SECT. I. Of the nature and division of the Lines of Measure in generall, for the performing of all the aforesaid Dimensions.

FOras [...]uch as to the due measuring of any Magni­tude or Quantity continu­all (in practicall Geome­try) there is required some certain Measure first to be given or appointed: Ther­fore 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 [...], [Page 2] which is as much as certain, definite, determinate, speak­able or expressible by voyce, 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 Schol. Mathemat. lib. 21 Adr. Met. Geom. pract. Par. poster. seu Gaeodaes. spoken or expressed, &c. But most of the Latine Geometers do call! such a line Ra­tionalis, a Rationall line, for that (as Ramus saith in the places here cited) it is rationall to it selfe, as are all magnitudes equall among themselves: and Clavius saith it is called Rationalis, be­cause Clavius in Def. 5, El. 10. it is alwayes 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 equall 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, rationall or irrationall; (but with this point we med­dle not here,) and so we here understand this Line to be most properly called Ratio­nall, as comprehending or containing in it self the dimensionall reason of allother lines measurable thereby. And therefore here we (for brevity) will with most Latine Translatours and Commentators, and also our H. Billingsley, a Citizen of Lon­don, and Lord [...]. 1596▪ English Translatour of, and Annota­tour [Page 3] on Euclid's Elements, understand any right line so first set, put or proposed, by the name of the Rationall Line (and this may be applyed 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 naturall Rationall line. And this we mean when any where we say simply the Rationall 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 artificiall and speedy mensuration of the aforenamed Fi­gures: And this Line we may (not unaptly) call the second supposed or artificiall Rationall Line, as being derived from the former, and so substituted in place thereof: like as in Numbers the Logarithmes are usually called Artificiall Numbers, as being substituted instead of the naturall num­bers, from which they are deduced, and whose place they supply in a most excellent and admirable manner, by per­forming all arithmeticall operations with that facility, ex­pedition, and compendiousnesse (and exactnesse also in some cases, as I shall afterwards upon occasion shew) which the naturall numbers themselves cannot, for that by these, the two most tedious and troublesome parts or species of Arithmetick (to wit, Multiplication and Division) are wholly avoyded and abolished, and that most difficult branch (or operation) thereof called the Extraction of Roots, is mightily abbreviated and facilitated: And so the Arithmetick performed thereby is usually called artificiall Arithmetick.

Now seeing that every continuall or continued Quanti­ty falling under Measure (in practicall Geometry) is refer­red and reduced to the discrete, that thereby its dimension [...] may be made more manifest to us; so that Geometry hath [Page 4] perpetually need of Arithmetick for the explicating and ex­pressing of its magnitudes in their dimensions: Surely no kinde 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 decimall solution, for that by this the practicall, instrumentall, or mechanicall part of measuring, or of the art Metricall, 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 partes dividitur, Geometriam, scil. & Geodaesiam: Areae nam (que) 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 greeke words, [...], do (poetically) signifie the same that the Latine 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 Mathematicall solution of unity or continuity, is of all others the most absolute and cer­tain, and the most perspicuous and rationall, 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 effe­ [...]ed by it. And what utility it hath brought to the Mathe­ [...]atiques in generall, may be sufficiently witnessed by that [...]ost noble and usefull part of Geometry, called Trigonome­irie, and that in the Radius of a Circle (which is the very Basis and Root of all Trigonometricall operations) where (to wit) first, that greatly renowned Mathematitian Johan­nes [Page 5] Regiomontanus, having for a long time used the Sexage­nary solution (as Ptolemy and others before him) did at last bethink himselfe of the Decimall, as being much better (see­ing that the Unit would perform a far greater Compendium then 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 & cō ­sequently other Mathematical arts, as Astronomy, Geography, &c. depending thereupon) did become, in the practice, far more facile and expeditious then before: for where the first proportionall Term of the Trigonometricall proposition, is the Radius or totall Sine (which very frequently happens, or may be so made for the most part, as the learned Pitis­cus excellently sheweth 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 Naturall Numb. conjunct or manifold Pitiscus, Trigo [...]. lib. 5. composition (or mutuall 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 in­tegrall, and the fractionall part thereof. And moreover the benefit of this solution of unity is excellently seen in that most excellent Arithmeticall operation, vulgarly called The Extraction of Roots, wherein (to wit) seeing the Roots of numbers not explicable or rationall (which the Alge­brists or Cossists commonly call Surd Numbers, and so their Roots surd Roots) cannot be exactly had, then those num­bers are reduced into some kinde of decimall parts (or parts of a great denomination, as Ramus termeth them) as C [...]n­tesms, [Page 6] Millesmes, &c. and that figurate, as Quadrate, Cu­bique, &c. that thereby their Roots may be had more certain and neerer to the truth, Ramus lib. Geom. 12 de Quadrato & 24 de Cubo. then they can by the naturall or vulgar ex­traction, as Ramus sheweth in the aforesaid Books, where he (the first) shewed this See Wingate A­rithm. 1 Book, [...]1 Chap. 3 Sect. kinde of Extraction which to some seems to have been the very foundation of Decimall Arithmetick, although Ramus hath no where else in his Mathematicall Works made any other use of this kinde of numbring, 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 Trigonometricall Numbers which now we use, may be termed Decimall, as they are derived from that Radius: For all the Sines to a Quadrant, and the Tangents to an Octant or semi-quadrant, are decimall parts or fra­ctions of the Radius, but indeed the least Secant is greater then the Radius: And so wee will here make use of this kinde 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 Geometricall Figures which we have here first named, and so for which we have first extracted these kinde of metricall lines (or linear numbers) are those foure which Archimedes himselfe more especially treated of, and which are as it were, the beginning and ground of all the rest, namely, the Circle, Spheare, Cylinder and Cone, and accordingly the like Lines may be extracted for all ordinate Planes and Solids whatsoever, as we afterwards shew; for that to these foure may be aptly refer­red all other regular and regular-like Figures; As to the Circle may be aptly referred all ordinate Planes, to the Sphear, all exactly ordinate Solids; to the Cylinder, [Page 7] all ordinate-based Prismes; and to the Cone, all ordinate­based 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 speciall superficiall part besides (whether the same consist of one entire plane only, as in the Cylinder and Cone, or of severall planes equall 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 artificiall metricall Lines, first extracted for the foure Figures first before named, ac­cording to any prime Rationall Line, and that to a Decu­millenary division of the same, are numerally thus.

• I. A Line for the most excellent superficiall mensuration (or Diametrall Quadration) of a Circle, is 1. 1284 [...]eré.
• II. A Line for the most exquisite Solid dimension (or Diametrall Cubation) of a Globe or Spheare, 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 Superficiall and So­lid, I shall afterwards shew in their severall places.

Every such second supposed, or artificiall Rationall Line, must be divided decimally, as is the first, true, or naturall [Page 8] Rationall Line, from which it is taken, according as the length thereof can conveniently beare: But if the first Ra­tionall Line cannot well admit of so many parts as are here set down, then 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 artificiall (or Diagonall) way, as hereafter we shall have occasion to shew. And so we have here set forth the Rationall line in parts of a large denomination, that so it might serve for more exactnesse 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 artificiall or supposed rationall line, may be said to be two fold, to wit, generall or univer­sall, and speciall or particular: Generall, 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 speciall in respect of the Fi­gure (Superficiall or Solid) to which it is appropriated and applyed, because that every such particular Figure as is mentioned in this Book, doth peculiarly and properly claim to it selfe (and that severall wayes) such a line for its more artificiall and expeditionall mensuration.

Now the use of the foure artificial Lines before-going, for the measuring of the foure first Figures aforenamed, we will deliver in the three practical or problematicall Propo­sitions following in the next Section; and withall the magnitudes and uses of all the other artificiall metricall Lines pertaining to the said foure Figures.

SECT. II.

SECT. III. Of the Superficiall Dimension of a Cylinder and Cone.

ANd as we have here shewed the most artifici­all 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 superficiall Measure. And the artificiall Lines for the performing hereof, I finde, 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 Sphear, by the only quadration of the Diameter, viz. 0.564, &c. 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 severall Lines being exactly set off from the prime Rationall Line, and then 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 artificiall Lines of Measure, and so multiplyed to­gether, their severall products shall be the superficiall con­tents of the Cylinder and Cone, according to the prime Rationall Line: where, by the superficies of a Cylinder [Page 46] 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, especialy Ramus, (in one word) Cylindraceum, and Conicum, that is as much as to say in English, the Cylindraceall or Cylindri­call, and the Conicall; and so the Superficies of a Spheare is called by them Sphaericum, viz. the Sphericall. But now for a briefe dilucidation of these superficiall dimensions in the Cylinder and Cone, I shall lay down an example in each of these three artificiall Metricall Lines (though here I do not draw them, but only expresse them by number) whereby the verity of these our dimensions also, may plain­ly appeare, to those that shall please to make tryall there­of. And first we may hereupon raise this practicall Pro­position, viz.

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 infolded together, do produce the Superficies of the Cylinder (without the two Bases) ex­actly, 264: whereby this Cylinder becomes absolute in all [Page 47] its dimensions; But by the other Cycloperimetricall terms, the Circumference was but 21.99 &c. (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 superficiall 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 multiplyed 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 excesse) but as 1/14 of a square integer or unit. And if the first or naturall Line of measure be made 1000, and so the second or artifi­ciall Line (being thereof 0.564) be also made 1000, then the said Diameter taken thereby, wil be 12.407, and the Side wil be 21.269, which two multiplyed 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 super­ficiarie dimension of a Cylinder by its Diameter and Side only, will further and fullier appear, if it be compared with the most naturall or vulgar way, in respect of the Side and Diameter only given: where, by the Diameter, the Cir­cumference being proportionally obtained, must then be multiplyed 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; M ^r. Ought­red in his Book entituled The Circles of Proportion, Part 1. Chap. 7. Sect. 10. delivers in this proportionall manner.

As 7 to 22, Or 1 to 3.1416.

So the Diameter and side multiplyed 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 &c. (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 severall ope­rations you see do consist of two severall Multiplications, and one Division, whereas ours consisteth of one Multipli­cation only, to wit, of the Diameter and Side together.

And from this Analogicall 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 equall 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 Proportionall between them, wil 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 &c.) which being put as the Semidiameter of a Circle, and so the Square thereof 84, the Area of the Circle wil be found 263.89378 &c. 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 equall to the Superficies of a right Cy­linder is the mean proportionall betweene the Side and [Page 49] (basiall) Dia meter of the Cylinder; and that the Analo­gie holds the same from the Quadrat of a Circles Semidi­ameter to the Circle it selfe, 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 Analo­gie is Lineall, and the other Superficiall.) Therefore the Analogie of the side and (ba­sial) Diameter of a right Cylinder infolded together, to the Superficies thereof (without the two Bases) wil 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 halfe the superficies of the Cylinder: for that the superfi­cies of a right or Isoskelan Cone is a rectangle Plane made of the basiall semi-periphery and the side (or the basiall periphery and the semi-side) and so the true Conicall super­fice or surface, will be 131.9469, (or 131 107/113, agreeing de­cimally with the former, or by the most common accompt, 132 exactly) And the finding of thetrue, genuine superficies of a right Cone by the side and basiall Diameter, the fore­said Mr. Oughtred also sheweth in the 8 Sect. of th [...] same Chapter before named, by this Analogie, viz.

As 7 to 22, Or 1 to 3.1416:

So the Semidiameter of the Base multiplyed 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 &c.

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 [Page 50] Conical. So that in both these wayes also, are two severall Multiplications and one Division, besides the bipartion or mediation of the basial diameter: or the same is performed by halfe the side with the whole Diameter, (for both come to one passe.)

And from hence you may observe, that Archimedes, lib. 1. de Sph. & Cyliud. prop. 14. (from whence the foresaid Analogie is deduced) demonstrateth the superficies of a right or Isoskelan Cone (without the Base) to be equall 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 proportio­nall between them is 6 6/13, or rather by immediate decimall production of the parts, 6.48074 &c. (whereas the other is by decimal reduction, but 6.461538 &c.) which being put for the semidiameter of a Circle, and so the Quadrat thereof 42, the Area wil be found 131.9469 &c. 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 &c. agreeing exactly with the true Conicall 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 (the Radius of the Base)

So 131 107/113, or 131.9469 (the true superficies)

To 38 219/45 [...] or 38.4845 the Base; [Page 51] (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 totall Superficies of the Cone, 170 195/452, or 170.4314 &c. which by the most common accompt, will be 170 1/2, the Conical being just 132, and the Base 38 1/2.

And the very same Analogie with the former, holdeth in the right, erect, or rectangle Cylinder, from the Super­ficies to the double of the Base, (or both the Bases joyntly) and contrar [...]ly; 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 &c. (the Superficies)

To 76 219/226, or 76.9690 &c. (the double of the Base, or the aggregate of the two Bases.) And so contrarily, will the Analogie hold from the double of the Base, or the two bases conjunctly (76 219/226, or 76.9690 &c.) to the Cylindrical Superficies (263 101/113 or 263.89378 &c.) 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 Cylin­dricall Superficies 263 101/113, or 263.8938 ferè, you adde the aggregate of the two Bases, 76 219/226, or 76.9690, you will have the total externall superficies of the Cylinder, 340 195/116, or 340.8628 ferè, which by the most vulgar ac­compt, [Page 52] 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 wil be found thereby, 15.04; and the basial Diameter (naturally 7) measured by the same Line, will be found 8.77, which two being multiplyed 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 wil 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 multiplyed together, do yeild 131.9488 ferè, for the Super­ficies of the Cone, which agreeth somwhat more neerly with the true Conical, viz. 131.9469, then that which was produced in this kinde 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 excesse or redundancie) 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 wayes 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 artificiall 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 naturall Line of measure, or the Prime Rationall Line it selfe, with the same expedition as that in the Cone by an Artificiall 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 Cylindricall super­fice. And so from the premisses 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 happen 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 diametrall and circum­ferentiall quadration (under a centesimall solution) will be found each of them severally; to be 6.20 (as I noted before in their solid dimensions) w ^ch squared, yeilds 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 Superficiall of a Cylinder and Cone (both theorically, and) practically, according to the Instrumentall part of Geometry; and that as well naturally [Page 54] (by way of metricall 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 analogicall (by way of comparison) from the naturall Measure to the artificiall; and that under an ample or numerous solution of the unity (according as I did in the Circle and Spheare) by means whereof, the artifici­all Measure may readily be deduced from the naturall; or the naturall measure be reduced to the artificiall: And first for the Cylinder,

Then for the cone;

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. Shewing briefly, the theoricall reason of the differen­ces happening between the naturall and artificiall Measure, in the superficiall and solid contents of Figures. And moreover, some obser­vations concerning the grounds and reasons of the Artifi­ciall Mensuration in generall.

