing that the Axis or Altitude of its compounding tetragonal Pyramid, is equal to half the side (or altitude) of its body; and which conjunctly with a trient of the Base (or the whole Base with a trient of that) will produce the Content of one of the composing Pyramids; whose sextuple will be the Content of all the Pyramids together, and so of the whole Hexahedrum: Or (more briefly) the said Pyramidal Axis conjunctly with the whole Base, will produce the Content of three of the said Pyramids, for half the Hexahedrum: And therefore hence it is, that the side of this Body, (which is its altitude, and so double to the altitude of its composing Pyramid) being enfolded with the Base (that is, cubed) produceth immediately the Content of all the 6 supposed Pyramids together, for the ●o●al Hexahedrum.  
And as the Hexahedrum; so also may the other four ordinate plain Bodies be artificially measured, both solidly and superficially, (or cubically and quadrately) by their other several lines of dimension, beside their Sides, (as I noted before) as either by their Axes, Diameters, or Diagonials, which in the Octahedron, Icosahedron, and Dodecahedron, are no other than the Axis or Diameter of their circumscribing, comprehending or containing Sphere, as in the Hexahedron, (as I noted occasionally in them before) or by their Altitudes, (which in the Dodecahedron and Icosahedron, are the same with the Diameter of their inscribed Sphere; as also in the Hexahedron, it being there the same with the side, and in the Octahedron, is the same with its Diagoniall, or Axis, being also the circumscribing Sphear's Axis, as I shown formerly.) Or by their Basiall or hedrall Diameters or perpendiculars; and in the Dodecahedron, by its basiall or hedral Diagony also, according as I shown in the Dimension of a pentagonal Pyramid, and also of a Pentagon itself simply. The artificial Lines, (or Lines of quadrature and Cubature) for some of which Dimensions, that are most material, (though indeed none of them are absolutely needful, the Dimensions of these Bodies being most easily and accurately performed by their Sides, as I have suffciently showed before) I shall here deliver, as I found them to be in relation to the prime Rational Line in general, under a decumillenary solution, (according to the reason of the 2d. Theorem, etc.) as followeth; beginning with solid dimension, as I have done in all the former Solids, as being the most considerable in these and all other solid Figures: the first of the two numbers belonging to each Body, noted with the letter c. expressing the quantity of the Line for cubical or solid dimension, and the second number, noted with q, the quantity of the Line for quadrate, or superficiary dimension.  
The artificial Lines of Measure, or Lines of Cubature, and Quadrature, are for the Axis of the Tetrahedron 1.6654 ferè. c. 0.6204. q. Octahedron 1.8172 ferè. c. √ c 6. 0.7598. q. √ qq 1/3. Icosahedron 1.4666 ferè. c. 0.6464 ferè q. Dodecahedron 1.4215. c. 0.6168, ferè, q. Altitude of the Icosahedron 1.1654. c. 0.5136. q. Dodecahedron 1.1296. c. 0.4901. q.  
By the first Section of which Lines (being duly set of and divided) the Axes of these Bodies being taken, the Cubes and Squares thereof, shall be their several solidities and superficialities, according to the nature of the Line by which they are measured: whereby it appears, that if the Axis of a Sphere be taken by these Lines (except those which are for the Tetrahedrum) its Cube and Quadrat shall be the solidity and superficiality of the inscribed Octahedrum, Icosahedrum, and Dodecahedrum, according to the respective Lines of measure, by which it is taken. And so the like may be done for the solidity and superficiality of the inscribed Tetrahedron, by the Diameter of its ambient Sphere: the Lines of measure for which purpose, I found to be of the Rational Line in general, 2.4980, for cubique dimension, and 0.9306, etc. for quadrate dimension.  
By the second Section of these Lines, the Altitude of an Icosahedrum and Dodecahedrum being taken; their Cubes, and Quadrats, shall be the solid and superficial Contents of their proper Bodies, according to the nature of the Line by which they are measured: And therefore, if the Diameter of a Sphere be taken by these Lines, the Cube and Quadrat thereof, shall be the solidity and superficies of the circumscribed Icosahedrum and Dodecahedrum, according to the respective Line, by which it is measured.  
And the like may be done in the Tetrahedrum and Octahedrum circumscribed; and in all the five Bodies both inscribed and circumscribed, may the same be done also, by the Circumference of the greatest or central Circle of the Sphere (circumscribing and inscribed) for finding their sold and superficial Contents: But the Lines of measure for these last-named Dimensions, as also for the cubique and quadrate dimension, (or the cubing and squaring, as I may so term them) of the four Bodies here last handled, by their hedral Diameters or Perpendiculars, and hedrall Diagony also in the Dodecahedrum, (as I have already showed in the Hexahedrum) I shall here omit, as needless and superfluous; and shall show the chief of these Dimensions, as also all the other beforegoing in these Bodies, which I have practically demonstrated by way of example, in an arithmetical manner, from their several artificial Lines of measure, (or Lines of Cubature and Quadrature) expressed only by Number; together first with the Linear dimensions of these Bodies, (in regard of their foresaid several lines of Dimension, except those here last of all named) leading thereunto; in a way of Proportion, after the most exquisite manner that may be, as from the natural Measure, according as I have done in all the precedent dimensions. But before I go to these, I think it very fit and expedient (for a conclusion of this Section) to give the practical Reader to observe by the way, for the more ease and conveniency in this kind of Mensuration, or metrical practice, (or more artificial kind of practical Geometry) which here we handle; & so to avoid multiplicity of artificial metrical Lines or Scales, arising by the manifold particular dimensions here considered and declared; what Lines of measure here already delivered and expressed by Number, do agreed, either in the whole throughout, or in part only, in their measure or magnitude from the prime Rational Line, so fare as is generally needful for ordinary use, which is to centesimal parts only of the Rational Line from which they are taken, and which division in that, for setting of the artificial Lines therefrom, and so also in the artificial Lines themselves for measuring thereby, is generally sufficient for ordinary use, as I have both said, and exemplarily showed in most of the precedent Dimensions; it having considerably failed but in three of all the foregoing practical demonstrative Examples, which was first, in the solid Content of the Sphere last handled, produced by the Line of Cubature pertaining to the Diameter: and secondly, in the solid content of the Dodecahedrum in the Pyramidal dimension, by the Line pertaining to a regular-based pentagonal Pyramid in general; where yet it did agreed sufficiently with the true, natural dimension, in the solidity of the compounding Pyramid itself; and therefore not so considerable in the Dodecahedrum: and than in the solid content of the Icosahedrum produced by the Line of Cubature belonging to its side: But two or three particular Examples are not to be regarded, in respect of the general; seeing that the like artificial dimensions of the same kind of Figures, may in other Examples hi● right enough; the reason of these defections having been showed sufficiently at first, especially in the last Section of the first Part. And therefore observe,  
I. That the artificial Line, for the quadrate dimension or squaring of a Circle, by its Diameter, (expressed first of all by Number, from a decumillesimal solution of the prime Rational Line, viz. 1.1284 ferè) and the Line for the cub●que dimension, (or cubing) of a Dodecahedron by its perpendicular-line of altitude (or inscribed Sphear's dimetient) under the same solution of the Rational Line, viz, 1.1296) will agreed in part, viz. in a centesimal solution of the Rational Line, being thereby 1.13 ferè, which I shown in the dimension of Circle, to be the Line C D, according to the primary Line A B.  
II. That the Line for the quadrate dimension of an ordinate Pentagon by its Diameter or Perpendicular, (expressed numerally, 1.1732 ferè) and the Line for the cubick dimension of an Icosahedron by its perpendicular of altitude (or inscribed Spheare's Axis) noted Numerally, 1.1654) will agreed sufficiently in part, viz. in centesimal parts of the Rational Line, being thereby, 1.17, the first complete, the second incomplete; which difference is not considerable.  
III. The Line for the dimension of a Tetragon by its Diagony, (or circumscribing Circle's Diameter) and for the artificial dimension of Triangles in general; and the Line for the solid dimension of an Hexahedron by its basial or hedral Diagony; and also the Line for the superficiary Dimension of a Cone by its Side and total basial periphery together, do agreed in the whole throughout, viz. 1.4142, etc. infinitely, being √ 2 of the entire prime Rational Line (as the unit or integer of Measure) in its power quadraticall: with which very nearly agrees the Line for the cubing of a Dodecahedrum by its Axis or Diagoniall, (or circumscribing Sphear's Dimetient) in centesimal parts of the Rational Line, viz. 1.42 completely.  
iv The Line for the Quadrate superficial dimension of a Sphere, (or the quadration of the Spherical) by its Diameter or Axis; and the Line for the square-like superficiary dimension of a Cylinder (or the rectangular parallelogrammation of its Superficies) by its Side and Diameter together, do meet in the whole throughout, viz. 0.56418 &c, infinitely.  
V The Line for the quadrate superficial dimension of a Tetrahedron (or quadration of its superficies) by its Side; and for the like dimension of an Octahedron by its Axis or Diagoniall (or comprehending Spheare's Diameter) and for the rectangular superficial dimension of a trigonal Pyramid, by the perpendicular of its base, and of its other or lateral triangular Plane together, do hap to be all one infinitely, in parts of the Rational Line, viz. 0.7598, etc. being √ 1/3 of the entire prime Rational Line, in its biquadratique potentiality, or capacity: with which agreeth in part, viz. to centesimal parts, the Line of quadrature for the Side of a Pentagon, (noted formerly by Number, 0.7624 ferè) viz. 0.76 complete, which in the Dimension of a Pentagon, was demonstrated by the Line P Q. in parts of the Line A B. So that here one and the same Line will serve thus fare, for these 4 several dimensions. And very nearly with this Line, agrees the Line for the cubing of an Icosahedrum by its side (noted formerly in parts of the Rational Line, 0.7710, etc.) taken centesimally, viz. 0.77.  
VI The Line for the square-like solidation (as I may term it) or rectangle-parallelepipedall dimension of a trigonal Pyramid by its Axis or altitude, and basial Diameter or perpendicular together; and for the solid dimension of an Hexahedron by its Axis or Diagoniall, (or ambient Sphear's dimetient) do agreed in the whole throughout, viz. 1.73205, &c, infinitely, being √ 3, of the integral Rational Line, taken in its power zenzicall or tetragonicall.  
VII. The Line for the rectangle-parallelipipedal Solidation of a tetragonall Pyramid by its Axis or Altitude, and basiall diagony or diameter tagether; and fo● the cubique dimension (or Cubation) of an Octahedron by its Axis or Diagoniall, (or containing Spheare's axis or diameter) are both one throughout, viz. 1.817, etc. infinitely being √ 6 of the prime Rational Line considered in its cubical capacity, or comprehensibility.  
VIII. The Line for the square-like, or rectangle-superficiary dimension of a tetragonall Pyramid by the side of its Base, and the perpendicular of its trigonal Plane together; and for the exact quadrate-superficiary dimension of an Hexahedron, (or the quadration of its superficies) by its Axis or Diagoniall (or circumscribing Spheare's Diameter) do agreed in the whole, viz. 0.7071, etc. being √ 1/2 infinitely, of the prime Rational Line, in its power or capacity tetragonical.  
IX. The Line for the rectangle-parallelepipedation, or parallelepipedal consolidation of a pentagonal Pyramid (as to an exact quadrate base) by the diagonal line, or angular subtense of its Base, together with the Pyramids Axis or altitude (noted formerly by number, 1.6589 ferè) and the Line for the exact Cubation of a Tetrahedrum by its Axis (noted in like manner, 1.6654 ferè) will sufficiently agreed in part; as to centesmes of the Rational Line, viz. 1.66; this latter complete, the former incomplete: but with so small a difference, as that one and the same Line may thus fare serve indifferently for both these dimensions viz. 1.66.  
X. The Line for the rectangle-superficiary dimension of a pentagonal Pyramid by the foresaid diagoniall of its Base, and the perpendicular of its trigonal Plane together, (viz. 0.8045 ferè) and the Line for the like dimension of a Cone, by the diameter of its base, and its side together, (0.7979 ferè) do agreed in part, as to centesmes of the Rational Line, viz. 0.80. the first complete, the second incomplete.  
XI. The Line for the exact quadrate supersicial dimension of a Tetrahedrum, by its Axis, (0.6204) and the Line for the like dimension of a Dodecahedrum by its Axis or Diagoniall (or ambient Sphear's dimetient) viz. 0.6168 ferè, wilsussiciently accord, as to centesimal parts, viz. 0.62; the first complete, the second incomplete.  
XII. Lastly, the Line for cubing of a Dodecahedron by its side, (0.5072) and the Line for the squaring (as I may so term it) of an Icosahedron, as to its superficial part, by its perpendicular of altitude (or inscribed Spheres dimetient) viz. 0.5136) will sufficiently concur in part, as to centesimal parts of the Rational Line, viz. 0.51; the latter complete, the former incomplete.  
So that here you may see, how that of all the several geometrical dimensions before particularly expressed, being in number 62, as requiring so many several lines of Measure, or Lines of quadrature, cubature, etc. the Lines for 28 of them, are contracted into 13, in respect of a centesimal solution of the Rational Line from which they are taken, but no further, (according to these 12 Notes or observations) And the Lines for 15 of the same dimensions, are contracted into 6, in regard of an infinite resolution of the Rational Line, according to the 3. 4. 5. 6. 7. 8 Notes. And so with the other 34 dimensions not here named, having their particular Lines of measure differing in the whole, (in respect of the fraction-part of the Rational Line, though not of the integral part) the artificial Lines for all the 62 dimensions aforesaid, will be contrived into 46, according to a centesimal partition of the prime or natural Rational Line. And the like agreement of Lines as is here demonstrated, may fall out between these and other the like Lines, for other Dimensions not here particularly expressed; and also between other Lines, which are none of them heresetdown: which I refer to the ingenious practiser to consider, according as he may have occasion offered.   

SECT. iv Expressing the manifold Dimensions in the five plain ordinate, or regular Bodies, Arithmetically, by way of Proportion, in the most exquisite manner that may be.  
ANd so having in the Section immediately beforeing, shown the Dimension both solid and superficial of the 5 plain ordinate (Pythagorean or Platonic) Bodies, according to our artificial way of measuring, in the most exquisite manner that may be: (or instrumental Cubature and Quadrature,) We shall in this Section, lay down the same Dimensions, with variety of others, in these Bodies, by Number, in the most exquisite Terms of Proportion that may be; such as have not yet been done (not more than the former artificial way of measuring the same) by any that I could ever meet with, or hear of: and which must needs very much exceed those, tedious, obscure, confused, Cossical Terms which Mr. Diggs in his forementioned discourse of these Bodies, hath Theorematically delivered; where yet, he hath left out the most material and useful ones, for the ready and speedy discovering of their solid and superficial Contents, as being inscribed in, and circumscribed to, a Sphere, in relation both to the Axis or Diameter, and the greatest Periphery of the Sphere, circumscribing and inscribed.  
And here I shall first begin with the Linear dimensions of these Bodies, in all the variety thereof, according to the forenamed several Lines of dimension belonging to them, as usually ascribed to them for their Dimensions, (as I did in the other ordinate Figures before-going, namely, the Circle, Sphere, Trigon, and Pentagon) and this, in relation to the first Dimension in Geometry, called in general, from the Greeks, Euthymetrie, or Mecometrie, and fromthe Latins Longimetrie; And which (in respect of the different kinds of Lines) I may call more generally from the Greeks, Grammemetrie; and from the Latins, Linemetrie.  
Therefore,  
1 The Tetrahedron's Side, is to its Axis, as 1. to .816497 ferè. 
√ 1/3 alike.  ambient Spheres Axis, as 1. to 1.224745 ferè. 
alike.   
2 The Tetrahedron's Axis is to its Side, as 1 to 1.224745 ferè. 
alike.  ambient Spheres Axis, as 1. to 1.5, subsesquialtera.  
3 The Axis of a Sphere, is to the inscribed Tetrahedron's Side, as 1 to .816497 ferè. 
√ 1/3 alike.  Axis, as 1 to .666667 ferè. viz. .666666 infinitely, ses-quialtera.  
4. The Side of the Octahedron, is to its Axis or Diagonal, or circumscribing Spheare's Dimetient, as 1, to 1.414214 ferè √ 2. Hexahedron, 1.73205. √ 3. Icosahedron, 1.902113. Dodecahedron. 2.802517.  
Contrarily,  
5. The Axis, Diagoniall, or angular Diameter (or the ambient Sphear's Dimetient) of the Octahedron, is to the Side, as 1 to .707107 ferè. √ 1/2. Hexahedron, .57735. √ 1/3. Icosahedron, .525731. Dodecahedron. .356822.  
6 The Diameter of a Sphere is to the Side of the circumscribed. Tetrahedron, as 1. to 2.44949 ferè. Octahedron, 1.224745 supdupla. Hexahedron, 1. aequalis, seu una. Icosahedron, 0.66158. Dodecahedron. 0.449028 ferè.  
Contrariwise.  
7. The Side of the Tetrahedron, is to the inscribed Spheare's Dimetient, as 1. to .408248. Octahedron, .816497. dupla. Hexahedron, 1. aequalis, as before. Icosahedron, 1.511522. Dodecahedron. 2.227033. ferè.  
And so the two last of these Proportions, are consequently of the Sides of those two Bodies to their Altitudes.  
And by the 6th. Section of Proportions, as also by the 5th. Sect. of the like proportions, in relation to the correspondent Circumference of a Sphere, you may observe, how that a Tetrahedrum and an Octahedrum being circumscribed to one Sphere, the side of the Tetrahedrum will be exactly double to the side of the Octahedrum: And so by the 7th Sect. beforegoing, you may observe contrarily, how that these two Bodies having one and the same side, the Diameter of the Sphere inscribed in the Octahedrum, will be exactly double to the Diameter of the Sphere inscribed in the Tetrahedrum. And the like with these, you may also observe afterwards in the 5 and 6 Sections pertaining to the correspondent Circumference of a Sphere. And the same proportion will the Tetrahedrum here hold to the Octahedrum, both for solid and superficial dimension, as it doth for lateral dimension, as I shall show afterwards.  
8. Tetrahedron's Axis is to it's inscribed Sphear's Axis, as 1. to .5 dupla. And therefore contrarily.  
9 The Axis of a Sphere is to the Axis of its circumscribing Tetrahedron, as 1. to 2. subdupla.  
10. The Axis of Tetrahedron's ambient or external Sphere, is to the Axis of its inscribed or internal Sphere, as 1, to .3333, etc. infinitely, viz. 3. to 1, tripla. And therefore conversly,  
11. The Axis of Tetrahedron's inscribed Sphere, is to the Axis of its circumscribing Sphere, as 1. to 3, subtripla.  
12. The Axis, Diagoniall, or angular Diameter of the Hexahedron, Or their circumscribing Sphear's Dimetient, is to their inscribed Sphear's Dimetient, as 1. to .57735. √ ●/3. Octahedron,  Dodecahedron,  Icosahedron. .794654.  
And so consequently, the latter of these two Proportions, is to be understood of the Axes of those two Bodies, to their Altitudes.  
Conversly.  
13. The Inscribed Spheare's Dimetient, is to the circumscribing Spheare's Dimetient, (or the Axis, or Diagoniall) of the Hexahedron, as 1. to 1.73205, √ 3. Octahedron,  Dodecahedron,  Icosahedron. 1.258401 ferè  
And so the latter of these two Proportions, is of the Altitudes of those two Bodies to their Axes.  
By which two Sections of proportions in these 4 Bodies, and by the two last proportions in the Tetrahedrum next before-going, viz. Sect. 10 and 11. it appeareth, that these five regular Bodies, in respect of their Spherical inscriptibility and circumscriptibility, do require three several distinct Spheres, circumscribing or containing, and inscribed or contained: That is, they being all severally inscribed within one Sphere, cannot than also be exactly circumscribed about (or cannot completely comprehend with in them) one Sphere, but three several Spheres; whereof that which is inscribed in the Tetrahedrum will be the lest, and that inscribed in the Hexahedrum and Octahedrum, will be equal, and that which is inscribed in the Dodecahedrum and Icosahedrum will be also as one, and the biggest of all.  
And so again contrarily, these five Bodies being severally circumscribed about (or comprehending in them) one Sphere, cannot than again be exactly comprehended or contained of one Sphere, but must have three several comprehending, containing or including Spheres; of which, that for the Tetrahedrum, will be the greatest; that for the Hexahedrum and Octahedrum will be alike; and that which is for the Dodecahedrum and Icosahedrum will also be of equal magnitude, and indeed the lest of all: (See M. Diggs his Discourse upon these Bodies, Probl. 17.) And the like to these may be observed in these 5 bodies, as being inscribed in, or circumscribed about, one Sphere, in respect of the Circles circumscribing their Bases, the Diameters of these Circles and of the body's circumscribing and inscribed Sphere, being compared together: And therefore.  
14. The Diameter of the Sphere circircumscribing the Tetrahedron, is to the Diameter of the Circle circumscribing their Base, as 1. to, .942809. Hexahedron, — Octahedron, .816497 ferè. Dodecahedron, — Icosahedron. .607062.  
And so the second and third of these Proportions, are also of the Axes or Diagonies of these 4 Bodies, to the diameters of their Base's ambient Circles: which in the Hexahedron, is of the corporal Diagony, to the basial or superficial Diagony.  
15. The Diameter of the Sphere inscribed in the Tetrahedron, is to the Diameter of the Circle circumscribing their base, as 1. to 2.828427. Hexahedron, — Octahedron, 1.414214. ferè. Dodecahedron, — Icosahedron. 0.763932.  
And to the second of these Proportions, is consequently of the Hexahedron's Side to its Basiall Diagony.  
The Converse of these are,  
16. The Diameter of the Circle circumscribing the Base of the Tetrahedron, is to the Diameter of their circumscribing Sphere, as 1. to 1.06066. Hexahedron, — Octahedron, 1.224745. Dodecahedron, — Icosahedron. 1.647278.  
And so the second and third of these Proportions, are also of the diameters of the basiall ambient Circles of these four bodies, to their Axes or Diagonies: which in the Hexahedron, is of the basiall or hedrall diagony, to the total corporal diagony.  
17. The Diameter of the basial or hedral ambient Circle of the Tetrahedron, is to their inscribed Spheres Diameter, as 1. to .353553. Hexhaedron, — Octahedron, .707107 ferè Dodecahedron, — Icosahedron. 1.309016.  
And so the second of these Proportions, is of the Hexahedron's basial Diagony or Diameter, to its Side.  
And the last of these Proportions, is of the basiall ambient Circle's Diameter, to the Altitudes of those two Bodies.  
So that you may here see by these 4 Sections of proportions, how that these 5 bodies being described either within or about one Sphere, have only three several circumscribing or containing Circles, for their Bases; whereof, that which is for the Tetrahedrum is the largest; that for the Hexahedrum and Octahedrum are both one, and the next to it; and that which is for the Dodecahedrum and Icosahedrum are also equal, and the lest of all; which Euclid E. 14. p. 5, and 21, speaks of, only in respect of these body's inscriptibility in one Sphere. As for the basial or hedral inscribed Circles of these 5 Bodies, whether inscribed in, or circumscribed about one Sphere; there is no such parity or agreement amongst them, but they are all different one from another.  
And as for the proportions of the sides of these Bodies to the Diameters of their hedral Circle's whether circumscribing or inscribed; and to their hedral Diameters or perpendiculars, etc. the same are to be had in the three ordinate Planes before handled, viz. the Trigon, Tetragon & Pentagon: those for the Tetrahedron, Octahedron and Icosahedron, out of the Trigon; those for the Hexahedron, out of the Tetragon; and those for the Dodecahedron, out of the Pentagon; But indeed, that of the Hexahedron's side to its hedral ambient Circle's Dimetient (or its hedral Diameter) & é contra; is also noted in the 15. and 17. Sections.  
And thus much for the linear proportions, in respect of the several lines of Dimension in these Bodies, being considered both simply or absolutely in themselves, and also as being inscribed and circumscribed, and that in relation to the Diameter of the Sphere circumscribing and inscribed. And from these we shall proceed to the like proportions, in respect of the Circumference answering to the said Sphear's Diameter, which I have not yet found touched upon by any man, in any kind whatsoever. And therefore.  
1. The Circumference of a Spheare's greatest, or central Circle, is to the Side of the inscribed Tetrahedron, as 1. to .259899 ferè. Octahedron, .225079. Hexahedron, .183776. Icosahedron, .167345. Dodecahedron. .113580. exactly.  
Viceversâ.  
2. The Side of the Tetrahedron, is to the circumscribing Sphear's greatest circumference, as 1. to 3.847649. Octahedron, 4.442883. Hexahedron, 5.441398. Icosahedron, 5.975664. Dodecahedron 8.804369.  
3. The Periphery of a Sphear's greatest Circle, is to in inscribed Tetrahedrum's Axis, as 1. to .212206. And so contrarily.  
4. The Axis of a Tetrahedrum, is to its ambient Sphear's greatest or true Periphery, as 1. to 4.712389.  
5. The Periphery of a Spheres greatest Circle, is to the side of the ambient. Tetrahedron, as 1. to .779697 ferè. subdupla. Octahedron, .389848. subdupla. Hexahedron, the same, as the Circumference to the Diameter, the Side of this body being equal with the Spheres Diameter. Icosahedron, .210589. ferè. I. Dodecahedron. .142930. D.  
Contrariwise,  
6. The side of the Tetrahedron, is to the inscribed Spheres greatest Periphery, as 1. to 1.28255. dupla. Octahedron, 2.56510. dupla. Hexahedron, the same as the diam, to the Circum●. for the reason aforesaid. H Icosahedron, 4.748589. Dodecahedron. 6.99643.  
7. The Periphery of a Sphear's largest Circle, is to the Axis of its ambient Tetrahedrum, as 1. to .63662:  
And again conversly,  
8. The Axis of a Tetrahedrum is to it's inscribed Sphear's greatest Periphery, as 1. to 1.570796.  
As for the proportions of the circumscribing Spheare's greatest or Diametral Circumference of these Bodies, to their inscribed Spheare's like Circumference, & contrà: And of the greatest Circumference of the Sphere both circumscribing and inscribed, to the Circumference of the basial or hedral circumscribing Circles of these Bodies, & contrà: they will be the same with those which are already expressed between the Diameters, viz. first in respect of the Sphere circumscribing and inscribed, between themselves; and than of both these Spheres severally with the Circles circumscribing the bases of these Bodies, as being described either within or without one Sphere (or several Spheres of one magnitude.)  
Having thus expressed the Linear proportions in these Bodies, or the proportions of linear dimension, as many as (and indeed many more than) are here absolutely needful for the measuring of them: I shall now come to show the proportions both superficial or quadrate, and solid or cubique deduced from thence, and both these conjunctly (for brevity sake) in reference to the other two Dimensions in Geometry, called from the Grecians, Embadimetrie (& Plat●metrie) & stereometry; and from the Latins, Plan●metrie & Solido-metrie: beginning with the latter of these as being most worthy and most considerable in solid Figures, as I have said before, the superficial dimension in them being not so useful or material, and also much more easily obtained; especially in these five Solids; whether the same be done naturally or artificially; And therefore first in respect of these Bodies considered simply and absolutely by themselves, without any inscriptibility and circumscriptibility, either spherical or mutual; the cubical proportions noted by the letter c, and the quadrate proportions by the letter q. will be as followeth; the bodies being placed in order according to their magnitudes increasing, both solidly and superficially.  
1. The Lateral Cube and Quadrat of the Tetrahedron, is to the Body's solidity and superficies, as 1. to .11785113. c. T. 1.7320508. q. T. Octahedron, .47140452. c. O. 3.464102 ferè q. O. Hexahedron. 1.000000. c. H. aequalis. sub-sextupla. 6.000000. q. H. aequalis. sub-sextupla. Icosahedron, 2.181695. c. I 8.660254. q. I Dodocahedron. 7.663119. c. D. 20.645729. q. D.  
Hence,  
2. The Side of the Tetrahedron, is to the Side of the Cube and Quadrat equal to the Body's solidity and and superficies, as 1. to .49028. c. T. 1.316074. q. T. Octahedron, .778346 ferè. c. O. 1.86121. q. O. Hex●●●dron, 2.44949. q. H. Icosahedron, 1.29697. c. I 2.94283. q. I Dodecahedron, 1.971523. c D. 4.543757. q. D.  
And hereby it appears, that these 5 Bodies having all the same side: the Octahedrum is quadruple to the Tetrahedrum in solidity, and double in superficiality: (& which I have showed before in their dimensions) The Hexahedrum is bigger than both those together, both in solidity and superficiality: the Icosahedrum is greater than those 3 together, in solidity, but lesser in the superficies: and is herein quintuple to the Tetrahedrum, and so double-sesquialter to the Octahedrum (as I have also shown before) And the Dodecahedrum is larger than all the other together, both in solid and superficial dimension.  
3. The Cube of Tetrahedron's Axis is to its Solidity as 1.10 21650635. Quadrat Superficies 2.59807621  
Hence,  
4. The Axis of the Tetrahedron is to the side of the Cube equal to its Solidity as 1. to 600468. Quadrat Superficies ●611855 ferè  
Than for the Tetrahedrum in reference to a Sphere, by way of inscriptibility therein, as to its Dimetient, it followeth.  
5. The Diametral Cube of a Sphere is to the inscribed Tetrahedron's Solidity as 1. to .0641500299. Quadrat Superficies 1.1547005  
Hence.  
6. The diameter of a sphere is to the side of the Cube equal to the inscribed Tetrahedron's Solidity as 1. to .4003123. Quadrat Superficies 1.0745699.  
Next for the other 4 Bodies, both simply in themselves, and also in relation to their ambient Sphere together, in respect of their common Axis or Dimetient, it followeth.  
7. The Cube and Quadrat of the Axis, or Diagoniall of the Octahedron, or of the circumscribing Spheres Dimetient, is to the Body's solidity and superficiality, as 1. to .166667 ferè c. O. 1.7320508 q. O. Hexahedron, .19245009. c. H. 2.00000. q. subdupla. H. Icosahedron. .3170189. c. I 2.393635. q I Dodecahedron, .348145. c. D. 2.628656 ferè. q. D.  
Hence,  
8. The Axis, Diagoniall, or angular Diameter of the Octahedron, or the circumscribing Spheres Axis or Diameter, is to the side of the Cube and Quadrat equal to the Body's solidity & superficiality as 1. to .550321. c. O. 1.316014. q. O. Hexahedron, .57735. c. the same as of the Spheres Diana. to the inseribed Hexabedrons' side, and so of its own Axis or corporal Diagony, to its Side, noted before. H. 1.41421. q. H. Icosahedron. .68186. c. I 1.54714 q. I Dodecahedron. .703483 ferè c. D. 1.621313. q. D.  
By which proportions, (as also by the 1. 2. and 5 Sections following of the like proportions, in relation to the greatest Periphery of a Sphere) it appears, that these five Bodies being all described within one & the same Sphere, do retain the same order and rank among themselves, in respect of their magnitudes or dimensions both solid and superficial, as they do, being of one common, lateral dimension, but not in that proportion and difference of dimension either solid or superficial; for that here the lateral Dimension being different in them all, and that by way of diminution from the lest body to the greatest, the superficial and solid dimension become thereby different in them all in like order, by way of augmentation; but nothing so much, as being all under the same lateral dimension. And hereby it also plainly appears,  
I That the Hexahedrum is triple the Tetrahedrum inscribed in the same Sphere, and so E 14. p. 32. And,  
II. That the Hexahedrum is to the Octahedrum within the same Sphere, solidly, as it is superficially, according to E 14. p. 27; and also as the Side of the Hexahedrum is to the Radius of the Sphere by the same Prop. And.  
III. That the Octahedrum is superficially sesquialter the Tetrahedrum, according to E 14. p. 14. and so (by the same Prop.) the Tetrahedrum is Basially or Hedrally, sesquitertian the Octahedrum. And,  
iv By the last named Sections of solid and superficial Proportions, together with the 3. and 5. Sections of Linear Proportions beforegoing, between the Diameter of a Sphere, and the sides of the 5 Bodies inscribed thererein; and also the first Section of Proportions, between the greatest Periphery of a Sphere, and the sides of the Bodies inscribed; it appeareh, how that the Octahedrum is to the Triple of the Tetrahedrum inscribed in the same Sphere, as its side is to the side of the Tetrahedrum; and so E, 14. p. 22. And,  
V That the Dodecahedrum is to the Icosahedrum in the same Sphere, both solidly and superficially, as the Hexahedrum is to the Icosahedrum laterally, according to E, 14. p. 9 and 11.  
Than for these Bodies in relation to a Sphere, in respect of their circumscriptibility about the same, as to its Dimetient; and so for the Hexahedrum, Dodecahedrum, and Icosahedrum, simply and absolutely in themselves also, in respect of their Dimetients of altitude, being all one with their inscribed Sphear's Dimetient: their proportions both cubatory and quadratory, will be as followeth: the Bodies being here placed in order according to their magnitudes both solid and superficial decreasing, which is according to their due hedral order, in respect of their denominations from the numbers of their bases or hedral Planes. And therefore,  
9 The diametral Cube and Quadrat of a Sphere, is to the solidity & superficies of its circumscribing Tetrahedron, as 1. to 1.7320508. c. T 10.3923048. q. T Hexahedron. the same as its Lateral Cube & Quadrat. H Octahedron, 0.8660254. c. O 5.1961524. q. O Dodecahedron. 0.693786. c. D 4.162719. q. D Icosahedron, 0.631757 c. I 3.79054 q. I  
And so the 4 last of these Proportions, are consequently, of the Cubes and Quadrats of the Altitudes of those two Bodies, to the solid and superficial capacities of the Bodies themselves.  
10. The Dimetient of a Sphere, is to the side of the Cube and Quadrat equal to the solidity and superficies of the ambient. Tetrahedron, as 1. to 1.200937. c. T 3.223710. q. T Hexahedron, the same as in Sect. 2. the side of this Body agreeing with the inscribed Spheres dimetient. H Octahedron, 0.953193. c. O 2.279507. q. O Dodecahedron, 0.885269. c. D 2.040274. q. D Icosahedron, 0.858058. c. I 1.946932. q. I  
And so the 4 last of these proportions, are of the Altitudes of those two Bodies, to the sides of the Cubes and Quadrats, equal to their solidities and superficieties.  
Whereby, as also by the 3, and 7 Sections of Proportions following, it plainly appears, how that these 5 Bodies being all described about one Sphere, the Tetrahedrum is the biggest of all, both in solid and superficial dimension, and is exactly double to the Octahedrum in both these dimensions (as it was showed before to be in its lateral dimension) and very near as big as the Hexahedrum and Octahedrum both together, in respect of both dimensions: And the Icosahedrum is the lest of all in both these dimensions; which is almost contrary to the former course holden in these Bodies, being described within one Sphere.  
N●w for the like Proportions, in relation to the Circumference of a Sphear's greatest Circle; and first, in respect of these Bodies spherall inscriptibility; it followeth,  
1. The Cube and Quadrat of the greatest or central Circumference of a Sphere, is to the solid & superficial capacity of the inscribed Tetrahedron. as 1. to .0020689369. c. T .1169956. q. T Octahedron, .0053752557. c. O .1754934. q. O Hexahedron, .00620681069. c. H .202642367. q. H Icosahedron. .01022434. c. I .2425259. q. I Dodecahedron, .01122822. c. D .266338. q. D  
Therefore.  
2. The circumference of a Sphear's greatest Circle, is to the side of the Cube and Quadrat equal to the solid and superficial content of the inscribed Tetrahedron. as 1. to .127423. c. T .342046. q. T Octahedron, .175173. ferè. c. O .418919. q. O Hexahedron, .183776. c. before for the side of the Hexahedron inscribed. .450158. q. before for the side of the Hexahedron, inscribed. Icosahedron. .2170427. c. I .492469. q. I Dodecahedron. .223925. c. D .51608 ferè q. D  
Than for the proportion's cubatorie and quadratarie, in relation to a Sphear's greatest, Diametral, or true Peririphery, in respect of these Bodies Spherall circumscriptibility, it followeth,  
3. The greatest Peripheriall Cube and Quadrat of a Sphere, is to the solidity & superficiality of its comprepending, containing, or including Tetrahedron, as 1. to .055861296. c. T 1.0529606. q. T Hexahedron, .032251534. c. H .6079271. q. H Octahedron, .027930648. c. O .526480314. q. O Dodecahedron. .2237567. c. D .4217715. q. D Icosahedron, .02037514. c. I .3840622. q. I  
Whereupon.  
4. The greatest, or diametral Pe●iphery of a Sphere, is to the Side of the Cube and Quadrat equal to the solidity & superficiality of the circumscribing or comprehending Tetrahedron, as 1. to .3822701. c. T 1.0261387. q. T Hexahedron .318309. c. the same as of the circumf. to the Diana. the side of this Body agreeing with the Spheres diameter. H .7796968. q.— H Octahedron. .30340798. c. O .7255896. q. O Dodecahedron .2817898. c. D .649439. q, D Icosahedron .2731284. c. I .619727. q. I  
Lastly, for these Bodies, and a Globe or Sphere compared wholly together, both solidly and superficially; and that according, both to their inscription and circumscription, the Proportions willbe as followeth, (the first or uppermost of the two Numbers belonging to each Body in the two first Sections, being for solid comparison, and the second for superficial)  
5. A Globe or Sphere, is to the inscribed Tetrahedron. Solidly and superficially together, as 1. to .12251753 T .3675526 ferè. T Octahedrens .318309886. agreeing with divers of the former proportions. .551328895. Hexahedron .3675526 ferè. agreeing with the superficial comparison in the Tetrahedron. . 6366●977. Icosahedron .6054614. I .76191789. I Docecahedron .6649087. D. .8367272. D  
Contrariwise.  
6. The Tetrahedron. Is to the circumscribing Globe or Sphere, solidly and superficially together, as 1, to 8.162097 T 2.720699 T Octahedron. 3.14159265, agreeing with many of the foregoing proportions. O 1.81379936— O Hexahedron 2.720699. agreeing with the superficial comparison in the Tetrahedron H 1.5707963— H Icosahedron. 1.6516327 ferè. I 1.312477 I Dodecahedron 1.503966. D 1.195133. D  
7. A Globe or Sphere, is to its ambient Tetrahedron both solidly and superficiarily together (in one and the same reason) as 1. to 3.30797337. Hexahedron 1.9098593. the same as of a Sphere to the Cube of its Axis, noted formerly; the side of this Body agreeing with the Axis of the Sphere. H Octahedron 1.65398668. subdu●plum Tetrahedri. Dodecahedron 1.325034. Icosahedron 1.206567.  

Conversly,  8. The Tetrahedron is to the inscribed Globe or Sphere, both solidly and superficially together in one, as 1. to .30229989 Hexahedron .52359877, the same as of the Cube of a Sphear's Axis, to the Sphere itself; the Cube of the Axis being here the ambient Hexahedrum. H Octahedron .604599788. double to the Sphere inscribed in the Tetrahedrum. Dodecahedron .7546973. Icosahedron .8287974.  
By the two last of which Sections of solid and superficial Proportions or comparisons between the 5 ordinate Bodies and a Globe or Sphere, it appeareth; that any of the said Bodies being circumscribed about a Sphere, the solid and superficial capacity thereof, will be one and the same numerally, (or in the number of a given Measure) viz. where the said two several Dimensions of the inscribed Sphere are alike in number: That is, two like Bodies exactly encompassing or environing two several, distinct Sphear's, whereof the solidity of the one, & the superficiety of the other, are numerally alike; there will the solid capacity of one of the like ambient Bodies, and the superficial capacity of the other, be also alike in the number of measure. And so likewise, two distinct Spheres being inscribed in two several like Bodies, whereof the solid measure of the one, and the superficial measure of the other, are numerally the same; there will the solidity of one of the inscribed Spheres, and the superficiety of the other, be also numerally the same: and which I have not found to be observed by any before.  
Many more Proportions might here have been raised, if they were needful; as namely of the Bodies among themselves in respect of their mutual inscription and circumscription; and those also which are the converse of many of the former, to wit, the Proportions of these Bodies solidly and superficially, to the Cubes and Quadrats of their Sides, and of their Axes and Altitudes; and so of the Diameter and greatest Circumference of their circumscribing and inscribed Spheres, whereby the sides, Axes and Altitudes of these Bodies, and so the Axes or Dimetients, and greatest Peripheries, of their circumscribing and inscribed Spheres, might be obtained, by having the solidities and superficialities of the Bodies only; and that after one Radical extraction, quadrate or cub●que according as I delivered in the Circle and Sphere, for the obtaining of their Diameters and Peripheries, by their superficial and solid Contents; and so in the ordinate Trigon for the finding of its side, and Diametral or perpendicular line; and in the ordinate Pentagon, for its side and diametral or perpendicular, and Diagonal-line, by their are all or superficial Contents: But that these kind of proporti 〈…〉 not so useful, being indeed more of curiosity than 〈◊〉 sigh the superficial and solid Contents of Figures are more usually inquired out by their sides, Diameters, and other▪ the Lines of Dimension, than these lines are by their superficial and solid Contents; for that the thing chief 〈…〉 in the dimension of all Figures, is their superficial 〈…〉 Contents (and in solid Figures, chief the so●●● Conte●● 〈◊〉 I said before) which must be obtained by 〈◊〉 of Dimension.  
〈…〉 ●●ving handled the five famous ordinate (Py 〈…〉 tonic) Bodies, or the angular, or recti 〈…〉 both geometrically, in an instrumen 〈…〉 our artificial way of measuring) 〈…〉 the most exquisite proportional 〈…〉 that may be, and that chief in reference to their solid and and superficial dimensions: I shall next come to the second theorematicall Proposition , in which, our more particular or special reason of our artificial instrumentary dimensions (or mechanical Cubature and Quadrature) of these Bodies, like as first of a Sphere, (as to the obtaining of the several artificial Lines of measure for performing the same) is contained.  

THEOR. II. Explicating particularly the foregoing artificial Lines for the cubick dimension, (or Cubing) of a Sphere; from our particular or special Reason of Dimension: And consequently, the Lines for the like dimension of all the other regular Solids.  
IF the Axis or Dimetient of a Sphere, equal to the Cube of the Parts of the Rational Line, be had; The same shall be the correspondent Line of Cubature, according to the said Parts. And the reason of that to the congrual Cubatorie Line, in reference to the whole entire Measure, will be as the reason of the Parts to the Whole; which is as the reason of their Cubes. And the like for the correspondent Periphery, or the Circumference of the greatest Circle.  
ANd the like in both these, for the superficiary (or quadrate) dimension of a Sphere (or quadration of the Spherical) only respect being here had to the quadrate parts of the Rational Line, as there to the cubique: and so indeed this may properly enough be referred to the first Theorem, the reason here, being the same with that.  
And the same reason holdeth in all the five forenamed ordinate plain or angular Bodies, both for solid or cubique, and for superficiaty or quadrate dimension, whether simply in themselves, or in relation to a Sphere, by way of inscription and circumscription, and that by any of their dimensional lines formerly named, as their sides, and Axes or Diagonials, or angular Diameters and Dimetients of altitude: And so by their circumscribing and inscribed Sphear's Dimetient, and also greatest Circumference, and other lines of dimension, according to the several Dimensions and dimensional Proportions beforegoing: And also in respect of their relations one to another, by way of mutual inscriptibility and circumscriptibility.    