ANd now again, as for the differences happe­ning between the superficiall and solid Con­tents of Figures, found by the naturall or vul­gar way of measuring, and our artificial way; we have formerly shewed, how small and inconsiderable they generally are▪ and also the practical, instrumental (or Geometricall) reason thereof; viz. that the severall Lines of dimension in the severall Figures (ei­ther naturally belonging to them, or artificially and com­monly abscribed to them) as namely, the Diameters and Circumferences of the Circle and Sphear; 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 artificiall Lines of Measure, are seldome or never exact and precise indeed in the parts of measure, but either deficient or redundant in the same, and so give the [Page 57] are all contents of those figures either a little lesser or grea­ter, then 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 continuall decimall augmentation (or subdecuplation) of the parts of those Lines, whereby the Contents were had still neerer to the truth; Which reason ariseth from (and so dependeth upon) the more true, natu­ral, theoricall (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 alwayes de­fective, so as that they being inverted, or drawn again into themselves, do not render 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 Cy­phers where there is need) the neerer 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 Generall,) viz. That thereby, all the tedious and troublesome operations of Arithmetick in Fractions, by the vul­gar way, are wholly avoyded: for that here all fractional numbers, whether comming alone by themselves, or toge­ther with integrall 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 excellencie of Deci­mall Arithmetick, as in reference to the work in hand: So that what errour (though inconsiderable) may at any time arise in our work, doth pro­ceed [Page 58] primarily and principally from the extraction of the Square and Cube-Roots; our artificiall Lines of measure giving immediately the Square and Cube-Roots of the Figures to which they are appropriated and applyed, (or the sides of their equall 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 regular­based Pyramids and Prisms: but else in all truly ordinate Superficies and Solids; as the Circle and Spheare before­going, 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 superficiarie 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 superfici­all dimension. And therefore if there were no defection in the aforesaid Radicall extractions, there would be none in our work; for that all our artificiall Lines of measure (or Lines of artificiall 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 equall Squares and Cubes of Figures, (in the naturall Mensurati­on) and also some other particular lines, or numbers of di­mension, in the artificiall Measure, deduced from the na­turall) are proportionally obtained severall wayes (accor­ding to unity) as from their severall lines of dimension before named (and which also are immediately given by [Page 59] our artificiall Lines of measure) are produced by Radical extraction, from the respective figures, to whose dimension (superficiall or solid) they serve, as being their Roots, or the sides of their equall Squares and Cubes, according to their foresaid severall dimensionall Lines, put unitly. But indeed, the greatest errour that can commonly arise here, (in respect of the difference of our measure from the true measure, whether superficiall or solid,) will be of no mo­ment, as I have shewed before (and shall also shew after) in severall examples. And if our measure agree with the true measure, but exactly to integers or units (as it almost always doth, and much neerer also; as even to small parts of an Integer or unit, (superficiall or solid) of the appoin­ted Measure) it will be sufficient in any matter of mecha­nicall Mensurations; for which, this our artificiall way of Measuring, was chiefely devised and intended.

But now further, as to the ground and reason, briefly, of this kinde of dimension (or of these artificial metricall Lines,) the same may be under­stood Concerning the grounds and rea­sons of the Ar­tificiall measure in generall. to be two-fold, to wit, generall or u­niversall; and speciall or particular; that consisting in Unity alone; this in the soluti­on of Unity: For the generall reason is by it selfe absolute, simple, and certain, without relation or limitation to any the proper, compounding, denominate (or other) parts of the Rationall Line, (as being the Integer or Unit of mea­sure) but considereth the same generally (and absolutely in it selfe) as some one entire or whole thing, (of which I shall (God willing) speake 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 chiefely in­sist and proceed upon) is limited and confined to the cer­tain, set, or commonly known parts of every particular rati­onall [Page 60] Line (they being considered discretely 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 minde think fit pri­vately to impose on the same for his use) and so commeth by them to that of the Generall. (Every particular, figurate Quantity or Magnitude, measurable in this artificiall 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 superficiall 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,

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 Propo [...]tion, Part 1 Chap. 9. Sect. 4. Palme (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 Palme and Digit in measuring, but only of the Foot and Inch (besides the Yard and Ell &c.) divide the [Page 61] Foot immediately into 12 parts, called by the Latines unciae & Pollices 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 Geometricall 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, usuall for Wood-land mea­sure, &c. 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 Eare: but a Barley-corn be­ing the least of all Measures (or rather no measure at all, being but the very begin­ning of measure, as an Unit is usually coun­ted no number it selfe, but only the beginning of Number) cannot be composed of any other.

And although Arithmetick in generall, 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 naturall Arithmeticall division into parts (or a di­vision into parts simply or arithmetically denominate) as halves, quarters, and the like; or more especially (and more exquisitely) as Decimall Arithmetick; into Tenths, Cen­tesms, &c. and so taketh the Contents of Superficies and Solids, to such parts thereof, Square and Cubick (or other­wise Superficiall and Solid) in generall; 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, &c. Yet this being understood only by Artists, and so not sufficient to sa­tisfie the vulgar: these simple or naturall Arithmeticall parts (or parts meerly divisionall) must at last be reduced [Page 62] to the proper compounding, denominate, or commonly known parts of the Measure given or appointed; (or the Geometricall or mensurall parts of the said Measure (as I may term them) they being by themselves alone, put as mea­sures certain, and intire, and so compounded of, (or dividu­all into) other the like kinde of parts, or inferiour 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, &c.

And now again, the speciall, particular, or partiall reason aforesaid, of this artificiall dimension (or of the Artificiall 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 Geome­tricall parts of every Measure first given or appointed (as the prime or naturall Rationall Line) only for the discovering or producing of the artificiall Lines of measure, as conside­red in the generall 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 artificiall Lines beforegoing, and also the other following, being expressed by Quantity discrete, or number, which shew their Magnitudes (or quantities in measure) from the intire, prime, or naturall Line of Measure in generall, 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 meerly to the foresaid parts of every particular Measure given (considered discretely or Arithmetically) for the constitu­ting of the said artificiall Lines of Measure so, as to give the [Page 63] superficiall and solid contents of Here note another the like artificiall 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, &c. (or the sides of their e­quall Squares and Cubes, &c.) from their severall lines of di­mension belonging to them, and by which severally they may be thus artificially measured; as the forementioned artificiall Lines do) only, (or for the most part) according to those parts, (being considered meer­ly in the nature of such parts of the prime or naturall 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 then are consi­derable in our generall 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 intire Measures themselves) will alter continually in every particular Figure, (in respect of quan­tity discrete, according to their Arithmeticall Indices or Exponents, which expresse 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 artificiall Lines before­mentioned do, according to their like Indices or Exponents before declared; especially for the most part; though it doth happen otherwise somtimes, as that one and the same ar­tificiall Line, is found either wholly throughout, or suffici­ently in part, (that is, in respect of the fractionall or deci­mall part of the naturall Line of measure, from which it is taken, shewed by the Arithmeticall Index or Exponent, [Page 64] how far soever the same be continued or extended decimal­ly) to serve unto severall Dimensions; as I shew afterwards, in Part 2. Sect. 3.) but also, in respect of each one and the same particular dimensionall line thereof, by it selfe alone, according as the said parts of the Measure proposed, do (arithmetically) alter: whereas the arithmeticall Exponents of the first mentioned Lines, relating generally to any whole Measure appointed (as I noted at the beginning) do conti­nue the same (without alteration) in every particular, di­stinct Dimension of one and the same Figure, according to its severall lines of Dimension aforesaid. And so these two severally mentioned (or supposed) sorts of artificiall Lines, will hereupon differ in every severall dimension of one and the same Figure, in respect of quantity discrete or Arithme­ticall, (according to their Indices or Exponents of measure) though they doe not, in respect of quantity continuate or Geometricall; or of Measure it selfe in generall, as I shall shew by and by: And then also they differ herein; That as the former artificiall Lines being immediately of the whole intire naturall Lines (either redundantly or defici­ently, according as their Arithmeticall Exponents do shew) considered without respect of parts, compounding or Ge­ometricall, (but only Arithmeticall, or meerly divisionall, as is alwayes necessarily required, for the exactnesse of mea­sure) and so giving the Areall contents of Figures accor­dingly; are themselves to be considered (in the artificiall Dimension) as whole intire 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 naturall Lines, from which they are (most properly) derived, and so [Page 65] which they do artificially represent, and consequently, ac­cording to which (chiefly) they give the superficiall and solid contents of Figures (as aforesaid.) And which Lines (for the reasons before alleadged, as also for distinctior­sake) we may well call, the particular or second artificiall Lines (or the Lines of parts) as the other may be called the prime or integrall artificiall Lines (or Lines of the Whole) And so the Measure arising there from may accordingly be called, the one, the prime or integrall Measure (or Measure of the Whole) as having its denomination simply and ab­solutely from the whole intire naturall Line of Measure ap­pointed: And the other the particular or partiall Measure (or Measure of the parts) as being denominated chiefly from the parts of the said Line of Measure.) And therefore, as those first Lines do artificially represent the respective naturall Lines from which they are taken, considered sim­ply and intirely in themselves, as the Integer (or Unity) of Measure severally: So these second Lines, do accordingly represent the said naturall Lines, as they are composed of certain denominate, or mensurall parts; (viz. of some in­feriour 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 shewed,) And so these second Lines are really none other then the first, divided into the like number of parts, as are the composing, constituting, (or Geometricall) parts of the respective naturall Lines from which they are deduced; and which parts of these (supposed) second artificiall Lines, we may conveniently call (by way of distinction from o­ther parts) their prime or Geometricall parts: and these being then divided severally into some certain number of parts, as is requisite for the exactnesse of Measure, as afore­said (according as the correspondent parts of the naturall [Page 66] Lines are) especially decimall; we may call the same, their second or Arithmeticall parts, (or the particles of measure, as being indeed only the parts of the other Parts.) So that these second Lines are never exactly decimall, as the prime Lines perpetually are, unlesse it happen, that the prime or compounding parts of the naturall Measure appointed be in a decimall number, for then as the naturall Line, so like­wise the correspondent particular, or secondary artificiall Line will be exactly decimall; 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, &c. As first (for example) in the naturall Mensuration; the most common, mechanicall 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 Geometricall parts (for so every measure is commonly said to be divided into the parts, of which it is prop [...]rly composed) and each Inch divided into some certain parts (as is necessary for exectnesse in measuring) which may best be decimall: 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 mixt measure (as they will for the most part) then must the same be first of all redu­ced into Inches, &c. before the content of the Figure, super­ficiall or solid, can be conveniently cast up, seeing that the Measure thus taken is mixt (as it were) of severall parts or kindes of measure, and so is of different denominations; and therefore must be reduced into one kinde of measure (or one mensurall denomination:) And then if the parts of the Inch be decimall, the work will be afterwards performed [Page 67] in a decimal manner, in Inch-measure only: But if the said lines thus measured, be found wholly in Feet (without Inches, &c.) then will the contents be had immediately in whole Foot-measure, w ^ch otherwise must be had in the like measure by Reduction from Inch-measure. And so seeing that the second artificiall Lines doe represent the naturall 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 superficiall or solid, by way of Qua­drature, Cubature, &c. and so artificially representing the Side of the equall Square or Cube, &c. be found mixtly, in Integers and prime parts, &c. of the same Lines; (as for the most part they will) then must the Measure so taken, be first reduced wholly into their said prime parts, &c. for the casting up of the content as aforesaid, (according to what was shewed before in the naturall Mensuration, after the most plain or vulgar way by a common naturall Line of parts) And then those artificiall prime, or mensurall parts, being divided Decimally; the dimension of the Figure proposed, will be performed in a Decimall manner, accor­ding to those parts only: But if the side, or other line of the Figure to be measured, doe fall out in Integers only of the said artificiall Lines, (viz. in whole Lines without any parts;) then will the Areall content be obtained immedi­ately, and exactly, in Integers of the naturall Measure ap­pointed (according to the reason of the prime artificiall Lines, or Lines of the whole Measure) which otherwise can be had in the integrall measure (or the measure denomina­ted only from the whole naturall Line) only, by way of re­duction from the primary, compounding, denominate, o [...] ­metricall [Page 68] parts of the same, in which it is first found, as be­fore was shewed.

And now therefore it appears from hence, that the ar­tificiall Dimension performed by these second or particular artificiall Lines, or Lines of parts, so called; or the artifici­all Lines, as they are considered meerly in the particular or speciall reason of the artificiall Dimension (or Reason of the Parts) taken in the latter respect; is not considerable in comparison of the artificiall Dimension performed by the prime or integrall (or more generall) artificiall Lines, or Lines of the Whole; or the artificiall Lines, as they are considered in the generall reason of the said artificiall Dimension (or Reason of the Whole) they giving the sides of the equall Squares, and Cubes, &c. of Figures (and so their superficiall and solid contents accordingly) imme­diately in In [...]egers (and decimall parts) of the prime Ra­tionall Line, of the naturall Measure appointed; which the particular Lines do give (for the most part) mixtly 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 kinde of Mensuration, I thought it would not be amisse to manifest thus much concerning this latter way; that so the ingenious Reader that shall please to exercise himselfe (practically or experimentally) in this artificiall way of measuring, may (by comparing the effects or re­sults of the generall and particular, or speciall reason there­of together, both in the extraction or production of the artificiall Lines themselves, & also in the said two severall wayes of working by them) receive a more full satis­faction, [Page 69] and the thing it selfe be accordingly confir­med.

And now seeing that the aforesaid parts of Measures given, are various (as in quantity Geometricall, so usually in quantity Arithmeticall) according as the Measures themselves are in magnitude various: therefore I shall first and principally prosecute our said speciall or partiall Reason of the artificiall dimension, and that chiefly in re­lation to the first, or more generall acception or considera­tion thereof, (namely, for the producing of the artificiall Lines, as they are considered in the generall Reason, by the parts of the Measure given (or the prime artificiall 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 naturall Line, according as the same is understood, either without or with the foresaid kinde of parts.)

And this I shall accordingly lay down in three spe­ciall Theorematicall Propositions (contained in the second part of this Work) answering to the three principall Pro­blematicall or practicall Propositions beforegoing (laid down in the Figures particularly handled in this first Part, namely the CIRCLE, SPHEARE, and CYLIN­DER, together with the CONE) and which will ge­nerally serve for all other ordinate or regular, and regular­like 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.

SECT. I. Proposing the foresaid Dimension in all right-lined regular Planes or Superficies in generall: And demonstrating the same particu­larly in two of the first of them.

SECT. II. Setting forth the Dimension, both solid and super­ficiall, of regular-based Pyramids in gene­rall, and their Compounds: And de­monstrating the same particularly in the three first kinds of them.

ANd now having sufficiently shewed our ar­tificiall Dimension in the three first recti­line or angular ordinate Planes in parti­cular; namely the Trigon, Tetragon and Pentagon, simply in themselvs, (but chief­ly 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 metricall reason: We shall next proceed to the like kind of dimension in them, as in order and relation to all Pyramidall 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 shew in the three first sorts of Pyramids, constituted or raised upon the three first Planes before named; and more especially [Page 107] for that, as those three Planes do concurre superficially, to the composition, (or to the superficiall 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 concurre in like manner solidly, to the composition, (or to the solid composition) of the said five Bodies, as I shall shew 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 trigonall, tetragonall, pentagonall, 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 equall ordinate Trigons compact together by solid angles (by E 11. d. 26) whih 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 concurre three superficiall 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 it's 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 trigonall Planes) are in number, subduple the trigonall sides constituting the same) And so any one of the said [Page 108] Trigonall 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 verticall angles) shal be the top or verticall point of the same; between which, and the Center of the Base, shall be adjudged the perpen­dicular altitude, (or the Axis) thereof: We shall first therfore shew 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 generall side of this Body) in answer to the foregoing quadrate dimension of an ordinate Trigon by it's side) and the Axis toge­ther; which we shall compare with the naturall or vulgar dimension, according as we have done in all the Figures beforegoing; whereby the same may withall be under­stood by such as are yet to learn.

And therefore first for solid dimension; Let the side of a Tetrahedron be of the Rati­onal Line in general, 12.00 (which is also the side of the base) so the trigonall Plane for the Base, wil be (as was found before in the dimension of an ordinate Trigon) 62.3538. Now the Axis, or altitude of the Te­trahedrum (and so of all right or isoskelan Pyramids whatsoever, The solid di­mensiō of a tri­gonal Pyramid. having regular bases) is had geometrically (in a Triangular manner) as was formerly [Page 109] shewed for the Axis of a Cone, according to E. 1. p. 47, &c. it being the greater containing side of the right angle in the rectangle Triangle made by the Axis of the said Py­ramid, and the basiall Ray, (which is the ray of the Circle circumscribing the ordinate trigonall base) for the lesser containing side; and by the side of the Pyramid, being the Hypotenuse to the right angle, in the Center of the Base: so that the Axis of the Pyramid, is the side of the residuall or differentiall Plane, (taken quadrately) between the La­terall Quadrat of the Pyramid, and the Radiall Quadrat of it's Base: which last Quadrat, (seeing that the side of the base, as being the side of an ordinate Trigon, is treble in power to the Ray of the Circle circumscribed to the base, by E. 13. p. 12) will be here found 48; by which the laterall Quadrat of the Tetrahedrum viz. 144, being dimi­nished, there will remain the Quadrat of the Axis, 96; ( viz. q 12—q 7 ferè) whose Root, irrationall or ineffa­ble, 9 15/19, or rather (according to a decimall extraction of the quadrate Gnomon) 9.7979, &c. (whereas the other is decimally, but 9.7894, &c.) is the true Axis or Altitude of this Tetrahedron: whereby it appeareth, that the Side of a Tetrahedron is potentially sesquialter the Axis of the same; and so E 14. p. 31. Or again, seing that the Side of the Tetrahedron is to the Diameter of the ambient Sphear, potentially sub-sesquialter, because the Diameter is po­tentially sesquialter the side, by E 13. p. 13, before cited; (and so the Diameter of the said Spheare is potentially quadruple-sesquialter to the semidiameter of the Circle circumscribing the Base of the said Pyramid inscribed,) therefore the Diameter of the said Spheare will come forth here potentially, 216; whose Root tetragonical, irra­tional or inexplicable, is vulgarly 14 20/29, which is by deci­mall [Page 110] resolution, 14.6896 &c. or the said Root is more truly, by immediate decimal production of the gnonomical additament, or additionall Gnomon, 14.6969, &c. for the Diameter of the ambient Spheare; whose subtriple dou­bled, viz. 9.7979, &c. is the altitude of this ordinate tri­gonall or Tetrahedrall Pyramid, exactly as before; which then being conjunctly compounded with the subtriple of the base, viz. 20.7846, &c. or the whole base with the subtriple of that, viz. 3.26598, &c. there will result the solidity of the Tetrahedron (to cube-centesmes of the Ra­tionall Line, by a sufficient extension or production of the two foresaid terms of consolidation) 203.646753 ferè. So that a Pyramid, is naturally the subtriple of a Prisme, ha­ving the same Base and altitude, by E 12. p. 7. as a Cone is the subtriple of a Cylinder, of equal base and altitude, by E 12. p. 10. and therefore if you triple the said Tetrahe­drum, you wil have a Prism, of an ordinate trigonal base, according to the said E 12. p. 7, being in content, 610.940259 ferè, and so of the same base and altitude with the Tetra­hedrum.

And though a Cone do somewhat resemble a Pyramid, and a Cylinder, a Prisme; yet a Cone cannot properly be called a Pyramid (as some do call it) nor a Cylinder a Pris­me; 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 it's selfe, any kind of Pyramid; and so a Cylinder to compre­hend any kind of Prisme; their Bases being the Circles circumscribed to the Base of the Pyramid or Prisme, and so their whole Bodies circumscribed to the whole Pyra­mid or Prisme, viz. their concave superficies just touch­ing [Page 111] the angular lines, or the sides of the respective Pyra­mid or Prisme inscribed. And contrarily, may any Pyra­mid be understood to cōprehend or include a Cone, & any Prisme 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, compleatly 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 artificiall Line for the so­lid dimension of a Trigonall Pyramid by the side of it's base and its Axis or Altitude toge­ther. for which purpose, I find (according to the reason of the 3 ^d. Theoreme, &c. fol­lowing) to be of the prime or natural Ra­tional Line in general, 1.9064, ferè; which being duly set off 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 artificiall base: And the Axis or Altitude of the Tetrahedrum, found geometrically before, 9.7979, &c. (which according to a centenarie solution of the unit, or of the Rationall Line, is 9.80 ferè.) will be found by the foresaid Line, 5.14, for the artificiall Axis; which being wholly infolded with the whole Base, will produce the rectangle regular-based Prisme, or Parallelipipedum, [Page 112] 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 Rationall Line cubed, or 1/3 of a cube-unit. And by a fur­ther solution of the two Lines of Measure, (viz. the natu­rall and artificial Rationall Line) the difference will be found much lesse, according to what I have said, and also plainly demonstrated in all the precedent Dimensions.

Now for the superficiarie dimension of this kind of Pyramid; if it be exact­ly Concerning the superficiall di­mension of a­trigonal Pyra­mid. ordinate, as the Tetrahedron, then 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 wil 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, wil 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 square­unit, or Integer.

But if the Pyramid be not exactly ordinate; then ought the superficies thereof to be considered first, and most pro­perly, as that of a Cone, called Conicum, which I noted formerly in the Dimension thereof; to wit, without the Base; to which must afterwards be added the base, to [Page 113] make up the whole Superficies. For so indeed the true Pyramidall superfice (what Pyramid soever it be) is most properly to be esteemed; it being comprehended in a re­ctangle Plane arising from the semperimeter of the Base, and the perpendicular-line of the isoskelan, or laterall tri­angular Plane of the Pyramid, which is the altitude of the said Plane, (not the angular line between the Base and the top, called the side of the Pyramid, made by the mee­ting together of two of the Planes, according to their sides, and so is no other then the side of the isoskelan Plane) bi­secting the same into two rectangle Triangles, upon the side of the base of the Pyramid (which side is also the base of the said isoskelan triangular Plane; the perpendicular falling thereupon from the top or verticall angle, by way of bisection or bipartition)

As here the perpendicular a, d of the Triangle a, b, c, representing the lateral Plane of the Pyramid, bisects the same into two rectangle Trian­gles, a, d, b, and a, d, c, right-angled at d, be­ing the middle of the side of the ordinate Base of the Pyramid, b, c, which is also the base of the said trian­gular Plane a, b, c; but the side of the Pyramid is the same with the side of the said Plane (as noted even now) viz. a, b, or a, c: and the rectangle Plane made by that and the semiperimeter of the base, will much exceed the Pyramidall superfice before mentioned; which we may [Page 114] conveniently demonstrate in the foregoing Tetrahedrum; whose side, or the side of whose Plane, (w ^ch, may here be represented by, a, b, or a, c,) being put naturally, 12; the perpendicular (w ^ch may here be a, d) wil be 10.3923, &c. (as was found formerly in the ordinate Trigon.) Now the side of this Pyramids base ( b, c) being the same with the other, ( viz. a, b, or a, c,) the perimeter of the Base wil be 36, whose half, 18, being augmented, by the said perpendicular. (or halfe the perpendicular, 5.19615 by the whole perimeter, 36) there will result the superficies of this ordinate Tetrahedrall Pyramid, without the Base as it were, or of three of the Planes, 187.0614; to which being added the fourth equall Plane, for the more proper and peculiar base, as it were, of the Pyramid, viz. 62.3538; the aggregate wilbe 249.4152, for the totall superficies, or surface of the Tetrahedrum, exactly as before, by the true, naturall dimension. But now the rectangle Plane made of the se [...]iperimeter of the base, 18, and the side of the Tetrahedrum, 12, would be 216, for the superficies of this Pyramid without the base (or of three of the equall Planes only) which exceeds the former, by 28.9386, and so being joyned with the base, 62.3538. would exceed the true totall superficies accordingly.

Now for a triall 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 agree with the rectangle Plane made naturally of the semi-perimeter of the Base, and the said hedrall perpendicular, for the true Pyramidall superfice, viz. without the proper base: The artificiall Line of Measure specially serving hereunto, I find (according to the reason of the 3 ^d. Theoreme, &c.) to be of the Ratio­nall [Page 115] Line in generall, deficiently, 0.8165 ferè, (which is Apotomally in Number, The artificial Line for the superficial Di­mension of a trigonall Pyra­mid, by the Side of its Base & the perpen­dicular-line of its laterall tri­gonall Plane, (or the alti­tude of the same) together. according to our generall reason of Mea­sure, 1—.1835 ferè) which being set off from the said Line, and the side of the fore­said Pyramid's 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 Pyramidall superfice, before-menti­oned ( viz. without the Base) which ex­ceedeth the like superficies formerly found most truly, 187.0614, not fully so much, as wil make in vulgar terms, 1/14 of a square-unit.

And thus much for the Dimension both solid and su­perficiall of the first kind of Pyramid in generall, whether ordinate or inordinate, so as the Base be ordinate: And so in speciall, of the first of the five ordinate plain Bodies, namely, the Tetrahedron; according to a Pyramidall di­mension only.

As for the first kind of Pyramidate, called in generall 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 it selfe.

Now for the second kind of Pyramidate, Cōcerning the Dimension of the Icosahedron in a Pyramidal way. called in generall a mixt Polyhedron, of which sort are usually reckoned three of the said five ordinate Solids, to wit, the Octahedron, Icosahedron, & Dodecahedron, [Page 116]

seing that the second of these is made up of the first kind of Pyramid (but not exactly ordinate,) in number 20, and equal and like, being comprehended under so many ordi­nate and equal Trigons, as their Bases, by which they are only eminent, their whole Bodies besides being latent, and meeting vertically in the Center of the Icosahedrum, (or of its circumscribing Spheare) which bases or trigonal Planes being composed together by solid angles, do there­fore comprehend or contain the whole Icosahedrum, ac­cording to E. 11. d. 29. and which solid angles (made by the connexion or concursion of the said Planes or bases, according to their angles) are, therefore in number sub­quintuple the plaine, superficial, or trigonal angles con­stituting the same, (so that to the constituting of one solid angle, here do concurre five superficial angles) unto all which from the Center of the Body, right lines being drawn, the whole Icosahedrum is thereby divided into 20 equal and like trigonal Pyramids, according to what I said before: and the angular lines called the Sides (made by the inclination of the said trigonal Planes according to [Page 117] their sides, and so equal with the same) are in number, halfe so many as the basial or hedral sides. Therefore I wil next briefly touch upon the dimension of this Body, how to perform the same, according to our artificial way of Py­ramidal Dimension before demonstrated; as it is vulgar­ly conceived to be composed of Pyramids: For the solidi­ty of one of the said compounding Pyramids being first obtained (which is the usuall way of measuring this, and also the two other ordinate mixt Polyhedrums) the solidi­ty of all the rest, (and so consequently of the total Icosahe­drum) is presently had, by the vigecuplation only of that one Pyramid.

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 speake) 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 then the Radius of the Spheare inscri­bed within the plain Solid) which way therefore I shal here pretermit, as being needless for me to demonstrate; the same being shewed by divers practicall Authours, especially Ramus and Clavius in Latine; and more abun­dantly in English, by our Countrey-man, M ^r. Digges long since, in his learned Discourse of geometrical Solids, annexed to his Pantometria, as a part thereof; to which Authours I therefore refer the diligent practizer for a full satisfaction in this poynt; my intention in this place, be­ing only to bring in our new or artificial way of Pyrami­dal [Page 118] Dimension, in the aforesaid ordinate Bodies, for the more easie and speedy obtaining of their solidities in a Py­ramidal way, the Hexahedron being excepted. And there­fore I shal here only shew, how, instrumentally or me­chanically to get the said Pyramidall Axis or altitude, by getting first the altitude of the whole body, (which [...] no other then the total Dimetient of the inscribed Spheare) being according to what I shewed before, for getting the altitudes of Cones and Pyramids in general. Therefore, if from the superiour Plane of the Icosahedrum, being pro­duced or extended, that is, from the inferiour or interiour superfice of any Plane placed upon the uppermost Plane of the Icosahedrum, a perpendicular be let down to the in­feriour or opposite Plane or Base thereof, in like manner produced, that is, to the superiour or exteriour superfice of a Plane placed under the Icosahedrum (and so, upon which the same lyeth) 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 Pyramidall altitude, is by this means quite ta­ken away, and the thing made very easie. 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 wil then be obtained with little labour, especially ac­cording to our foregoing artificial pyramidal Mensuration; For the said Axiscr altitude of the Icosahedron's cōposing Pyramid, being taken by the foresaid artificial Line for the solid dimension of a Trigonal Pyramid, and the same be [Page 119] tiplied into the Quadrat of the side, taken by the same Line; the product shal be the solid content of the said compounding Pyramid; whose vigecuple wilbe 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 shew 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 totall superficies of this Solid, wil readily be had by the Line for the Lateral quadrature of an ordinate Trigon (according as I shewed before in the superficies of a Tetrahedron) for the side thereof being measured there­by, its Quadrat wilbe the area of one of the bases or Planes, whose vigecuple wilbe the total Icosahedral su­perficies.

Now for the second kind of Pyramid, to wit, the Tetragonal, upon a regular The Dimensi­on of a tetra­gonall Pyra­mid, shewed in the Octabe­dron. base, we shal next shew our artificial di­mension thereof, both solid and superficial, and first in the other trigonal ordinate mixt Polyhedrum, or the second trigonal ordinate Body, called Octahedron, seing that the same is composed of two equal and like tetrago­nal Pyramids of equal altitudes (according to E. 14. p. 16.) meeting in their bases, and so making one common base, which is a Quadrat described by the side of the Octake­drum, whereupon this regular Solid is comprehended un­der twice so many equal ordinate Trigons, as is the Te­trahedrum, being composed together by solid angles, which therefore are in number, subquadruple the superfi­cial or trigonal angles constituting the same, (so that four [Page 120]

superficial angles do here meet to make up one solid angle) And so this Body may also be cōsidered under another py­ramidal composition, to wit, as composed of 8 equal and like trigonal Pyramids, (according to the number of its trigonal bases, which shal also be the bases of the said Py­ramids) concurring vertically in the Center thereof (or of its circumscribing Spheare) like as the Tetrahedron (though that be an intire Pyramid of it selfe, and of the most simple kind of all, to wit, the Trigonal) may be con­ceived to be composed of 4 equal and like trigonal Pyra­mids, (according to its 4 bases or Planes) meeting vertically in the Center of its body, or of its comprehending or con­taining Spheare. And so we shal here shew the artificial dimension of this Body in a pyramidal way, as being taken in the first pyramidal composition, by shewing the di­mension of its tetragonal Pyramid; which we wil first perform by the natural way of its dimension for a confir­mation of our artificial way, according as we have done in all the precedent Dimensions.

And therefore first (for example-sake) let the Side of [Page 121] the Base of a tetragonal Pyramid, as also the true side it self of the Pyramid, be of the Rational Line in general, the same with that of the foregoing trigonal Pyramid, or the Tetrahedron, viz. 12.00: then the Diagony of the base, (which is also the Diameter of its circumscribing Circle) wilbe 16.97; and thence, the Axis of the Pyramid

8.485, as being equal to the semi-diagony of the base, (or semidiameter of its circumscribing Circle.) For seing first, that the Diagony of the Base, is double in power to the Side thereof, by E, 1. p. 47, &c. the power of the Diago­ny wilbe 288, whose Root irrational, 16.97, &c. wilbe the Diagony it self; whose half therefore, 8.485, &c. is the se­midiagony (or Radius of the circumscribing Circle.) And then again, seing that the Axis, and basial semidiagony, as being the two sides about the right angle of the rectan­gle Triangle made by the said two lines, and the side of the Pyramid, are both together but equal in power to the said Pyramidal Side, as being the Hypotenusal side of the said Triangle, by E, 1. p. 47. before-cited; if therefore [Page 122] the Lateral power of the Pyramid, viz. 144, be dimi­nished by the semi-diagonial power, viz. 72; (which is just half the lateral power,) there must needs remain the same for the power of the Axis; and so the Axis it selfe, the same with the basial semi-diagony, viz. 8.485, &c. which (sufficiently produced) being infolded with a tri­ent of the Base, viz. 48; there wil result the true solidity of the Pyramid, (to cube-centesmes) 407.293506: and so half the solidity of an Octahedrum composed of two such Pyramids; which therefore doubled, wil give 814.587013 for the whole solidity of the Octahedrum. Now for the performing of this simple Pyramidal Dimension, by an ar­tificial Line of Measure proper thereun­to, which, I find (according to the reason The artificiall Line for the so­lid dimension of a tetragonal Pyramid, by the side of it's Base, and it's Axis or Alti­tude together. of the 3 ^d. Theoreme, &c. following) to be of the Rational Line in general, 1.4430, &c. The side of the Base of the foregoing tetragonal Pyramid put naturally 12.00, being measured thereby (under a centesi­mal solution) wilbe found 8.32, whose Quadrat is 69.2224 for the artificial base: and the Axis of the Pyramid, being natu­rally, 8.48, wilbe found by the same Line, 5.88; which two artificial numbers of dimension being wholly mul­tiplied together (that is, the whole base with the whole axis) there wil arise 407.027712, for the artificial solidity of the Pyramid; which wanteth of the true, or natural solidity, only 1/4 ferè, of a cube-unit: and this then being doubled, wil give, 814.055424 for the artificial solidity of the Octahedrum, wanting of the true, natural solidity be­fore declared, only about 1/2 of a cube-unit, which is no ve­ry considerable matter, And if the said artificial Line of [Page 123] Measure be made 1000 parts, and then the Axis of the Pyramid, and the side of it's base, be measured thereby; they wil produce the solidity stil nearer the truth.