SECT. V Showing the Dimension of all exactly ordinate, or regular solid Bodies, artificially, for Gravity or Weight, as is for solid Measure: And demonstrating the same particularly, in the first ordinate Solid here handled, namely a Sphere.  
ANd the like reason of Dimension to that before-going, will hold in a Sphere, and the five plain regular (Platonic) Bodies, for gravity, or Quantity ponderall (according to any Metal & Weight assigned) as for solid magnitude, or Quantity dimensional; there being generally the same mathematical reason of these two Quantities, in so much, as that they are usually by Mathematicians compared together in several kinds of Bodies: Or divers kinds of bodies are compared together among themselves in this twofold reason of Quantity; to wit, Magnitude or dimension, and Gravity, or Ponderosity, as you may see in Archimedes, Ghetaldus, and others: And so these two do proportionally answer each other, in so much, as that one may be deduced from the other; as Gravity from Magnitude, and Magnitude from Gravity; or Weight from solid Measure, and contrà: And therefore the gravity or ponderosity of each one of the foresaid regular solid bodies taken in some certain magnitude or bigness, being first known according to some certain Metal, Weight, and Measure appointed; there may be artificial Lines of measure extracted for every several kind of Body, according to the said particular Metal, Weight and Measure, (and that according to the foresaid several Lines of Dimension in those Bodies, by which they have been showed to be artificially measured) so as that any one of the said dimensional lines of each Body in any magnitude whatsoever, being measured by its proper artificial Line or Scale for this purpose, & cubed, the same shall be the weight of the metalline Body proposed: And which will therereupon hold in a Sphere, and all the 5 plain regular Bodies, not only as considered simply and absolutely in themselves alone, but also as in relation one to another, by way of inscription and circumscription; and so in the said 5 bodies, not only in the said relations to a Sphere, as being inscribed in, or circumscribed to the same, but also mutually one to another, as was said before for solid (and superficial) measure: And so the Diameter or Circumference of a Sphere, being taken by their proper, respective artificial Lines of Measure for this purpose (according to any certain Body, Metal, Weight, and Measore assigned) the several Cubes thereof, shall be the weight of the respective Body inscribed, or circumscribed, according to which is proposed; And the first of these, is the same (in respect of a Sphear's Diameter) with the artificial dimension of the four greatest of the said 5 regular plain bodies, by their Axes, Diagonies or angular Dimetients, (these being all one with their circumscribing Sphear's Dimetient, as I have showed before) And the latter agrees with the like dimension of the three last of those bodies, by their Altitudes, (being the same with their inscribed Sphear's Dimetient; and is also in the Hexahedron, the same with its Side, as I have likewise shown before) And so the reason of the artificial Lines for this cubical dimension of the foresaid Bodies, for weight, as for solid measure, may be partly referred to the foregoing 2ds. Theorem; the difference being, that respect must be here had to the (compounding, denominate) parts simply, of the Weight proposed, in such manner, as is there to the cubical parts of the Measure proposed; So that these artificial Lines may be easily produced therefrom; And therefore I shall not need here to raise a particular Theorem upon the same. All which might be performed also Arithmetically by way of Proportion, from the natural Measure appointed, according as all the former Dimensions: And both these ways I shall here particularly demonstrate in the first regular Solid beforegoing, to wit, a Sphere, or Globe, simply in itself, as coming most in use; and that in the most usual and useful Metal for this purpose. But first I will give the Proportions or comparisons of all the usual or principal (or commonly received) kinds of Metals, according to the experiments and observations of Marinus Ghetaldus, in his Archimedes promotus (who is generally supposed to have come the nearest to the truth herein, of any man that hath ever yet written hereof) according as they are there delivered by him in the second Table of that Tractate, next after after Theor. 9 or prop. 17: which Mr. Gunter in the 5th. chap. of the 3d. Book of his Sector, hath expressed in the same terms, but more decimally, and that in whole numbers, by changing the natural or vulgar fractions of those numbers into decimal, and so expressing his natural mixed or heterogeneal numbers, in whole numbers, after a decimal manner, putting the first number 10000, whereas Ghetaldus puts it but 100; which I have here collected orderly into this Table following, in proportion direct and reciprocal, in respect of the equal Magnitudes and gravities of like Bodies of different Metals.  
In like Bodies of several metals and equal magnitude, having the weight of the one, to found the weight of the rest. Gold. 10000 3895 ☉ In like Bodies of several metals and equal weight, having the magnitude of the one, to found the magnitud of the rest. The converse of the former. Quicksilver. 7143 5453 ☿ Lead. 6053 6435 ♄ Silver. 5439 7161 ☾ Bresse. 4737 8222 ♀ Iron. 4210 9250 ♂ Tin. 3895 10000 ♃  
Which meralline Proportions or comparisons. Mr. Oughtred in his foresaid Book of the Circles of Proportion, Part. 1. chap. 10. hath expressed in the Terms following, being deduced from Ghetaldus his first Table of Comparisons (if I much mistake not) which is from Prob. 5, and 6, or Prop. 12, and 13, of that Treatise.  
Gold 3990 Quicksilver 2850 Led 2415 Silver 2030 Brass 1890 Iron 1680 Tin 1554   geometrical diagram 
As suppose here again the former bullet of 9li. avoirdupois, whose diameter being put 4.00 inches, the Circumference will be 12.57 inches ferè which by its proper Line of Cubature, will be found 2.08, as the Diameter before by its proper Line, and therefore by Cubation, must needs produce the same ponderosity as before.  
But here note, that an Iron-Sphear of 4 inches the diametral magnitude in Ghetaldus his measure, will weigh with him, 12li. 2. ounc. 1 scrup. 14 gr. and 2/37 of a grain; for so I found it to be by the former proportional numbers, by comparing this Sphere with a Sphere of Tin of the same diametral magnitude, according as Ghetaldus showeth in his foresaid book, and from him Mr. Oughtred in his forenamed book: and which is also expressly noted by Ghetaldus in a Table, wherein he hath set down the weights of a Sphere, in all the foresaid Metals, from 1/4 of an inch the diametral magnitude, to 12 inches, or the whole Foot, proceeding all along by quarters of inches. But now 9li. avoirdupois is in our Troy-weight, by Assize or Goldsmith's weight (according to the commonly received proportion of the Pound-avoirdupois to the Pound-troy, 60 to 73) but 10 57/60 pounds, or 10.95li. exactly; which is 10 pounds, 11 2/5 or 11.4 ounces; or according to the common division of the ounce-troy by penny-weights, is 10li. 11 oun. and 8 penny-weights exactly.  
Now Ghetaldus divides a Pound into 12 ounces; an ounce into 24 scruples; and a scruple into 24 grains: so that his ounce weigheth 576 grains, and his pound, 6912 grains: And we divide our Pound-troy, into 12 ounces (as he doth his pound) but the ounce-troy we usually divide into 20 penny-weights, and a pennyweight into 24 grains, so that our ounce-troy weigheth but 480 grains, and consequently our Pound-troy 5760 gr. All which showeth, that either his Weight, or Measure, or rather both of them, do differ from ours, as I shall further show: As for the Measure which he useth, he saith it to be the ancient Roman Foot, divided into 12 unciae, or inches as ours is, and which by his description and delineation thereof, in his forementioned book, seemeth to differ but very little from our English Foot, (if any thing at all) and that deficiently, and which Mr Oughtred in his forenamed book also observeth: But Mr. John Greaves sometimes Professor of Geometry in Gresham College London, and now Professor of Astronomy 
Concerning the magnitude of the Roman Foot.  in the University of Oxford, hath in his Discourse of the Roman Foot, etc. (published by him in English, Anno 1647; and deduced, not only from divers Authors, but also from his own observations and experiments, which in that his learned discourse he seemeth to have made with greatpains and industry, in his travails in foreign parts, but especially in Italy, and there at Rome) more clearly expressed the Dimension or magnitude of the true Roman Foot; having done the same not only linearly (by the half thereof, as Ghetaldus hath done) but also Numerally, in comparing it with the standard measures of England, and divers other Nations: For the draughts or delineations thereof in Books, cannot give us the true length, in respect of the divers accidents happening to the paper whereon it is imprinted, and especially the contraction or shrinking of it after the impression, (as both these authors do give us to observe) which while it was moist in the Press, received the true Measure, but afterwards being dried, loseth somewhat thereof; and so Ghetaldus in the last page of his foresaid book, giveth us to note particularly in his draught there of half the Roman Foot, and how much is there to be added to it, to make up the true measure. For although Mr. Greaves doth conclude upon the same Measure of the Roman Foot, which (he saith) Ghetaldus doth, (for the truest Measure) yet if we compare their draughts or delineations of the half Foot, together, (in their foresaid books) we shall found that of Mr. Greaves, to be shorter than that of Ghetaldus, by almost 1/8 of Ghetaldus his Inch, as it is there set out; or 1/10 of our English inch exactly. And such like disagreement is to be found among other Authors in their delineations or draughts of the same Roman Foot, for the reason aforesaid.  
Now the Foot which the said Mr. Greaves (among such a diversity of opinions concerning the true Roman Foot, as are to be found, and so many Feet as are taken to be Roman) pitcheth upon for the most genuine and true Roman Foot, (being led not only by the anthorities of Ghetaldus, and divers other learned and judicious men, (as he saith) but also by his own observations and experience) is that which is commonly called by writers, Pes Colotianus, from the place where it is (or sometime was) to be found, namely, in hortis Colotianis in Rome, upon the Monument of Cossutius (which now he saith to be removed thence) 
The Roman and English Foot compared together.  which Foot he comparing with our English Foot, which he took from the iron-Yard, or Standard of 3 Feet, at the Guild-hall in London (for there is no single Foot-standard) findeth to be 967 such parts, as the English Foot contains 1000; and so the English Foot to contain 1034.13, such as the said Roman Foot contains 1000: whereby this Roman Foot should be 11.6 (or 11.604 exactly) such parts as our English Foot is 12; viz. 11.6 (or 11.604) inches, and so wanting of the English Foot, only 0.4 inch, or 0.396 inch exactly. But there are two other Roman Feet reckoned by him, which (by his account) do come nearer to our English Foot; the first whereof is that on the Monument of Statilius, in hort is Vaticanis in Rome, which he observed to be 972 such parts as the English Foot is 1000, (and to be 1005.17 of the Pes Colotianus, being 1000) whereby this Foot will be 11.7 ferè, of the English Foot, being 12, viz. 11.7 inches ferè, it being 11.66; which wanteth of the English Foot, but, 0.34 inch. And the other Foot is that of Villalpandus, deduced from the Congius of Vespasian in Rome, which he saith to be 986 parts of ou● English Foot containining 1000, (and 1019.65, of Pe● Colotianus, being 1000) and so is 11.8 of the English Foot being 12, viz. 11.8 inches, which wants of the whole English Foot, only 0.2 inch. But the ancient Greek Foot doth by his observation more nearly agreed with our English Foot, than this last Roman Foot, being (by his collation) 1007.29, of the English Foot 1000, which is hardly .09 of an inch above a Foot English, it being 12.087 inches english.  
As for the Weights used by Ghetaldus in his forenamed Treatise, which he saith to be the Weights used in his time (which is not very long since) they are surely the Roman weights (for he lived at Rome as I take it, when he wrote that book) which have continued the same for many ages without alteration, as some writers do attest: And which Mr. Greaves in his Discourse of the Denarius (annexed to that of the Roman Foot) which he puts as an undeniable principle and foundation from whence the weights of the Ancients may be deduced, as the Roman Foot for the Principle of their Measures) 
The Roman Weight, and our Troy or Goldsmith's Weight, compared together.  having collated with the Troy-weights from our English Standard for Gold and Silver, by grains thereof, saith, that the Roman Pound both ancient and modern containeth 5256 such grains, and so the Roman Ounce both ancient and modern, 438 of the same grains, the Troypound containing 5760 grains (as I noted before) and so the Troy-ounce, 480: whereupon the 
* Thomasius in the end of his Dictionary, reducing the weights and measures, etc. of the Ancients, to those in use with the English nation; saith, that the Roman Pound is 10 oun. and an half Troy: and so the Roman ounce is 3 quarters and an half of an ounce-troy. By which account, the Roman pound will contain but 5040 english grains, or such as our troypound contains, 5760, which comes short of Mr. Greaves his account by 216 grains And so the Roman Ounce will contain but 420 such grains, which falleth short of Master Greaves his observation, by 18 grains. But herein we rather give credit to the later and exacter observations and experiments of Mr. Greaves. But yet how our Troy-weight may have been altered since Thomasius his time, I know not.  Roman Pound and Ounce should be (in the lest terms) but 73/80 of our English pound and Ounce Troy-weight, and so the proportion of the Roman pound & ounce to our Troypound & ounce (for the conversion or commuration of the Roman weight to our Troy-weight) as 80 to 73 (which is the commonly received proportion of the Ounce- avoirdupois to the Ounce- troy, for the like conversion, as I shall show afterwards, and which is decimally, as 10000 to 9125 exactly.) As for the division of the Roman ounce immediately by scruples, in number XXIV which Ghetaldus useth, Mr Greaves speaketh not of it, but only by drams, in number VIII, as the Ounce is commonly divided by the Physicians of all Countries, and the physical or medicinal weights which we use in England, are the same with the Troy-weights for Gold and Silver; only the Ounce-troy is commonly by our Goldsmiths divided into 20 penny-weights (as I shown before) a pennyweight consisting of 24 grains: and the Ounce by Physicians is universally divided into 8 drachmas; a Drachma into 3 scruples; and a scruple (by an evil custom received in shops) into 20 grains, which aught to have (according to the ancient custom) 
Morellus in cap. 1. Prolegom. ad composit. medicament.  24 grains, and so be equal with a pennyweight; so that the Number of grains in the medicinal Pound and Ounce, is the same with that in the Troy: For the scruple being 20 grains, the Dram will be 60, the Ounce 480, and so the Pound 5760, as before-noted: whereas else the Scruple being made 24 grains, (as anciently it was) the dram would be with us, 72 gr. the Ounce 576, and so the Pound 6912 gr. as Ghetaldus hath it; but the number of grains contained in the Roman Pound or ounce, with the Romans themselves, Mr. Greaves showeth not, whereby we might found what the difference is between their grain and ours: But collating the number of grains contained in the Pound or Ounce used by Ghetaldus (which we take to be the same which Mr. Greaves noteth for the Roman-pound and Ounce, both ancient and modern) with the number of grains from the English-Standard for Gold and Silver, contained in the said Roman-pound and Ounce, (which we shown even now from Mr. Greaves) we shall found the Grain (such as the Roman-pound contains 6912, and the Ounce 576, according to Ghetaldus, as we lately shown) to be (in the lest terms) but 73/96 of the English grain, (such as the Roman pound contains 5256, and the Roman Ounce 438, according to the observations and experiments of Mr. Greaves) So that the Romanegrain should be to the Troy grain, from the English-Standard (for the conversion of Roman grains to our Troygrains) as 96 to 73; & so consequently the Roman-weights in general (as pounds and ounces) reduced into the proper grains, will be to our Troy-weight in grains, accordingly.  
And according to these collations and proportions of these two Weights the one to the other; the former iron-Spheare weighing with Ghetaldus 12 lively 2. oun. 1 scrup. 14 2/37 gr. or 84134 2/37 grains Roman (according to his measure of the Spheare's Diameter by inches of the Roman Foot) will be in Troy-weight from the English Standard, 11 lively 1 oun. 2 drams, 16 1●4/111 gr. (or according to the vulgar Division of the ounce-troy by penny-weights, 11 lively 1 oun. 5 penny-weights, and 16 1●4/111 gr.) or 63976 104/111 gr. which is very near 63977 gr. For first, I say,  
As 80 to 73, So 12 lively 2 oun. 1 scrup. 14 gr. Roman, (viz. 12 lively 2 oun. 38 gr.) or 146 38/576 oun. Roman, to 133 6●●1/23●40 ounces-troy; which is 133 oun. and 137 gr. ferè: and these reduced into libral weight, are 11 lively 1 oun. 2 dr. or 5 p. w. and 17 gr. ferè, as before.  
Than secondly; As 96 to 73, So are 84134 2/37 gr. Roman, to 63976 104/111 gr. Troy, as before.  
Which exceeds the former troy-weight of this Sphere (deduced from its avoirdupois-weight, according to the common proportion, 60 to 73) viz. 10 lively 11 oun. 3 drams, and 12 gr. (or 10 lively 11 oun. and 8 pennyweight, as before) or 63072 gr. by 1 oun. 7 dr. and 5 gr. or 1 oun. 17 p. w. and 17 gr. or 905 gr. which difference in the Troy-weight here, may arise not only from the difference between Ghetaldus his Weight and Measure, and ours, but also from some difference in the Metal itself, which I shall speak of afterwards. Now if we shall reduce his measure of the Spheare's dimetient, being 4/12, or 4 inches of the Roman Foot, (which we shown before, to be the Pes Colotianus, as being most approved of by him, for the true ancient Roman Foot) to inches of our English Foot (according to Mr. Greaves his foresaid collation of these two Feet together) the said Spheares Dimetient will be less than 4 inches of the English Foot, viz. but 3.868 inches-english (the whole English Foot, or 12 inches. English, being 12 〈…〉 inches Roman, according to Mr. Greaves his pro 〈…〉 Foot to the English Foot, 1000 to 〈…〉 before noted) and so the former 〈…〉 the Sphere, 12li. 2 oun. 1 scrup. 14 2/37 gr. with Gheta●dus, or 11 lively 1 oun. 2 dr. 17 gr. ferè, with us in Troy-weight, (which according to the common proportion of the Troy librall weight, to the avoirdupois librall weight, 73 to 60, is 9 lively 2 oun. and 1.057 dr. avoirdupois) answering to the Spheare's diametrali magnitude of 4 inches upon the Roman Foot, will answer to 3.868 inches upon the english Foot. Or again; if we shall reduce our english measure of the said Spheares dimetient, 4 inches (and so commonly holden to weigh 9 lively avoirdupois, which is 10.95 lively troy, as we shown before) to Roman measure in inches, we shall found the same (according to the former Pedal Collations) to be more than 4 Roman inches, viz. 4.136 inches, and the gravity of an iron Sphere of this Diameter, will be in Ghetaldus his weight, 93044.86 grains, or 13 lively 5 oun. 4 dr. 20.86 gr. (or according to Ghetaldus his Division of the ounce) 13 lively 5 oun. 12 scrup. and 21 gr. ferè) which is in our Troy-weight (according to the former Collations of these two weights) 70752.86 gr. or 12 li, 3 oun. 3 dr. 12.86 gr. (or by the common division of the ounce-troy by penny-weights, 12 lively 3 oun. 8. p. w. and 0.86 gr.) 
* Archimed. promote. theor. 9 prop. 17.  For seeing that Spheres of the same kind, are among themselves in gravity, as the Cubes of their Dimetients are in magnitude: Therefore the weight of the iron-Spheare, whose Diameter is 4 inches Romane-measure, or 3.8 68 inches english-measure, being found as before; the weight of the other Sphere of the same kind, whose diameter is 4 inches english-measure, or 4.136 inches Roman-measure, will be found also as before: Or more readily, by having the weight of such a Sphere, whose Diameter is one inch, which by Ghetaldus his Measure and weight, is 1314 22/77 grains. or 2 oun. 6 scrup. 18 22/●7 gr. and so in our troy-weight, 999 71/111 grains, or 2 oun. 1 p. w. and 16 gr. which weight therefore will answer to an iron-Spheare whose Diameter is 0.967 inch, english-measure, for this answers to one inch-Roman; and so the weight of such a Sphere who●e Diameter is one inch-english (which is 1.034 inch-romane) should be by this account, 1105.51 graines-English, or 2 oun. 2 dr. 25.5 gr. or 2 oun. 6 p. w. and 1.5 gr. troy (which are in Roman weight, 1453.83 gr. or 2 oun. 4 dr. 13.8 gr. or by Ghetaldus his division of the ounce; 2 oun. 12 scrup. and 13.8 gr.)  
Or (here briefly to show the use of the former proportional Numbers for Metals among themselves) in respect of two Spheres of different kinds of Metals, and like magnitude) the same weight of the foregoing Sphere of Iron, of 4 inches english-measure, or 4.136 inches Roman-measure, the diameter maybe produced as before, by the weight of a Sphere of any other Metal first had, being of the same magnitude: As for example, a Sphere of Tin, (for so Ghetaldus commonly deduceth the weights of other metalline Spheres from a stanneal Sphere, or a Sphere of Tin) whose diametral magnitude is 4 inches-english, or 4.136 inches-Romane, I found to weigh by Ghetaldus, 86066.5 grains (or 11li. 5 oun. 10 scrup. and 2.5 gr.) which is with us in Troy-weight, 65446.4 gr. (or 11li. 4 oun. 2 dr. 46.4 gr. or 11li. 4 oun. 6p. w. 22.4 gr.) Now therefore, according to the foresaid propo●●onall Numbers for Tin and Iron, I say; As 3895 to 4210 (which is with Ghetaldus in his  second Table of the Comparison of divers kinds of Bodies in gravity and magnitude, as 38 18/19 to 42 1/19) or more accurately (the former terms being incomplete and unabsolute) As 1554 to 1680 (which is with Ghetaldus in his foresaid first Table of the like Comparisons, as 1 to 1 3/37, that is, as 37 to 40, and which is decimally, as 1. to 1.081,081, infinitely; or integrally, 1000 to 1081, or 1000,000 to 1081, 081 completely) So is 86066.5 gr. (or 12li. 5 oun. 10 scr. 2.5 gr.) the Sphere of Tin in Ghetaldus his weight, or 65446.4 gr. (viz. 11li. 4 oun. 2 dr. 46.4 gr.) the same Sphere in our Troy-weight, to 93044.86 gr. (or 13li. 5 oun. 12. scr. 21 gr.) the Sphere of Iron in Ghetaldus his weight; or to 707.2.86 gr. (viz. 12li. 3 oun. 3 dr. 12.86 gr.) the same Sphere in our troy-weight, as formerly: which being converted into Avoirdupois-weight, (according to the former terms of proportion between these two Weights) will be 10.0961. lively viz. 10li. 1 oun. 8.576 dr. according to the common division of the Pound-avoirdupois into 16 ounces, and so of the ounce into 16 drams, and under which division or parts of weight, they go not; this kind of weight serving with us for all kind of coarser or grosser Commodities, it being the common Mercatory weight, (and called by a French name, Avoirdupois) and so for the weighing of all Metals, but Gold and Silver, to which only the Troy-weight is assigned: Neither indeed is there any need of such exactness in the weight of a bullet or any other Body of Iron, or other like base Metal, as to grains, but only to satisfy Art itself in the curiosity thereof.  
But now whether the common Tenent of an iron-bullet of 4 inches the diameter, to weigh just 9 li avoirdupois, were deduced from any certain and exact experiment or not, is a question here to be made; Nor was it my purpose to try the same, having not conveniences and accommodations thereunto: but only having the true weight of a Sphere or Bullet of Iron of any magnitude, in a certain measure given (as Inches) thereby to show a way for the exact and most speedy obtaining of the weight of a Sphere or bullet of any magnitude whatsoever, which I have partly declared & demonstrated already, and shall more by & by: And looking in several books of Gunnery, wherein are set down the weights of iron-bullets (or round shot, as they call it) fitted to all the usual pieces of Ordnance with us in England, according to the diameters of their bore, mouths, or concavities (abating usually 1/4 of an inch of the Gun's said diameter, for the diameter of the bullet) I found in one of them, the weight assigned to an iron-bullet of 4 inches diameter or crassitude, to be just 9li. and in another book this weight assigned to a bullet of 4 1/2 inches the diameter and so in other books, other weights assigned to a bullet of the same magnitude: And therefore finding such a discrepancy among our Masters of Gunnery in this thing, so that I could not discover from them any certainty in the weight of an iron-Sphear or bullet, at a certain magnitude or crassitude: I repaired one day in August 1648, unto Mr. John Reynolds, one of the Clarks of the Mint in the Tower of London, (and sometime Assay-master at Goldsmiths-hall) a man much noted by artists for his industry and ingenuity in the Mathematics; (and so indeed I found him to be) who very courteously entertaining me at his house in the Tower, with discourse about many exexperiments made by him in the Mathematics, and showing the same to me (as he had once done in some of them, long before) among which were those concerning the weights of Metals, and their proportions one to another: I desired of him to be resolved in this point of art concerning the weight of a Sphere or bullet of Iron, according to some certain magnitude or crassitude, knowing him to have all accommodations fit for the finding out of the same; and he thereupon produced me an experiment made by him not long before, upon a large bullet, which he than shown me; but I observed the same to be very unfit to ground an experiment upon, being not only much rusty, but also having several holes and cavities therein, (which seemed to be chief from the antiquity of it) which might well hinder the finding of the true weight, according to its diametral magnitude, and which himself than doubted much of: whereupon I importuned him for another the like experiment, which might be exact; And so both for his own satisfaction, and mine also, he soon after, got another bullet, which was very sound & solid, and clear enough from rust, and also as round every way, as could well be imagined; whose Diameter we thereupon took by a pair of Callaper Compasses (conceiving it to be a surer way to found the same, than by the Circumference) as precisely as possibly we could, and measuring the same upon a line of inches, found it to be 4.8 inches; and than for my further satisfaction, I took the Circumference of the bullet with a thread, by means of the small crease or Circle which encompassed the bullet exactly in the middle, being made by the Mould in which it was cast, and measuring the thread upon the said line of inches, we found it to be 15.25 inches, which by Cycloperimetricall proportionality, gives the diameter as before; so that we might well conclude, the magnitude of this bullet to be rightly found by us. Than for the weight thereof, we tried the same with all possible preciseness, both by Avoirdupois-weights, and Troy-weights, and found it to weigh, 15li. 12 1/4 oun. avoirdupoiz; and 19li. 1 25/32 oun. Troy, viz. 19li. 1 oun. p. w. 15 gr. According to which experiment, an iron-bullet (made of cast iron, such as bullets are usually made of, and which weigheth much lighter than forged iron) of 4 inches the diameter, will weigh upon the point of 9li. and 2 oun. avoirdepois, which differs not much from the common Tenent.  
And so according to this our experiment (to which we will adhere for the finding out of the weight of any Sphere or bullet made of cast-iron) the artificial Line of measure, or cubatory Line of gravity, for the speedy discovering of the weight of any Sphere or bullet whatsoever, made of cast-iron, by the Diameter or Circumference thereof, as was formerly showed, will be 1.91 inch, for the Diameter, and 6.01 inches for the Circumference; and these in respect of librall weight, as the two former Lines, viz. 1.92 inch, and 6.04 inches, deduced from the former common Tenent of the weight of an iron bullet of 4 inches diameter, which differ but very little one from another, as they are here set, from a centesimal division of the Inch. Now these two latter Lines being divided as the former, and the Diameter and Circumference of the foresaid Sphere or bullet measured thereby, the same will be found severally, to be 2.09 ferè (which by the other Lines were 2.08) which cubed, yields 9.129329 ferè. for the weight of the bullet, which is 9 lively 2 oun. 1 dr. averdepoiz, differing from the true weight (by way of excess) only 1 dr. which is not considerable.  
But now what sort of Iron, Marinus Ghetaldus in his experiments upon the same, for the weight thereof, and so its proportion to other Metals, meaneth, whereby a Sphere of 4 inches diameter, english-measure, (or 4.136 inches Roman-measure) should weigh according to him, 10.096 lively avoirdupois (as being deduced from 12.283 lively troy and that from 13.461 lively Roman) we know not, if Mr. Greaves his foresaid collations and comparisons of the Roman weight and Measure (used by Ghetaldus) with the English, be true, (as we are willing to believe they are) and also the foresaid proportion of the Troy-pound-weight to thè Avoirdupois-pound-weight, as it is commonly holden, viz. 73 to 60; although the aforenamed Mr. Reynolds will have it to be, as 17 to 14; deducing the same from a general 
The Troy and Avoirdupois Pound compared together.  Maxim, and undeniable Principle (as he saith) that 136 pounds-Troy, and 112 pounds' Avoirdupois are equilibral, or equivalent in weight, which are in the lest terms of proportionality, 17 and 14; and which indeed I found by experience to be the truer, as I shall afterwards show: And this is to be understood properly (as the Terms here stand, from the greater to the less) for the reduction of Troy-pound-weight (or Troy-pounds) to Avoirdupois-pound-weight (or Avoirdupois-pounds) in as much as 73li. Troy, make but 60li. Avoirdupois; or rather 17li. troy make but 14li. avoirdupois: and so the Avoirdupois Pound, is (in the lest terms) 73/60, or rather 17/14 the Troypound; whereas else, if we will precisely compare the Pound-troy simply, as the less, with the Pound-avoirdupois, as the greater; than must the Terms be rather taken the contrary way: And thereupon the Pound-troy will be to the Pound-avoirdupois, as 60 to 73, or rather, 14 to 17: And so the Pound-troy will be (in the lest terms) 60/73, or rather 14/17 of the Pound-avoirdupois. And which several Terms of proportion, though they seem, to differ but very little in the Reason thereof itself; yet may the difference of the weights of things, produced severally thereby (by way of conversion or reduction of one kind of weight to the other) be many times considerable; and the more, the greater that the quantity of the weight so reduced, is; as I shall plainly show afterwards: And so I will here first compare these two several Proportions together (and that according to the common acception of the Terms, from the Pound-troy to the Pound-avoirdupois, viz. the greater Term as antecedent, to the less, as consequent, and so the reasons of the Terms will be of the greater inequality; and the rationality of the first, or common Terms, will be (by prolation, from the Parabole of the said Terms, or Quantity of the Reason) super-tredecupartiens-sex agesimas, viz. superpartient 13/6●: and of the other Terms, supertripartiens quartas-decimas, viz. superpartient 3/14, and so the difference of the Reasons, or the differential Reason, but that of 1. to 420, viz. 1/420, according to a plain or simple subduction, or differencing of Reasons, by reduction of them to one common or conjunct Consequent, which imitates the subduction 
Rom. Arithm. l. 2. c. 2 & 3. à Laz. Schon. Et Clau. de Proport. composit. in fine 9 Elem. Eucl.  of fractional Numbers, by reduction to one common or conjunct Denominatour; and which is the most genuine, proper, and rational subduction, or discrimination of Reasons (both as Clavius, Ramus and divers other of the best Authors do teach) thus;  
Or more plainly thus, after the manner of Fractions.  
And which may also be seen by the like operation in the superpartient terms only, thus:  
And not according to that operation of Reasons or Proportions, which some do falsely and improperly call the subduction or subtraction of Reasons (as Clavius also saith in the place forecited) being indeed the true division or Resolution of Reasons, and which is as the division of Fractions; whereby they make the quotall Reason to be the differential or residual Reason; and which would than be here by prolation, of the greater inequality, viz. 1 1/510, according to the terms 511/510 (whereas the other or true differential Reason, is of the less inequality, by very much) but approaching very nigh unto a Reason of equality, or unity, or a Reason singular and individual; and this latter is also unexpressible or indenominate as the former, and is had by a conjunct composition only of the alternate or heterologall terms, thus.  
Anteced. 73. 17 1022 511 Quotient. Conseq. 60 14 1020 510 Quotient.  
Or more plainly thus,  
Or yet more plainly and readily, by permutation of the terms of the dividing or resolving Reason, in respect of their places; for so the work of Resolution will be changed into a Composition of the terms, according to the ordinary Multiplication of Fractions, thus,  
And the latter of these Proportions between the Troy and Avoirdupois-pound, makes the Pound Troy to be more of the Pound Avoirdupois, or to come nearer the same in quantity, than the other or common Proportion doth; as may be more plainly discerned, by reducing the said several Proportions into decimal Terms, for so, the Proportion of the Pound-troy, to the Pound-avoirdupois from the Terms of 73 to 60, will be, as 1.0000. to .8219, and from the terms of 17 to 14, as 1.0000 to .8235. Or by taking the Terms the contrary way, (as for the conversion or reduction of Avoirdupois-pounds to Troy-pounds) the Pound-avoirdupois will be to the Pound-troy from the terms of 60 to 73, decimally, as 1.00000, etc. to 1.21666, etc. infinitely; and from the Terms of 14 to 17, as 1.00000, etc. to 1.21428, etc. and so here the Pound Avoirdupois is by these latter Terms, less of the Pound Troy, (and so comes nearer the same) than by the other Terms. And the Reason of these latter Terms (as they here stand, from the less to the greater,) will be greater than the Reason of the first or common Terms (according to the exact comparing of Proportions or Reasons together, as you may see in the forecited places of Ramus and Clavius) in regard that the two several Reasons being reduced to one common Consequent (as aforesaid) the new, compound or correspondent Antecedent of the latter Proportion, will be greater than the like Term of the first; or the foresaid Antecedent of the latter Proportion, will be more of the common Consequent, than the like Antecedent of the first Proportion (as in the former operations, the contrary happened, where the Terms of Proportion were put the contrary way, viz. from the greater Term as antecedent, to the less, as Consequent) as you may here plainly see by these several subsequent operations.  
Or more plainly thus, in a fractional manner.  
Or again, the same will appear by a contrary operation, which is by reducing the two several Proportions (as the terms be here put) to one common Antecedent; for so the new, compound, or correspondent Consequent of the latter Proportion or Reason, will be less than the like Te●● of the first or common Proportion, (and thereby the Reason of these latter Terms, will accordingly be greater than that of the other Terms (according to the forecited Authors) as you may plainly see by this next operation following; the compound Antecedents of the operations next beforegoing, being here changed into the like Consequents.  
And here the difference of the Reasons (or the differential Reason) will be that of 2 to 1241, viz. 2/1241, as you may see in the first of these two operations before-going.  
And now according to these latter termer terms of Proportion of the Po●nd-troy to the Pound-avoi●dupois, viz. 17 to 14; the foresaid Sphere of Iron will be in avoirdupois weight, 10.1158 li, which (by conversion of the fraction-part into the proper parts of this weight) is 10 lively 1 oun. 13.6 dr. whereas by the common terms, it was 10 lively 1 oun. 8.6 dr. and so the difference of weight, 5 dr. exactly. But I conceive (as in all probability and reason I should) the Iron used by Ghetaldus in his experiments, to be the finer sort of iron, or forged Iron, which weigheth heavier than cast, coarse, or drossy iron, the proportion of weight between them, being (as I have deduced it from the experiments of Mr. Reynolds; and partly of myself also, upon this Metal) in general, such as is between 1000000 and 951832 arguing from forged Iron to cast iron, as from the more to the less: and so contrarily from cast-iron to forged iron, as from 
The Proportion between forged iron and cast-iron.  the lesser to the greater, the proportion of weight, will be such as of 1000000 to 1050605: For the said Mr. Reynolds found the weight of a cube-inch of fine Iron, which had been kept in the Treasury of the Tower of London, ever since King Henry the 7th. his time or longer (and that very neatly in a velvet-Case) to be 4.169 ounces Troy, viz. 4 oun. 3 p. w. & 9.12 gr. troy. which is 0.3474li. troy: And from the former Bullet of cast-iron of 4.8 inches the Diameter, which we found to weigh 19 pound, 1 oun. 15 pennyweight, and 15. grains troy, or 229. ounces, 15 p. w. and 15 gr. (which is 229 25/32 ounces troy) we gather the weight of one cube-inch of cast-iron to be 3.968 oun. troy; which is 0.33068li. troy. And so according to this experiment, a Sphere or bullet of one inch Diameter, made of fine, or forged Iron, will weigh a .18288 oun. troy, which is 0.1819li. troy: and so a Sphere or bullet of 4 inches Diameter, made of the same metal, will weigh 11.642 poundstroy (or 11li. 7 oun. 5 dr. and 38.17 gr. or 11li. 7 oun. 14 p. w. and 2.17 gr.) which being compared with the weight of an iron-Sphear of 4 inches Diameter (●nglish-measure) decuced f●om Ghetaldus before into our Troy-weight, viz. 12.283li. etc. troy, or 12li. 3 oun. 3 dr. 12.86 gr. or 12li. 3 oun. 8 p. w. and 0.86 gr.) it will be found to want thereof, 7 oun. 5 dr. 34.11 gr. or 7 oun. 13 p. w. and 22.11 gr. (according to the difference between 12.283483 li and 11.642044 li. being 0.641339 li.) And so the weight of a Sphere of fine Iron of 1 inch diameter, here found 2.18288 oun. troy, viz. 2 oun. 1 dr. 27.78 gr. or ● oun. 3 p. w. 15.78 gr. being compared with the weight of a ferreall or iron-Sphear of the same Diameter, deduced formerly from Ghetaldus into our troy-weight, viz. 2.30315 oun. or 2 oun. 2 dr. 25.5 gr. or 2 oun. 6. p. w. and 1.5 gr. it will be found to want thereof 57.7 gr. viz. 2 p. w. 9.7 g. which operations agreeing so nearly one with another, it is thereby manifest, that the Iron meant by Ghetaldus, is the finer sort of Iron, or purely forged iron; But that we cannot make his experiments and ours exactly to agreed in a Sphere of this Metal; may hap not only in respect of the difference between his weight and measure and our●, and the uncertainty of the proportions between them, whereby the one might be exactly reduced to the other; but also in respect of the difference between the Metal used by him and us; for that all Iron (or other Metal) of the like sort, is not always of the same gravity or ponderosity precisely because all is not of a like finesse or coarsnesse, and so that which is finest, will still be heaviest.  
Now Ghetaldus beginning to found out the weights of metalline Spheres, according to the usual known Weights with him, found that no diligence or industry of man could make a Sphere so exact as it aught to be, and therefore he procured a Cylinder to be made, and that of Tin, equal in height to the Diameter of its Base (which were of a certain magnitude in inches of the Roman Foot) for that this might be turned in a Lathe much exacter, and more easily than a Sphere: and hereby found, that a Cylinder made of Tin, being of one inch or 1/12 of the Roman Foot in its crassitude and altitude, did weigh 3 oun. 4 scrup. which reduced into grains of his weight, is 1824 gr. whose 2/3 being 1216 gr. is the weight or gravity of a Sphere of the same Metal, whose diameter is equal with the diameter or height of the Cylinder; according to the demonstrations of Archimedes, lib. 1. de Sph. and Cyl. prop. 32. It being there showed by him, that a Cylinder, whose Base is equal to the greatest Circle in a Sphere, and its altitude equal to the diameter of the Sphere (or of the said Circle) is sesquialter the said Sphere. And so Ghetaldus having found the weight of one Sphere of Tin (at a certain magnitude) from thence he deduceth the weight of any other Sphere, and that not only of the same Metal, but also of any other metal, by the proportions of Tin to the other Metals (in like Bodies of equal magnitude) found out by him at first.  
Now the foresaid weight of 1216 gr. (or 2 oun. 3 scr. 16 gr.) for a Sphere of Tin of 1 inch the diameter with Ghetaldus, will answer to the like kind of Sphere whose diameter is 0.967 inch from the English Foot, (according Mr. Greaves his foresaid comparison of the Roman Foot with the English) which being converted into our Troy-weight (according to Mr. Greaves his foresaid comparison of the Roman weight with our Troy-weight) is 924 2/3 gr. (viz. 1 oun. 7 dr. 24 2/3 gr. or 1 oun. 18 p. w. 12 2/3 gr.) and hence we gather the weight of a stanneall Sphere, whose diameter is 1 inch, or 1/12 of the English Foot, to be 1022.6 gr. english, (viz. 2 oun. 1 dr. 2.6 gr. or 2 oun. 2 p. w. 14.6 gr.) whereby we may easily obtain the weight of any other Sphere of the same Metal, and also of any Sphere of any the other metals, by means of the foregoing proportional Numbers between Tin and those other Metals: For so a Sphere or Bullet of Gold (supposed fine) of one inch diameter English-measure, will be found, (according to the proportion of 3895 to 10000, which is with Ghetaldus in his second Table of the comparison of several sorts of Bodies in gravity and magnitude, 38 18/19 to 100) or rather (the former terms being uncompleat) of 1554 to 3990 (which is with Ghetaldus in his first Table of the comparison of the same Bodies in gravity and magnitude, 1 to 2 21/37, and that's as 37 to 95 in the lest rational and absolute terms) to weigh 2625.6 gr. english, (which is 5.47 oun. ●roy, viz. 5 1/● ferè, being 5 ou. 3dr. 45.6 gr. or 5 oun. 9 p. w. 9.6 gr.) And so the like Body of fine Silver, of the same magnitude, will be found, (according to the proportion of 3895 to 5439, which is with Ghetaldus in his foresaid second Table of the Comparison of several Bodies in gravity and magnitude, 38 18/19 to 54 22/57) or more accurately (the former terms being not absolute or complete) of 1554 to 2170, (which is with Ghetaldus is his forementioned first Table of the like Comparisons, 1 to 1 44/11●, and that is as 111 to 155, in the lest terms rational and absolute, and which are by decimal numeration, as 1. to 1.396, 396 infinitely; or integrally, 1000 to 1396, or 1000, 000 to 1396, 396 exactly) to weigh 1427.95 gr. english (viz. 2 oun. 7 dr. 47.95 gr. troy, or 2 oun. 19 p. w. 11.95 gr.) In like manner will be deduced the weight of Iron and the other base sort of Metals, from Tin●e (in Bodies of like form and magnitude) But because the other Me●als besides Gold and Silver, as usually weighed with us, by the common Mercatory weight aforesaid, called Avoirdupois. Therefore for the more immediate and speedy obtaining of their weights from Tin (in spherical bodies) it is best to have the weight of the foregoing Stanneall Sphere of one inch diameter, viz. 1022 ●/5 gr. english, or 2.1304 oun, troy, (viz. 2 oun. 2 p. w. 14.6 gr. as before) reduced into avoirdupois unciall weight, and that decimally, whereby the weight of the like kind of Sphere of any other magnitude may be readily obtained; and so consequently the weight of any other metalline Sphere therefrom, in its proper weight of Avoirdupois. Which said weight therefore of 2.1304 oun. troy, will be inavo●rdupois unciall weight (according to the commonly received proportion of the Troy ounce-weight to the avoirdupois ounce-weight, 73 to 80) 2.3347 oun. (viz. 2 oun. 5.355 dr.) But according to Mr. Reynolds his proportion of these two weights the one to the other, which is, as 51 to 56 (being deduced from his proportion of the Troy-pound-weight to the Avoirdupois-pound-weight noted before, and which proportions of weight are the nearest aod truest that may be, as I shall straightway show) the same sphere will be 2.33928 oun. avoirdupois (viz. 2 oun. 5.428 dr.) which exceedeth the former weight only .073 of a dram, which in this kind of weight is altogether inconsiderable.  
Or for the ready obtaining of a Sphere of Tin, of any magnitude, (and so of any other Metal therefrom) in librall-weight, it is convenient to have the foresaid Sphere of Tin of 1 inch diameter, in libral-weight, both Troy and Avoirdupois, and that decimally; which will here be in Troy-weight, 0. 17753li. and in Avoirdupois-weight, (which is here chief to be regarded) according to the commonly received proportion of the troy-weight to the avoirdupois weight) 0. 1459189li. and according the other (and better) proportion, 0. 146205li. the difference of which from the other (by way of excess) being hardly, 0. 0003li. viz. 3 parts of a pound-avoirdupois, divided into 10000 equal parts; Which difference of weight, although it be here of little or no value (as also that in the operation immediately preceding) in regard of the smallness of the Body here handled, whereby the difference between the common Proportions of the Troy-weight to the Avoirdupois-weight, and the other proportions, may seem to be very small, and inconsiderable: Yet if we go to to reduce a metalline (or other) Body of a greater magnitude or dimenon, out of one of these Weights into the other (according to the said several proportions beforegoing) we shall found the difference of weight (in one and the same kind) to be still greater, and the greater or weightier that the Body is, the greater will be the difference of the weight produced by the said different terms of proportion, in respect both of librall and unciall weight, insomuch as that the difference of weight will many times be considerable: As in the foregoing Sphere of iron, of 4 inches the diameter, (english-measure) whose weight being found from Ghetaldus, to be 13. 461li. Roman, and from that, to be in Troy weight (according to Mr. Greaves his collations of these two kinds of weight the one with the other) 12. 283li. and than the same converted into Avoirdupois weight, according to the two several proportions between these two weights; the difference of weight was there found to be 5 drams avoirdepois, which is almost 1/3 of an ounce. And greater differences of weight (in this respect) than this, I shall show by and by, and also more afterwards.  
Now therefore, for the proportion between the Troy-ounce, and the Avoirdupois-ounce 
The Troy and Avoirdupois Ounce compared together; as also the several Proportions assigned between them.  according to the Terms 51 and 56 (deduced from the foresaid Maxim or Principle of librall weight, that 136li. Troy, and 112li. Avoirdupois are eqvilibral, or equiponderant, and so accordingly from thence, 1632 oun. Troy and 1792 oun. Avoirdupois, which are in the lest terms of Proportion, 51 and 56) the same being compared with the foresaid common terms, 73 and 80, will seem indeed to differ but little therefrom in the Reason itself, and that deficiently, as the terms here stand, from the less to the greater, like as the correspondent Terms before for librall weight, did from the common terms, being taken from the greater term to the less, viz 7● to 60, and 