And if the solidity of an Octahedrum be immediatly re­quiree; then the total Axis or Diagony thereof (which is double to the Axis of the foresaid Pyramid, and so no other then the Diagony of it's base, or the Diameter of the Circle circumscribing the same, and also the Axis or Dia­meter of the Sphear circumscribing the Octahedron) being infolded with a trient of the compounding Pyramids base, (which is no other then the Octahed [...]on's Lateral Quadrat, as was shewed before) or the whole base wi [...]h a trient of the said Axis or Diagonial; there wil immediatly result the solidity of the Octahedrum: For so the true Axis or Diagonial of the foresaid Octahedrum, 16.97, &c. suffi­ciently produced by Radical extraction, or otherwise, viz 16.97056274, &c. being infolded with a trient of the aforesaid pyramidal base, viz. 48. (or the whole base 144 with a trient of the axis, viz. 5.65685424, &c.) 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 infolded with the foresaid total common base of the two compounding Pyramids, produced by the same Line, 69.2224; there wil also immediatly result the solidity of the Octahedrum, 814.747648 ferè, which comes much neerer the true soli­dity, then the former dimension; this differing there from (now by way of excesse) 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 Bodyes having here one and the [Page 124] 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, shal compleatly 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 shewed 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 joyned together in their bases, as I said be­fore.

As for the superficiarie dimension of this kind of Pyra­mid which now we have in hand, the The superfici­all dimension of a tetragonal Pyramid; And the artificiall Line for per­forming the same by the Side of the base, and the perpendicular line of the tri­gonall 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 it's tri­angular Plane together. And the artificial Line of Measure for this purpose, I find (according to the reason of the same The­oreme) to be of the Rational Line in gene­ral, 0.7071, &c. √ 1/2, which is Apotomal­ly in Number, as 1—.2929 ferè) which Line being set off there from, and divided in a due manner, and then the two forena­med Lines of the Pyramid commensurable by the same, be [Page 125] accordingly measured thereby, the product arising by their mutual multiplication, shal 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 wil 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, wil be 10.39, &c. as was shewed in the super­ficiarie 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 wil 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 it's proper Line, for superficial measure, wil be found artificially, the same that the Diagony or Diameter of its base is natu­rally, viz. 16.97, &c. and the foresaid perpendicular wil 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 square­unit 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 [Page 126] Octahedrum, wil 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, wil be the total superficies of the Octahe­drum: And so the hedral perpendicular of the foresaid Octahedrum being taken by this Line, and doubled, wil be 29.39. which multiplyed by the side of the Octahe­drum, found by the same Line, 16.97; the Plane produ­ced therefrom, wil be, 498.7483, for the superficies of the Octahedrum; which comes a little nearer the true super­ficies, then 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, shal be the true superficies of the Octahedrum; being no other then 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, &c. (or 12, and 20.78, &c.) being now the single product of 24, and 20.78, &c. Or the side being quadrupled, and the said perpendicular taken single, shal together produce the same; being no other then 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 heer the product of 48, and 10.39, &c. Or the superficies of this Solid, may be had againe artificially by the Lines [Page 127] of quadrature pertaining to the Trigon, as I shewed be­fore for the superficies of a Tetrahedrum; for so, one of the Planes or bases being had; the Octuple thereof shal be the total Octahedral superficies.

Now for the third sort of Pyramid, to wit, the penta­gonal, upon an ordinate Base; I shal next briefly shew our artificial Dimension thereof, and that in the third ordinate mixt Polyhedron, or The Dimensi­on of a penta­gonal Pyramid demonstrated in the Dodeca­b [...]dron. the pentagonal ordinate Body, namely the Dodecahedron; seing that the same is com­posed of this sort of Pyramid, in number 12, equal and alike, which are compre­hended under so many equall ordinate Pentagons, as being their Bases, whereby they are only

eminent, their whole bodies besides being hidden, & meet­ing vertically in the Center of the Dodecahedrum, (or of its ambient Spheare) as those of the Icosahedrum, before­mentioned; the structure, or fabrick of these two bodies being much alike: Which bases therefore, or Pentagonal Planes, being compact together by solid angles, do com­prehend [Page 128] the total Dodecahedrum, by E, 11. d. 28. and these solid angles (made by the inclination or connexion of the said Bases or Planes according to their angles) are thereupon, numerally subtriple the plain, superficial, or pen­tagonal angles composing the same, which are just so ma­ny as the trigonal angles of the Icosahedrum, the sides of these two Bodies, or the sides of all their bases together, being numerally the same; (so that three superficiary an­gles do here meet together, for the composition or con­stitution of one solid angle, as in the Tetrahedrum; and this solid angle is the greatest of any of the other Bodies, being contained or included by 3 3/5 plain right angles; and so is in quantity 1 1/5 solid right angle, as its composing, or hedral angle, is 1 1/5 plain or superficial right angle.) And so wee shal here consequently shew our artificial Dimension of this ordinate Solid in a pyramidal way. And this Py­ramidal dimension I shal perform by the side of the base of the compounding Pyramid, and its Axis together, as I did before in the Pyramid trigonal and tetragonal, in reference to the Tetrahedron & Octahedron. And the Line of mea­sure The artificiall Line for the so­lid d [...]mension of a pentago­nal Pyramid, by the side of it's base, and Axis together. for this purpose, I find (according to the reason of the 3 ^d. Theoreme, &c.) to be of the Rationall Line in generall, 1.2036, &c. Therefore admitting the side of the base of a Dodecahedron's compounding Pyramide (which is also the side of the Dodecahedrum it self) to be of the Rational Line in general, 6.00; then the Axis or altitude of the said Pyra­mid (being the semi-altitude of the Dodecahedrum, or the semi-axis, or semi-dimetient of it's inscribed Spheare) wil be found, (according to the proportion of the side of a [Page 129] Dodecahedrum to the Axis or Dimetient of its inferibed Sphear, noted afterwards) to be 6.68; which two Pyra­midal lines being measured by the foresaid artificial Line, (in a centenary solution) wil be found, the first, 4.98, for the artificial side of the Pyramids base, and the other, 5.55,

for the Pyramids artificial Axis; Now 4.98, being squa­red, yields, 24.8004, for the artificial Base; which aug­mented by 5.55, yields 137.642220 for the artificial soli­dity of the compounding Pyramid singly; and which augmented by the number of the compounding Pyramids, yields, 1651.706640, for the solidity of Dodecahedrum; which indeed differs somwhat considerably (by way of defect) from the true natural solidity, in respect of the total Dedecahedrum, though not of its compounding Pyramid alone. For the side of the Dodecahedrum, or of its Py­ramids base, being naturally 6, the true area of the base, wil be found (according to the trigonometrical operations in the Pentagonal Dimensions before going) 61.9371, &c. [Page 128] [...] [Page 129] [...] [Page 130] whose subtriple, 20.6457, &c, being infolded with the whole axis or altitude of the Pyramid, naturally 6.68, &c. there wil arise the true, natural solidity of the Pyramid, 137.936159 ferè; which our measure wants of, not 1/3 of a solid integer or unit; and the duodecuple hereof, 1655.233908 ferè, is the true solidity of the Dodecahedrum. Or the Base and Axis of the Pyramid being wholly in­creased together, there wil result the Pentagonal Prisme, 413.808476, containing three of the compounding Py­ramids, (according to the reason of E, 12. p. 7. before ci­ted) being of equal base and altitude with the Pyramid.) and so a quarter of the Dodecahedrum; whose quadruple therefore, 1655.233904, is the total Dodecahedrum; which agrees with the other natural measure thereof, without any sensible difference.

Or again, seing that the rectangle Solid ( Prisma or Parallelepipedum) contained under the Perpendicular from the Center of any regular plain Body, to any of it's bases or Planes, and a trient of the total superficies, comprehends the solidity of the whole body (according to what I shew­ed formerly in the dimension of a Sphear, 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 w ^ch 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 then the Axis of the compounding Pyramid, and the semi-axis of the inscribed Sphear) 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 [Page 131] 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 Center of any rectiline regular Plane, or superficial Figure, to any of it's sides, and the semi-perimeter of the same, comprehends the Area of the whole Figure (as I shewed formerly in the Pentagon, and which answereth to that of the dimension of a Circle, for the producing of it's 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 then 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 it self.) Therefore the Base of the Dodecahedrum being 61.9371, &c. the total superficies thereof, wil be 743.2462 &c. whose subtriple, 247.7487, &c. being augmented by the foresaid perpendicular (or axis of the compounding Pyramid) 6.68, &c. there wil result the total Dodecahe­dral [Page 132] solidity, 1655.2339, &c. 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 Pyramid's Base, 6.

[...]

Again, 2 ^ly.

[...]

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 then 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) wil 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) wil be found thereby, 5.551 ferè; which two by a conjunct composition, wil 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 excesse) 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 wil 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 wil be the total Dode­cahedral superficies. But a farre better way for the dimen­sion both solid and superficial of this, and the other plain regular Bodies, I shal shew in the next Section, which wil be wholly taken up about the said five Bodies.

As for the superficial Dimension of a pentagonal Pyramid seing it is but the Concerning the superficial dimension of a Pentagon all Pyramid; And withall, the ar­tificiall 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 shew­ed 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 laterall Plane together: Therefore I shal not need to insist upon the same in this last Pyramid here particularly handled, by way of exem­plary illustration; But shal only give the artificial Line of measure for the perfor­mance thereof, as I find it to be from the natural Line of measure, or prime Rational Line in general, deficiently, 0.63245, &c. (that is Apotomally in number, from our general reason of Measure, as 1—.36754, &c. the said artificial Line being √ 2/5 of the natural Line, in it's power quadratick.

And thus may the Dimension both solid and super­ficiall of this and all other Pyramids, be performed artifi­cially [Page 135] 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 artificiall 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 generall, in a decu-millenary 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.

All which Pyramidal Dimensions beforegoing, in a Li­near, instrumental, or geometrical way, I shal next briefly expresse 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 naturall Measure be reduced to the artifi­ciall. Therefore,

¹

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 superficiall) of Prisms upon the like bases, by artificiall 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 shewed formerly.) But these I shal here passe by, leaving them to the industry of the ingenious Practitioner, that shal please to exercise himself therein; and the rather for that the artificial Lines of measure pertaining to the Pyramids, wil also serve (if need be) for the dimension of the corresponding Prisms, (as I shewed 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, wil 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, wil serve for the pentagonal, or hep­tahedral [Page 138] Prism; and so forward: For as the Pyramid be­gins 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 ob­lique or inclined Pyramid or Prism upon a regular base, as readily as of a right or upright one, accor­ding Concerning the dimension of oblique or inclined Pyra­mids & Prisms. as I shewed formerly for oblique Cones and Cylinders; the true altitudes of these Bodies (not their Axes) being con­sidered; seeing that every such oblique 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 speciall and peculiar arti­ficiall way of measuring both solidly and superficially, the four plain ordinate Bo­dies, or rectiline regular Solids be­fore handled, then was before; & this, after the most exquisite manner that may be. Toge­ther with the like arti­ficiall dimension of the other like regular Body, severall waies.

HAving now shewed our artificial Dimen­sion several waies, in Pyramids, and their Compounds, or Pyramidates; and consequently of the five fore-named or­dinate plain Bodies, or rectiline regular Solids, in a Pyramidal way: (the Hexa­hedron excepted.) We shal next come to shew the di­mension of the said Bodies in a farre 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 exactnes, and [Page 140] so in the very same manner, as that of the Circle, Trigon, God [...], &c. And for solid measure, as that of a Cube, by the natural or vulgar way of measuring the same (which weshal also here shew by the like artificial way with the rest) And this only by squaring and cubing any one of their dimensional lines (as in the Sphear) 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 Platonicall 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 fabrick 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 Mathematiques, 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 Countreyman [Page 141] Billingsley, in his learned Annotations upon Euclids Ele­ments in English, after the 25 ^th. defin. of the 11 ^th. 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 wayes, both naturally and artificially; I resolved (by Gods assistance) to prosecute my metrical conceits and invention herein, as far as in any other Figure.

And seing that the sides of these Bodies may more rea­dily and accurately be taken by a Line of measure, then any other their lines of dimension adscribed 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 shal here briefly demonstrate their dimensions both solid and superficiall (by way of example) artificially by their sides only; and withall, shal by the way, give artifi­cial Lines for the like dimension of them, by some other of their dimensional lines adscribed 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 find, by the reason of the 2 ^d. Theoreme, &c. following, to be of the prime Rational Line in general, (in a decu-millenary solution,) as followeth.

Which Lines being duly set off and divided, as the for­mer, and so the Sides of these bodies measured thereby; the Cubes thereof, shal be the solid contents of the same, accor­ding to the dimensional reason of the prime Rational Line: which I shal 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 shal here add the like for the Ico­sahedron.

Therefore first; the Side of the Tetrahedrum before handled, being put natu­rally, 12.00, the true soli­dity thereof, was there found, by the naturall Pyramidall Dimension, 203.646753, and by our artificial Pyramidall Di­mension, 203.359474. Now the Line of Cuba­ture [Page 143] for the side of a Tetrahedrum being first put under a centenary solution only, and the side of the foresaid Te­trahedrum measured thereby, the same wil be found 5.88, whose Cube is 203.297472 for the solidity of the Tetrahe­drum; which wanteth of the true solidity, about as much as the foregoing artificial Pyramidal Dimension, viz. 1/3 of cube-integer or unit only; which is near enough for any ordinary use: But however proceeding by a decuple of the former parts, in the Line of Cubation, viz. 1000; the solidity of this Tetrahedrum wil be found thereby, 203.608800, &c. which now wants of the true solidity, hardly so much as 1/26 of a cube-integer; the side of the Tetrahe­drum being now by this Line, 5.883.

II. The Side of the foregoing Octahedron being put na­turally 12.00, the solid content thereof was found 814.587012, by the natu­ral pyramidal Dimensi­on, and 814.055424, by the artificial. Now the Line of cubical Dimensi­on for the side of an Octa­hedron, being put in a cen­tesimal partition; the side of the aforesaid Octahe­drum, wil be found there­by 9.34, which cubed, gives 814.780504, for the solidi­ty of the Octahedrum, which exceeds the true solidity, scarcely 1/5 of a cube-integer or unit. And this very mea­sure would be produced by the artificial Line for the solid dimension of a tetragonal Pyramid, by the diagony of its base and its Axis together: it being the same with the [Page 144] Line for cubing an Octahedron by its Axis or Diagonial, as I shal shew afterwards; and the Axis of this Body be­ing double to the Axis of its compounding tetragonal Py­ramid, and so equal to the Diagony of the said Pyramids base, as I shewed formerly.

III. The side of the Dode­cahedron formerly handled, being put naturally, 6.00; the solidity thereof, was found by the natural Pyram. dimension, 1655.233905 fe­rè; and by the like artificiall dimension (in a millenary solution of the proper Line of Measure) 1655.323188 ferè. Now the side of this Dodecahedrum being taken by its Line of cubation (in a centesimal solution) wil be found thereby, 11.83 ferè, whose Cube is, 1655.595487 ferè, for the solidity of the Dodecahedron, differing in­considerably from the true solidity, being (by way of ex­cesse,) but about 1/3 of a cube-integer. But however if the said Line of Lateral Cubatnre be made 1000, the solidity of this Dodecahedrum wil be produced thereby, much nearer the true one, viz. 1655.175675, &c. which now differeth from the true solidity (by way of defect) hardly 1/17 of a cubique integer or unit: the side of the Dodecahe­drum being now by this Line, 11.829.

IV. The side of an Icosahedron being put naturally, 12.00, the solidity thereof wil be found by the natural Pyrami­dal Dimension, upon the very point of 3770: For the Ax­is of the compounding Pyramid, wil be found (by the pro­portion [Page 145] of the side of an Icosahedrum to the Axis of its inscribed Sphear, or altitude of its own body, hereafter declared) to be 9.069, &c. which with a trient of the base (being the same with that of the foregoing Tetrahedrum and Octahedrum, viz. 62.3538, &c.) viz. 20.7846, &c. wil produce the solidity of the compounding Pyramid, 188.498398. (which by the proper artificial Pyrami­dal Line, wil 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.

[...]

Again, 2 ^ly.

[...]

Or again, 3 ^ly.

[...]

Or 4 ^ly. (to come towards our present way of measuring here proposed) the same wil 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 it self, 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 5 ^ly. 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 immediatly after the former Proportion; (to which doth answer exactly, our present artificial Mensuration, o [...] organical, or mechanical Cubation) As,

[...]

Which seueral Logarithmical operations, do agree wel 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, wil fal between 3769 and 3770.

And with these several artificial Numerations, wil be found to agree very nearly, our present artificial Mensura­tion: For the side of this Icosahedrum being measured by [Page 148] its proper Cubatorie Line, (taken in a centenary solution) wil 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 wil 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; then the side of this Icosahedrum measured thereby, wil be found 15.564 ferè, whose Cube is 3770.193726, &c. 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 superf [...]cial dimension of the 4 regular Solids before­going, by way of exact Qua­drature. of these Bodies, the artificial Lines of qua­drature for this purpose, I find (according to the reason of the first and 2 ^d. Theoremes &c.) to be of the prime Rational Line in ge­neral (under the former solution) as fol­loweth.

By which Lines duly set off and divided, the sides of these Bodies being measured; their Quadrats shal [...] be the total superficies; which I shal 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 totall 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, wil be found, 15.79, whose Quadrat is 249.3241, for the total superficies of the Tetrahedrum, which wanteth of the true superficies in [Page 150] 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 wil be double to that of the Te­trahedrum, (as I shewed before) viz. 498 8306. Now the side of this Octahedrum being measured by its proper Line of quadration (in a centesimal solution) wil 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 wil give the Octahedron's side, 22.335 ferè, w ^ch squared, giues 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 wil 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, wil be found 35.31, whose Quadrat is 1246.7961, for the superficies of the Icosahedrum; which wants of the true superficies, scarc [...]ly 1/3 of a square-integer. But yet the side of the Icosahedrum being measured by its said Line under a millesimal solution, wil 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 totall superficies thereof, was formerly found 743.2462, &c. 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, wil 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.

And so again, the side of a Dodecahedrum being double to the former, and so the same with the sides of the three Bodies before going; the basial or hedral area, wil be the same with the area of the Pentagon formerly hand­led, viz. 247.7487, &c. and so the total superficies (ac­cording to what I have said of the composition of this Body) wil be upon the point of 2973, viz. 2972..9844, or more exactly, (by a further extension or production of the Pentagonal area) 2972.9849; which is quadruple the superficies of the former Dodecahedrum: the base of this being quadruple the base of that. Now the side of this Dodecahedrum being measured by its foresaid Line of Qua­drature, wil be, 54.52; which squared, gives 2972.4304, for the Superficies of the Dodecahedron, which wanteth of the true Content only about 1/2 of a square integer: But however, proceeding in the partition of the Line of Qua­drature, but one degree of parts further, viz. to 1000; the Dodecahedron's superficies wil be produced thereby, 2972.975625 (the side being 54.525) which now wanteth of the true superficies, being 2972.984949 (in vulgar accompt) hardly 1/107 of a quadrate unite or integer of the measure first assigned. [Page 152] As for the other ordinate plain Body, name­ly the Hexahedron, (or orthogonial Iso­hedron, Concerning the dimension of the Hexahe­dron. or ordinate Parallelepipedon) the same cannot more readily be measured, (especially for solid Measure) then by the natural way, which is by cubing it's side, being taken by the prime or natural Rational Line:

And so almost as readily for su­perficial also, by squaring its side, taken by the same Line; whose sextuple wil be the total super ficies: But else however, both these Dimensions may be per­formed artificially, either by the Diagoniall, Diameter, or Axis of its Body (which is no other then the axis or Dimetient of its circumscribing or comprehend­ing Sphear, and which in the Hexahedron cannot imme­diatly be taken, unlesse the same be concave, and one side or base open) or else by the Diagony or Diameter of its Base (which is the same with the Diameter of the Circle circumscribing the said Base, as I shewed formerly in the Dimension of a tetragonal Pyramid, and before that, of a Tetragon it self,, by its Diagony.) And so, whereas the Diagoniall or Axis of the Hexahedron, is naturally triple in potencie or possibility, to the side thereof, by E, 13. p. 15, (as the Diagony or Diameter of its base is naturally double in the same manner, to the Side, by E, 1. p. 47, &c. which I shewed before,) here it becoms artificially equal (imme­diately in act, and so in power, quadrate and cubique) to the side taken naturally, or by the prime or natural Line of [Page 153] measure: And the artificial Line of measure for this pur­pose, in reference to solid measure, I find (according to the reason of the 2 ^d. Theoreme, &c.) to be of the prime Rational Line in general, the same with The solid dimension of an Hex­ [...]hedron by its Axis or Diagoniall according tocubatiō: And the artificiall Line for perfor­ming the same. that which was formerly found for the solid dimension of a trigonal Pyramid by it's ba­siall Diameter or perpendicular; and its Ax­is together, viz. 1.7320, &c. √3. By which the Diagoniall or Axis of an Hexahedrum being taken, wil be found to agree with the Side thereof taken by the naturall Line of Measure, and so being cubed, must needs pro­duce the same Content: Whereby it appears, that if the Axis or Diameter of a Sphear be taken by this Line, the Cube thereof shal be the inscribed Hexahedron.

II. Then for the superficial dimension of this Body by its said Axis or Diagoniall; I find 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 superfici­all dimension of the Hexahe­dron artificially by its Axis or Diagony, ac­cording to qua­dration. base, and the perpendicular of its trigonall plane together, viz. 0.7071, √ 1/2: By which the said Axis or Diagony being ta­ken; the Square thereof shal be the total Superficies of the Hexahedrum: And therefore if the Diameter of a Sphear be taken by this Line, the Quadrat thereof shal 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 find to be the very same with that which [Page 154] 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 wil be found to a­gree with the side taken by the prime Rati­onal Line; and so being cubed, must needs produce the same Hexahedral solidity.

IV. Then for the superficial dimension of this Solid, ar­tificially, by its said basial Diagony or The superficial dimensiō of the Hexhaedron by its basial or he­dral Diagony, according to quadrature ar­tificially. Diameter; the Line of quadrate dimen­sion for this purpose, I find (according to the reason of the 1 & 2 ^d. Theoremes, &c.) to be of the Rational Line in general, (de­fectively) 0.57735, &c. which is Apot [...] ­mally in Number, 1—.42265, &c. 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 shal 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 it's superficiety, or su­perficial part; as also the two former waies by the Diagonial of the Base & of the total Hexahedron it self) and the Line of qua­drature convenient for this purpose, I find (by the reason of the foresaid Theoremes) to be of the prime rational Line in gene­ral [Page 155] (deficiently) 0 4082, &c. the Apotomal segment, or parts of diminution, being .5917, &c. the Line it self 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 shal 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, then the vulgar or natural way, by squaring the side taken by the natural Line of measure, and then sextuplating that square, as being the Base. And here it appears, that if the Diameaer of a Sphear be taken by this last artificial Line, the Quadrat thereof shal 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 before­going, under a Pyramidal cōposition & resolutiō (as al plain Solids generally are; even Pyramids thēselves; as I shew'd before in the first and most simple kind of all, to wit, the trigonal, and that in the Tetrahedrum it self) 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 center of the hexahedral Body; (or of its ambient Sphear,) 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) angles, which are in number, sub­triple the plain, superficial, or hedral (right) angles consti­tuting or including the same, as in the Tetrahedrum and Dodecahedrum. And hereupon this regular Solid wil also [Page 156] admit of a Pyramidal dimension, for the obtaining of its solid Area, like as the other; which wil be most easily perform'd, seing 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) wil pro­duce the Content of one of the composing Pyramids; whose sextuple wil 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, wil 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 infolded with the Base (that is, cubed) produceth imme­diatly 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 then the Axis or Diameter of their cir­cumscribing, comprehending or containing Sphear, 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 Sphear; 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 shewed formerly.) Or by their Basiall [Page 157] or hedrall Diameters or perpendiculars; and in the Dode­cahedron, by its basiall or hedral Diagony also, according as I shewed in the Dimension of a pentagonal Pyramid, and also of a Pentagon it self 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 shewed before) I shal here de­liver, as I find them to be in relation to the prime Rational Line in general, under a decumillenary solution, (according to the reason of the 2 ^d. Theoreme, &c.) 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.

By the first Section of w ^ch Lines (being duly set off and divided) the Axes of these Bodies being taken, the Cubes and Squares thereof, shal 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 Sphear be taken by these Lines (except those which are for the Tetrahedrum) its Cube and Quadrat shal be [Page 159] 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 Sphear: the Lines of measure for which purpose, I find to be of the Rational Line in general, 2.4980, for cubique di­mension, and 0.9306, &c. for quadrate dimension.

By the second Section of these Lines, the Altitude of an Icosahedrum and Dodecahedrum being taken; their Cubes, and Quadrats, shal 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 Sphear be taken by these Lines, the Cube and Quadrat thereof, shal 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 Sphear (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 shewed in the Hexahedrum) I shal here omit, as needlesse and superfluous; and shal shew the chief of these Dimensions, as also all the other beforegoing in these Bo­dies, [Page 160] which I have practically demonstrated by way of example, in an arithmetical manner, from their several ar­tificiall 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 practicall Reader to observe by the way, for the more ease and conveniency in this kind of Mensuration, or me­tricall practice, (or more artificial kind of practical Geo­metry) w ^ch here we handle; & so to avoid multiplicity of artificiall 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 agree, either in the whole throughout, or in part only, in their measure or magnitude from the prime Rational Line, so farre 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 off the artificial Lines therefrom, and so also in the artificial Lines themselvs for measuring there­by, is generally sufficient for ordinary use, as I have both said, and exemplarily shewed 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 Sphear last handled, produced by the Line of Cubature pertaining to the Dia­meter: and secondly, in the solid content of the Dodecahe­drum [Page 161] in the Pyramidal dimension, by the Line pertaining to a regular-based pentagonal Pyramid in general; where yet it did agree sufficiently with the true, natural dimensi­on, in the solidity of the compounding Pyramid it self; and therefore not so considerable in the Dodecahedrum: and then 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; seing 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 shew­ed 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) wil agree in part, viz. in a centesimal solution of the Rational Line, being thereby 1.13 ferè, which I shew­ed 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) wil agree sufficiently in part, viz. in centesimal parts of the Rational Line, being thereby, 1.17, the first compleat, the second incompleat; 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 agree in the whole throughout, viz. 1.4142, &c. infinitely, being √ 2 of the intire prime Rational Line (as the unit or integer of Measure) in it's 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 compleatly.

IV. The Line for the Quadrate superficiall dimension of a Sphear, (or the quadration of the Sphaerical) 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 happen to be all one infinitly, in parts of the Rational Line, viz. 0.7598, &c. being √ 1/3 of the intire 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, [Page 163] 0.7624 ferè) viz. 0.76 compleat, 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 wil serve thus farre, 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, &c.) taken centesimally, viz. 0.77.

VI. The Line for the square-like solidation (as I may term it) or rectangle-parallelepipedall dimension of a tri­gonall 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 agree in the whole throughout, viz. 1.73205, &c, infinitely, being √ 3, of the integrall Ra­tional Line, taken in its power zenzicall or tetragoni­call.

VII. The Line for the rectangle-parallelipipedal Soli­dation of a tetragonall Pyramid by it's 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, &c. 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 agree in the whole, viz. 0.7071, &c. being √ 1/2 [Page 164] infinitely, of the prime Rational Line, in its power or ca­pacity tetragonical.

IX. The Line for the rectangle-parallelepipedation, or parallelepipedal consolidation of a pentagonall 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) wil suffici­ently agree in part; as to centesmes of the Rational Line, viz. 1.66; this latter compleat, the former incompleat: but with so smal a difference, as that one and the same Line may thus farre serve indifferently for both these dimensions viz. 1.66.

X. The Line for the rectangle-superficiary dimension of a pentagonall 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 agree in part, as to centesmes of the Ra­tional Line, viz. 0.80. the first compleat, the second in­compleat.

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 compleat, the second incompleat.

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 Sphears di­metient) [Page 165] viz. 0.5136) wil sufficiently concurre in part, as to centesimal parts of the Rational Line, viz. 0.51; the lat­ter compleat, the former incompleat.

So that here you may see, how that of all the several geometrical dimensions before particularly expressed, being in nūber 62, as requiring so many several lines of Measure, or Lines of quadrature, cubature, &c. 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, wil 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 referre 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 imme­diatly beforeing, shewed the Di­mension both solid and superficial of the 5 plain ordinate (Pythago­rean or Platonick) Bodies, accor­ding to our artificiall way of mea­suring, in the most exquisite man­ner that may be: (or instrumentall Cubature and Quadrature,) We shal 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 (no more then 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 usefull ones, for the ready and speedy discovering of their solid and superfi­ciall [Page 167] Contents, as being inscribed in, and circumscribed to, a Sphear, in relation both to the Axis or Diameter, and the greatest Periphery of the Sphear, circumscribing and in­scribed.

And here I shal first begin with the Linear dimensions of these Bodies, in all the variety thereof, according to the fore-named several Lines of dimension belonging to them, as usually adscribed to them for their Dimensions, (as I did in the other ordinate Figures before-going, namely, the Circle, Sphear, Trigon, and Pentagon) and this, in relati­on to the first Dimension in Geometry, called in general, from the Greeks, Euthymetrie, or Mecometrie, and fromthe Latines Longimetrie; And which (in respect of the different kinds of Lines) I may call more generally from the Greeks, Grammemetrie; and from the Latines, Line­metrie.

Therefore,

Contrarily,

Contrariwise.

And so the two last of these Proportions, are consequent­ly of the Sides of those two Bodies to their Altitudes.