Thomasius in the forecited place of his Dictionary, saith, that the Goldsa●ith, (or Troy) Ounce hath to the Avoirdupois Ounce, proportion ●●squiund●cimal, So that (to wit) 11 ounce troy are exactly equal to 12 ounces avoirdupois: and which (he saith) he proved by a m●st just and exact Balance. But this Proportion is somewhat greater than either o● the other two here noted, & makes the ounce Averd. to be more of the ounce Troy than either of those tw●.  
Mr. wingate in his A●ithm. l. 1 c. 1. Sect. 34. saith, that the avoirdupois. P●und is composed of 14 ●un. troy, and 12 p. w. viz. 14.6 oun. troy; which is according to the commonly r●ce●ved Prop●●tiors between the Avoirpois and Troy-weight: But according to the other proportions, it will be most truly but 14 oun. 11 p. w. and 10 2/7 gr. troy.   17 to 14: the Reason of the common terms, as they here stand from the less as the Antecedent, to the greater, as the Consequent, being by prolation, of the lesle inequality, or inequality of the less, sub-super-septupartiens septuagesimas-tertias, viz. subsuperpartient ●/73; and of the other Terms, (standing correspondently) sub-super-quintupartiens quinquage simasprimas viz. subsuperpartient 5/51; and so the difference of the Reasons, or the differential or residual Reason (according to the plain, simple and proper subduction of Reasons before shown) that of 1 to 560, vix. 1/56●; And this, for the conversion or reduction of Troy-unciall weight to the like Avoirdupois-weight; and so the Ounce-troy will be (in the lest terms) 80/73 or more truly, 56/51 the Ounce-avoirdupois: Or else, the Terms, in this position, from the less to the greater, might seem rather to be taken contrarily (according to an exact comparison) for the proportion of the Avoirdupois-ounce, (as the less) to the troy-onnce (as the greater) for so the Avoirdupois-ounce, will be 73/8●1 or rather 51/56 of the Troy-ounce. And these latter Terms of proportion, do make the Ounce Troy more to exceed the Ounce-avoirdupois, than the first or common Terms do, and so make the Ounce-avoirdupois to come short of the Ounce-troy (or be less of the same) accordingly; as may more easily be seen, by reducing of the foresaid Terms of proportion into decimal terms; for so, the proportion of 73 to 80, will be as 1.00000 to 1.09589, and of 51 to 56, as 1.00000 to 1.09804 ferè: And contrarily; by permutation of the Terms, the proportion of 80 to 73, will be decimally, as 1.0000 to .9125 exactly (as I shown formerly, upon another the like occasion,) and of 56 to 51, as 1.0000 etc. to .9107 etc.  
And therefore, now to show (by the way) the difference between the foresaid 
The two foregoing several Proportions between the Troy and Avoirdupois Weights compared, and examined experimentally, by the Balance, in the foregoing Iron Bullet.  several proportions of the Troy and Avoirdupoiz weights both librall and unciall, the one to the other, by comparing them with the weight of some Body, taken both by Troy and Avoirdupois-weights, and that especially in one and the same Balance, and so converting it out of the one kind of weight into the other, by the said two several kinds of proportions between them, whereby may be known which of these are the nearer and truer, as most agreeing with the Balance itself: We will here take the foregoing Cannon-bullet of cast-iron, of 4.8 inches the diameter, whose weight we found (as I have noted before) in the Tower of London, by an exact Balance, with Weights both Troy and Avoirdupois, to be 19li. 1 oun. 15 p. w. 15 gr. Troy (which is 19li. and 1 25/32 oun. or wholly in librall-weight, 19 29/12● or 19. 1484375li. exactly) and 15li. 12 1/4 oun. Avoirdupois (which is wholly in librall-weight, 15 49/64 or 15. 765625li. completely) Now if we shall convert the said Bullet from its Troy-weight to Avoirdupois-weight, according to the common proportion of the Troylibrall-weight to the Avoirdupois-librall-weight, viz. 73 to 60, we shall found the same to be (in the lest terms) 15 1725/2336 or 15. 73844li. Avoirdupoiz, which wanteth of the true weight from the Balance, 0. 02718li. which is by reduction or conversion into the proper denominate, compounding parts of this weight 0. oun. 6.958 drams avoirdupois, which is upon the point of 7 dr. and that's almost half an ounce avoirdupois. But according to the other proportion of the Troy librall-weight to the Avoirdupois librall weight, viz. 17 to 14, the said bullet will be found to weigh (in the lest terms) 15 837/1088 or 15. 7693li. Avoirdupoiz, which exceedeth the weight, not fnlly one dram, being but 0.94 dr. as will appear by reduction of the fractional terms into the proper denominate parts of this weight, and which is not considerable.  
Again; if we will reduce the said Bullet taken wholly in Troy-unciall-weight, being 229 25/32 or 229.78125 oun. exactly, into Avoirdupois unciall-weight, according to the said several proportions of the one to the other, we shall found the very same differences necessarily to hap: for according to the common proportion of the Troy-unciall-weight to the Avoirdupois, viz. 73 to 80, it will be found (in the lest terms) 251 119/146 or 251.815 oun. avoirdupoiz, which wanteth of the true or ballance-weight, (being 252.25 oun.) 0.435 oun. which by reduction, gives 6.96 drams as before: But according to the other proportion of the Troy unciall-weight to the Avoirdupoiz, viz. 51 to 56, it will be found 252 21/68 oun. or 252.3088 oun. avoirdupoiz, which differeth from the ballance-weight (by way of excess) only 0.0588 oun. which is by reduction, 0.94 dr. as before.  
And so again contrarily, if we work from the Avoirdupoiz-weight to the Troy-weight, we shall found the like proportional differences of weight accordingly. For first if we convert the foresaid bullet, out of its true avoirdupois-weight from the Balance, 15 40/64li. or 15.7656, etc. into Troy-weight, according to the vulgarly received proportion of 60 to 73, for librall-weight (being the converse of the former) we shall found the same to be (in the lest terms) 19 697/384●, or 1815104li. Troy, which differeth from the true weight of the Balance (19 19/12● or 19.1484375 li) by way of excess, 0.0330729li. which is by conversion into the proper, compounding denominate parts of this weight 0. oun. 3 dr. 10.5 gr. troy, or 0. oun. 7 p. w. ●2. 5 gr. But according to the other proportion of 14 to 17, it will be found (in the lest terms) 19 129/896, or 19. 1439732li. Troy, which differeth from the true weight (by way of defect) only 0. 0044643li. which is by a continual reduction into the lest parts of weight, but 25.7 gr. or 1 p. w. and 1.7 gr. which is of little or no value in this thing.  
And so likewise if we reduce the said Bullet out of its true avoirdupois weight taken wholly in ounces (according to the Balance) being 252 1/4 or 252.25 oun. into Troy weight by ounces; first, according to the common proportion of 80 to 73, for unciall-weight (being the converse of the former) we shall found the same to be (in the lest terms 230 57/320 oun. troy, or 230.178125 completely, which exceedeth the true weight, 229 25/32 oun. troy, or 229.78125 exactly, by 0.396875 oun. troy exactly, which is by reduction into the lest parts of weight, 3 dr. 10.5 gr. or 7 p. w. 22.5 gr. as before from librall weight: But according to the proportion of 56 to 51, it will be found to be (in the lest terms) 229 163/224 oun. troy. or 229.727678, which is deficient from the true weight, only 0.053572 oun. which by the like reduction, is but 1. p. w. and 1.7 gr. as before from librall weight.  
By which operations it is sufficiently evident, that these latter Proportions between the Troy and Avoirdupoiz weights, are the truer, (and indeed the nearest and truest that may be found) and which I shall (upon the like occasion) further confirm afterwards by a double experiment from the weight of a liquid body, in the measuring of Vessels.  
And thus much by the way concerning Weight and Measure in general, in reference to the work here in hand, (being the like artificial Dimension of metalline regular Bodies, for the speedy discovering of their gravities or weights, and more particularly of a ferreall or iron Sphere, as was formerly of a Sphere in general for solid measure) being induced thereunto by Ghetaldus in his foresaid work of Metals, in which he differeth from us in both these, as we have abundantly showed.  
And so these operations beforegoing in the particular metalline Spheres aforenamed, for the weight thereof, are from the experiments of M. Ghetaldus, & reduced from his weight and measure to ours, according to the observations and experiments of our Countryman Mr. Greaves upon the same, and his collations of them together, as aforesaid; And also in one of them, from my own experience, according to our English weights and measure; with which we must rest contented, till some other experiments be produced, both in these, and also in the other Metals, from our English weight and measure, and which we may expect from Mr. Reynolds aforesaid, who hath taken great care and pains, and used much industry therein.  
As for the weights of Metals compared in Spheres of one and the same magnitude, set down in the latter part of Mr. Ponds Almanac, in Troy-weight (where also are noted the foresaid common proportions between Troy and Avoirdupois weights) they are arcording to the experiments of Ghetaldus, being deduced from his proportional or comparative numbers, into the parts of Troy-weight, though not very precisely; which therefore I have here put most correct and exact thus; supposing (with him) first a Sphere of Gold, to weigh just one pound-troy, and from thence, the other metalline Spheres of the same magnitude, to weigh accordingly, as followeth.  
  oun. p. w. gr. mi. gr.   1. G. 12. 00. 00. 00. 5760. ☉ 1 2. QS. 8. 11. 10. 6. ferè 4114 2/7 ☿ 2 3. I. 7. 5. 6. 6. 3486 6/19 ♄ 3 4. S. 6. 10. 12. 12. 3132 12/19 ☾ 4 5. B. 5. 13. 16. 8. 2728 8/19 ♀ 5 6. I. 5. 1. 1. 5. 2425 5/19 ♂ 6 7. T. 4. 13. 11. 7. 2243 7/19 Jupit; 7  
But now to return a little to our foregoing work of the artificial dimension, or diametral and circumferential cubation of a Sphere of cast-iron, for the weight thereof, which as we shown before by the Integer of weight itself (or by librall weight) So we will next show how to perform the same by the composing, denominate parts thereof immediately, viz. ounces avoirdupoiz (or uncial weight) And the Line or Scale of equal parts for this purpose, in respect both of the foresaid common Tenent of the weight of an iron-Sphear or bullet, and also our own experiment, I find (according to the reason of the precedent second Theorem, and also our general reason) to be for the Diameter of such a Sphere, (as to a centesimal partition of the measure given) 0.76 inch, which is but very little above 3/4 of an inch: and the Line for the Circumference, according to the common Tenent, to be 2.40 inches ferè, and according to our experiment, 2.39 inches ferè, (so that here also one and the same Line of measure may indifferently serve in both) which two Lines being divided as the former, and than the Diameter or Circumference be taken by their proper respective Line or Scale, and cubed, the same shall be the weight of the Sphere or Bullet in ounces and decimal parts immediately: For so the Diameter of the foresaid bullet of 9li. and 2 oun. ferè avoirdupois, (viz. 9.123626 lively ferè) being 4 inches, and the Circumference, 12.57 ferè, will be found each of them, by their proper cubatorie Line or Scale, for unciall gravity, (being made 100 parts) to be 5.27 ferè, which cubed, yields 146.363183 ferè, for the weight of the Bullet in ounces, which exceedeth the true weight being 145.978009 oun. by 0.385174 oun. ferè, which by conversion or reduction, yieldeth about 6 drams, and which in a thing of this nature is not considerable: But yet if we will stand more precisely upon the weight of this Bullet, if than we divide the two foresaid artificial Lines of measure for this purpose into more parts, as 1000 (for the natural Line, or the Inch being so divided, the artificial Line will be thereof, for the Diameter, 0.760 ferè, and for the Circumference, 2.387 ferè) & so measure the Diameter & Circumference of this Sphere or bullet thereby, we shall found the same to be severally, 5.265, which cubed, gives 145. 946984625 ounces for the weight of the bullet, which wants of the true weight aforesaid, hardly half a dram, which in this kind of weight is as near as need be desired.  
And so again in the other Bullet of 15li. & 12 1/4 oun. avoirdupoiz (or 252.25 ounces) weighed by us; if the Diameter or Circumf. noted formerly, be taken by these Lines under a centesimall division, they will be found each of them to be 6.32 ferè, whose Cube is 252.435968 oun. ferè, which exceedeth the true weight, only 0.185968 oun. which by conversion, gives near upon 3 drams, and which is not considerable: But being measured by the same Lines under a millesimall partition, they will be found each of them, 6.318, which cubed, affords, 252.196389. etc. ounces, which now wants of the true weight, not fully one dram being but 0.86 dr.  
And so also if the Diameter or Circumference of this Bullet be taken by their proper respective Lines of measure for librall weight beforegoing (viz. 1.91 inch, for the Diameter, and 6.01 inches, for the Circumference) the same will be found severally (according to a centesimal partition of the Lines) 2.51 ferè, whose Cube is 15. 813251 ferè for the weight of the bullet, (which by reduction is 15li. 13 oun. and 0.19 dr. avoirdupois) exceeding the true librall weight, viz. 15.765625 lively (or 15 lively and 12.25 oun.) only 0.047626 lively ferè, which by conversion into the proper parts of this weight, gives about 3/4 of an ounce, viz. 12.19 dr. But being measured by the said Lines in a millesimal partition, (for the natural Line, or the Inch, being 1000 parts, the artificial Line will be thereof for the Diameter, 1.914, and for the Circumference 6.014 ferè) they will be found severally, 2.507, which cum bed, yields 15.756617, etc. (viz. 15li. 12 oun. 1.69 dr) wanting now of the true weight only 2.3 drams.  
Now if any shall upon occasion, make use of the Troy-weight in a Sphere of this 
The artificial Lines of measure, for the most speedy discovering of the weight of a Sphere or Bullet of cast-Iron, in Troy-weight, both librall and unciall.  metal (though indeed this kind of weight is not usual for any Metal besides Gold and Silver, as I have noted before) than the artficial Lines of measure for the speedy discovering of the weight thereof by the diameter, will be 1.794 inch for librall-weight, and by the Circumference, 5.636 inches: And the Lines for unciall weight, will be for the Diameter, 0.784 inch ferè, and for the Circumference, 2.462 inches.  
And now albeit no art or industry of man, can make a Sphere of metal or other matter, so very exact and precise indeed, as it aught to be (according as I noted before from Ghetaldus) Yet considering that the weight of this and the other coarser kind of Metals in any thing, is not so precisely stood upon as the weight of Gold and Silver, (being precious Metals) So the exactness of a Spherical body made of any of them for ordinary use (as a Cannon-bullet, which is commonly made of cast-iron, as I noted before) is not so much stood upon, so as it come something near the same: and indeed there is hardly any Cannon-bullet, or other bullet for shooting, but is so round (being cast in a Mould) as that the Eye can hardly adjudge or discern it to be not exactly orbicular or spherical; and so the weight thereof can be very little mistaken, being obtained by the diameter or Circumference of the same, taken in a certain set Measure, either natural, as Inches, or the like; or artificial, as deduced therefrom, according as I have here showed at large. And which several dimensions with many other the like metrical Conclusions pertaining hereunto, and the Converse of the same (and all chiefly in reference to Gunnery) I will next express proportionally by Number, (from our foregoing experiments) according to the aforesaid Metal and Measure commonly used in this thing, and the Weight both Avoirdupoiz and Troy, and in each of these, both by the librall and unciall-weight together, which in the several Sections of proportions following, are (for brevity-sake) noted by the letters of distinction, l, and u.  
1. The Cube of the Diameter in measure, is to the Sphere itself in weight, as 1. to .1425566. l. Avoirdupoiz. weight. 2.280906. u .17314487. l. Troy-weight. 2.0777384. u Circumfer. .00459767. l. Avoirdup. .07356273. u .0055841877. l. Troy. .06701025. u  
Conversly.  
2. The Sphere in respect of weight, is to the Cube of its Diamet. in respect of measure, as 1. to 7.0147552. l. Avoirdup. cubique measure. 0.4384222. u 5.7755104. l. Troy. 0.4812925. u Circumf. 217.50144. l. Avoirdup. 13.59384. u 179.07707. l. Troy. 14.923089. u  
Hence,  
3. The Diamet. is to the side of the Cube equal to the Sphere in weight, as 1 to .522391. l. Avoirdup. Linear measure. 1.316343. u .5573609. l. Troy. 1.276038. u Circumf. .16628227. l. Avoirdu. .41900507. u .1774135. l. Troy. .4061755. u  
Than for the speedy discovering of the weight of a spherical or any other body whatsoever made of this Metal, by the solidity thereof, in the Measure aforesaid: and contrariwise the solid content by the weight thereof, both Avoirdupoiz and Troy, and in each of these both librall and unciall weight, the Proportions will be as followeth.  
1. As 1. to .27226314. l. Avoirdupois. Weight. 4.3562103. u .33068234. l. Troy. 3.968188. u  
Contrariwise.  
2. As 1. to 3.672917. l. Avoirdup. Solid measure. 0.229557. u 3.02405. l. Troy. 0.25200418. u  
And so likewise for discovering the gravity of any Body of fine or forged Iron by the magnitude, in the foresaid Measure, and contrá: the proportions will be from the forementioned experiment made upon this Metal, as followeth.  
1. As 1. to 0.3474. l. Troy. Gravity. 4.169. u 0.2861. l. Avoirdupois. 4.5777. u  
Conversly.  
2 As 1. to 2.878, l. Troy. Magnitude. 0.2399 ferè u 3.495. l. Avoirdupois. 0.218. u  
And thus having showed our artificial Mensuration in regular Solids, aswell for gravity or ponderosity, as for solid (and superficial) measure; I shall now close up this Section and Part, with our third theorematicall Proposition, (answering to the third principal problematical or practical Proposition in the first Part) and the practical demonstration thereof, in which our more particular, or special reason of the like dimension of all regular-like Solids (as particularly of a Cylinder) in the several respects aforesaid (in reference to the producing of the artificial Lines of measure, for performing the same) is contained.  

THEOR. III Expressing particularly, the artificial Lines for the solid dimension of a Cylinder, from our particular or special ground and reason formerly declared: And consequently, the Lines for the like dimension of a Cone, and all other regular-like Solids in general.  
IF the two proper Dimetients aforenamed, of a (right or erect) Cylinder, exactly adequate to the cubical, (composing or resolving) Parts of the Rational Line, shall be obtained according to an exact congruency or congruity; The same shall be the proper, respective artificial Line of Cyhnoricall solidation, according to the Parts: And th●●e will be of them to the correspondent or congruall▪ Lines or d●m●nsion according to the whole Measure, the Reason that is of the Parts to the Whole; which is as the Reason of their respective Cubes.  
THe like reason holdeth for the cylindrical Circumference conjunctly with the Dimetient of altitude: and also for both these in the Cone, in respect of its basiall Dimetient and Periphery with the Dimetient of altitude, for solid dimension, and with its Side for superficiary dimension; and also for the basiall Dimetient of a Cylinder with its Side, in respect of superficial, dimension: And so the like with these, for all Pyramids and Prismes constituted upon regular Bases, in respect both of solid and superficiary dimension, according to the several ways formerly declared and demonstrated in all these Figures: only respect being had in all these, to the quadrate parts of the Rational Line for superficiary dimension, as is here to the cubique parts for solid dimension.  
And the like reason will here hold in all these solid Figures, in respect of weight or gravity in any Metal whatsoever, as for their solid measure, according to what I shown before in the Sphere, and consequently in the other re-Bodies, being made of Metal (respect being here had to the parts of the weight proposed, as in the Theorem itself, is to the parts of measure, as I shown before in the like case upon the 2d Theorem) so as that the like artificial Lines of Measure being extracted for these Figures severally in reference to weight, according to any Metal, Weight, and Measure proposed; their gravities will be thereby obtained in the same manner as their solid magnitudes or measures, according to the several ways formerly declared and demonstrated for the same.  
And now because of the two several Dimetients (or dimensional Lines) here continually concurring, there occurreth some more variety, than in the two preceding Theorems: Therefore I will here give a full demonstration or illustration of this, in an Arithmetical manner; by which those two (with all the things necessarily depending on them) as well as this, may be plainly understood, seeing they be all grounded upon one and the same reason.  
THerefore, let the Rational Line (in respect of its properly composing denominate parts) be R 12. So the Cube thereof, CR, 1728; to which be conceived or constituted a Cylinder, exactly agreeable for magnitude or dimension, & so, as that its two foresaid Dimetients, be in exact congruency, or Congruity (such as the Greeks (Euclid & Proclus) call 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉; the Latins, (as Congruentia, or Congruitas: & so the greek 
Eucl. Axiom. come. 8. Ram. Geom. lib. 1. ●l. 9 & Scholar mathemat. lib. 8. prop. 4.  words do generally signify) which therefore we are here next to inquire. Therefore seeing than that like Solids in general, do mutually hold in a triplicate or cubique reason of their homologal Terms by E 8. p. 19 & 27; and more particularly, E 11, p. 33. and E 12. p. 8. and so in special, Cylinders of their basial Dimetients (or other like Terms) by E 12. p. 12. To this I assume another Cylinder, whose two Dimetients are already given in the same kind, with those supposed in the former Cylinder, which (in the first Numbers from an unit, absolute, making the whole Cylinder absolute in all its dimensions, according to the more ancient and vulgar Cyclometricall terms (in respect of the Base) but not the newer) let be (under the general notion of a Binomiall in relation to the Rational Line taken as before) A = D R 12+2; whereupon will arise the Cylinder, rational and absolute (according to the said vulgar Cyclometrical Terms, in respect of the Base, and binomially in reference to the Cube of the Rational Line taken as before) CR, 1728+428; but according to the newer, and most approved ●etragonismall Terms, in respect of the base, and first in the more vulgar or Metian expression noted formerly; for the performing of this operation after the more usual or vulgar way of Numeration, (and for that this expression is finite and limited in itself, whereas the decimal expression runs infinitely) it will be irrational and inabsolute, CR, 1728+427 15/113 (which is by decimal conversion of the fractional terms, CR, +427.132743, etc. very nearly agreeing with that which is produced by the proper decimal Cyclometricall terms, in respect of the base of the Cylinder; and that both from themselves simply or naturally, and also artificially, from their ●ogarithme, as shall be showed by and by) than by E. 12. p. 12 aforesaid, and also by the reason of ●. 7. p. 19 it holdeth in a triple or cubical reason of the aforesaid Cylindrical Dimetients conjunctly, as the Terms homologal, thus;  
CR 1728+427 15/113. CR 1728:: CR 1728+1016 (viz. CA = D. R 12+2 in CR 1728+427 15/113): CR 1728+472 38416/24353● or (by their greatest common Divisour, or number of exact commensuration, 686) 472 56/355 in the lest terms of the same reason (viz. CA = D in CR 1728) which is decimally, CR 1728+472. 157746, etc. whose Root or Side irrational or ineffable, (and binomiall in reference to the Rational Line, taken as before, and according to a decimal extraction of the cubical Gnomon) R 12+1. 0062, etc. is A = D in CR 1728, being the thing first sought for; or the artificial Line according to the parts of the natural Line of measure proposed: which from the most admirable invention of the Logarithmes, is most speedily performed by a simple Composition and Resolution of Numbers, together with a Trichotomie of the cubical Logarithme; the first or greater Cylinder, being raised from the most exact Cyclometricall Logarithme, (in respect of the Base) answering to the most exact decimal Terms aforesaid.  
 
Or again, seeing that between two like Solids, there do necessarily intercede two mean Proportionals, by E. 8, p. 12, and 19; the two Means between these two Cylinders, will be found (binomially, in relation and comparison to the foresaid Rational Cube, and according to a decimal enumeration of the cubick Gnomon) CR 1728+274. 15289, etc. and CR 1728+132. 03231, etc. most compendiously and exactly, by logarithmical Computation, thus;  
CR 1728+427. 13256, etc. 3, 3334739884 S CR 1728 3, 2375437381 S Dif. 0, 0959302503 A 1/3 dif. 0, 0319767501 A CR 1728+132. 0323, etc. 3, 2695204882 dat Mean. CR 1728+274. 152, etc. 3, 3014972383 greater Mean.  
Or the same may be obtained more plainly, (though not so readily) in a dis-junct manner, at two distinct operations, by the Analogy which holds from the Cube of one Solid to the Cube of the mean Proportional falling next to it, as doth from that Solid to the other propounded, between which the two mean Proportionals are required.  
Than seeing the greater Mean serveth here our present purpose; it followeth according to E 7. p. 19 thus; CR 1728+427. 13256, etc. CR 1728+274. 15289, etc.:: A = D, R 12+2: A = D, R 12+1. 0062, etc. as before: which appeareth plainly and briefly, by logarithmical or artificial Numeration, thus;  
CR 1728+427. 13256, etc. 3, 3334739884 S CR 1728+274. 15289, etc. 3, 3014972383 A A = D, R 12+2 1, 1461280357 A Sum. 4.4476252740 A = D, R 12+1. 0262, etc. 1, 1141512856. Dif.   
Agreeing exactly with the first logarithmical operation.  
Or again, the said A = D in CR 1728 required; found more briefly yet, after this manner.  
CR 1728+427. 13256, etc. 3, 3334739884 S CR 1728 3, 2375437381 S Dif. 0, 0959302503 1/3 dif. 0, 0319767501 S A = D, R 12+2 given, 1, 1461280357 S A = D, R 12+1. 00622, etc. inquired. 1, 1141512856 Dif.  
Which being found out these three several ways; it followeth lastly, according to the consectary of the third Theorem, in this simple Analogy. R 12: R 1:: A = D, R 12+1. 0062, etc. A = D, R 1+. 0838, etc.  
Which last found Number, is for the said congruall Dimetients of a Cylinder, according to the unity of measure; and so for the artificial (or second Rational) Line according to the whole entire natural (or prime Rational) Line in general.  
And what manner of working is used here, for the investigation of this artificial Line in the Cylinder; the same is to be used for the like in the Cone: and which (though needless) I will here further illustrate in the same.  
Therefore, suppose here the foresaid Rational Line (taken in its composing, denominate parts) R 12. and to its Cube, C R 1728, a (right, or Isoskelan) Cone exactly adequate (as the Cylinder before) having ●ts two like proper Dimetient-lines exactly congruall (as those of the Cylinder) in the investigation or inquisition whereof conjunctly, consisteth the first operation. Therefore seeing than, that as Cylinders (and other like Solids) so Gones, are respectively one to another, in a triple or cubical Proportion of their Basiall Dimetients (or other correspondent Terms of dimension) by E 12. p. 12 aforesaid: To this be supposed another like Cone, of given or known Dimetients, and exactly congruall, as the former; which let he the same with those of the foregoing Cylinder, included in the Binomie, A = D, R 12+2; whereupon this Cone will be subtriple that Cylinder, viz. according to the vulgar Numeration, from the Metian Cyclometricall-Terms, as to the composition of the Base (under the common acception of an Apotome, in reference to the foregoing Rational Cube) irrational or unabsolute, C R 1728-1009 211/339 (which is by decimal conversion of the ●raction-part, 1728-1009. 622418, etc. and which very nearly agrees with that which is produced by the most exact (decimal) Cyclometrical terms in regard of the Base, and that both naturally or simply from themselves, and also artificially from their correspondent Logarithme; being C R 1728-1009. 622480. as will appear in the logarithmical operation following) Than by E 12. p. 12, and also by the reason of E 7. p. 19 before cited, it holdeth in a triplicate or cubique reason of the aforesaid conical Dimetients conjunctly, as the Terms homologal, thus;  
C R 1728-1009 211/339: C R 1728:: C R 1728+1016 (viz C A = D, R 12+2 in C R 1728-1009 211/339): C R 1728+4872 115248/24353●; or rather in the lest homologal fractional terms (by their greatest common Measure, or Number of exact Symmetry, 686) C R 1728+4872 16●/35●, viz. C A = D in C R 1728; which by decimal dinumeration, is C R 1728+4872. 473239, etc. whose Root or side cubical, and irrational or inexplicable, is (binomially in reference to the foresaid Rational Line, and according to a decimal eradication, or resolution of the cubique Gnomon, R 12+6. 7723, etc. for A = D in C R 1728, being the thing first inqu●red; or the artificial Line of measure, according to the parts of the natural Line proposed: And which may most easily and readily be obtained by logarithmical or artificial Numeration, by a simple Composition and Resolution, or Prosthaphereticall suppu●ation only; and that the several ways which I shown before in the Cylinder; whereas the working out of these things by the natural Numbers (or the vulgar way of Numbering) is most laborious, and intricate, by reason of the many large and tedious Multiplications, Divisions, and Radical extractions, both quadrate and cubique, arising from; which notwithstanding I have also done both in this of the Cone, and that of the Cylinder, and likewise in other Figures, where the same might be conveniently performed after this manner, as in the Circle and Sphere, for the finding out of their respective artificial Lines of Mensuration before set forth.  
But now to the operation of our present question in the Cone, by the artificial or logarithmical Logistique; according to a cubical proportion of the aforesaid homologal Terms, thus;  
 
Which differeth from the number found by the natural operation, in the decimal fraction, but very little, that giving it .7723 etc. this .7582 etc. which breedeth no sensible difference in the Conclusion of the work; but however the number produced by the logarithmical operation is the truest; the second assumed Cone (though the first in place here) being produced from the exactest Cyclometricall Logarithme, as to the composition of its Base, upon which it is raised and constituted.  
Or again secondly, by finding out the two Mean Proprotionals between these two Cones, which is most speedily and accuratly performed also by virtue of the Logarithmes, as before in the Cylinder; and so they will be found (under the two common Apotomies, in reference to the Rational Cube, and according to a decimal Analysis of the cubical Gnomon) C R 1728-765. 465285 etc. for the lesser Mean; and 1728-438 .3256268 etc. for the greater Mean, thus,  
 
Or the same may be found more disjunctly, according to the Analogy before declared in the Cylinder, for the finding of the two Means; the reason being the same here.  
Now the lesser Mean serving here our purpose; it followeth thence Analogically thus: C R 1728-1009, etc.: C R 1728-765. 465, etc.:: A = D in C R 1728-1009, etc. viz. R 12+2: A = D in C R 1728, viz. R 12+6. 7582, etc. as before; as appeareth briefly and plainly, by this subsequent Logarithmeticall Calculation.  
 