And by the 6 ^th. Section of Proportions, as also by the 5 ^th. Sect. of the like proportions, in relation to the corre­spondent Circumference of a Sphear, you may observe, how that a Tetrahedrum and an Octahedrum being circum­scribed to one Sphear, the side of the Tetrahedrum wil be exactly double to the side of the Octahedrum: And so by the 7 ^th Sect. beforegoing, you may observe contrarily, how that these two Bodies hauing one and the same side, the Diameter of the Sphear inscribed in the Octahedrum, wil be exactly double to the Diameter of the Sphear 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 Sphear. And the same proportion wil the Tetrahedrum here hold to the Octahedrum, both for solid and superficiall dimension, as it doth for lateral dimension, as I shal shew afterwards.

8. Tetrahedron's Axis is to its inscribed Sphear's Axis, as 1. to .5 dupla. And therefore contrarily.

9. The Axis of a Sphear is to the Axis of its circum­scribing Tetrahedron, as 1. to 2. subdupla.

10. The Axis of Tetrahedron's ambient or externall [Page 171] Sphear, is to the Axis of its inscribed or internal Sphear, as 1, to .3333, &c. infinitely, viz. 3. to 1, tripla. And therefore conversly,

11. The Axis of Tetrahedron's inscribed Sphear, is to the Axis of it's circumscribing Sphear, as 1. to 3, subtri­pla.

And so consequently, the latter of these two Proporti­ons, is to be understood of the Axes of those two Bodies, to their Altitudes.

Conversly.

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 Sphericall in­scriptibility and circumscriptibility, do require three seve­ral distinct Sphears, circumscribing or containing, and in­scribed or contained: That is, they being all severally in­scribed within one Sphear, cannot then also be exactly cir­cumscribed about (or cannot compleatly comprehend with in them) one Sphear, but three several Sphears; where­of that which is inscribed in the Tetrahedrum wil be the least, and that inscribed in the Hexahedrum and Octahe­drum, wil be equal, and that which is inscribed in the Dodecahedrum and Icosahedrum wil 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 Spheare, cannot then again be exactly comprehended or contained of one Spheare, but must have three seve­rall comprehending, containing or including Sphears; of which, that for the Tetrahedrum, wil be the greatest; that for the Hexahedrum and Octahedrum wil be alike; and that which is for the Dodecahedrum and Icosahedrum wil also be of equal magnitude, and indeed the least 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 Sphear, in re­spect of the Circles circumscribing their Bases, the Diame­ters of these Circles and of the Bodie's circumscribing and inscribed Sphear, being compared together: And there­fore.

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 superficiall Diagony.

And to the second of these Proportions, is consequently of the Hexahedron's Side to its Basiall Diagony.

The Converse of these are,

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 totall corporall diagony.

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 Sphear, 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 least of all; which Euclid E. 14. p. 5, and 21, speaks of, only in respect of these bodie's inscriptibility in one Sphear. As for the ba­sial or hedral inscribed Circles of these 5 Bodies, whether inscribed in, or circumscribed about one Sphear; 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, &c. 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 severall 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 Sphear circumscribing and [Page 176] inscribed. And from these we shal 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.

Vice-versâ.

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.

Contrariwise,

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 its 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 Sphear both cir­cumscribing [Page 178] and inscribed, to the Circumference of the ba­sial or hedral circumscribing Circles of these Bodies, & contrà: they wil be the same with those which are already expressed between the Diameters, viz. first in respect of the Sphear circumscribing and inscribed, between them­selves; and then of both these Sphears severally with the Circles circumscribing the bases of these Bodies, as being described either within or without one Sphear (or several Sphears 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 then) are here absolutely needful for the measuring of them: I shal now come to shew 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 Geometrie, called from the Grecians, Embadimetrie (& Plat [...]metrie) & Stereometrie; 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. wil be as followeth; the bodies being placed in order according to their magnitudes increasing, both solidly and superficially.

Hence,

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 shewed before in their dimensions) The Hexahe­drum is bigger then both those together, both in solidity and superficiality: the Icosahedrum is greater then 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 shewed before) And the Dodecahedrum is larger then all the other toge­ther, both in solid and superficial dimension.

Hence,

Then for the Tetrahedrum in reference to a Sphear, by way of inscriptibility therein, as to its Dimetient, it followeth.

Hence.

Next for the other 4 Bodies, both simply in themselves, and also in relation to their ambient Sphear together, in respect of their common Axis or Dimetient, it follow­eth.

Hence,

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 Sphear) it appears, that these five Bodies being all described within one & the same Sphear, do retain the same order and rank among themselves, in respect of their magnitudes or dimensions both solid and superficiall, as they do, being of one common, lateral di­mension, but not in that proportion and difference of di­mension either solid or superficiall; for that here the late­ral Dimension being different in them all, and that by way of diminution from the least body to the greatest, the su­perficiall 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 Sphear, and so E 14. p. 32. And,

II. That the Hexahedrum is to the Octahedrum with­in the same Sphear, 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 Sphear 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 Sphear, and the sides of the 5 Bodies inscribed thererein; and also the first Section of Proportions, between the great­est Periphery of a Sphear, and the sides of the Bodies in­scribed; [Page 184] it appeareh, how that the Octahedrum is to the Triple of the Tetrahedrum inscribed in the same Sphear, 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 Sphear, both solidly and superficially, as the Hexahedrum is to the Icosahedrum laterally, according to E, 14. p. 9 and 11.

Then for these Bodies in relation to a Sphear, 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, wil 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,

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 superficiall capacities of the Bodies themselves.

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 Sphear, the Tetrahe­drum is the biggest of all, both in solid and superficiall di­mension, and is exactly double to the Octahedrum in both these dimensions (as it was shewed before to be in its la­teral dimension) and very near as bigg as the Hexahe­drum and Octahedrum both together, in respect of both [Page 187] dimensions: And the Icosahedrum is the least of all in both these dimensions; which is almost contrary to the former course holden in these Bodies, being described within one Sphear.

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,

Therefore.

Then for the proportions cubatorie and quadratarie, in relation to a Sphear's greatest, Diametral, or true Peri­riphery, in respect of these Bodies Spherall circumscriptibi­lity, it followeth,

Whereupon.

Lastly, for these Bodies, and a Globe or Spheare com­pared wholly together, both solidly and superficially; and that according, both to their inscription and circumscripti­on, the Proportions wilbe 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 superficiall)

Contrariwise.

By the two last of which Sections of solid and superfici­all Proportions or comparisons between the 5 ordinate Bodies and a Globe or Sphear, it appeareth; that any of the said Bodies being circumscribed about a Sphear, the solid and superficiall capacity thereof, wil be one and the same numerally, (or in the number of a given Measure) viz. where the said two severall Dimensions of the inscribed Sphear are alike in number: That is, two like Bodies [Page 195] exactly encompassing or environing two severall, distinct Sphear's, whereof the solidity of the one, & the superficiety of the other, are numerally alike; there wil the solid capa­city of one of the like ambient Bodies, and the superficiall capacity of the other, be also alike in the number of mea­sure. And so likewise, two distinct Sphears 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 wil the solidity of one of the inscri­bed Sphears, 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 needfull; as namely of the Bodies among themselves in respect of their mutuall 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 Sphears, whereby the sides, Axes and Altitudes of these Bodies, and so the Axes or Dime­tients, and greatest Peripheries, of their circumscribing and inscribed Sphears, 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 Sphear, 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 Diame­trall or perpendicular, and Diagonal-line, by their are all [Page 196] or superficial Contents: But that these kind of proporti­ [...] not so usefull, being indeed more of curiosity then [...] sith the superficial and solid Contents of Figures are more usually inquired out by their sides, Diameters, and other▪ the Lines of Dimension, then these lines are by their superficiall and solid Contents; for that the thing chiefly [...] in the dimension of all Figures, is their superfici­all [...] Contents (and in solid Figures, chiefly the so­ [...] Conte [...] [...] I said before) which must be obtained by [...] of Dimension.

[...] [...]ving handled the five famous ordinate (Py­ [...] tonick) 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 chiefly in reference to their solid and and superficiall dimensions: I shal next come to the second theorematicall Proposition before-mentioned, in which, our more particular or special reason of our ar­tificial instrumentary dimensions (or mechanicall Cubature and Quadrature) of these Bodies, like as first of a Sphear, (as to the obtaining of the severall artificiall Lines of mea­sure for performing the same) is contained.

SECT. V. Shewing 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 Sphear.

ANd the like reason of Dimension to that before-going, wil hold in a Sphear, and the five plain regular (Platonick) Bodies, for gravity, or Quantity ponderall (ac­cording to any Metal & Weight assign­ed) as for solid magnitude, or Quantity dimensional; there being generally the same mathematicall 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, [Page 199] 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 bignesse, being first known according to some cer­tain Metall, Weight, and Measure appointed; there may be artificiall Lines of measure extracted for every severall kind of Body, according to the said particular Metall, Weight and Measure, (and that according to the foresaid severall Lines of Dimension in those Bodies, by which they have been shewed 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 shal be the weight of the metalline Body proposed: And which wil therereupon hold in a Sphear, 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 Sphear, as being inscribed in, or circumscribed to the same, but also mutually one to another, as was said before for so­lid (and superficiall) measure: And so the Diameter or Circumference of a Sphear, being taken by their proper, respective artificial Lines of Measure for this purpose (ac­cording to any certain Body, Metall, Weight, and Mea­sore assigned) the several Cubes thereof, shal be the weight of the respective Body inscribed, or circumscribed, accor­ding to which is proposed; And the first of these, is the [Page 200] same (in respect of a Sphear's Diameter) with the artifici­all 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 shewed 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 shewed 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 2d. Theoreme; 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 Theoreme 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 waies I shal here particularly demonstrate in the first regu­lar Solid beforegoing, to wit, a Sphear, or Globe, simply in it self, as comming most in use; and that in the most usual and usefull Metall for this purpose. But first I wil give the Proportions or comparisons of all the usuall or principall (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 [Page 201] after Theor. 9 or prop. 17: which Mr. Gunter in the 5 ^th. chap. of the 3 ^d. 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 mixt or heterogeneall numbers, in whole numbers, after a deci­mall 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 reciprocall, in respect of the equal Magnitudes and gravi­ties of like Bodies of different Metals.

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.

And for a further explana­tion Here, note that the number for Silver should not be 2030 but 2170; which might be so set down by the negligence or oversight of the Printer, though I find it not noted by Mr. Oughtred's Translatour, who set forth his Book in En­glish, among the typogra­phicall errours collected by him in the end of that Book; but else all the other Num­bers do correspond with the Numbers of the former Ta­ble. and use of both these pro­portional Tables, I referr the Reader to the forenamed seve­ral Authors, in the places fore cited (though I shall partly shew it afterwards my self) And now of these Metals, se­ing that Iron is most used it the Body w ^ch here we inten [...] chiefly to treat of and handle, namely a Globe or Sphear, and that chiefly in reference to [...] Bullet, especially the greater, or Cannon-bullet (for the use of Gunners) which is com­monly made of the foresaid metall, (and which now a daies is too much used among us) We wil therefore here shew its cubicall dimension in the said Metall, according to its proper kind of Weight with us, w ^ch is Avoirdupois-weight, beginning first with the Sphear's Diameter. And the Line of equall parts for this purpose (according to the common esteem of the gravity or weight of an iron-Sphear or Bul­let with us, at a certain magnitude, in a certain Measure gi­ven, expressed by & by) I find by the reason of the second Theoreme beforegoing, in respect of the proper compo­sing, denominate parts of the integer of the weight propo­sed, (or according to our general reason of Measure, in re­spect of the Integer of the weight it self; the same reasons holding here for weight, that were formerly exgressed for Measure, according to what I said before) to be of the common denominate parts of a Foot with us, (in a centesimal solution) 1.92. Now because we cannot distnct­ly divide the Line of measure first given for this purpose, [Page 203] viz. an Inch or Pollicar, (as the prime Rational Line) into 100 parts, (as is requisite to do) according to the sim­ple, vulgar, or natural division of a right Line:

Therefore we wil here divide it by the artificial way, commonly cal­led the Diagonal or decimal way; according to the rec­tangle Parallelogram, A B C D, whose length A C or B D being the Inch divided into 10 equal parts, and so its breadth A B or C D (taken at pleasure) divided in like manner: the whole Inch wil be divided equally into 100 parts, according to the smal quadrangles or parallelogrās, made by the several lines of division, drawn direct accor­ding to the length of the Pa­rallelogram, and oblique or diagonal, according to the breadth; and these latter lines may also be drawn ex­actly rectangular or tranverse to the former lines. And to this Parallelogram is annex­ed another, viz. C D E F, whose length C E or D F, is A C, or B D, 1.92, for the Line of ponderal Cubation, (as I may term it) or of cubical gravity, proper and peculiar to the Diameter of a Sphear or Bullet of Iron, in the fore­said kind of weight, according to the common Tenent here [Page 204] following: which being likewise divided as the prime Line, according to the rectangle Parallelogram C D E F, and so the Diameter of an iron Sphear or Bullet be taken thereby; the Cube thereof shal be the weight of the bullet in its foresaid weight Avoirdupois, according to the Inte­ger of weight, viz. the Pound. As for example: An iron­bullet of 4 inches the Diameter, is commonly said with us, to weigh 9 ^li. avoirdupois. Now for a trial of this by our Cubatorie Line of weight, or Line of cubical gravity; I measure the diameter of this Bullet or Sphear (being of the Line A C, 4.00) by the Line C E, or the diagonal Scale C D E F and find it to be thereof, 2.08, for the artificiall Diameter, whose Cube is 8.998912 ^li. for the weight of the bullet, which reduced into the proper parts of the fore-said kind of weight, the same will be 8 pounds, 15 ounces, 15 drams, and very near 1/4 of a dram, so that it wants of 9 ^li. but about 1/4 of a dram, which in such a thing is as nothing. But because the diameter of such a Sphear, cannot be first measured, but must be had by means of the Circumference, first obtained by a Line of measure, (unles there be a pair of Callaper Compasses ready at hand) whereupon a Line for cubing the Sphear by its Circumfe­rence, must needs be generally more convenient for use, then that for the Diameter, according as I noted formerly in the generall dimension of a Sphear: Therefore, I will here also deliver a Line of weight for cubing an iron-Sphear by the Circumference, which (by the reasons aforesaid) I find to be of the Line A C, 6.04, (viz. 6.04 in­ches) which being divided either diagonally into 100 parts, as the Line A C, or C E, or rather only according to the length thereof simply, seing it may sufficiently bear it: and so the Circumference of an Iron-Sphear or Bullet [Page 205] be taken thereby; the Cube thereof shal be the weight of the Bullet.