Exactly agreeing with the first logarithmical operation.  
Or again thirdly and lastly, the same Dimetients in the Cone, C R 1728, found out conjunctly and congrually, most briefly of all, thus,  

C R 1728-1009. 6224 etc. 2,8563527 S  
C R 1728-3,2375437 S  
Dif. 0,3811910  
1/3 0,1270637 A  
A = D, R 12+2 given 1,1461280 A  
A = D, R 12+6. 7852, inquired 1,2731917. aggreg.   
Which being had these several ways; it followeth lastly thereupon, according to the reason of the Consectary of the foregoing Theorem, by way of Analogy, in these Terms; viz,  
R 12: R 1:: A = D, R 13+6. 7852 &c: A = D, R 1+. 56318 etc. Or R 12: R 12+6. 7852 etc.:: R 1: R 1+. 5632 ferè. Which last found Number is for A = D in the Cone answering to C R 1; and so for our artificial Line of rectangle parallelepipedation (according to an exact quadrate Base) or parallelepipedall consolidation of a Cone desired, in reference to the whole entire natural Line of measure in general.  
And after the same manner may be found cut the like artificial Lines for the consolidation of a Cone, and Cylinder, by their basiall Peripheries and altitudinary Dimetients together, as hath been here done, by their basiall and altitudinary dimetients together: (which basiall Dimetient and Periphery do continued the same throughout the Cylinder, it being of equal crassitude, but continually altar in the Cone) And these Lines we shown before in the practical demonstration and use of the other two Lines of solidation pertaining to these two Bodies, to be for the Cylinder, of the Rational Line, 2+. 3266 etc. and for the Cone 3+. 3531 etc. and these by the natural way of working, from the Metian Cyclometricall terms, in respect of the Base of the Cylinder and Cone: and by the artificial or Logarithmeticall way, I found the same to be for the Cylinder 2+. 32489 etc. (very little differing from the other) and for the Cone, exactly agreeing with the former: And therefore I shall not need (I conceive) to make any more ado about the demonstration hereof: And the same way holdeth for the extraction of the like artificial Lines for the Superficial dimension of the Cylinder and Cone, and also for the dimension both solid and superficial of all regular-based Pyramids and Prisms: And for all these Figures, not only in respect of solid and superficial measure, but also of gravity or weight, according to any Metal proposed; as I have noted before upon the 3d. Theorme.  
And all these things, together with the like in the other Figures, , may be yet more readily obtained, and that according to our general ground and reason of this artificial Dimension (or of our artificial metrical Lines) which was formerly said to consist in unity alone. For as in our particular, partial, or special Reason thereof, every particular Measure propounded (as the prime or natural Rational Line) being considered according to its Parts, (composing or constituting, or otherwise dividing) every particular kind of Figure, or figurate Magnitude, measurable in this way, was consequently said to be considered in the first or second Power of the same (as in the powers of Number) according to the nature and kind of the Dimension; for the producing of the artificial Lines, (according to the three foregoing Theorems) So here in our general Reason, or Reason of the Whole, or of Unity; every Measure being considered in itself simply and absolutely, (as was formerly declared) every such figurate Magnitude (as aforesaid) is to be considered accordingly, in the like Powers of Unity (as of the whole entire Measure in general) according to the nature of the Dimension. And so again, as in the performance of these Dimensions, by the natural Measure proposed (or the prime Rational Line in general) in an Arithmetical manner, by way of Proportion (according as hath been showed in the several dimensions beforegoing, among the dimensionall Proportions) for the more immediate producing of the sides of the Squares and Cubes, etc. equal to the Figures proposed to be measured, and which are immediately given by our artificial Lines of measure; we begin with Unity simple, and proceed from that to Number simple or linear, being some Root of a Figurate Number superficial or solid; & so to the figurate Number itself; as from some dimensional line of a Figure (by which the same is proposed to be measured in a quadratary or cubatory manner, according to the nature of the Dimension; or by which the side of the equal Quadrat or Cube of the Figure is to be produced, either naturally in an arithmetical analogical manner, as aforesaid; or artificially, by a Line of measure convenient for the purpose) to the side of the equal Quadrator Cube; and so come to the Content of the Figure itself? Or we come to the same more immediately, and also most naturally and properly; proceeding from Unity taken in the power thereof, prime or second, (according to the nature of the dimension propounded) to Number figurate, and than from that to the correspondent Root thereof; as from some dimensional Line of a Figure, in the aforesaid powers thereof, to the Figure itself; and than from that to the side of the equal Quadrat or Cube. So in this (for the immediate producing of the artificial Lines) we proceed from Unity figurate in the first or second power thereof, according to the nature of the Dimension, to a Number figurate or potential, accordingly, and from that to the correspondent Root thereof; as from every kind of Figurate Magnitude in general, falling under our artificial Dimension, to the like powers of some one of its dimensional lines by which it is proposed to be thus measured (or squared and cubed, or otherwise artificially measured, according to the nature and kind of the Magnitude propounded, and of the Dimension itself) and so from thence to the said dimensionall line itself of the Figure, for the artificial Line of measure required: which as in all exactly ordinate or regular Figures, it is but one such line simply; so in all regular like-Figures it is of two together according as we have lately showed. And so we might hereupon raise a 4th. and general Theorem, if it were needful; but that which I have already said concerning the same, is very sufficient.  
And what hath been here spoken for solid Figures, in reference to their solid measure, is to be understood in the same accordingly, for gravity or weight (according to any Metal, or other like ponderous matter proposed) For as before, according to our particular or special reason, every Integer of weight whatsoever proposed, was said to be considered in its parts taken simply in number; and so every kind of metalline, or other like ponderous (regular and regular-like) Body, was accordingly to be considered in the same, as in the like parts of Unity: So here the Integer of weight propounded, is to be considered simply absolutely and entirely, as in the nature and reason of unity itself; and so every kind of metalline (or other like ponderous) regular and regular-like body, is to be considered accordingly, for the producing of the artificial Lines of measure therefrom, for the speedy discovering of the gravity or ponderosity of any such Body proposed, in any magnitude whatsoever.     

PART III. Containing that kind of Dimension, or metrical practice, which is commonly called, the Gauging of Vessels; after a most artificial, exact, and expeditionall manner.  

SECT. I. Concerning the measuring or gauging of Vessels, in general.  
ANd what hath be●● here done for the solid Dimension of a Cylinder (or the dimension of a solid Cylinder) by artificial Lines found out for the same, according to any Measure appointed: the like may be done for the liquid dimension of a concave Cylinder (or cylindrical Vessel) for the more speedy finding out of the liquid Content, according to any Liquor and Measure given. As our Vessels for Wine and Ale, or Beer (which too, most commonly come to be measured) and other liquid things, which though they be not absolute Cylinders in themselves, yet may be (and commonly are) conceived or supposed as Cylinders, (being reduceable by Art thereunto) for the better measuring of the same, by finding out and taking the Mean between the Diameter at the Head, and the Bung of the Vessel (or otherwise the Mean between those two Circles) and so obtaining the liquid Content thereof; which is commonly called Gauging of Vessels; the Measure by which these Vessels are thus valued or estimated, being usually a Gallon, and which is the greatest of our liquid Measures, and but the beginning, as it were, of Vessell-Measure; but in Wine, and Ale or Beer, holdeth not one and the same; but differeth very sensibly, as is commonly known: And therefore we will next briefly show the measuring or gauging of these cylindrical Vessels (or Spheroidall, as some will have it, this kind of Vessel being more commonly taken for a Sphaeroides, having the two ends equally cut of; though for mine own part, I conceive this kind of Vessel may more properly be termed a Cylindroides, by the same reason that a Sphaeroides and Conoides are so called; this having the same similitude or resemblance to an exact Cylinder, that those have to an exact Sphere and Cone) by Lines of measure peculiarly appropriated and applied to them (as we did before in the Cylinder in general for solid measure) according to the different kind and quality of the Liquor, and so the different quantity or magnitude of the liquid Measure given, in relation to solid Measure in inches.   

SECT. II. Setting forth the Quantities of the Wine and Ale-gallons, in reference to the gauging of Vessels.  
IT is generally holden by Artists about the City of London, that a Wine-Gallon containeth in its concave Capacity, 231 cubical or solid inches, or is insensibly differing therefrom: But for the Ale or Beer-Gallon, I find the same to be as generally controverted among them. Mr. William Oughtred, a reverend Divine, and most eminent Mathematician, beforenamed, after some experiments made by him to found out the solid content of this Gallon in inches, besides the experiments of some others which came to his sight, finding some difficulty therein, in regard both of the irregularity which he observed to be usually in the several Standard-Gallons which he met with, and also their disagreement one from another in their Contents, as himself confesseth and declareth in his foresaid book of the Circles of Proportion, Part 1. chap. 9 looketh back there to the first ground and principle of our English Measuring from Barley-corns: and so at length he cometh to a rational conjecture of the Ale-gallon (and that very neatly, and pretty nearly also, as I shall straightway show) in cubique inches, according to the number of the square-parts or Feet in the common Statute-Pertch or Pole, viz. 272 1/4, as you may see in the place forecited.  
But Mr. John Reynolds aforenamed also, (who seemeth to have been as industrious in this, as in many other mathematical experiments) will have this Gallon to contain 288 3/4 cubique inches; holding that the Wine-gallon (which he strongly affirmeth to be 231 inches) is to the Ale-gallon in such proportion precisely, as 4 to 5; or rather for the reduction of Wine-measure to Ale-measure, as 5 to 4; which is according to Mr. John Goodwin long ago, in his little Tract entitled, A Table of gauging, published above 50 years since, and dedicated to the than Lord Major and Aldermen of the City of London: wherein he showing how to reduce Wine-measure into Ale-measure, & contrà; saith, that 5 gallons of wine-measure make but 4 gallons of Ale-measure: with which very nearly agrees the opinion (not certain experiment perhaps) of some others, who will 
This Mr. Goodwin was Master in the Mathematics to Mr. Reynolds, as himself hath told me.  make this Gallon to contain just 288 inches upon this ground, that a cube-Foot should hold in its concave capacity just 6 Ale-gallons, and so consequently one ale-gallon must contain just 288 inches, which that learned gentleman Mr. Edm. Wingate, a Barrister of Grays-inn (a man eminent for his mathematical abilities) first declared to me by word of mouth, and soon after I found the same noted in his book of the use of his Rule of Proportion, chap. 10. And these two jump so nearly together, as if one were borrowed from the other: but I declaring this to Mr. Reynolds at my first seeing of him, he said that he had not observed this thing.  
Now as for Mr. Oughtred's Ale-gallon of 272 1/4 inches, the said Mr. Reynolds indeed alloweth of such a Gallon-measure, but not for any liquid thing, but for dry things, as Corn, Coals, Salt, and other dry things measurable by this kind of Measure; and so calleth it the dry Gallon-measure: And thereupon he will have to be 3 several Gallons (or other like Measures) one for Wines, (which also serveth for oils, strong-waters and the like) Another for Ale and Beer, and a third for Corn, Coals, and the like; and this he maketh lesser than the Ale-measure, whereas surely it should rather be greater, if there be any difference at all between them: And these three several Gallonmeasures, he compareth together, or differenceth by these three Numbers, viz. 28, 33, 35. as to show their proportions one to another: viz. the Wine-Gallon (231) to the dry Gallon-measure (272.25) as 28 to 33. which is so in the lest terms, rational or absolute; but otherwise in the lest proportional terms, irrational or unabsolute, and finite or limited, I found them to be as 7 to 8.25. And the said Wine-Gallon to his Ale-Gallon (288.75) as 28 to 35, which in the lest rational terms, is indeed as 4 to 5; but otherwise in the lest terms irrational (but finite or limited) I found them to be as 1. to 1.25. and which again is integrally rational, 100 to 125: And the dry-Measure to the Ale-measure, as 33 to 35, which cannot be abbreviated in terms rational.  
And surely, evil Custom seemeth to have brought up three such distinct measures (and which the foresaid Mr. Wingate hath also expressed to me) For at the Guild-Hall in London, where is generally holden to be the true Standard for these Measures, and so from which all others of the like kind throughout the Kingdom, are usually derived, there are but two such distinct Measures only (as we have been there informed for a certain truth) viz. one for Wines (and so for strong-waters, oils, and the like) and the other for Ale, Beer, and dry things, as Corn, Coals, Salt, and the like; which latter is commonly called the Winchester Measure, and from this are taken the bigger dry Measures, as the half-Peck and Peck, and so on to the Bushel, which is the greatest of our dry Measures: Which said Standard-Measures at the Guild-hall, the foresaid Mr. Reynolds confessed to me (going to him on purpose to receive some satisfaction from him about the Wine and Ale or Beer-measures (which was in June 1646, and than he gave me in writing under his hand, the solid content of the Ale-gallon to be 288 3/4 inches, and so its Proportion to the Wine-gallon, to be exactly as 5 to 4 (or for the reduction of Ale-measure to wine-measure, as 4 to 5) that he had never made any trial of them, (neither could I found that any other had, or if they had, it was surely to small purpose) but only of those Measures in the Tower of London (which he pleads for to be the most ancient and true standard Measures) and at Cowpers-Hall, and some other such places, which seem to be but some particular Customary measures, differing from the generally received Standard-measures at the Guild-hall.  
And therefore to be fully satisfied in this point, concerning the true Wine, and Ale or Beer Measures, according to the common Standards, (and more especially about the Ale or Beer-measure, finding such a diversity of opinions concerning the same, and in so vast a difference, as that between Mr. Oughtred's and Mr. Reynold's Ale-gallon, being 16 1/2 cube-inches) Myself and one Mr. Baptist Sutton, (a man well known in the City among artists) did agreed to go together to the Guild-hall, where he was well acquainted with the keeper of the Standard-measures and Weights, who otherwise I found to be very nice and scrupulous in showing of them; and for our further satisfaction herein, we made known our intention to the foresaid Mr. Wingate, who much approved of the same, expressing his desire also of it: And so August 9th. 1645; we repaired together to the Guild-hall, carrying along with us two large square glasse-vials, which we first weighed in a Goldsmith's Balance by Troy-weights, (as being the best) which were supoosed to be exact enough; and afterwards filling the two brasse-standard-Gallons for Wine, and Ale or Beer, with fair water from the Cistern, and that with all possible preciseness, we poured forth the same with the like accurateness, into the said two glass vials, and than weighed the Glasses with the water in them by the same weights: and so comparing the weight of each Glass alone, with the weight of the glass and water together, we found the Wine-Gallon of water to weigh 117 ●/4 ounces-Troy, and the Ale-gallon of water 140 9/16 ounces, (which last, according to the common division of the Ounce-troy by penny-weights, is 140 ounces, 11 penny-weights, and a quarter) which do hold in proportion (from the lesser to the greater) as 10000 11937, which comes very near, as 5 to 6. Than seeing that Weight and solid measure do hold in proportion one to another, so as that one may be deduced from the other, as I have showed before; if we compare these two Gallon-weights of water, with several experiments made by myself, and Mr. Reynolds, severally (and conferred together) for the finding out of the true weight of water in relation to its solid measure in inches, (or for the comparing of its gravity and magnitude together, which thing is most admirable and excellent use, as I shall show more afterwards) we shall thereby discover the solid capacity of the said two Gallon-vessels in inch-measure; which is the very groundwork of Gauging.  
Now as to the foresaid experiments; the said, Mr. Reynolds did first (amongst other things to this purpose) 'cause a Vessel to be made of Wood, by an exact 
* Mr. John Thomps●● in Hosiar Lane.  Workman, in the form of an oblong rectangle Parallelepipedum, (or long Cube as some term it, though improperly, as they call an oblong rectangle Parallelogram a long Square) whose Base was 4 1/2 inches square, and the height, depth, or length, (which you will) 14 inches, and so the solid capacity thereof, 283 1/2 inches; and which was closed up at both the ends or bases, saving that in the middle of one end, was made an hole for the pouring in of water, and which was no bigger, than that he might guess in the filling thereof to a drop or two of water, more or less: which Vessel therefore being precisely filled with fair settled Rain-water (as being the fittest, as I shall show afterwards) and than as precisely weighed by Troy-weights, he found the water thereof alone to weigh 12 lively and 5 1/4 oun. or 149 1/4 ounces troy. And he not being contented with this own experiment, he caused such another Vessel to be made, every way like and equal in its dimensions with the former, and that by the same Workman; which he filling with the like water, found it to agreed in weight exactly with the former. But yet he not resting fully satisfied with these two experiments, he procured such another Vessel to be made, by another Workman, of the very same Dimensions with the former; which he filling with the like water as aforesaid, found the weight of the said water alone to be 12 lively and very near 6 oun. Troy, (or 12 1/2 pounds-Troy ferè) exceeding the former weight about 3/4 of an ounce, and which he conceived to be the truer, (notwithstanding the exact agreement between the two former experiments) by comparing these experiments with some other of the like kind, which had been made before by himself, or some other body; And this difference of weight seemeth to proceed chief from some difference of measure in the Inch, by which the two first Vessels, and the last were made, being done by two several Workmen. And therefore (considering the difficulty in a work of this nature, in respect of the nicety and curiosity of the experiment) he comparing these with some observations which I had than made by the buy, to this purpose; we concluded together at length, that the nearest and indifferentest weight of the water exactly filling up the foresaid Vessel of 283 1/2 cubick or solid inches, would be 12 lively and 5 1/2 oun. Troy (or 149 1/2 ounces) and this to stand good.  
And than after this, I got an exact cubical Vessel to be made of throughly seasoned wood, with all the accuratnesse & preciseness that could be, being 6 inches the inside, (or the base thereof exactly 6 inches square) and so the whole Cube in its concave capacity, exactly 216 inches; and which than, to keep it from sucking in water in any part, or any water to soak into it, was well primed all within, with a thin oile-colour (yet of a sufficient body) having afterwards a Cover put on it, with a little hole in the middle thereof, about 3/4 of an inch wide, as the foresaid Vessels of Mr. Reynolds had: And which cubical Vessel I than filling with all the exactness and preciseness that might be, with fair settled Rain-water, at Gold-smiths-Hall; and so having the same as exactly weighed by the Standard Troy-weights; I found the weight of the water alone (deducting the weight of the empty Vessel itself first of all had, from the weight of the vessel and water together) to be upon the very point of 114 ounces, or 9 lively & 1/2. without any considerable difference therefrom: and thus I found it to be, at two several trials. Now according to the two first observations of Mr. Reynolds, aforesaid; 216 inches of the forenamed water, should weigh 113.7 oun. Troy, viz. 113 oun. and 14 p. w. which comes short of our observation, by about a quarter of an ounce; and according to his last observation the same should weigh 114.3 oun. ferè, viz. 114 ounces, and near upon 6 p. w. which exceeds thè weight found by us, just so much as the other wants of the same; So that the weight of this cubical body of water produced by our experiment, falleth directly in the middle between the several weights of the same deduced from his foresaid experiments, upon one and the same kind of vessel. And according to this our most exact observation; the weight of 283.5 cubick or solid inches of the foresaid water, (being the content of each of Mr. Reynolds his th●●e foresaid vessels) will be 149.625 ounces troy exactly, which is 149 oun. 12p. w. and an half. And this is the very arithmetical Mean between his two first observations, agreeing one with another, being 149.25 oun. and his last, being 150 oun.  
And so now according to this experimental Conclusion of mine own; I shall proceed exactly in the subsequent operations upon the Wine and Ale-Gallons: For so the weights of the two several Gallons of water aforesaid, being compared severally with this experiment, the solid capacity of the Wine-gallon, will be found 223.105 inches, and of the Ale or Beer-gallon, 266.329 inches.  
But I not resting fully satisfied with this one experiment in the said standard-Gallons (though we conceived the same to be performed with as much care and diligence as might be) and so desirous to try the same thing over again, to see how nearly two several trials would agreed to confirm the matter; knowing that two testimonies upon any thing are much better than one; I again moved the said M. W●●gate and Mr. Sutton (whom I still desired as witnesses to what was done) for another trial of this thing, and that divers times; but could not accomplish my desires herein, till about two years after: And so in July 1647, I and Mr. Sutton went together again 
2d. experiment upon the wine & ale-gallons.  to the Guild-hall (Mr. Wingate having promised to go along with us, but was hindered by other occasions) carrying along with us two other great glasse-Vials like the former, into wh●ch first pouring the foresaid Standard-gallons of water exactly filled, and than weighing the said Glasses with the water in them severally, by the great standard-Ballance there, with Avoirdupois-weights, and afterwards the empty Glasses severally (being well dried first) by the same weights, and so comparing them together as before; we found the weight of the Wine-gallon of water alone, to be 8 lively 1 oun. 3 dr. avoirdupois (or 129 3/16 ounces avoirdupois) and of the Ale-gallon of water, to be 9 lively 9 oun. 12 dr. (or 153 3/4 ounces avoirdupois) which are in Troy-weight (according to the most exact Proportions of the Avoirdupois weight to the Troy-weight before noted, viz. 14 to 17. and 56 to 51) 9 lively 9 oun. and 5.223 dr. or 13 p. w (which is 117.65 oun. troy) the Wine-gallon; wanting of the first observation or experiment (viz. 9li. and 9.75 oun. or 117.75 oun.) only 0.1 oun. troy, which is 2 p. w. or 4/5 of a dram-troy, which difference is of no moment. And for the Ale-gallon, 11li. 8 oun. and 5/28 of a dram-troy, or 25/56 of a p. w, viz. 10 5/7 gr. (which is 140 5/224 oun. or 140.0223 oun. troy) wanting also of the first experiment (viz. 11li. 8 oun. and 11 1/4 p. w. or 4 1/2 dr. or 140 9/16 oun. or 140.5625 oun. troy, (about half an ounce-troy, and which difference is of small moment in the matter of gauging; but yet this latter experiment is the truer, as more nearly agreeing with the other experiments and observations following.  
Now these two Gallons of water in this second experiment, are in proportion (from the lesser to the greater) as 100000 to 119013, which comes near the former proportion: And being compared with the foresaid experimental Conclusion made by me, (for the weight of water in reference to solid inch-measure) will give the solid content of the Wine-gallon, 222.9 inches, and of the Ale-gallon, 265.3 inches; which wanteth of the former solid measure in the Wine-gallon, not fully 1/5 of an inch, and in the Ale-gallon, about 1 inch. With difference between these two experiments, especially in the Ale-gallon, though in the matter of Gauging, the same can breed no sensible error or difference, as will afterwards plainly appear, when we come to show our gauging-Lines: yet for my further and fuller satisfaction in this nice and curious piece of art, so much handled and controverted by Artists, as I said before; and that I might come as near the matter as possibly might be; I urged again for another trial: A●d thereupon in November next following, I again 
3d. experiment upon the Wine & Ale-gallons.  repaired to the Guildhall, carrying ●●●h me two other large glasse-vials, (differing much in form from the other before used, though indeed this be nothing material to the purpose,) which I first caused to be weighed severally by the standard- avoirdupois weights there; and than (with the help of the keeper of the Standards) filled both the standard-Gallons with fair water from the Cistern, with all the accuratnesse that might be, as before; and which with the like accurateness being poured out into the said two Glasses: I caused the Glasses with the water in them, to be weighed severally by the same weights and Balance, and that as exactly as might be, and thereupon found (by the Comparison aforesaid) the Wine-Gallon of water to weigh alone, 8li. 1 oun. 11 dr. avoirdupois (or 129 11/16 oun. avoirdup.) and the Ale-gallon of water, to weigh 9li. 9 oun. 15 dr. avoirdupois, (or 153 15/16 ounces) which do exceed the second observation, in the weight of the wine-Gallon 8 dr. or half an ounce avoirdupois, and in the weight of the Ale-gallon, only 3 dr. or 3/16 of an ounce avoirdup. And these two last Gallon-weights of water, do hold in proportion (from the less to the greater) as 100000 to 118699 ferè, which is very little less than the former proportions. And these also being collated with our foresaid experiment of 216 cube-inches of water, to weigh 114 ountroy (or 9li. and an half) which is in Avoirdupois-weight according to the nearest proportions of the Troy-weight to the Avoirdupois, before declared and demonstrated, 125 9/51 oun. (or 7 14/17 li.) do give the solid capacity of the Wine-Gallon, 223.784 inches, and of the Ale-gallon 265.629 inches; which do exceed the first experimental observation, in the Wine-gallon, by 0.679 inch only; & the second, by 0.86 inch; and doth want of the first observation in the Ale-gallon, only 0.7 inch; and exceeds the second, by 0.3 inch only; which last is very little.  
But I being desirous to be further satisfied in the weight of these two last Gallons of water: So soon as I had performed the same at the Guild-hall by the Avoirdupoiz weights there; I caused the said Glasse-Vials with the Gallons of water in them, to be straightway carried unto Gold-smiths-Hall, to be tried by the great Standard-Ballance of Troy-weight there (as being the most exact kind of weight) where first weighing each Glass together with its water, and afterwards each Glass alone (being throughly dry) I found (by comparing the one with the other as before) the weight of the Wine-gallon of water alone, to be 118 1/16 oun. troy, viz. 118 oun. and 1/2 dr. or 1 1/4 p. w. (which make 9li. 10 oun. 0.5 dr. or 1.25 p. w.) and the weight of the Ale-gallon of water alone, to be 140 oun. and 4 1/4 p. w. or 1.7 dr. (which is 11li. 8 oun. and 4.25 p. w. or 1.7 dr. Troy) And these two Gallon-weights do hold in proportion (from the less to the greater) as 100000 to 118761, which exceeds that which was produced by the avoirdupois-weight, (viz. 118699 ferè) by 62 parts of 100000. And being conferted with our foresaid experiment for finding the proportion between the ponderall and dimensionall quantity of water, or its gravity and magnitude; will give the solid content of the Wine-gallon, 223.697 inches, and of the Ale or Beer-gallon, 265.666 inches: which exceeds the first experiment, in the wine-gallon, by 0.59 inch; and the second by 0.776 inch; and wants of the third experiment in the same, from the avoirdupois-weight by the balance, only 0.087 inch. And it wants of the first observation in the Ale-gallon, 0.66. inch; and exceeds the second by 0.36 inch only; and this third by avoirdupois-weight from the balance, only by 0.04 inch ferè, which is as much as nothing.  
And here having a fit occasion and opportunity, I shall (by way of Digression) 
The foresaid several Proportions between the Troy and Avoirdupois Weights, compared again together, & examined by the Balance upon another Experiment.  speak somewhat more concerning the Proportions between the Troy and Avoirdupois weights, for a further confirmation & verification of what was said formerly, and also demonstrated upon a Cannon-bullet concerning the same: And therefore the weight of these two last Gallons of water, taken first by Avoirdupois-weight at the Guild-hall, (viz. 8li. 1 oun. 11 dr. or 129 11/16 oun. the Wi●e-gallon, and 9li. 9 oun. 15 dr. or 153 15/16 oun. the Ale-gallon) being converted into Troy-weight, and that first, by the more common terms of proportion, viz. 60 to 73 for poundweight, or 80 to 73 for ounce-weight, will give the Wine-gallon, 9li. 10 oun. 6 p. w. and 19.125 grains exactly, or 118 oun. 6 p. w. and 19.125 gr. or 118 oun. 2 dr. and 43.125 gr. troy: & the Ale-gallon, 11li. 8 oun. 9 p. w. 8.62 gr. or 140 ounces, 9 p. w. 8.62 gr. or 1●0 oun. 3 dr. & 44.62 gr. troy, which do exceed the true weight taken from the Balance at Goldsmiths-hall, by 5 p. w. and 13.125 gr. or 2 dr. and 13.125 gr. in the Wine-gallon, and by 5 p. w. and 2.62 gr. or 2 dr. and 2.62 gr. in the Ale-gallon. And than by the other terms of proportion, viz. 14 to 17 for librall weight, or 56 to 51, for unciall-weight, the Wine-gallon will be 9li. 10 oun. 2 p. w. and 3.96 gr. or 118 oun. 2 p. w. and 3.96 gr. or 118 oun. 0.5 dr. and 21.96 gr. troy: and the Ale-gallon, 11li. 8 oun. 3 p. w. and 20.7 gr. or 140 oun. 3 p. w. and 20.7 gr. or 140 oun. 1 dr. and 32.7 gr. troy; which differ from the true weight of the Balance, (by way of excess) in the Wine-gallon, but 21.96 gr. or 22 gr. ferè; and (by way of defect) in the Ale-gallon, only about 9 gr. which differences are very inconsiderable, the greatest of them not amounting to a pennyweight, whereas the lest of the differences produced by the vulgar proportions, is above a quarter of an ounce-troy. And hereby it is again most manifest, that these latter Proportions between these two kinds of weight, are much the truer, and surely the nearest and truest that may be found, and are therefore generally to be received.  
And these our experiments beforegoing for the discovevering of the solid Contents of the foresaid Wine and Ale-gallons at the Guild-hall, may be confirmed by some other experiments which I afterwards made upon the same. For before my second observation of this thing, by the weight of the Gallons of water, I caused a concave Cube to be made of Brass, of 4 inches the Side exactly, and so the whole Cube, 64 inches; 
4th. experiment upon the Wine and Ale-gallon.  into which (being set level) myself, and Mr. Sutton aforenamed, together pouring out the two Gallons of water in the Glasses which we had from the Standard-Vessels at the Guild-hall, as aforesaid, with all the accurateness and preciseness that might be (at several times) we found first the Wine-gallon to fill the Cube three times, and than half the Cube (as nearly as we could possibly measure it) which being computed, do make 224 cube-inches exactly, for the solid content of the Wine-gallon; And than we found the Ale-gallon to fill the Cube 4 times, and moreover to arise to such an height of the said Cube, (viz. 0.7 inch) as being computed, did make 11.2 cube-inches; all which together do give 267.2 cube-inches for the solid capacity of the Ale-gallon. Both which do so nearly agreed with the former experiments (especially in the Wine-gallon) as that this experiment may sufficiently confirm the former; this being as plain and demonstrative an experiment as ●ay be. And if it had been performed by a cubical Vessel so large, as might have received into its concave capacity each of the Gallons of water wholly at once (which indeed I afterwards wished had been done, but my forementioned cubical Vessel of wood was made long after the trial of this Experiment) than the same might probably have yet come nearer the truth; for that the of●en filling of this small Cube, might 'cause some small error, which by a more large or capacious Vessel, might have been avoided, wherein the solid content of each Gallon might have been had at once, by the height (or depth) of its water; though indeed the difference between this which we tried, and some of our former experiments (in the Wine-gallon) is in a manner insensible, and between all of them, in respect of both Gallons, is altogether inconsiderable. And this experiment by the brass Cube, I afterwards tried again by myself, and found it to differ as much as nothing from the former. But indeed to have the solid Contents of the Wine and Ale-gallons so very exactly and precisely by their liquid Contents, as can be imagined according to the strictness of Art, is (I may say) impossible, unless the Standard-Vessels were so narrow-mouthed, as that in the filling thereof, one might be able to guess at a few drops of water, which in the Standard Vessels at the Guild-hall (and I think inall others) cannot be done, they being so wide at the mouths or tops, as that a spoonful of water more or less in the filling thereof is hardly discernible, or so much more, as might breed the difference of half an ounce more or less, in the weight of the water, and so consequently of one inch more or less, in the solid content; seeing that one inch cubick or solid of water weigheth (as we shall show anon) half an ounce-troy, at lest: And yet in all these several experiments and observations compared together, the greatest difference of solid measure is but about one inch and an half, and that in the Ale-gallon: but yet setting aside the experiment made by the brass Cube, (which is too large;) the greatest difference in the same will be but one inch solid, which will breed no sensible error in the matter of gauging, as I, said before. And which may show how very near the matter we have come, for the discovering of the true Contents of the common standard-Gallons for Wine and Ale or Beer, and surely as near the truth as can well be gone.  
But yet that there might no likely way be left unatrempted, for the discovery of this thing, I will add to the former, one experiment more, 
5th. exper●iment in the Ale-gallon alone, by taking its Dimensions.  being very demonstrative, which I made last of all in the foresaid brass standard Ale-gallon at the Guild-hall, by taking the proper linear Dimensions thereof, it being indeed an exact segment of a Cone (or Calathoidall) the internal or concave superficies from the top to the bottom, being very strait and smooth, aswell as the external or convexe Superficies, and also exactly circular throughout; only a little shelving or arching at its meeting or connexion with the bottom, and this not precisely plain, but rather a little hollowish, yet not so much as to make any sensible error in giving the solid capacity of the Vessel, as will straightway appear, by comparing the same with the observations before-going, and also one other observation following after. Which standard-gallon Vessel, as it is some what like in form to one of those which Mr. Oughtred speaketh of in the forecited place of his Circles of Proportion, concerning Gauging (as being showed 
Circles of Proportion. Part 1 chap. 9 Sect. 4.  unto him, and also the measures thereof first given him, by that great Antiquary Mr. William Twine of Oxford, whom he faith to have undergone great pains and charge in finding out the true Contents of our English Measures (and whom I well knew at Oxford, being of the same House with him) So also it comes very near it in all its dimensions: For by the Diameters of the top and bottom, and the height of that Vessel which they together measured, (and which you may see in the  book) they found, that the same would contain in its concave capacity, 268.85 cubick inches, which exceedeth the measure of our Gallon-Vessel, produced by the experiment of the brass Cube aforesaid, but little more than one inch and a half; and the measure deduced from the first experiment by weight, about two inches and an half; and that which was deduced from the second experiment, by weight, (being the lest of all) by about three inches and an half: But indeed, beside that the sides of that standard Ale-gallon were a little arching (as Mr. Oughtred saith) he observed divers other irregularities in the said Vessel, which might well hinder the discovering of the true Content thereof, by some few inches.  
Now the linear dimensions of the Standard Ale-Gallon at the Guild-hall (which I took as exactly as I could, and I believe insensibly differing from the truth) I found to exceed those of Mr. Twine's Vessel (taken both by him and Mr. Oughtred together, as I noted before) in the Diameter of the top, but 0.33 inch; and in the Diameter of the bottom, but 0.1 inch; and to want of that in the perpendicular height (or the depth) 0.8 inch; by which I found the solid dimension of this conical Vessel (or of this decurtate or detruncate Cone) to be 265.5 inches, according to a multiple or conjunct composition of the aggregate of the two Bases, and their mean Proportional (produced most exactly by logarithmical supputation) with a trient of the Altitude, thus,  
 