As suppose here again the former bullet of 9 ^li. avoirdu­pois, whose diameter being put 4.00 inches, the Circum­ference wil be 12.57 inches ferè which by its proper Line of Cubature, wil 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, wil weigh with him, 12 ^li. 2. ounc. 1 scrup. 14 gr. and 2/37 of a grain; for so I find it to be by the former proportional numbers, by comparing this Sphear with a Sphear of Tinne of the same diametrall magnitude, according as Ghetaldus shew­eth in his foresaid book, and from him Mr. Oughtred in his fore-named book: and which is also expresly noted by Ghetaldus in a Table, wherein he hath set down the weights of a Sphear, 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 9 ^li. avoirdupois is in our Troy-weight, by Assize or Gold­smiths weight (according to the commonly received pro­portion of the Pound-avoirdupois to the Pound-troy, 60 to 73) but 10 57/60 pounds, or 10.95 ^li. 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 10 ^li. 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 [Page 206] into 20 penny-weights, and a penny-weight into 24 grains, so that our ounce-troy weigheth but 480 grains, and consequently our Pound-troy 5760 gr. All which sheweth, that either his Weight, or Measure, or rather both of them, do differ from ours, as I shal further shew: As for the Measure which hee useth, he saith it to be the ancient Romane 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 Professour of Geometry in Gresham Colledg London, and now Professour of Astronomy Concerning the magnitude of the Romane Foot. in the University of Oxford, hath in his Discourse of the Roman Foot, &c. (publish­ed by him in English, Anno 1647; and deduced, not only from divers Authours, 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 forraign parts, but especially in Italy, and there at Rome) more clearly expressed the Di­mension or magnitude of the true Romane Foot; ha­ving done the same not onely 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 authours do give us to observe) which while it was moist in the [Page 207] Presse, 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 shal find that of Mr. Greaves, to be shorter then 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 Authours 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 Foote, as are to be found, and so many Feet as are taken to be Romane) 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 remov'd 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 Guild­hall 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 [Page 208] said Roman Foot contains 1000: whereby this Romane 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 Romane 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 wil 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 agree with our English Foot, then 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 [Page 209] 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 Gold-smiths 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 moderne containeth 5256 such grains, and so the Roman Ounce both ancient and modern, 438 of the same grains, the Troy-pound 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, &c. of the Auncients, 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 ac­compt, the Roman pound wil contain but 5040 english graines, or such as our troy­pound contains, 5760, which comes short of Mr. Greaves his accompt by 216 graines And so the Romane Ounce will containe but 420 such graines, which falleth short of Master Greaves his ob­servation, by 18 grains. But herein wee 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. Romane Pound and Ounce should be (in the least terms) but 73/80 of our English pound and Ounce Troy-weight, and so the pro­portion of the Roman pound & ounce to our Troy-pound & 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 shal shew after­wards, and which is decimal­ly, as 10000 to 9125 exactly.) As for the division of the Ro­mane ounce immediately by scruples, in number XXIV [Page 210] which Ghetaldus useth, Mr Greaves speaketh not of it, but only by drams, in number VIII, as the Ounce is common­ly divided by the Physitians of all Countreys, and the phy­sicall 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 shewed before) a penny­weight consisting of 24 grains: and the Ounce by Physi­tians is universally divided into 8 drachmes; a Drachme into 3 scruples; and a scruple (by an evil custome received in shops) into 20 graines, which ought to have (according to the ancient custome) 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 medicinall Pound and Ounce, is the same with that in the Troy: For the scru­ple being 20 grains, the Dram wil 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 Romanes themselves, Mr. Greaves sheweth not, whereby we might find 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 shewed even now from Mr. Greaves) we shal find the Grain (such as the Roman-pound con­tains [Page 211] 6912, and the Ounce 576, according to Ghetaldus, as we lately shewed) to be (in the least terms) but 73/96 of the English grain, (such as the Romane pound contains 5256, and the Romane 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 Romane grains to our Troy­grains) as 96 to 73; & so cōsequently the Roman-weights in generall (as pounds and ounces) reduced into the proper grains, wil be to our Troy-weight in graines, 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 li. 2. oun. 1 scrup. 14 2/37 gr. or 84134 2/37 graines 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 li. 1 oun. 2 drams, 16 1 [...]4/111 gr. (or according to the vulgar Division of the ounce-troy by penny-weights, 11 li. 1 oun. 5 penny-weights, and 16 1 [...]4/111 gr.) or 63976 104/111 gr. which is very neer 63977 gr. For first, I say,

As 80 to 73, So 12 li. 2 oun. 1 scrup. 14 gr. Roman, (viz. 12 li. 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 li. 1 oun. 2 dr. or 5 p. w. and 17 gr. ferè, as before.

Then 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 Spheare (deduced from its avoirdupois-weight, according to the common proportion, 60 to 73) viz. 10 li. 11 oun. 3 drams, and 12 gr. (or 10 li. 11 oun. and 8 penny-weight, as before) [Page 212] 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 it self, which I shal 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 shewed 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 Spheare's Dimetient will be lesse then 4 in­ches of the English Foot, viz. but 3.868 inches-english (the whole English Foot, or 12 inches. English, being 12 [...] inches Romane, according to Mr. Greaves his pro­ [...] Foot to the English Foot, 1000 to [...] before noted) and so the former [...] the Sphear, 12 ^li. 2 oun. 1 scrup. 14 2/37 gr. with Gheta [...]dus, or 11 li. 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 li. 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 againe; if we shall reduce our english measure of the said Spheare's dimetient, 4 inches (and so commonly holden to weigh 9 li. avoirdupois, which is 10.95 li. troy, as we shewed before) to Roman measure in inches, we shall find the same (according to the former Pedal Collations) to be more then 4 Roman inches, viz. 4.136 inches, and the gravity of an iron Sphear of this Di­ameter, will be in Ghetaldus his weight, 93044.86 graines, [Page 213] or 13 li. 5 oun. 4 dr. 20.86 gr. (or according to Ghetaldus his Division of the ounce) 13 li. 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 li. 3 oun. 8. p. w. and 0.86 gr.) Archimed. promot. theor. 9. prop. 17. For seing that Spheares 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 Spheare 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 Spheare, whose Diameter is one inch, which by Ghetaldus his Mea­sure and weight, is 1314 22/77 graines. 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 Spheare who [...]e Diameter is one inch-en­glish (which is 1.034 inch-romane) should be by this ac­compt, 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 shew the use of the former proporti­onal Numbers for Metals among themselves) in respect of two Spheares of different kinds of Metals, and like mag­nitude) the same weight of the foregoing Spheare of Iron, [Page 214] of 4 inches english-measure, or 4.136 inches Roman-mea­sure, the diameter maybe produced as before, by the weight of a Sphear of any other Metal first had, being of the same magnitude: As for example, a Sphear of Tinne, (for so Ghetaldus commonly deduceth the weights of other me­talline Sphears from a stanneal Sphear, or a Sphear of Tinne) whose diametrall magnitude is 4 inches-english, or 4.136 inches-Romane, I find to weigh by Ghetaldus, 86066.5 grains (or 11 ^li. 5 oun. 10 scrup. and 2.5 gr.) which is with us in Troy-weight, 65446.4 gr. (or 11 ^li. 4 oun. 2 dr. 46.4 gr. or 11 ^li. 4 oun. 6p. w. 22.4 gr.) Now therefore, according to the foresaid propo [...]onall Numbers for Tinne and Iron, I say; As 3895 to 4210 (which is with Ghetaldus in his fore-mentioned 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 termes being incompleat 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, infinitly; or integrally, 1000 to 1081, or 1000,000 to 1081, 081 com­pleatly) So is 86066.5 gr. (or 12 ^li. 5 oun. 10 scr. 2.5 gr.) the Sphear of Tinne in Ghetaldus his weight, or 65446.4 gr. (viz. 11 ^li. 4 oun. 2 dr. 46.4 gr.) the same Sphear in our Troy-weight, to 93044.86 gr. (or 13 ^li. 5 oun. 12. scr. 21 gr.) the Sphear of Iron in Ghetaldus his weight; or to 707.2.86 gr. (viz. 12 ^li. 3 oun. 3 dr. 12.86 gr.) the same Sphear in our troy-weight, as formerly: which be­ing converted into Avoirdupois-weight, (according to the former terms of proportion between these two Weights) wil be 10.0961. ^li. viz. 10 ^li. 1 oun. 8.576 dr. according to the common division of the Pound-avoirdupois into 16 [Page 215] 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 al 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 exactnesse in the weight of a bullet or a­ny other Body of Iron, or other like base Metall, as to graines, but only to satisfie Art it self in the curiosity thereof.

But now whether the common Tenent of an iron-bullet of 4 inches the diameter, to weigh just 9 ^li auoirdupois, were deduced from any certain and exact experiment or no, is a question here to be made; Nor was it my purpose to trie the same, having not conveniences and accomodations thereunto: but only having the true weight of a Sphear or Bullet of Iron of any magnitude, in a certain mea­sure given (as Inches) thereby to shew a way for the exact and most speedy obtaining of the weight of a Sphear or bullet of any magnitu [...]e whatsoever, which I have partly declared & demonstrated already, and shal more by & by: And looking in severall 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 9 ^li. 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 [Page 216] of the same magnitude: And therefore finding such a discre­pancie 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 ingenu [...]ty in the Mathematiques; (and so indeed I find him to be) who very courteously entertaining me at his house in the Tower, with discourse about many ex­experiments made by him in the Mathematicks, and shew­ing the same to me (as he had once done in some of them, long before) among w ^ch were those cōcerning 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 Sphear or bullet of Iron, according to some cer­tain magnitude or crassitude, knowing him to have all ac­comodations 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 then shewed me; but I observed the same to be very unfit to ground an experiment upon, being not only much rusty, but also having severall holes and cavities therein, (which see­med to be chiefly from the antiquity of it) which might wel hinder the finding of the true weight, accord­ing to its diametrall magnitude, and which himselfe then 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 [Page 217] 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 find the same, then 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 then for my further satisfaction, I took the Cir­cumference of the bullet with a thred, 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 thred 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 wel conclude, the magnitude of this bul­let to be rightly found by us. Then for the weight there­of, we tried the same with all possible precisenesse, both by Avoirdupois-weights, and Troy-weights, and found it to weigh, 15 ^li. 12 1/4 oun. avoirdupoiz; and 19 ^li. 1 25/32 oun. Troy, viz. 19 ^li. 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 then forged iron) of 4 inches the diameter, wil weigh upon the point of 9 ^li. and 2 oun. avoirdepois, which dif­fers not much from the common Tenent.

And so according to this our experiment (to which we wil adhere for the finding out of the weight of any Sphear 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 Sphear or bullet whatsoever, made of cast-iron, by the Diameter or Circumference thereof, as was formerly shewed, wil 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. [Page 218] 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 Sphear or bullet measured thereby, the same wil 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 li. 2 oun. 1 dr. averdepoiz, differing from the true weight (by way of excesse) 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 Sphear of 4 inches diameter, english-measure, (or 4.136 inches Roman-measure) should weigh according to him, 10.096 li. avoirdupois (as being deduced from 12.283 li. troy and that from 13.461 li. Roman) we know not, if Mr. Greaves his foresaid collations and comparisons of the Romane weight and Measure (used by Ghetaldus) with the English, be true, (as we are willing to beleeve they are) and also the foresaid proportion of the Troy-pound­weight 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 generall The Troy and Avoirdupois Pound compa­red together. Maxime, and undeniable Principle (as he saith) that 136 pounds-Troy, and 112 pounds Avoirdupois are equilibral, or equi­valent in weight, which are in the least terms of proporti­onality, 17 and 14; and which indeed I find by experience [Page 219] to be the truer, as I shall afterwards shew: And this is to be understood properly (as the Terms here stand, from the greater to the lesse) for the reduction of Troy-pound­weight (or Troy-pounds) to Avoirdupois-pound-weight (or Avoirdupois-pounds) in asmuch as 73 ^li. Troy, make but 60 ^li. Avoirdupois; or rather 17 ^li. troy make but 14 ^li. avoirdupois: and so the Avoirdupois Pound, is (in the least terms) 73/60, or rather 17/14 the Troy-Pound; whereas else, if we will precisely compare the Pound-troy simply, as the lesse, with the Pound-avoirdupois, as the greater; then 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 least terms) 60/73, or rather 14/17 of the Pound-avoir­dupois. And which several Terms of proportion, though they seem, to differ but very litle in the Reason thereof it selfe; 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 shew afterwards: And so I will here first compare these two severall 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 lesse, 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 differen­tiall [Page 220] 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 Clav. de Pro­port. composit. in fine 9 Elem. Eucl. of Fractionall 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 Authours 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 termes only, thus:

And not according to that operation of Reasons or Pro­portions, which some do falsly and improperly call the sub­duction or subtraction of Reasons (as Clavius also saith in the place fore-cited) 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 differentiall or residuall Reason; and which would then be here by prolation, of the greater inequality, viz. 1 1/510, according to the termes 511/510 (whereas the other or true dif­ferentiall Reason, is of the lesse 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.

Or more plainly thus,

Or yet more plainly and readily, by permutation of the termes of the dividing or resolving Reason, in respect of their places; for so the work of Resolution wil be chan­ged into a Composition of the termes, 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, then the other or common Proportion doth; as may be more plainly discerned, by reducing the said severall Proportions into decimal Termes, for so, the Proportion of the Pound-troy, to the Pound-avoirdupois from the Termes of 73 to 60, wil 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 Troy­pounds) the Pound-avoirdupois wil be to the Pound-troy from the termes of 60 to 73, decimally, as 1.00000, &c. to 1.21666, &c. infinitely; and from the Terms of 14 to 17, as 1.00000, &c. to 1.21428, &c. and so here the Pound Avoirdupois is by these latter Terms, lesse of the Pound Troy, (and so comes nearer the same) then by the other Terms. And the Reason of these latter Terms (as they here stand, from the lesse to the greater,) wil be greater then the Reason of the first or common Terms (according to the exact comparing of Proportions or Reasons together, as you may see in the fore-cited pla­ces of Ramus and Clavius) in regard that the two severall Reasons being reduced to one common Consequent (as a­foresaid) the new, compound or correspondent Antecedent of the latter Proportion, wil be greater then the like Term of the first; or the foresaid Antecedent of the latter Pro­portion, wil be more of the common Consequent, then 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 lesse, as Consequent) as you [Page 224] may here plainly see by these severall subsequent operati­ons.

Or more plainly thus, in a Fractionall manner.

Or again, the same wil appear by a contrary operation, which is by reducing the two severall 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, wil be lesse then the like Te [...] of the first or common Proportion, (and thereby the Rea­son of these latter Terms, wil accordingly be greater then that of the other Terms (according to the fore-cited Au­thours) 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 differen­tiall Reason) wil 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 termes of Pro­portion of the Po [...]nd-troy to the Pound-avoi [...]dupois, viz. 17 to 14; the foresaid Sphear of Iron wil be in avoirdupois weight, 10.1158 li, which (by conversion of the fraction­part into the proper parts of this weight) is 10 li. 1 oun. 13.6 dr. whereas by the common terms, it was 10 li. 1 oun. 8.6 dr. and so the difference of weight, 5 dr. exactly. But I cōceive (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 then cast, coarse, or drossie iron, the proportion of weight be­tween them, being (as I have deduced it from the experi­ments of Mr. Reynolds; and partly of my self 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 lesse: 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, wil be such as of 1000000 to 1050605: For the said Mr. Reynolds found [Page 226] 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 7 ^th. 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 0.3474 ^li. 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 penny-weight, 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 0.33068 ^li. troy. And so according to this experiment, a Sphear or bullet of one inch Diameter, made of fine, or forged Iron, wil weigh a .18288 oun. troy, which is 0.1819 ^li. troy: and so a Sphear or bullet of 4 in­ches Diameter, made of the same metal, wil weigh 11.642 poundstroy (or 11 ^li. 7 oun. 5 dr. and 38.17 gr. or 11 ^li. 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. 12.283 ^li. &c. troy, or 12 ^li. 3 oun. 3 dr. 12.86 gr. or 12 ^li. 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 Sphear 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 wil be found to want thereof 57.7 gr. viz. 2 p. w. 9.7 g. which operations agreeing so nearly one with ano­ther, [Page 227] 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 a­gree in a Sphear of this Metall; may happen 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, wil still be heaviest.