Which comes very near the solid Content of this Vessel produced by all our former experiments; especially the second and third, from which it differs as much as nothing; the one of them, giving 265.3 inches, and the other 265.6 inches:  
And all these our experiments in the Ale-gallon, 
6th. Experiment in the Ale-gallon, taken from the half-Peck: and that from the Bushel.  may yet be further confirmed by another observation or experiment being taken from the half-Peck, which they hold at the Guild-hall, to be equal in Content to the Ale-gallon, as being taken therefrom (and so I found it to be) and which I shall deduce from the Bushel, according to the Dimensions thereof, established by an Act of the common-council of the City of London, and yearly published by authority of the Lord Major, which ordains the breadth or wideness of a Bushel to be 19 inches, and the depth 7 2/2 inches; which being cast up according to a Cylindrical dimension, will be found to contain in in its concave capacity, 2126.5 cubick or solid inches, whose 1/8 for the half-Peck, is 265.8 inches, from which the solid Measure of the Ale-gallon, found by most of our former experiments, doth insensibly differ in a manner, especially that of the third experiment (by weight both avoirdupoiz and Troy, from the balance) which gave the same, 265.6 inches; or somewhat nearlier, that from Troy-weight, 265.7 ferè. And if we shall mediate between that of the first observation, being the greatest (except that of the brass Cube, which is too large) viz. 266.3 inches, and that of the second, being the lest, viz. 265.3 inches, (as is usually done in such like cases, where several observations or experiments made upon one and the same thing, do a little differ, and as they for the most part will, let all the art and industry be used that may be) the Mean between them, will be just 265.8 inches, for the solid capacity of the Ale-gallon, exactly agreeing with that of the half-Peck.  
And nearly agreeing with the foresaid Bushel, I found the Content of a standard-Bushell of Queen Elizabeth, which I was informed to be in the hands of the City-Founder, which was made of Brass, in the last year but one of her reign viz. Anno 1601 having her Inscription or Title about it; and somewhat resembling a segment of a Cone, it being wider at the top than at the bottom, whose dimensions there fore I took with all the accuratnesse that might be, and found the Diameter of the top (or upper base) to be one foot, and 7.5 inches, or 19.5 inches: the Diameter of the bottom (or lower base) one Foot and 5 inches, or 17 inches: and the depth (or height) 8 inches (or more accuratly, 8.1 inches) which being cast up, according as the Ale-gallon beforegoing; the solid capacity thereof will be found, 2122.165 inches; thus,  
 
Which wanteth of the Londonbushell, (being 3126.465 inches) 4.3 inches: But indeed this Bushel was made a little turning or winding outwards near the edge or top thereof, (which was made very sharp or thin) which might give so much more in the solid content thereof, as to make it equal with the City-bushell; and so I suppose this Londonbushell was first intended to be made equal with that of Queen Elizabeth, as nearly as might be; And indeed the true Diameter of the top (from edge to edge) I found to be 19.6 inches, according to which if the solid capacity of the Bushel should be computed, the same would be found, 2134 inches, but that is too large: and the other comes nearer the truth: And the 8th. part of this last, for the half-peck, and so for the Ale-gallon, will be but 266 3/4 inches.  
Another measure of an Ale-gallo●, Mr. Oughtred 
There came into the hands of the City-Founder, together with the foresaid standard-Bushell of Queen Elizabeth, a standard A●e-gallon of the same Queen, made of Brass in the very same Yea ●f her Reign, having also her Epigraph about i●: & which, I went with an intent chief to have measured; but indeed before my coming thither, it was sold away to a Town in Yorkshire called Whitby: But the same having been compared with the standard Ale-gallon at the Guild-hall, by the measure of water; was found to agreed there with, without any considerable difference, both as one of the Founder's men, and also the underkeeper of the Standards for the City, told me, who were present at the trial thereof: which may be a means to confirm that at the Guild-hall: And these two standard Gallons, as they were both of a like capacity, so also of a like form.  makes mention of in the place forecited, as being presented to him also by the foresaid Mr. Twine, it being (as he calleth ●t) a Standard-Gallon of Queen Elizabeth, which the said Mr. Twine had tried by another Vessel made of brass, in manner of a Parallelepipedum, whose base was exactly six inches square and the Sides divided into inches, and twentieth parts: into which he pouring out the said standard-Gallon filled with water, found it to arise un●o such an height therein (viz. 7.6 inches) as being computed, wouldgive 273.6 cubique inches for the solid content of the said Ale-gallon. And hereabout Mr. Oughtred conceives might be the true Content of the other Ale-gallon measured by him and Mr. Twine.  
All which experiments and observations aforegoing, (with some others made by Mr. Oughtred) may sufficiently demonstrate, the true Ale-gallon not to be of so large a capacity, as to contain 288 cubick inches, and upwards, as Mr. Reynolds, (and some others) will have it.  
As for the discovering of the solid measure of the Standard Wine-Gallon at the Guild hall, in a geometrical manner, by the Linear dimensions thereof, as before of the Ale-gallon; the truth is, I attempted not the same, in regard of the irregularity which I found in that Vessel, by the much arching or curvity of its Sides (whereby it is much like to that Ale-gallon which Mr. Oughtred and Mr. Twine thus measured; only this is wider at the bottom than at the top, whereas that was wider at the top than at the bottom, as the Ale-gallon at the Guild-hall) which makes it to differ from a segment of an exact Cone, and so may rather be taken for a ●egment of a Conoid.  
But having by our two last experiments (especially the 5th.) most plainly and manifestly discovered in a geometrical manner, the dimensionall quantity of the Ale-gallon in inch-measure, as nearly as may be (which two most plain and demonstrative experiments do not considerably differ, the difference between them being but .3 of an inch-cubique) and than by the 2d. and 3d. experiments the ponderall quantity thereof, in respect of the weight of the water exactly filling the same, (which too experiments also do not considerably differ; the difference being but 3dr. avoirdupois, as we there shown, or about 1 1/3 dr. troy, being near 1/6 of an ounce-troy) especially the 3d. experiment (which I take to be the most exact for this Gallon, in this respect) we may thereby (collating them together) be able to discover that which was experimented by me, from my large cubical Vessel of 216 inches: and also confirm the same; and so consequently likewise confirm the solid Contents of the Wine and Ale-gallon, first of all discovered or produced thereby: For the solid measure of this Ale-gallon, being found most plainly and demonstratively by our two last observations or experiments, to be very nigh 266 inches: and the weight of the water exactly contained in it, found most nearly (by the 3d. experiment) to be (in Troy-weight) 11 lively 8 oun. and near about 4 p. w. or 140.2 ounces: the weight of the water exactly contained by our foresaid cubical Vessel, will be found thereby (according to geometrical proportionality) 113.86 oun. which wants of that found by us, only 0.14 oun. which is but about 3. p. w. And so likewise may the same be very nearly confirmed by the Wine-Gallon, according to our 4th. and most plain demonstrative experiment for the same (by the brass concave Cube) by which the solid measure thereof being found 224 inches; and by the three former experiments (conferred together) the weight of the water contained exactly in the said Gallon-vessel, very near about 9 li. and 10 oun. Troy, or 118 oun. the weight of the water exactly contained in the foresaid large cubical Vessel of 216 inches, will be found thereby (according to the foresaid proportionality) very nearly as before, viz. 113.78 oun. which wants of the true weight found by us, but 0.22 oun. and which is only about 4 p. w. or 1/5 of an ounce-troy.  
Moreover for a further confirmation of this thing, and that very nearly, I shall produce another experiment, which I made upon the foresaid brasse-Cubicall Vessel of 64 inches: For so soon as I and Mr. Sutton aforenamed, had measured out the Standard Wine and Ale-Gallons thereby, as aforesaid; we first weighed the said concave Cube by a small Balance of his, with Troy-weights, and than again filled the same with fair water, as precisely as possibly we could; which we also weighing together, found the additional or differential weight (take it which way you will) to be (as nearly as we could possibly guess) 33.5 ounces, for the water alone; And afterwards I being desirous to make a further and exacter trial of this experiment, I carried this Cube unto Goldsmiths-hall, where I procured the same to be weighed first alone, by the exact Standard Troy-weights, with all the accuratnesse that might be; and than filling the Cube with fair settled water, as precisely as possibly I could, had the same weighed together, with the like accuratnesse; and so by collating these two several weights together, found the weight of the 64 inches of water alone, to be exactly in a manner as before, viz. 33.5 oun. (or but some few grains over) and this the Assay-master judged to be the nearest weight thereof that could be given, considering the wideness of the Vessel, to be such, (and that fully open on one side) as that some few drops of water more or less in the filling thereof, were not very discernible. Now according to our former experiments, the weight of this cubical body of water, will be found, 33.7 oun. troy; (or rather 33.8 ferè) which comes pretty near the other: but indeed this last being deduced from our experiments made by Vessels of a much greater capacity (or by a greater quantity of water) must needs be the truest; so that the other is somewhat wanting of the true weight. And these our experimental Conclusions upon the weight of water in reference to its solid measure; or the comparing of its quantity ponderall with its quantity dimensionall, I shall also afterwards as nearly confirm by a manifest experiment made last of all by me, upon a solid body of a known magnitude in inch-measure, in respect of its gravity taken both in the air and in the water, and the same compared together.   

SECT. III. Containing the practice of Gauging, according to our artificial way of measuring: together with the natural Dimension (by way of comparison) for a confirmation of the same.  
HAving by the several ways and means before-going, discovered the nearest quantities of the common standard-Gallons for Wine and Ale or Beer, in solid inches; and that of the Wine-Gallon, to be at the most but 224 inches, and of the Ale or Beer-gallon, but 266 (which too are in proportion very nearly as 5 to 6; being as 5 to 5 15/1●) We shall now proceed to the practice of Gauging itself; or the discovering of the liquid Contents of Vessels for Wine and Ale or Beer in Gallon-measure; and that in the most easy and speedy manner that may be, according to our artificial way of measuring, being here, as the solid dimension of a Cylinder, before declared and demonstrated, (as I have already noted) by the like artificial Lines of measure: though indeed the reason thereof cannot be exactly referred to the 3d. Theorem before-going, which is for cylindrical Dimension in general; nor yet to our other, or general Reason of measure, fully expressed soon after that; according as the prime Integer of Vessel-measure (a Gallon) is here considered; to wit, in the nature of a solid (cylindrical) body, expressed in some certain solid measure, according to the capacity thereof: whereas else the same being considered either simply, absolutely, and entirely in itself, as such a liquid Measure, or in its Parts compounding, etc. (as being by themselves apart taken for other lesser or inferior liquid Measures) like as the Integer of measure (or of Weight) in general, was formerly; than will the Reason of this Dimension (as in reference to the quantities of measure in the artificial gauging-Lines) be the same with that of Measure (or Weight) in general; especially according to our first or general acception or consideration thereof; but will exacty agreed with that of weight, in both the respects of Reasons aforesaid. And thus the quantity of the artificial gauging-Line for Wine-measure (according to the Gallon of 224 inches) will be 6.58 inches; and for Ale or Beer measure, (according to the Gallon of 266 inches) 6.97 inches. But yet considering the loss that happens in Wine by the dregss or leeses thereof, and much more in Ale and Beer, by the frothing and otherwise: therefore we have thought it meet and convenient, to allow one inch more at lest in the measure of the Wine-gallon, and 4 inches in the Measure of the Ale or Beer-gallon; and so make the Wine-gallon to be 225 inches, and the Ale-gallon 270 inches; which we conceive to be the most near and indifferent measures (in a general respect,) for these two Gallons, that may be: And these two do hold in proportion, exactly as 5 to 6: And in this proportion we might suppose they were first intended, considering how very near the same, the true Contents thereof do come: And therefore according to these two measures, we shall proceed in the work of Gauging. And the length of our artificial gauging-Line serving hereunto, will be for Wine-measure, 6.59 inches, and for Ale or Beere-measure, 7.00 inches: which being divided decimally (as all the former Lines) into 10, or rather 100 parts, or more, (and so turned over several times upon a Rod or ruler of a convenient length) and than the Diameters and lengths of Vessels taken thereby; their Contents shall be obtained in Gallon-measure immediately, as the solid Contents of Cylinders were formerly had, according to any Measure appointed.  
But seeing these Vessels are not exactly cylindrical (as I said at first) but Cylindroidall (or as others will have them, Spheroidall) and so the Diameter altereth between the middle and the end of them, or the bunghole, and the head; Therefore must a way be first had for the finding out of such a mean Diameter, as may reduce this unequal Body into an absolute Cylinder, or so near the same as possibly may be; that so the true Contents thereof may be had, or very near the same: By which mean Diameter, must not here be understood a Mean either arithmetical or geometrical, but a third sort of Mean differing from them both: the discovering of which, is the first and main thing in the practice of Gauging, next after the discovering of the Contents of the Wine and Ale-gallons in solid inch-measure. And which we shall show, most easily and speedily how to perform, by this first Rule next ensuing.  

RULE I How to found the mean or equated Diameter of any Vessel.  
Augment the difference of the two Diameters, by .7, and add the Product to the Diameter at the Head: the aggregrate shall be the mean or equated Diameter.  
 
But Mr. Oughtred neglecting, and indeed wholly rejecting a mean Diameter, hath regard to the mean Circle (as it were) between that at the Bung, and that at the Head, being composed of 2/3 of the Circle at the Bung, and 1/3 of the Circle at the Head, and thereby reduceth the Vessel to a Cylinder; and 
See Circ. of proport. part. 1. chap. 9  which two portions of those two Circles, he showeth how to found most readily, by this twofold Analogy, viz.  

1. As 1. is to 0.5236: So is the Square of the Diameter at the Bung, to 2/3 of the Circle at the Bungler  
2. As 1. to 0.2618: So the Quadrat of the Diameter at the Head, to 1/3 of the Circle at the Head.   
Or, for the more exactness in working, these Proportions might better be somewhat extended in the Terms; And so the first of them, will be, as 1. to. 523599 ferè: And the second, as 1. to. 261799.  
And so accordingly in this our Example, the Square of the Diameter at the Bung, being 1085.7025; the 2/3 of the Circle at the Bung, will be 568.47, &c: and the Square of the Diameter at the Head, being 694.8496; the 1/3 of the Circle at the Head, will be 181.91, etc. and which will also be had, by finding the whole Circles at the Bung and Head, according to their Diameters: For the whole Circle at the Bung, will be found, 852.7087, whose 2/3 is 568.4725: and the whole Circle of the Head, will be found 545.7336, whose 1/3 is 181.9112 as before: which two portions or sections of these two Circles being added together, do make, as it were, the mean Circle, between them, 750.3837 which must stand for the equated Base, as it were, of the cylindrical Vessel, supposed to be reduced by this means to an absolute Cylinder: the Diameter of which Circle therefore, may not unfitly be called the mean or equated Diameter; and which will be found to differ but very little from ours, it being 30.910 ferè, and ours is 30.973, and so the difference only about 0.06 inch, which in my slender judgement is scarcely considerable: And therefore I see no just cause why Mr. Oughtred should so much inveigh against Mr. Gunter, for his mean Diameter, as he doth, in the forecited place of his Circles of Proportion: though I must needs confess this way proposed by Mr. Oughtred, for finding a mean Circle, is the most ●ationall and demonstrative. And which way of Dolial Dimension, is (amongst others) laid down as the most sacile, though a little in another manner, by that great Geometrician, of famous memory. Mr. Henry Briggs, in his Arithme●ica Logarithmica, (first written and published by him in Latin) cap. 24. viz. by finding the Circle at the Bung (or of the middle crassitude of the Vessel, as he calls it) and at the Head, by the Diameters thereof; & so raising from them and the height, (or length) of the Vessel, two exact Cylinders, and than taking the difference between them, whose 2/3 being added to the lesser Cylinder; or the 1/3 subtracted from the greater; shall give the solid capacity of the Vessel; as I shall show by and by, among the other operations of this kind.  
Now the arithmetical Mean between the Diameter at the Bung, and the Head, in this our example, is 29.655. and the geometrical Mean (or the mean Proportional) is 29.47, which is less than the other, and that is less than the aforesaid mean Diameter.  
Having thus shown most briefly how to found out the mean or equated Diameter; I shall next show as briefly how to found out the Content of any Vessel in Gallons, either of Wine or Ale and Beer, according to the common or natural measure by Inches, and that by a twofold Aualogisme, in reference to the two aforesaid Gallonmeasures.   

RULE II. How to found the Content of any Vessel in Gallons.  
As 286.5 ferè (if Wine-measure) or 343.8 ferè (if Ale or 
The like Analogisme for Wine-measure, according to the Gallon of 231 inches, will be by the Number 294: and for Ale-measure, according to the Gallon of 272 1/4 inches, by the number 346.6. And here from our Gallonmeasures, the gauge point (according to M. Gunter) would be for Wine-measure, 16.93 inches; and for Ale-measure, 18.54 inches: But with these points we meddle not.  Beer-measure) to the length of the Vessel in inches: So is the Square of the equared Diameter, to the Content in Gallons.  
The reason of these two Numbers, or of this Analogisme, is deduced from that of a Circle to its circumscribing or Diametral Quadrat, which is vulgarly, as 11 to 14 (but more accurately, 1000000 to 1273239, or in the nearest rational terms, according to Metius, as 223 to 284; but more truly, according to his Cycloperimetricall terms, it will be in terms irrational, as 88.75 to 113) Wherhfore if the cubick inches contained in a Gallon, be augmented by the consequent term, and the product be resolved by the antecedent, the Quotient shall yield the first analogismall term.  
Now for a trial of this Rule: Suppose a Vessel to be in length, 39.54 inches; the Diameters at the Bung and Head as before: the Square of the mean or equated Diameter (found by our way) is 959.326729, which multiplied in the length, gives 37931.77886, etc. cubique or solid inches, which divided by 286.5 (if Wine-measure) giveth in the Quotient, 132.40688, etc. gallons of Wine: or being divided by 343.8, (if Ale or Beer-measure) giveth 110.339 gallons of Ale or Beer. For the mean or equated Diameter, being by us, 30.973 inches, the Circle answering thereto (for the equated base of the Vessel) is 753.45345 square or superficial inches, which multiplied into the length, gives the solid content of the Vessel (as if it were a just Cylinder) 29791.54945, etc. cubique or solid inches; which being divided by 225 (if Wine-measure) gives in the Quotient, 132.4068, etc. gallons of wine, as before; or by 270 (if Ale-measure) gives 110.339 gallons of Ale or Beer, as before also. But now according to Mr. Oughtred, the mean Circle between that at the Bung & that at the Head (for the equated base of the Vessel to be reduced to a Cylinder) being but 750.3837 inches, the same multiplied by the length of the Vessel, will produce the solid Content thereof, but 29670.17149 inches. And which will also be produced by Mr. Briggs his way before declared: For so the Circle at the Bung, or middle of the Vessel (as the greatest Circle of the Vessel) viz. 852.7087. etc. inches, being multiplied by the length of the Vessel, 39.54 inches, will produce the solid content of the greater of the two Cylinders , 33716●10396, etc. inches. And the Circle of the Head, or end of the Vessel (as the lest Circle of the Vessel) viz. 545.7335, etc. inches, being multiplied by the same length of the Vessel, will produce the solidity of the lesser Cylinder aforesaid, 21578.30654 etc. inches. All which are most accurately and easily produced by the Logarithmical Logistique, and not otherwise; or else not without a great deal of pains and trouble of Calculation; and either of which here to show, is altogether needless and superfluous. Now the difference of these two Cylinders, is 12137.79742, etc. inches, whose 1/3 viz. 4045.93247, etc. being deducted out of the greater Cylinder, or 2/3 thereof, viz. 8091.86494, etc. being added to the lesser Cylinder; there will either of these ways, result 29670.17149 inches, for the solid capacity of the Vessel, as before: and which you may see performed all together, in this subsequent operation;  
Which divided by 225, (for Wine-measure) affordeth 131.8674, etc. gallons of Wine: or by 270 (for Ale-measure) yieldeth 109.8895 gallons of Ale or Beer, which our account exceedeth in the Wine-measure, about 0.5395 gallon, which is somewhat more than half a gallon: and in the Ale-measure, 0.4495 gallon, which by conversion or reduction into the proper parts of liquid Measure, is about 3 pints, and an half, and which differences in so big a Vessel, arescarcely considerable.  
But now to apply these natural operations in vessel-measure, to our artificial way of Gauging, that thereby the use of our gauging-Lines may be the better understood, and the verity thereof demonstrated; I will lay down the several dimensions of the fore-supposed Vessel, according to the same, delivering first a brief Rule for the use thereof, which is thus;  
First, take the diameter at the Bung and Head of the Vessel, by these Lines, and thereby get the mean Diameter as before is showed: Than, multiply the Square thereof by the length of the Vessel, taken by the same Lines; and the Product shall be the Content in Gallons.  
1 Example, in Wine-measure.  
Which fraction above 132 gallons, gives somewhat more than half a gallon, viz. 4.32 pints exceeding our former measure of 13●. 40688 gallons, by about one pint: and the measure of 131.867 gallons, (deduced from the solid content of the Vessel in inches, found by Mr. Oughtred's, and Mr. Brigg's way;) by somewhat more than 5 pints. But indeed the Diameters and the length of this Vessel, here set down from the artificial gauging-Line, are a little larger than what they really should be; according to the quantities of the said dimensional lines of the Vessel, laid down before from the natural or vulgar Measure, being arithmetically compared with the said artificial Line taken in its true quantity from the natural Measure, according to a more ample or numerous partition thereof: and so they give the liquid content of the Vessel in gallon-measure, somewhat more than otherwise they would. But these falling so very nearly upon whole parts, or Integers of the artificial gauging-Line; therefore for more plainness and readiness of working, I thought good to express them artificially in whole Numbers only.  
And the liquid content of this Vessel, would be according to the solid content of the Wine Gallon, commonly taken to be 
* The artificial gauging-Line will be according to this Gallon, 6.65 inches.  231 inches; but 128.4423, gallons; which is 128 gallons, and somewhat more than 3 1/2 pints: and which wanteth of the true measure, (being 131.8674 gall. according to the foresaid Standaru-Gallon of 225 inches) 3.425 gall. which by reduction of the parts, is 3 gall. and 3 2/5 pints.  
2 Example, in Ale or Beere-measure.  
Which parts above 109 gallons, do yield 7 pints, and near upon 1/5 of a pint: and which doth want of our former measure of 110.339 gallons, about 3 pints and an half; but exceedeth the measure of 109.8895 gallons (deduced from the foresaid solid content of the Vessel found by Mr. Brigg's and Mr. Oughtred's way) only about 0.08 pint, which is altogether inconsiderable. But yet the more parts these Lines are divided into, the more exactly still will be produced thereby, the content of any Vessel in gallon-measure, according to what was said and demonstrated in all the other Dimensions beforegoing, upon the several artificial Lines of measure: So that this kind of Gauging-Line, is as exact as any whatsoever.  
And as I formerly showed in the solid dimension of a Cylinder, how the same might be performed artificially, not by the Diameter & the height (or length) thereof together, b● also by the Circumference with the height: So here in like manner, may the Content of any Vessel be had in Gallon-measure, not only by the mean Diameter with the length thereof; but also by finding a mean Circumference between that 〈◊〉 the Bu●g and that at the Head; whose Square being multiplied in the length of the Vessel, shall g●●● the liquid capacity thereof immediately in gallons: And the quantity of the artificial Line of measure, serving hereunto, will be for Wine-measure (according to to our foresaid Gallon) 14.14 inches, and for Ale or Beer-measure, 15.03 inches. But because the usual way, by the Diameters, is the easiest and readiest, and also that the Circumferences of the insides or Concaves of Vessels (especially at the middle or bung) can hardly be taken; therefore I will use not more words about this thing.  
But here it may perhaps be expected, that I should show the ready manner of taking the Diameters of Vessels at the Bung and Head, and theirlengths, by our gauging-Lines or Rods; but this being a thing easily understood by the pregnant Practitioner, and the same also fully showed in books particularly for Gauging, long since published; I shall pass them by, having said sufficient for the practice of Gauging; it being not mine intent and purpose, here to set down every particular Circumstance pertaining to Gauging; but briefly to show the making and use of these new artificial gaving-lines or Scales, and that according to our new experiments and observations for the measure of Wine and Ale, or Beer.      

A-TACTOMETRIA. OR An Appendix, for the dimension, (or the discovering of the solid quantity or capacity,) of any irregular kind of Body whatsoever; after the most exquisite manner that may be.  
HAving now fully finished the Dimension of regular and, regular-like Magnitudes in general, according to our artificial way of measuring, (or more artificial kind of practical Geometry) at first proposed; together with the natural or vulgar, by way of metrical comparison: and of the Solids both regular and regular-like, aswell in respect of gravity, or quantity ponderal (according to any Metal assigned) as of magnitude, or quantity mensurall: And than in the last Part, the like dimension of that irregular kind of concave Body, (or Vessel) which is chief used for the containing or keeping of our most common potable Liquors, viz. Wine, Ale, and Beer; I shall now at last (by way of an Appendix) add somewhat, for the most easy and exact discovering or obtaining of the solid capacity, or dimensionall qnantity, of any other kind of Body whatsoever, how irregular soever it be; whose Dimensions can in no wise be taken by a Line of measure (especially so as it be but of a reasonable bigness) and so whose solid content cannot be obtained in any certain Measure proposed, after the manner of geometrical Mensuration. And this I shall show from our forementioned experiments made upon the most common liquid Body, for the comparing of its gravity and magnitude, or ponderall and dimensionall quantity together; drawing first from thence some apt and brief Notes or Conclusions, for the more easy and ready performing of this kind of Dimension. And so from thence we gather, the nearest and truest weight of one Inch cubique or solid of clear or fair settled water in general, to be 0.527 oun. troy, or rather, 0.528 ferè, being 0.52777, etc. which is but little more than half an ounce, it being (by conversion or reduction into the proper parts of this weight) 4 dr. or 10 pennyweight and 13 gr. precisely in a manner. And yet afterwards, for further curiosity in this thing, I caused to be made of brass, a concave Cube of one inch, open on one side, but having a lid or Cover made very exact, to clap close on it, with an hole in the middle thereof, not bigger, than to receive through it a small drop of water, that so I might be able to fill the same precisely to the lest drop: (but indeed this Cube was made too scanty, and so wanted of its true magnitude, as I found by comparing it with other exact cube-inches; and it being filled with fair water, and weighed by a very curious Balance at Goldsmiths-hall, as precisely as might be; the weight of the water alone, was but half an ounce troy, & about 2 gr. over, and this I tried twice over: though yet for all this, I well knew that the true weight of one inch of water (or of any other humid or liquid body) could not be so exactly deduced from a Vessel or Body of one inch only, as from a Vessel or body containing many inches; in regard of the exceeding nicety and curiosity in the making of so small a Vessel, and than in the filling & weighing of the same; over there is in a larger Vessel: & for that in the filling of so small a Vessel, the lest drop of liquor more or lesle, is of some moment, whereas in a large Vessel, a few drops more or less are not considerable, so as to make any sensible or considerable difference in the weight of one inch being deduced therefrom. (Nor is our foresaid cubical Vessel of 64 inches, so very sufficient for this purpose, as to ground so nice & curious an experimental Conclusion thereupon.) And therefore we must proceed herein, à majori ad minus, from a greater quantity to a less, both for the discovering of the weight by the measure (as before) and also of the Measure by the weight: and not à minori ad majus, from a less to a greater; for that will not hold good in this thing. But indeed in a solid Body, of what matter soever it be, the thing will hold just contrary to the former, in both the foresaid respects: For herein, we must rather go à minori ad majus, than otherwise: And so the weight of one inch cubick or solid, may more exactly be had from a Body of one inch cubick only (being exactly made, and which I found is hard to do) than from a Body of many inches; and consequently, the gravity or ponderosity of any bigger body (of the same matter) may be had most exactly and speedily therefrom: For thus Ghetaldus in his experiments , for the comparing of sundry Metals together, in gravity and magnitude, and so to determine exact proportions between them; began upon small quantities, and from thence deduceth greater quantities: And so in particular, for finding the gravities of spherical Metalline bodies by their diametral magnitudes, he first began with a Sphere of a small magnitude; Or rather, because he could not have such a body exactly made (as I noted formerly) he procured a metalline Cylinder to be made, (which might be more exactly done) of equal Diameter and altitude; and the same to be but two inches from the Roman Foot; and thereby he obtained the gravity of a Sphere, of the same metal, whose Diameter was equal with the Diameter, and altitude of the Cylinder, etc. and from this he deduceth the gravity of any other Sphere of the same Metal, of what magnitude soever, as I formerly showed. And so likewise have I seen our Countryman Mr. Reynolds beforenamed, in making the like experiments, for finding the proportions of gravity, etc. in Metals, perform the same by Bodies of one Inch cubique only, holding it a surer way, so to obtain the weight of one inch cubick or solid of any Metal (or other solid substance) than by a body of a greater magnitude: Though yet (for mine own part) I conceive there may not be altogether so much curiosity or difficulty in this, as in the former; but that the weight of a solid body of one inch in magnitude, may as (if not more) nearly be deduced from the weight of a body of a greater magnitude or dimension, (of the same matter) as the weight of a liquid or humid body of one inch in magnitude or bulk, can be deduced from the weight of a liquid body (of the same kind) of a greater bulk or crassitude; by how much, the weight of any solid body in general, may be had more exactly than the weight of any liquid body; sigh that a solid body is weighed immediately by itself in the Balance or Scale, without the help of any other thing to contain it; whereas a liquid body (especially one so thin and fluid as water is) cannot well be weighed by itself (unless it be in a small quantity) but by means of some Vessel to hold it, whose weight must also be had first or last by itself (though indeed, it is not absolutely material or requisite to take notice of the same, as to any certain, known, regular, denominate weights, unless the Vessel be last of all weighed by itself, for otherwise any irregular indenominate weights will serve to poise it, if it be first weighed alone, before it receive the liquid Body whose weight is required. But this by the way.  
Now one inch of water weighing, 0.5277, etc. oun. Troy, viz. 0.528 oun. ferè the same will be found in Avoirdupois-weight, to be 0.5795 oun. (which are in librall or poundweight, 0.04398 lively troy, and 0.03622 lively avoirdupoiz) And 
The dimensionall and ponderall quantity of water compared together several ways.  from the same experiment I found, that a body of water of 40 inches, will weigh 21 oun. troy almost exactly, and 23 oun. avoirdupois almost as nearly; and that 36 inches of water will weigh 19 oun. troy exactly, but which will be in avoirdupois-weight, 20 44/5● oun. or 20.86, etc. And so contrariwise, arguing from the gravity or ponderall quantity of water, to its magnitude or dimensionall quantity; the solid measure of one ounce- troy, will be (in unciall or inch-measure) 1.8947 inch; and of one ounce avoirdupois, 1.72556 inch; (and consequently in Pedal or Foot-measure, 0.001096, for Troy-weight; and 0.00099859 for Avoirdupois-weight.) And so the solid measure of one pound-troy of water, will be, Uncially, 22.7368; and of one pound-avoirdupois, will be Uncially 27.609: and the same will thereupon, be Pedally, 0.013158 ferè, for troy-weight, and 0.015977, for avoirdupois-weight. And so from our foresaid experiment made by the large cubical Vessel, which was an Octant of a Foot cubique; the weight of a Body of water of a just Foot in magnitude, (or the weight of a cube-Foot of water) will be eactly, 912 oun. Troy, and which is just 76 lively Troy: And this will be in Avoirdupois-weight 1001 21/51 oun. or 1001.41176 oun. and so 62 10/1● lively or 62.588 lively  
But indeed, of all these terms of proportion or comparison, between the gravity and magnitude, or the ponderall and dimensionall quantity of water, those of ●1 ounces, to 40 inches (in respect of Troy-weight, which is here the best) are the efittest for a generalluse: for that I found them to answer indifferently to all the several experiments that have been made by me, and Mr. Reynolds severally, for the foresaid purpose; which said several experiments, though in the weight of one Inch of water, and so in the solid measure of one Ounce, they may make some small difference, so as some what to altar the decimal numbers, by which the same are expressed (the said numbers being produced or extended beyond two or three figures in the fraction; and in which all our best experiments do concur, viz. as to make the weight of one cubick inch of water, 0.527 oun. troy, or very near thereabout; and so the solid measure of one ounce-troy of water, about 1.895 cube-inch) and so to make some little difference in the operations performed thereby, especially the greater that the quantity of water used, is; yet they all agreed in this, as to give to 40 inches of water the weight of 21 oun. troy, (and so contrarily, to 21 oun. troy of water the solid measure of 40 inches,) without any considerable difference: And than besides that these two numbers thus answering each other reciprocally, they may also more easily be borne in memory, than the decimal numbers, which denote the gravity of one inch of water, and the magnitude of one ounce of water: And the like will be for 23 ounces to 40 inches, where the operation is performed by Avoirdupoiz-weights, for want of Troy-weights. But yet again after all this, considering the foresaid experiment of the large cubick Vessel, made by me, was as exact, (I may be bold to say) as well may be, or as need be desired; than the foregoing quantities of 19 oun. troy of water to 36 inches of the same, (which are those of 114 oun. and 216 inches, in the lest Terms) may be generally used, (being as easily born in memory as those of 40 inches and 21 oun.) to express the Proportion between the gravity and magnitude, or the ponderall and dimensionall quantity of water; and so for the producing or discovering of the solid measure of any irregular body, as I shall next of all show.  
Having therefore thus discovered the weight of water in general, in relation to its solid measure, etc. (and that as nearly we conceive, as may be, or need to be) or the proportion between the gravity or ponderall quantity, and the magnitude or dimensionall quantity thereof, and the same laid down here in all the fittest terms that may be, for a general and speedy use; we may thereby be able easily and exactly to discover the solid content of any solid Body whatsoever, how inordinate 
How to found out the solid capacity of any irregular Body, by the weight of water.  or irregular soever it be; by conferring it with the foresaid common liquid Body. according to magnitude; and that is, by finding out or discovering such a body or quantity of water, as is equal in bulk, magnitude or crassitude, which the solid body, whose measure is required: Which water being exactly weighed, and than the dimensionall quantity or solid measure thereof, deduced from its weight, by means of the foregoing terms of proportion for that purpose: the same shall be also the solid measure of the solid body required. And this may be performed by any irregular kind of Vessel whatsoever, by filling the same exactly with fair water, 
How to find the exact quantity of water (or other liquid body) which is equal in magnitude or dimension to any solid Body given.  and than sinking in the solid body, so low at lest, as wholly to cover it: for so the water forced out of the Vessel by the solid body, shall be equal in bulk or magnitude thereunto; the solid body now taking up the same room or space in the Vessel, which that did. And the quantity of water so forced out of the Vessel, may be found, either by first weighing the Vessel full of water, and than the Vessel with the remainder of the water left therein, after the taking out of the solid body, (for here the weight of the Vessel alone need not be had) whose difference of weight shall be the weight of the water effluxed: or else (which is much easier and exacter, especially if the Vessel be so great, as that it cannot conveniently be weighed) by making a small hole near the top of the Vessel, or a nick or notch in the very edge, brim, or top thereof; and so filling the Vessel exactly up to the said hole or notch, through which the water forced out by the solid body may run, and so be received into some other smaller Vessel; which may be done exactly to a drop, if there be a quill or some such like things as a pipe or spout, closely fastened in the said hole or notch, whereby to convey the water cleary away into the other Vessel, without spilling any; for so the weight of this water (which is equal in magnitude to the solid body) being taken; and than the solid dimension thereof produced thereby, (according to some of the foresaid proportional Conclusions for this purpose) the same shall be the solidity of the body, given to be measured. And herein, it maketh no matter, whether the solid body be sunk into the Vessel of water, any lower than just to cover it, or not; for it comes all to one pass, seeing that it still takes up but the same room or place in the water; and so consequently causeth but the same quantity of water to flow out. And so may the solid measure of any part assigned, of any solid body be had, by sinking that part only into the Vessel of water, and than observing the quantity of the water effluxed thereby; for this shall be of equal bulk or magnitude with that part of the solid body; and therefore the solid content thereof being found out by its weight (as before) the same shall be the solid content of the said assigned part of the solid body, as was required.  
And thus also may the capacity of any irregular concave Body or Vessel whatsoever, be discovered in solid measure, by the weight of the water exactly filling the same (or the solid measure of any part thereof assigned, by the weight of the water exactly filling that part) if the bigness thereof hinder not the convenient weighing of it; according as I formerly showed for discovering of the concave capacities of the City-Standard-Gallons for Wine and Ale or Beer, in solid inch measure; and both which Vessels were of an irregular form, especially the Wine-Gallon, whose dimensions could not be rightly taken by a Line of measure, as I than noted.  
Or again, the weight of the water equal in magnitude to any irregular solid body, whose measure 
How, to found the gravity of the Water (or other liquid body) which is equal in magnitude to solid body given, by the gravity of the solid body only. And so the manner of weighing a solid body in Water.  is disired, may be had by the weight of the solid body only; the same being first exactly weighed in the Balance or Scale, after the usual manner; and afterwards out of the Scale, in water; which is (according as 
* Archimed. promote. post exempl. 1 prob. seu 8 prop.  Ghetaldus teacheth) by hanging the solid body by one of the Scales of the Balance, into the water, so as it may hung freely therein, being covered therewith, and that by an horse-haire, (which hesaith to be, in a manner, of equal weight with water, and so neither addeth to, nor taketh from the gravity in the Body to be weighed) or many such hairs joined together, if the gravity of the Body require them; but with due consideration had of such part thereof (in respect of weight) as hangs out of the water, that is (as he saith) as hangs out of the Scale to which the Body is fastened, even to the Body itself; by putting the like quantity of hair in the other Scale where the weights must be put; l●st that the many hairs together, should add some weight to the Body, (and so neglecting that part of the hair which is wound about the body, or with which the same is bound up, because that is supposed to be equally weighty with the water, as being with the Body in the water, and so to have no gravity therein) for so the Scales shall become equiponderant; and than the Body hanging freely in the water (and so as neither of the Scales touch the water) the same shall be ponderated or poised, as if it hung in the air; and will be in the water so much lighter than it is in the air, as is the gravity of that portion of water, which is equal in magnitude or bulk to the solid body: and which Archimedes demonstrateth, and from him Ghetaldus in the 
Archimed. lib. 1. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 (〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉) seu de iis quae insident aut vehuntur in humido, sive liquido, (Velure 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 in aqua, ut vulgò dicitur) prop. 7.  forecited prop. of his Archimed. promote. And therefore the difference between the gravity of the solid body taken in the air, (which is the true gravity thereof) and the gravity in the water, shall be the gravity of so much water, as is equal in magnitude to the solid body. But this must be understood of such a solid body or Magnitude, as is heavier than water; that is, being put therein, will sink, or be carried down into the same, of its own accord: Whereas else, if it be a solid body which is lighter than the water, that is, being put into the same, will not sink, or descend down of itself; than the gravity of such a portion of the water, as is equal in magnitude to this solid body, must (in this way) be had by the adjection or apposition of some other solid body, which is weightier than water, (whose gravity both in the air and water are first had, as also the gravity of the lighter body in the air) so as these two several Solids being joined together, may make up (as it were) one Body; which being let down into the water, may voluntarily sink or descend: And both which Solids thus put together in one, will be lighter in the water, than the heavier Solid alone, (as Ghetaldus showeth in the forenamed place) And will be also lighter therein, than they are in the air, (both their true weights being taken together) by so much, as is the gravity of the water equal in magnitude to them both, (according to that of a solid body alone which is heavier than water) And this shall exceed the gravity of the water equal in magnitude to the lighter body alone, by so much, as is the gravity of the water equal in magnitude to the heavier body; and therefore the gravity of the water equal in magnitude to the lighter body (which was required) is thereby immediately obtained. And what hath been here spoken concerning Water, the like understand of any other liquid or humid body, in which a solid body may be thus weighed. And indeed this latter way for the finding of the gravity of the water (or other liquid body) which is equal in magnitude to a solid body given, by the gravity of the solid body only (& thereby to discover the quantity or capacity of the solid body, according to a certain measure appointed, as before; which is in my judgement, the mostexcellent and beneficial use that can be made of it) is much more neat, than the former, but withal much more curious and difficult in the performance; and therefore I shall rather refer the practitioner to the first way, as being very plain, easy, and exact.  
But now, that I might be found not more wanting in the practice and experience of these things, than of the other before going (for with the bore speculation of things I could never yet well content myself, though they were never so likely in themselves, (as many men do, and so take up all things upon trust) I will here produce a manifest experiment made by myself, of both these ways. under one, in reference to the discovering 
A double experiment upon the foregoing Conclusions. Apr. 12.1649.  of the dimensionall quantity of a solid body, by such a body of water as is equal in magnitude thereto, whereby the verity of all these mathematical Conclusions may plainly appear. And therefore to this end and purpose, I sought to get some solid body of Metal or Stone, of a reasonable bigness, which might not suck or receive water into it, and be also wrought into some such regular form, as that the solid content thereof might be exactly obtained by a Line of measure, according to the plain and usual way of measuring, and also such an one, as might easily be hanged in a string; without any fear of slipping out when it was in the water; as some Prismall or cylindrical body, or the like; but which (though with much seeking up and down) I could not meet with ready to my hand, nor otherwise easily obtain: And so after all this ado, I could procure none that might any way serve the turn, but one which was of a spherical form, and that was a Marble-bullet, made as round and smooth as might well be imagined, (and which therefore was the most troublesome for the performance of this experiment, that could be, being the most difficultly hung in a string, of any other kind of body) whose diameter I therefore first took most exactly by a Callaper Compass, and applying the same to a Diagonal Scale of an Inch, I found it to be 4.95 inches, and from thence the solid Content, 63.5 inches: which being done; I immediately carried the Bullet to Goldsmiths-hall (where I was to make my experiment) and there got the same exactly weighed in the Scale; and so found the weight thereof, 89 oun. 2 p. w. (which are 7 lively 5.1 oun. troy) Than for the discovering of the weight thereof in water, as also for the performing of the other Conclusion, for the finding out of the quantity of water equal in magnitude thereunto, both under one: I got presently an earthen Vessel, very broad and deep, which held about a good Pailful and an half, or more; causing first a small hole to be drilled through it near the top, and than a pipe of Tin to be fitted thereto, and fastened therein so close, as that no water might pass through the hole beside the pipe; and than I next prepared several small lines or strings of horse-tail hairs twisted together, with which I girted the Bullet several times about, till it would hung firmly, and not slip out (and which I found very tedious and troublesome to do, by reason of the smoothness and weightiness of the body, and the slipperiness of the horse-haire together) continuing the said strings or lines out together from the bullet to a convenient length, and tying them together, whereby the bullet might hung down in the water from the Balance: which done; I fastened the said strings (not to one of the Scales, as Ghetaldus seems to intimate, but) to one end of the Beam, where the Scale is hung, which I conceived to be the best way; especially considering the weightiness of the Body: And than because in that respect, it required a strong Beam whereon to hung, and that the Scales belonging to the great Beam would be troublesome, and an impediment in the way, in respect of their largeness: therefore I, and the Assay-master of Gold-smiths-hall together, conceived it best to put in their place a small pair of Scales, which might be sufficient to hold the Weights; and so the Body might hung freely down by the Scale without interruption; which being done; I set the foresaid earthen Vessel under that end of the Beam on which the Body hung, and than filled the same with fair water exactly up to the hole therein, till the water entered the pipe, (letting it run a few drops till it stayed of itself, for so I was sure of the exact filling thereof to a due height) under which I than set a glasse-viall with a little Funnel in it, which might receive the water forced out of the Vessel by the bullet, to a drop; and than sinking the said Body gently into the water, (affording it so much string therein, as might keep it fully covered all the time) I had the Weights put into the other Scale, together with near about such a quantity of horse-haire, as was contained in the several small strings joined together, by which the bullet hung (neglecting that part thereof which was wound or wrapped about the bullet in the water, for the reason before alleged) and so the bullet hanging freely in the water, we found it to weigh 55 ounces and 3 quarters, Troy (which is 4 lively and 7.75 oun.) and thus I let it hung in the water duly poized, till all the water was run out into the Glass, which it could make to flow out; for this water thus effluxed, was of equal magnitude with the said Bullet. And this being finished I repeated one part of my experiment the next day, by filling up the Vessel again to the hole thereof exactly as before; & than setting another Glass with the Funnel in it, under the pipe, I sunk the said bullet into the water, letting itabide therein, till it had forced out all the water that it could, to the last drop (not one drop running beside the pipe) & than afterwards weighing the said several Glasses of water effluxed out of the Vessel by means of the bullet, I found them to differ insensibly; the weight of the water alone being 33 ounces, and 8 p. w. or 33.4 oun. troy, (which nearly agrees with our experiment made upon the brass cubical Vessel of 64 inches aforesaid, whose water we found, to weigh 33.5 oun. troy) Now the difference between the weight of the Bullet in the air, or the Scale, 89.1 oun. (or 89 oun. 2 p. w) and its weight in the water, 55.75 oun. (or 55 oun. and 15 p. w.) is 33.35 oun. (or 33 oun. 7 p. w.) for the weight of the water equal in magnitude with the bullet; which you see differs (defectively) from the weight of the water forced out of the Vessel by the bullet, only 1 p. w. which is not considerable: And yet had I been so curious in weighing the bullet in water, as to have put in the Scale where the weights were, the same quantity of horsehair, ●s precisely to an hair, as were contained in the lines or s●rings by which the bullet hung, except those which were about it in the water (according as Ghetaldus directeth) than haply might the weight of the water equal in magnitude with the bullet, produced thereby, have exactly agreed with the weight of that which was forced out of the Vessel by the bullet; but indeed I think there wanted about so much hair in the weight-Scale as might have made the bullet to weigh in the water, 1 p. w. less: But I conceived there was no need of so great curiosity in such an experiment as that was, especially upon so great a body; neither do I conceive it absolutely necessary & requisite to perform the same by horsehair, but that silk or thread might serve the turn, putting so much in the Scale with the weights, as is used about the body to be weighed, and this can breed no sensible error, unless it be for the performing of some very nice and curious experiment indeed; as to found the exact difference and proportion between the weight of a Metal (as Gold or Silver especially) other thing, in the air, and in the water, and the like; than indeed 
Ghetaldus in A●chimed. promote. post prop. 19 speaking of his new artifice, whereby he would found out the quality of Gold from the gravity only which it hath in the air, and in the water; saith, that the weight of pure Gold, which is in the air 19, will be in the water 18. And the weight of Silver, which in the air is 31, will be in the water 28: And so the weight of air, which is in the air 9, will be in the water 8.  to use horse-haire, and in that strictness and preciseness, as Ghetaldus speaks of, will be altogether requisite, for the reason before delivered: And now you see how these two experimental Conclusions do manifestly confirm one another, & so both of them do confirm our former experiments for the weight of water in relation to its solid measure, (and so for the solid measure thereof in reference to its weight) for thereby the weight of 63.5 inches of water, (the same as the solidity of the Marble-Sphear aforesaid) will be upon the point of 33.5 ounces troy: And so (to come to the point in hand, which is the discovering of the solid capacity of this spherical body in inch-measure, by the weight of the water which is of equal magnitude with the same) the solid measure of 33.4 oun. troy of water (the quantity of water agreeing exactly in magnitude with the Bullet) will be thereby 63.3 inches, for the solid content of the bullet, which wanteth of the solidity found at first by the Diameter 63.5 inches, only 1/5 of an inch. And indeed the solid Content thus found by water, I may adjudge to be the truer measure, in regard there was a small floating in one place of the bullet, which might well make the solid content thereof lesser by so much than it was, being taken according to a full rotundity or sphericalnesse every way, without any flatting in the same, and according to which the Diameter was taken. And therefore in any solid body, of how regular a form so ever it be, where there is any such flatting, dent, or hollowishnesse, or other like defection in any part of the superficies thereof, which may diminish somewhat of the true solid dimension which it naturally obtains and aught to have, according to such a form (and which yet will be fully deduced from its linear Dimensions, according to a plain way of measuring, as if there were no such defection) there the solid Content (in the state the body than is) will be most truly discovered by water (as I have here shown two several ways) for that the water will search out the true quantity of those defections, which a Line of measure cannot, and so give the solid content of the Body accordingly.  
And this way for discovering the measure of any irregular Solid, must needs be generally better than that which Mr. Diggs hath delivered in his 
* Pantomet. lib. 3. cap. 14.  Stereometry; to be performed by an exact Cubical, or other rectangular prismal or parallelepipedal Vessel, with water; and that in the usual way of measuring; by taking the Dimensions of the Vessel, as to the finding of the superficial content of its rectangle Base, taken according to the concavity thereof; and than noting the ascent of the water, in the Vessel, when the solid body lies covered therein, and also when it is out, whose difference being enfolded with the superficicies of the water (or of the foresaid rectangle Base of the Vessel) gives the solid measure of the water equal in bulk or magnitude with the said Body, and so withal of the Body itself; 
Clau. Geom. pract. l. 5. c. 11.  and which way Clavius also speaketh of, as being the way vulgarly noted and used by Artists: whereas the way delivered here by us, may be performed by any kind of Vessel whatsoever, how irregular soever it be (as was the Vessel by which I performed my foregoing experiment upon the Marble-Sphear or Bullet, for the discovering of its solid measure by water) and not only for the discovering of the Content of any irregular solid or gross body, but also of any irregular concave body, according to a solid dimension, as I have both said and also experimentally showed before. But however that common way delivered by Mr. Diggs and also by Clavius and some other Latin Authors, need not to be tied or confined wholly to such a kind of Vessel as they speak of, but the same may be as well performed by any other kind of prismal, or any Cylindrical Vessel, such as a Pail or other circular Vessel, being exactly made: but that indeed the Bases of these Vessels, will not be altogether so readily obtained as those of an exact cubical, or other rectangular Vessel, unless it be by our artificial way of measuring, where the base of the Vessel is equilateral or exactly circular. And of this way I would have produced an experimental example, aswell as of the other, and that by the same regular body, to have compared them together, if I could have met with ever a Vessel fit for the purpose: but however the thing being so very plain and perspicuous of itself, there needs no example either from experiment or otherwise, to illustrate it.  
But now whereas after all this, it may 
Objection. concerning the difference of gravity in Water, in reference to the foregoing Dimension of Bodies.  be objected, that all water weigheth not alike; but that different kinds of water (as Rain-water, Fountain-water and Riverwater) are of a different gravity; and therefore our foregoing experimental Conclusions for the weight of common water in general, in relation to its solid measure (or for the comparing of its gravity and magnitude, or ponderal and dimensional quantity together) cannot hold generally true: To this I answer; that albeit these several waters 
Answer.  do usually differ somewhat in gravity, (which I cannot deny, but must needs acknowledge, and that not only from natural reason itself, but also from my own experience, which I shall now come to show,) yet not so much, as to make any notable, or considerable difference in the solid content of a Body produced severally thereby (especially Rain and Riverwater) for which end those Conclusions were aymedat, and intended by us; but that any one of them may be indifferently used in this thing, without the committing any considerable error, as I shall show by and by, in discovering the gravities of several waters and comparing them together. And Marinus Ghetaldus in all his several experiments made upon Water, for the comparing of it and divers other bodies both solid and liquid, together, in respect of gravity and magnitude, speaketh only of water in general, without any difference or distinction.  
Now the water by which Mr. Reynolds made his experiments upon his 3 several parallelepipedall Vessels of 283 1/2 inches, formerly mentioned, to found the true weight thereof, was (as he told me) fair settled Rain-water; And such was also the water, by which I made my experiment upon the cubical Vessel of 216 inches, as I noted formerly, it being the most simple kind of water, and so generally the best for that purpose: for so that learned Mathematician W. Snellius going about to discover the true weight of the Rhynland-Foot, in its cubical capacity (which he will have to be exactly equal to the old 
* Suel. in Eratosth. Batav. l. 2. c. 2. And where he pitcheth upon the same Roman Foot (by name,) for the truest, which Mr. Greaves and most others do; namely, Pes Colotianus aforementioned: but differeth in the magnitude thereof from Mr. Greaves; for that he makes it greater than the English Foot; whereas Mr. Greaves makes it lesser. For he saith that the London-Foot, which is generally used throughout England, and whose measure was taken from the Iron Standard of 3 Feet, kept in the Guild-hall; and so transmitted to him, is 968 parts, such as the Leyden-Foot, vulgarly called there the Rbynland-Foot, (which he makes exactly equal with the foresaid Roman-Foot) is 1000, Eratosth. But. l. 2. c. 1. whereas Mr. Greaves makes the very same measure of the English-Foot to be 1034.13, such as Pes Colot. is 1000: and the Rhy●land-Foot (or that of Snellius) to be 1068.25 of the same parts. And so this Foot to be 1033 such parts as the said english Foot contains 1000 (and the foresaid Roman-Foot to be but 967 such parts as was formerly showed) and consequently the English Foot will be from thence, 968 such 〈◊〉 ●s the Rhynland Foot is 1000, which ●●tees exactly with the observation of Snellius. But indeed Snellius takes the dimension of Pes Colot. chief from the bore description and delineation thereof made by Ph●l●nder in his Commentaries upon Vitruoius, Architect. l. 3. c. 3. (who there faith, that he found this Foot ●o agreed with that of Statisius) presuming upon that typograp●icall ssenent, which he had received about the contraction of letters and Lines upon the paper, after their impression, by a 60th part. For so saith he in Eratosth. Bat. l. 2. c. 1. before-cited, Charta uda dum praelo subiicitur & typum patitur, ipsá pressurá & humour quem anteà imbiberat, non nihil extenditur & seipsan fit amplier, qua post modùm siccate, iterùm contrabitur, & simùl linearum measuras quas receperat, junsto exhibet minores. Pars enim sexagesima typorum & formoiuns longitudini excusis decedit, quemadmodùm à diligentibus & peritis Typographis sciscitando edoctus sum. Which last must needs be erroneous and uncertain, and cannot hold, generally true, as reason itself (besides the experience of myself and others) may plainly demonstrare, according to what I formerly: said concerning this point.  Roman-Foot) in water, that thereby he might (as it seemed to him) transmit the measure thereof so much the more certainly and easily to posterity, (like as many eminent men together had done before at Rome, for the determining and establishing of the exact measure or quantity of the said Roman Foot (as Ciaconius out of Lati●us Lati●ius his * observations of the Roman Foot reporteth, who was one of the eight that made the experiment together at Rome) and so two others, by name, Lucas Paeius, and Villalpandus, by two other Vessels; conceiving it to be a surer way to discover (or recover) the same, than by hairs, grains of Corn, digits, palms, and the like) did above all other waters, choose Rain-water, because that (saith he) being fallen from the 
See M. Greaves his Discourse of the Roman Foot. pag. 12. 13.  heaven, brings down with it no taincture of any earthy dregss; and moreover for that it seemed to be alike in a manner, in all places; and this he used after many days settling, being thereby made very pure and clear.  
And than, the water with which I measured the Standard- Wine and Ale-gallon at the Guild-hall, to found the weight thereof, and thereby the solid Content in inches, was from the new River of Ware (commonly called Middleton's River, and which water is in a manner of equal weight with Rain-water, as I shall show by and by) as also that at Goldsmithe-hall, with which I measured my two brass concave Cubes, the one of 64 inches, and the other of 1 inch, to found the weight thereof: (and the less Cube also another time with rain-water) and also by which I performed my last experiment there, upon the weighing of a solid body in water, etc.  
As for Rain-water compared with Fountain, Spring, or Well-water, 
Est in aequali mole ratio aquae plaviae ad distillatam, quemadmodùm 1000000 ad 997065 pluviae au●èm ad putealem, ut 1000000 ad 1007522. Eratosth. Bat. l. 2. c. 5. The last of which I found indeed to be so, according to the Weights of those two waters set down by him, or more completely, as 10005000 to 10075216. But the first of these I found according to his weights of the waters in the Vessel by which he made the experiment, (and so of his cube-Foot of the same waters, which I exactly deduced from thence) to be as 1000000 to 997117. being more completely, as 10000000 to 9971167.  in respect of gravity; Snellius observed the proportion to be, as 1000000 to 1007522; and of Rain-water natural, to the same artificial, or distilled, to be as 1000000 to 997065: For that he might have his Rain-water exactly defecated, or cleansed & purified from all earthy dregss or grounds, it seemed good to him (as he saith) to use for the experiment of the exact weight of a cubical Foot of water, chief that which might again be collected into itself from vapours and exhalations, and so betook himself to Chemical distillation, according to that form, which they usually call by a technicall or artificial term or expression, Balneum Mariae, seu Maris, because that is in no wise forcible or violent, but very gentle; And than moreover he used pure Fountain-water, to try what his cubical Foot might altar in gravity, in these three waters: And this experiment he performed by a cylindrical Vessel made of brass with all the accurateness that might be, having its altitude and Diameter equally semi-pedall, from whence he deduced the weight of a cube-Foot in Water.  
But if we will be so very curious concerning the difference of gravity in several waters; than we may as well question, whether all water of the same kind; be of the same gravity or not (except Rain and Snow-water, which we conceive to hold alike in all places, without any sensible difference) as whether all Fountain, and all Riverwater do weigh alike, especially the first of these two, but that it may altar and differ in gravity as well as in other things, according to the different nature and quality of the Earth where it is engendered, and of the veins and passages thereof, through which (as it were through Channels) it runs, and so the various matter wherewith it is mixed; and more especially, if one water be a mere simple water, and another be a mineral (for so Naturalists do usually distinguish waters) for than these will more sensibly differ in gravity; and so will several mineral waters among themselves, being of a different nature: And indeed most of our Riverwater seems to come from Fountains or Springs; and also for that Aristotle gives to them both, the same original of generation, 
Aristos. l. 1. Meteor. c. 13.  to wit, from vapours and fumes (or air) in the caves and passages of the earth, condensed and concreted into water by the coldness of the Earth. And therefore seeing that the generation 
Magir. Physiol. peripatet. l. 3. c. 4. Com. & l. 4. c. 7.  of fumes and vapours. (and thence of water) under the earth, is continual; it followeth, that the flowing, and the course of Rivers is perpetual: so that a River is, as it were, no other thing than the water of a perpetual Fountain or Spring, continually running on in a great body or bulk. 
Keckerm. System. physic. l. 2. c. 15.   
But now as for the difference of gravity in waters homogeneal, (or of the like kind and denomination) I shall prove the same by experiment, from four several Fountains, or Springs (and partly from two Rivers) in comparing them with Rain-water several times: and which is as much as (if not much more than) any other hath done in this kind, that I could ever yet hear of: And the waters which I first tried by way of ponderall comparison; were first, Rain-water, as being the Base of the Experiment, in regard of what hath been said there of before; and so to which all the other waters are here compared; and which I received into a clean Vessel as it fell from the Clouds, that so I might have it as pure as might be, without the mixture of any earthly matter: And the Fountain-waters which I now used, were from the Conduit in Grays-inn Fields, commonly called by the name of Lambs-Condait, which is a pure Spring of itself; and from the Standard in Cheapside, which is conveyed under ground by pipes, from a Spring at Paddington, being a Village about 3 miles distant from London to the NW. (as also is the water of the other two Conduits at the two ends of the said Street, from the same Springhead) and yet is received as pure and clear at the said Conduit, as if were taken at the Springhead itself: And than the River-waters were from the Thamès, and from the new River of Ware, or Midleton's River aforesaid. And with these I tried Snow-water, (which was pure Snow as it fell, having never touched the ground nor other thing, beforeit came in the Vessel where it was put, and there dissolved.) Which other several Waters aforesaid, I took the pains to see always fetched from the right places, thereby to avoid all errors and mistakes which might hap by trusting to any messenger alone; and so had them carried to Goldsmiths-hall (where I was to make this experiment also) letting them stand there a settling for 2 days: (though indeed the Fountain-waters being so very pure and clear, needed not so much settling) And the Vessel, by which (as a Standard or Gage) I first tried these six several waters, was the same by which Mr. Roynolds had formerly made the like experiments (though not upon so many waters, unless aritificiall waters, as distilled strong-waters, and wines, & some other liquid substances, as he told me) and that was a glasse-Phiall which held almost 3 quarters of a Wine-pinte, having the neck thereof done about with lead or pewter, and a top or Cover of the same metal made to screw on, which upon the proposal of my intention to him concerning this experiment, and his good liking and desire of the same, he courteously offered to lend me, and I as courteously accepted; for I could not than meet with one so fit for that purpose as that was, save those which were too small; for this was at the smallest. And so the 21th of March 164 8/9, I made my experiment at Goldsmiths-hall by the said Glass with all the exactness that possible I could, 
I experiment for the gravities of several waters, and the same compared with Rain-water. Mar. 21. 1648/9.  both in filling the said Glass with the several waters one after another as they were weighed (beginning with Rain-water) still screwing on the said top, to keep in the water from falling out, as nearly as I could, & than in the weighing of them; & thereby I found first the Snow-water to 2 grains less than the Rain-water, and the two River-waters to weigh each of them I gr. more than Rain-water; and so also the Standard-water in Cheapside, (and thereupon these 3 to be of equal weight) & the Lambs-Conduit-water to weigh 4 gr. more than Rain-water. But indeed finding this Glass to be very uncertain for the performing of this so nice and curious an experiment, in regard both that as I still screwed on the top or Cover thereof after the filling it, to keep in the water that none might fly out, there still issued forth some water, and besides also that the mouth of the Glass was somewhat too wide, (considering the smallness of the Glass) for the filling of it with every several water exactly alike to the lest drop as was requisite to do in do: small a Vessel; sigh that I observed, that one ordinary drop of water more or less would altar the weightfull 2 grains more or less; which made me continually to iterate the experiment by the said Glass upon every several water, by filling up the Glass again, and than drying the outside thereof before I put it again into the Scale●. But however being doubtful of the same, and that I might make this experiment with all the exactness that might be; I did thereupon for my further satisfaction, repair to the Glass-house in Broad-street, and there cau●●d a Glasse-Phiall to be presently blown before me, which might hold a Wine-pinte, as near as could be guessed I for this I conceived would be a convenient bigness, so ●s it being filled with water, might be conveniently weighed in a small Balance that would turn upon some small part of weight, as that was which we had used for the other Glass, being a very nimble and subtle Balance, that would turn upon the 4th. or 5th. part of a grain, which was as much as could well be expected from a balance to weigh a Vessel of this capacity being filled with water or other liquor,) and was also blown round, somewhat after the manner of an Urinal, but made flat at the bottom, so as to stand upright, and very smooth and even at the top, and with so small a mouth or orifice, as that it might be always filled alike to the lest drop; for that, when I came to fill the same, not one drop could be shaken out, till a little was first sucked out with a quill, (insomuch as that one might have carried it full of water in their pocket, with the mouth downwards, and not have spilt one drop therein:) and which being thus exactly made for the purpose; I weighed the very same waters therein, which I had done before by the other Glass, having let them stand all the while in their several Glasses, with an addition of two other Fountainwaters, which in the mean space I had been told of, as being generally accounted the two finest waters abo●t London; the one near the Post●rne-gate on Tower-hill, called the Postern-Spring; the other on the backside of St. Giles-Church at Cripplegate, called there by the name of Crowders Well; both which are commonly accounted exceeding good for all manner of soar eyes; and have a very pleasant taste, like that of new-milke, (especially that of 
* Upon the weighing of this water, the Assay-master of Gold-smiths-Hall, by name, Mr. Alexander Jackson (who was pleased with much ccurtesie, humanity, and patience, to assist me in these and all other the experiments which I th●re made; as to the work of the Balance, or the matter of weighing, and which he performed with his own hands, with all the accuratnesse that might be) told me that he once knew an ancient man in this City, who whensoever he was sick, would drink plentifully of this water, and was thereby immediately made well: and so, being overcome with Drink (as he often would be) would presently drink of this water to make him sober, as finding it to be the most speedy remedy.  Crowders-Well, which much exceeds the other for sapour and gravity) and which therefore for their virtue and gravity above the rest, may be taken for mineral. And so by this new Glass, and the foresaid 
2d. Experiment, for the gravities of several waters etc. Mar. 31. 1649.  Balance, I found first the Snow-water to weigh one grain lighter than Rain-water; as also the new Riverwater (and so these two to weigh alike) and the Thames-water not to differ sensibly in gravity from Rain-water, and the Conduit-water in Cheapside to be 2 gr. heavier than Rain; and and the Lamb's conduit-water to be 3 gr. heavier than Rain; and than, the Postern-Spring-water, to weigh 7 gr. more than Rain; and St. Giles-water, (or that of Crowders-Well) to weigh 12 gr. or half a pennny-weight more than Rain-water; and so to be weightier then the Postern-Spring-water by 5 gr. And so as I weighed each water, I continually iterated my experiment upon the same, by putting out a few drops, and than filling up the Glass again; and so afterwards still drying the Glass throughly on the outside, I again committed the same to the Balance, and so found the several waters to weigh as before; save only the Thames water now weighed one gr. more than Rain-water, which whether it was in the filling of the Glass, or in the weighing, I cannot justly say: But soon after this it happening to rain, I took some pure fresh rain-water, and withal, such other of the aforesaid Waters as were near the place of observation, fresh again, viz. the Conduit-water in Cheapside and the two foresaid River-waters, and (after due settling) tried them over again by the same Glass and Balance; as also the Snow-water 
3d. Experiment for the gravities of several waters. Apr. 7th.  which I had used before (for fresher I could not than get) and found them all to weigh exactly as at first by this Glass; and so the Thames water to be of equal gravity with Rain-water.  
And than I having a small Glass standing by me, whose mouth was rather less than that of the pinte-glasse and held little more than 2 ounces-troy and an half; I thought good to make an experiment by the same, upon some of the foresaid waters, 
4th. Experiment for the gravities of several waters. Apr. 21.  viz. Rain & Snow-water, & the two River-waters: (because these did neerliest agreed one with another) and thereby found the Snow-water to be half a grain lighter than Rain-water; and the two River-waters, to be each of them of equal weight with the Rain-water as before. And this was performed by a very small and subtle Balance, which would easily be turned with one mite, or the 20th. part of a Grain.  
And thus having discovered the difference of gravity in several waters by smaller Vessels, (or quantities of water) I conceived it very convenient after all this, to make one experiment or observation more upon all the foresaid several waters, by a much larger quantity; considering that these smaller quantities were not so sufficient to discover the difference of gravity, as to ground or determine Proportions of gravity thereupon; and so that the greater the quantity of the waters was, the greater & more apparent still would be their difference of gravity; and so to see how this would agreed in Proportion with the other: And to this end I got the largest vitreall Vessel or Vial that I could meet with fit for the purpose; which was one that held near about five wine-pintes and an half, having a very small neck, done about with lead or pewter, and a ●op or Cover to screw on very close (like the Glass by which I made my first observation in this kind) and which in the screwing on, would not force the lest drop of water out of the Glass, being exactly filled; and the mouth of the Glass not being half an inch wide: And so having provided all the foresaid eight waters fresh again (except the Snow-water, which could not be had fresher than that which I used before) and the same duly settled; I found by this Glass (from the great 
5th. experiment upon several waters, in respect of gravity. May 3d.  Standard-ballance, which would sensibly turn opon one grain) first the Snow-water to weigh 8 gr. less than Rain-water; and the two River-waters to be equi-ponderant with the Rain-water: and the Conduit-water in Cheapside, to weigh 14 gr. more than Rain; and the Lambs-conduit-water to weigh 24 gr. (or 1 p. w.) more than Rain: and the Postern-Spring water to be heavier than Rain-water, by 2 p. w. and 9 gr. (or 57 gr.) and last, the water of Crowders-Well, to be weightier than Rain-water, by 3 1/2 p. w. or 84 gr. and so to exceed that of the Postern-Spring in gravity, by 1 p. w. and 3 gr. viz. 27 gr.  
So that now from these several exact experiments and observations, it is manifest, that all waters homogeneal (or of the same kind) are not of the same gravity, but do sensibly differ therein, as we have here proved from four several Fountain or Spring-waters, all of them differing in gravity one from another; and that in the very same continued order and proportion in a manner, from all the several experiments by which they were tried: as that of the Conduits in Cheapside (from Padington-Spring) to be the lightest; and the next above that, the water of Lamb's Conduit (in Gray's-inns fields) and than the next to exceed that, the Postern-Spring water (on Tower-hill) and than the heaviest of all, the water of Crowders-Well at St. Giles. Cripplegate; and which thing I was very desirous to demonstrate▪ And therefore consequently, that no certain, positive Proportion can be determined between Rain and Fountain, or Pluviall and Puteall water, as Snellius hath done; and which I cannot but wonder, that he (so intelligent an artist) should do, from one single experiment or observation only, having used but one Fountain-water (by what I can perceive from him) as if he took all Fountain or Puteall waters to be of the same gravity. And that which he used in that experiment, seemeth to be much heavier than the heaviest of those fountain-waters which we have here made use of; as appeareth by his comparing it with Rain-water the proportion being (as I shown before from him) as 1000000 to 1007522; whereas the proportion of Rain-water to the weightiest of, the Fountain-waters which we have here experimented, will be (by our l●st and largest experiment, which is the best for that purpose) but as 1000000 to 1002104 ferè, and from which the Proportion between these two waters, deduced from the observation made by my new-pint-Glasse, will but little differ, (and that by way of defect) it holding there from Rain-water, to St. Giles-Well-water, as 1000000 to 1001925 ferè, both which by a millenary contraction, will be in a manner the same, viz. 1000 to 1002; and by a decumillenary contraction or abbreviation, the one will be as 10000 to 10019, and the other (and better) as 10000 to 10021. and both which by arithmetical mediation, will be as 10000 to 10020; and so between 1001925 and 1002104, the intermedian proportion arithmetical, will be as 1000000 to 1002014.  
But indeed, if we should have yet further tried the same by a larger quantity of the waters: than probably might the proportion have still risen higher; and so have come somewhat nearer that of Snellius, according as we have still observed from our several experiments; that as the vessel or quantity of each particular water hath been greater, so the higher hath risen the proportion of gravity between them, (though very little) according as the difference of gravity hath become somewhat greater than perhaps it should: For that surely, whether the same be experimented by a lesser or larger quantity; yet the same proportion of gravity should arise, according as the the difference of gravity should be proportionably the same; and which hath nearly happened in these several observations by several quantities of waters, except those of the first experimene●●● the imperfect Glass. And which would continually hap, in case the gravity of each particular water could still be taken in every several Vessel, or Quantity, so very precisely to the lest part of weight, as might be imagined: but which, considering that every Balance (as an Instrument or Organ, consisting of divers parts) is continually subject to mutation upon every small occasion, (especially those smaller, nice, nimble, and subtle Balances, such as we used in most of our experiments of this kind, which by the ordinary breath of one's mouth or nostril (any thing near at hand) or the lest motion of the air, or noise, or sound, will be sensibly diverted from their due course and positure, to which they tend) and so not infallible; therefore the same cannot well be expected, though notwithstanding we here continually used all the sedulity and solicitude that could be; taking continually the gravity of each particular water, upon an exact equilibrity or equiponderancy of the Scales, according to the most precise perpendicularity or rectitude of the Index, or Tongue of the Balance; as nearly, as by the sight might possibly be adjudged; insomuch as that we could not perceive any sensible error or mistake to be committed therein. And moreover for a further avoiding of errors herein; I performed each particular experiment upon the several waters (according as they are set down orderly before) still at one time, according to one and the same setting, or rectifying of the Balance; conceiving it to be more convenient and sure so to do, than at sundry times apart, for that there are hardly any Scales to be met with, but at several times, will require a several setting or rectification.  
But for the determining of a certain, positive Proportion between Rain and any other 〈◊〉, (or between any other two particular waters) it is best surely to use as large a quantity of those waters, as can conveniently be weighed, (consideration being had of the bigness and ponderousness of the Vessel to contain them) But the vitreall Vessel or Vial, by which I made my last observation, would hold well-nigh as much as the aereal cylindrical Vessel by which Snellius made his observation, as I have computed it, by comparing his Weight and Measure with ours.  
And by most of our several observations, we found the two River-waters aforesaid to be of like gravity with Rain-water, without any sensible difference, and so to be equiponderant in themselves. And for Snow-water, we found the same by every particular observation (even from the lest vessel or quantity of water, where the difference of gravity was least discernible) to be sensibly lighter than Rain-water, & so the lightest of all; & in which, our experimental observations will agreed with natural reason itself, which showeth Snow to be a much lighter substance than Rain; and in which all 
* Zanard. Disput. de Vniverso el●memari, part. 3. qua●st 22. Aliara dispositionem habet nubes nivis ab ea quae est aquae: nam nubes pluviae habet, quod sit magis densa, magis obscura, & magis unita in anum locum: Nivis autem nubes est tenuis & subtilis, & cùm multum de aereo continet, est clara & quasi alba, & per aerem sparsa, etc.  Philosophers do agreed; that the Cloud or matter of which Snow is engendered, is more hot and dry than that of which Rain is, and also more airy, and so the humour or moisture resolved out of Snow, is airy, and very light, and as it were a froth or some, whereupon it so nourisheth and cherisheth the Earth: And because it contains much of air, it behooveth that it should contain also much of warmth and moisture, although yet it is drier than water. And so Aristotle likeneth Snow to Froth or Foam (& also calleth it so) in respect of its whiteness, which he saith to arise chief by means of those parts thereof which are more airy; for that for 

Tenet multum de calido aereo nubes nivosa etc. Zanard. ibidèm.  
Aristor. lib. 2. de generat. animal. cap. 2.  
Plin. nat. hist. lib. 17. cap. 2. & Keckerm. Syst. phys. lib. 6. cap. 9 Theor. 1. & 2. de nive.  
Colleg. Conimbr. in Aristot. meteor. Tract. 7. cap. 5.  
Magir. physiolog. p●ripat. lib. 4. cap. 6.   this reason also, Froth (saith he) is white; and so water having oil mingled with it. And so also Pliny calleth Snow aquarum coelestium spumam, which his english Translator Dr. Holland interpreth, the foam or froth of Rain-water from Heaven, concerning which see Keckerman. And the Conimbricensians (according to Aristotle, lib. 1 Meteor. cap. 11. and lib. de mundo, c●p. 4) say, that Snow is a Cloud conglaciated, or frozen together in a friable density; and which Cloud obtains so much a greater ficcity than that which is changed into water, by how much it congeals or grows together by the power or efficacy of the more prevalent Cold; sigh that the Cold while it bindeth, doth express, or force out the moisture, etc.  
Seeing therefore that the matter of Snow, is more hot, and dry, and more airy thin and subtle than that of Rain; and consequently more light; and that Snow-water is no other than Snow dissolved, and so still retains the nature of Snow: therefore also will Snow-water be necessarily lighter than Rain-water.  
As for the Proportion between these two Waters, I may from all our several 
The Proportion of Rain-water to Snow-water.  observations and experiments beforegoing, conclude the same to hold generally from the heavier to the lighter, as 10000 to 9998. For by 2 or 3 several observations made by the new Glass of almost a wine-pinte, where the gravities of these two waters (and so the difference of gravity) were found still the same; the proportion will be as 100000 to 99984, ferè: and by the last observation, being made by the great Glass of about 5 1/2 wine-pintes, holding about 7 poundstroy of water, it will be as 100000 to 99980 ferè (and which I take to be the truer,) and if we will take the intermediate proportion arithmetical, the same will be 100000 to 99982 ferè. Or the former proportion will nearly hap, if we shall mediate between that which▪ will be produced from the lest Glass of all, (which was that of about 2 1/2 oun. troy) being as 10000 to 9996 completely, and that which we have here produced from the greatest Glass of all, viz. 10000 to 9998 in a manner completely; which will be 10000 to 9997.  
And now from these our several experiments before-going, for discovering, for discovering the gravities of the several kinds of common, simple, or natural Waters, as in reference to the finding out of the solid quantities or capacities of Bodies altogether inordinate or irregular, which will not in themselves admit of an ordinary or regular kind of Dimension, but their contents must be obtained by some extraordinary or unusual kind of way, as we have lately showed; it is manifest, that the way here propounded by us for the same, may be performed by any such kind of Water in general, as we have here tried, according to the experiments formerly laid down by us for that purpose, from Rain-water, (and so upon which this our Atactometricall practice is grounded, as being the most indifferent water for a general use in this kind:) for that in our last observation, which was made by the greatest quantity of the several Water's asorenamed, weighing about 84 oun. or 7li. Troy, the difference of gravity between Rain-water and the heaviest of the Fountain or Spring-waters, was but 3 1/2 p. w. or 84 gr. And which said quantity of water is much more than double the quantity of that which was found to be of equal magnitude with the foresaid spherical stone-body, and so from whose gravity we obtained the content of the said solid body, in the measure propounded; & therefore had the same been thus inquired from both those Waters, that is, the lightest (except S●owwater) and the heaviest, (though the Water by which it was done, we have showed to be of like gravity with Rain-water) there could have been no considerable difference therein.  

THE CONCLUSION. For the more spèedy absolving or expediting of all the foregoing Dimensions in general.  
ANd now for the more easy and speedy performance still of all the Dimensions, and metrical Conclusions contained and mentioned in this Book, I shall subnect this one thing as a Coronis to the whole Work, by way of advertisement and advice to the practical Reader that is not yet acquainted with the most compendious ways of arithmetical Calculation, which is, that he would use continually all along with his geometrical Lines, or Lines of measure, (whether natural or artificial) that most excellent artificial arithmetical Line (as I may so term it) or Line of Numbers (so called by the author and contriver thereof, Mr. Edm. Gunter, being deduced by 
See also Mr. Edm. Wingates Rule or Scale of proportion, of the like kind. And, Mr. Oughtred's Circles of Proportion; or rather the same converted into ● spiral Line.  him from the Logarithmical or artificial numbers, and so being, as it were, a Line or Scale of Logarithmes) by which all Mathematical Questions and Conclusions to be wrought by Proportion, (as indeed what are there, which may not be reduced to a way of proportion) are cast up, or computed in a most compendious manner (as it were geometrically or mechanically) merely by Scale and Compass, without the labour of the pen: For all those our artificial Dimensions, which consist merely upon the squaring of some one number, or of one simple quadrature only, (such as are of all ordinate or regular Planes or Superficies, and whether taken simply in themselves alone as Figures, or many of one kind conjunctly, as constituting the total Superficieties of regular Solids,) will be absolved by one extent of the Compasses only, being doubled, or once repeated upon the said Line, according to the Analogism of Multiplication. And that will be, as from an Unit upon the said Line of numbers, to the linear number or term taken by the artificial Line of measure, or Line of quadrature, (for some dimensional line of a Figure whose superficial Content is required therefrom, as representing artificially the side of the equal Square) so from that, to the quadrate number. for the superficies of the Figure.  
And all those artificial Dimensions which depend merely upon the cubing of some one number or term (such as are of all ordinate or regular Solids, for their solid measure) will be absolved by one and the same extent of the Compasses being trebled, or twice repeated upon the said Line of artificial Numbers, according to the foresaid Analogisme (in a compound or double Multiplication) which will be, as from an Unit, to the linear number or term taken by the artificial Line of measure, or Line of Cubature, (for some dimensionall line of a regular Solid given to be measured thereby, it representing artificially the side of the equal Cube) so from thence to the Square thereof; and from that to the Cube, for the solid content.  
And all those artificial Dimensions which consist not of cubing, but yet of a twofold Multiplication; the one whereof is a Quadrature, (such are of all regular-like Solids, as Cylinders, Cones, and the like) will be absolved by two several extents of the Compasses, each of them being doubled or once iterated upon the said Line of numbers; which will be first, as from an Unit to the quadratary term taken by the artificial Line of measure, representing some dimensionall line of the ordinate Base; So from that, to the Quadrat thereof, for the artificial Base: Than, as from an Unit to that Quadrat; So from the other number or term taken by the same artificial Line of measure, for the Axis or Altitude of the solid Figure to be measured; to the solid area of the Figure. Or it will be, As from an Unit, to the number for the Axis or altitude; So from the quadrate number for the Base, to the rectangular solid number, for the whole solid Figure itself. And here, if the two foresaid lines of a regular-like Solid, to wit, the Axis, & the basiall line (whatsoever it be) do hap to be equal (as they very rarely do) than will the Dimension be cubical, as that of an exactly ordinate Solid by some one of its dimensional lines; and so be absolved upon the Line of Numbers accordingly. Than lastly, for the superficial Dimension of these regular-like Solids, consisting of one Multiplication only, and that commonly of two unequal Terms, and therefore not an exact Quadrature; the same will be absolved by one extent of the Compasses, once repeated upon the said Logarithmeticall Line, or Line of Proportion, according to the Analogism of Multiplication; which will be, as from an Unit to one of the two linear Numbers or terms taken by the artificial Line of measure (which do represent some one dimensionall line of the Base, and the side, or other line upon the superficies of the solid Figure, according to the nature thereof,) So from the other, to the rectangle superficiary Number▪ for the superficial Area of the solid Figure (the Base▪ thereof being secluded) And here also, if the said two dimensionall lines of the regular-like Solid, do hap at any time to be equal (as they v●ry seldom will) than this Dimension will be exactly quadratary, as the superficial dimension of an exact regular Solid, by some one of its dimensionall lines.  
And the like with these, understand for the more spèedy computing of the gravities, or ponderall quantities of regular and regular-like metalline bodies, being inquired out artificially in the same manner as their solid measures: and therefore, as there the last proportional number or term upon the said Line or Scale of Numbers, denoteth solid measure, here it will denote gravity or ponderosity.  
And so likewise in the work of Gauging, for the speedier computing of the liquid Contents of Vessels in Wine or Beer (as was of the solid content of a Cylinder, or any other regular-like Solid, from our artificial way) where, after the mean Diameter of the Vessel being had, according to the artificial Gauging-Lines; it will hold upon the foresaid Logarithmical Line (or Line of Numbers) by a twofold extent or opening of the Compasses, thus; viz. first, as from an Unit to the mean Diameter, So from that to the Square thereof; Than, as from an Unit to that Square; So from the length (or height) of the Vessel to the liquid content thereof in Gallon-measure. Or as from an Vnit to the length of the Vessel, so from the Quadrat of the mean or equated Diameter, to the liquid measure aforesaid.  
And so with the like expedition by this Line, understand all Atactometrical operations to be absolved, in computing the solid quantities of irregular bodies from the gravity of the equal body of water, according to the Terms of Proportion or comparison between the gravity and magnitude, or the ponderall and dimensional▪ quantity of this liquid Body, delivered by us in our Atactometrical Discourse.  
And so all metrical operations arising here ●●ō the natural Measure, will be thus expedited, according to the dimensional Proportions delivered in this Book, both for measure and weight (and all others of the like kind, not here particularly expressed) according to what I said before. And thus I put a Period to these my mathematical Contemplations and Exercitations.  
Soli Deo universipotenti, qui omnia (ut loquitur Sapiens) mensurâ, numero, & pondere disposuit, sit gloria, honos & laus, in saecula, & omnem sempiternitatem. AMEN.