Now Ghetaldus beginning to find out the weights of metalline Spheares, according to the usuall known Weights with him, found that no diligence or industry of man could make a Spheare so exact as it ought to be, and therefore he procured a Cylinder to be made, and that of Tinne, equall in height to the Diameter of its Base (which were of a certaine magnitude in inches of the Roman Foot) for that this might be turned in a Lathe much exacter, and more easily then a Spheare: and hereby found, that a Cylinder made of Tinne, 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 graines of his weight, is 1824 gr. whose 2/3 being 1216 gr. is the weight or gravity of a Spheare of the same Metall, 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 shewed by him, that a Cylinder, whose Base is equall to the greatest Circle in a Spheare, and its altitude equal to the diameter of the Spheare (or of the said Circle) is sesqui­alter the said Spheare. And so Ghetaldus having found the [Page 228] weight of one Spheare of Tinne (at a certaine magnitude) from thence he deduceth the weight of any other Spheare, and that not only of the same Metall, but also of any other metall, by the proportions of Tinne 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 Spheare of Tinne of 1 inch the diameter with Ghetaldus, will answer to the like kind of Spheare 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 Troy­weight (according to Mr. Greaves his foresaid comparison of the Romane 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 Spheare, 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 obtaine the weight of a­ny other Spheare of the same Metall, and also of any Sphear of any the other metals, by means of the foregoing propor­tionall Numbers between Tinne and those other Metals: For so a Sphear 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 termes 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 least rationall and absolute termes) 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. [Page 229] 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 severall Bodies in gra­vity and magnitude, 38 18/19 to 54 22/57) or more accurately (the former termes being not absolute or comp [...]eat) 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 least term [...]s rationall and absolute, and which are by decimall numeration, as 1. to 1.396, 396 infinitly; 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 baser 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 Tinne (in sphericall bodies) it is best to have the weight of the foregoing Stanneall Spheare 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 Spheare of any other magnitude may be readily obtained; and so consequently the weight of any other metalline Spheare therefrom, in its proper weight of Avoirdupois. Which said weight therfore of 2.1304 oun. troy, will be inavo [...]rdupois unciall weight (according to the cōmonly received proportiō 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 [Page 230] 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 shew) the same spheare 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 Sphear of Tinne, of any magnitude, (and so of any other Metal therefrom) in li­brall-weight, it is convenient to have the foresaid Sphear of Tinne of 1 inch diameter, in libral-weight, both Troy and Avoirdupois, and that decimally; which wil here be in Troy-weight, 0.17753li. and in Avoirdupois-weight, (which is here chiefly 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 excesse) 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 im­mediatly preceding) in regard of the smalnesse of the Body here handled, whereby the difference between the common Proportions of the Troy-weight to the Avoirdupois­weight, 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 severall proportions beforegoing) we shal find the difference of weight (in one and the same kind) to be [Page 231] stil greater, and the greater or weightier that the Body is, the greater wil be the difference of the weight produced by the said different termes 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 Sphear of iron, of 4 inches the diameter, (en­glish-measure) whose weight being found from Ghetal­dus, to be 13.461li. Romane, and from that, to be in Troy weight (according to Mr. Greaves his collations of these two kinds of weight the oue with the other) 12.283li. and then the same converted into Avoirdupois weight, ac­cording to the two severall proportions between these two weights; the difference of weight was there found to be 5 drams auoirdepois, which is almost 1/3 of an ounce. And greater differences of weight (in this respect) then this, I shall shew 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 severall Proporti­ons assigned be­tween them. according to the Terms 51 and 56 (deduced from the foresaid Maxime 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 least terms of Proportion, 51 and 56) the same being compared with the foresaid common termes, 73 and 80, wil seem indeed to differ but little therefrom in the Reason it self, and that deficiently, as the terms here stand, from the lesse to the greater, like as the correspondent Terms before for librall weight, did from the common terms, being taken from the greater term to [Page 232] the lesse, viz 7 [...] to 60, and Thomasius in the fore-cited place of his Dicti [...]n [...]y, 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 equall to 12 ounces avo­irdupois: and which (he saith) he proved by a m [...]st just and ex­act Balla [...]c [...]. But this Prop [...]rti­on is som [...]what greater then ei­ther o [...] the other two here no­ted, & makes the ounce Averd. to be more of the ounce Troy then 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 lesse as the Antecedent, to the grea­ter, as the Consequent, be­ing by prolation, of the less in [...]quality, or inequality of the lesse, sub-super-septu­partiens septuagesimas-ter­tias, viz. sub-super-parti­ent [...]/73; and of the other Terms, (standing correspō ­dently) sub-super-quintu­partiens quinquage simas­primas viz. sub-superpar­tient 5/51; and so the diffe­rence of the Reasons, or the differentiall or residuall Reason (according to the plain, simple and proper subduction of Reasons be­fore shewed) 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 wil be (in the least terms) 80/73 or more truly, 56/51 the Ounce-avoirdu­pois: Or else, the Terms, in this position, from the lesse to the greater, might seem rather to be taken contrarily (according to an exact comparison) for the proportion of the Avoirdupois-ounce, (as the lesse) to the troy-onnce (as the greater) for so the Avoirdupois-ounce, wil be 73/8 [...]1 or rather 51/56 of the Troy-ounce. And these latter Terms of [Page 233] proportion, do make the Ounce Troy more to exceed the Ounce-avoirdupois, then the first or common Terms do, and so make the Ounce-avoirdupois to come short of the Ounce-troy (or be lesse of the same) accordingly; as may more easily be seen, by reducing of the foresaid Terms of proportion into decimall termes; 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 shewed formerly, upon another the like occasion,) and of 56 to 51, as 1.0000 &c. to .9107 &c.

And therefore, now to shew (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 Bal­lance, in the foregoing Iron Bullet. severall 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 Ballance, and so converting it out of the one kind of weight into the other, by the said two se­verall kinds of proportions between them, whereby may be known which of these are the neerer and truer, as most agreeing with the Ballance it selfe: 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 Ballance, 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 librall­weight, 19 29/12 [...] or 19.1484375li. exactly) and 15li. [Page 234] 12 1/4 oun. Avoirdupois (which is wholly in librall-weight, 15 49/64 or 15.765625li. compleatly) Now if we shall con­vert the said Bullet from its Troy-weight to Avoirdupois­weight, according to the common proportion of the Troy­librall-weight to the Avoirdupois-librall-weight, viz. 73 to 60, we shall find the same to be (in the least termes) 15 1725/2336 or 15.73844li. Avoirdupoiz, which wanteth of the true weight from the Ballance, 0.02718li. whcih 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 halfe 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 least terms) 15 837/1088 or 15.7693li. Avoirdupoiz, which exceedeth the weight, not fnlly one dram, being but 0.94 dr. as will appeare by reduction of the fractionall termes into the proper denominate parts of this weight, and which is not considerable.

Againe; 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 severall proportions of the one to the other, we shall find the very same differences necessarily to happen: for ac­cording to the common proportion of the Troy-unciall­weight to the Avoirdupois, viz. 73 to 80, it wil be found (in the least termes) 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, [Page 235] which differeth from the ballance-weight (by way of excesse) 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 shal find the like proportional differences of weight accordingly. For first if we convert the foresaid bullet, out of its true avoirdupois­weight from the Ballance, 15 40/64li. or 15.7656, &c. into Troy­weight, according to the vulgarly received proportion of 60 to 73, for librall-weight (being the converse of the for­mer) we shal find the same to be (in the least termes) 19 697/384 [...], or 1815104 ^li. Troy, which differeth from the true weight of the Ballance (19 19/12 [...] or 19.1484375 ^li) by way of excesse, 0.0330729 ^li. 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 wil be found (in the least termes) 19 129/896, or 19.1439732li. Troy, which differeth from the true weight (by way of defect) only 0.0044643li. which is by a continuall reduction into the least 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 Ballance) 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 find the same to be (in the least terms 230 57/320 oun. troy, or 230.178125 compleatly, 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 least parts of weight, 3 dr. 10.5 gr. or 7 p. w. [Page 236] 22.5 gr. as before from librall weight: But according to the proportion of 56 to 51, it wil be found to be (in the least 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 generall, 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 Sphear, as was formerly of a Sphear in generall 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 shewed.

And so these operations beforegoing in the particular metalline Sphears 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 Countreyman 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, til some other experiments be produced, both in these, and also in the other Metals, [Page 237] 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 therin.

As for the weights of Metals compared in Sphears of one and the same magnitude, set down in the latter part of Mr. Ponds Almanack, 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 Sphear of Gold, to weigh just one pound-troy, and from thence, the other metalline Sphears of the same mag­nitude, to weigh accordingly, as followeth.

But now to return a little to our foregoing work of the artificiall dimension, or diametral and Circumferentiall cu­bation of a Sphear of cast-iron, for the weight thereof, which as we shewed before by the Integer of weight it self (or by librall weight) So we will next shew how to perform the same by the composing, denominate parts ther­of immediatly, viz. ounces avoirdupoiz (or uncial weight) [Page 238] 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 finde (according to the reason of the precedent second Theoreme, and also our generall reason) to be for the Diameter of such a Sphear, (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 then the Diameter or Circumference be taken by their proper respective Line or Scale, and cu­bed, the same shal be the weight of the Sphear or Bullet in ounces and decimal parts immediatly: For so the Diame­ter of the foresaid bullet of 9li. and 2 oun. ferè avoirdu­pois, (viz. 9.123626 li. ferè) being 4 inches, and the Cir­cumference, 12.57 ferè, wil 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 wil stand more precisely uppon the weight of this Bullet, if then we divide the two foresaid artificiall Lines of measure for this purpose into more parts, as 1000 (for the naturall Line, or the Inch being so divided, the artificial Line wil be thereof, for the Diameter, 0.760 ferè, and for the Circum­ference, 2.387 ferè) & so measure the Diameter & Circum­ference of this Sphear or bullet thereby, we shal find the [Page 239] same to be severally, 5.265, which cubed, gives 145.946|984625 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 wil 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 wil be found each of them, 6.318, which cubed, affords, 252.196389. &c. 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 wil be found severally (according to a centesimal partition of the Lines) 2.51 ferè, whose Cube is 15.813|251 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 li. (or 15 li. and 12.25 oun.) only 0.047626 li. 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 artificiall Line wil be thereof for the Diameter, 1.914, and for the Circumference 6.014 ferè) they wil be found severally, 2.507, which cu [Page 240] bed, yields 15.756617, &c. (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 Spheare of this The artificiall Lines of mea­sure, for the most speedy discovering of the weight of a Spheare or Bullet of cast-Iron, in Troy­weight, both li­brall and unci­all. metall (though indeed this kind of weight is not usuall for any Metall besides Gold and Silver, as I have noted before) then the artficial Lines of measure for the speedy discovering of the weight thereof by the diameter, wil be 1.794 inch for librall­weight, and by the Circumference, 5.636 inches: And the Lines for unciall weight, wil 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 Spheare of metall or other matter, so very exact and precise indeed, as it ought 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 preti­ous Metals) So the exactnesse of a Sphericall 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 somthing neer 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 discerne it to be not exactly orbicular or sphericall; and so the weight there­of can be very litle mistaken, being obtained by the diame­ter or Circumference of the same, taken in a certaine set Measure, either naturall, as Inches, or the like; or artifici­all, as deduced therefrom, according as I have here shewed [Page 241] at large. And which severall dimensions with many other the like metrical Conclusions pertaining hereunto, and the Converse of the same (and all cheifly in reference to Gun­nery) I will next expresse proportionally by Number, (from our foregoing experiments) according to the aforesaid Metall 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 severall Sections of proportions following, are (for brevity-sake) noted by the letters of distinction, l, and u.

Conversly.

Hence,

Then for the speedy discovering of the weight of a sphe­ricall or any other body whatsoever made of this Metall, by the solidity thereof, in the Measure aforesaid: and con­trariwise [Page 243] 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.

Contrariwise.

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 Metall, as fol­loweth.

Conversly.

And thus having shewed our artificiall Mensuration in regular Solids, aswell for gravity or ponde [...]osity, as for solid (and superficiall) measure; I shall now close up this Section and Part, with our third theorematicall Proposi­tion, (answering to the third principall problematical or practicall Proposition in the first Part) and the practicall demonstration thereof, in which our more particular, or speciall reason of the like dimension of all regular-like Solids (as particularly of a Cylinder) in the severall re­spects aforesaid (in reference to the producing of the arti­ficiall Lines of measure, for performing the same) is con­tained.

SECT. I. Concerning the measuring or gauging of Vessells, in generall.

ANd vvhat hath be [...] here done for the solid Dimension of a Cylinder (or the dimension of a solid Cylin­der) by artificiall 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 Cylindricall Vessell) for the more speedy finding out of the liquid Con­tent, according to any Liquour and Measure given. As our Vessells for Wine and Ale, or Beer (which two, most [Page 263] 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 Meane between the Diameter at the Head, and the Bung of the Vessell (or otherwise the Meane between those two Circles) and so obtaining the liquid Content thereof; which is commonly called Gauging of Vessells; the Mea­sure by which these Vessells 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 Beere, holdeth not one and the same; but differeth very sensibly, as is common­ly known: And therefore we will next briefly shew the measuring or gauging of these Cylindricall Vessells (or Spheroidall, as some will have it, this kind of Vessel being more commonly taken for a Sphaeroides, having the two ends equally cut off; 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 Spheare 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 Liquour, 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 cubi­call or solid inches, or is insensibly diffe­ring therefrom: But for the Ale or Beer-Gallon, I finde the same to be as gene­rally controverted among them. Mr. William Oughtred, a reverend Divine, and most eminent Mathematician, be­fore-named, after some experiments made by him to find 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 severall 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 commeth to a rationall conjecture of the Ale-gallon (and that very neatly, and prety nearly al­so, as I shal straightway shew) in cubique inches, accor­ding [Page 265] 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 fore-cited.

But Mr. John Reynolds aforenamed also, (who seemeth to have been as industrious in this, as in many other ma­thematicall experiments) wil 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 Goodwyn long agoe, in his little Tract entituled, A Table of gauging, publish­ed above 50 years since, and dedicated to the then Lord Major and Aldermen of the City of London: wherein he shewing 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 wil This Mr. Good­wyn was Ma­ster in the Ma­thematicks 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 Ale­gallons, and so consequently one ale-gallon must contain just 288 inches, which that learned gentleman Mr. Edm. Wingate, a Barrester of Grayes-Inne (a man eminent for his mathe­maticall 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 seing 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 Gallon­measure, but not for any liquid thing, but for drie things, as Corne, Coals, Salt, and other dry things measurable by this kind of Measure; and so calleth it the drie Gallon­measure: And thereupon he wil have to be 3 severall Gal­lons (or other like Measures) one for Wines, (which also serveth for oiles, strong-waters and the like) Another for Ale and Beer, and a third for Corne, Coales, and the like; and this he maketh lesser then the Ale-measure, whereas surely it should rather be greater, if there be any difference at all between them: And these three severall Gallon­measures, he compareth together, or differenceth by these three Numbers, viz. 28, 33, 35. as to shew 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 least terms, rationall or absolute; but otherwise in the least proportionall termes, irrationall or unabsolute, and fi­nite or limited, I find 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 least rationall termes, is indeed as 4 to 5; but otherwise in the least termes irrationall (but finite or limit­ted) I find them to be as 1. to 1.25. and which again is integrally rationall, 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 Custome 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 Kingdome, are usually deri­ved, [Page 267] 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, oiles, and the like) and the other for Ale, Beer, and drie things, as Corn, Coales, Salt, and the like; which latter is commonly called the Win­chester Measure, and from this are taken the bigger drie Measures, as the half-Peck and Peck, and so on to the Bushell, which is the greatest of our drie 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 then 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 triall of them, (neither could I find 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) My self and one Mr. Baptist Sutton, (a man well known in the City among artists) [Page 268] did agree to go together to the Guild-hall, where he was wel acquainted with the keeper of the Standard-measures and Weights, who otherwise I found to be very nice and scrupulous in shewing 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 9 ^th. 1645; we repaired together to the Guild-hall, carrying along with us two large square glasse-vials, which we first weighed in a Gold-smiths Ballance 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 Cisterne, and that with all possible precisenesse, we powred forth the same with the like accuratenesse, into the said two glasse vials, and then weighed the Glasses with the water in them by the same weights: and so comparing the weight of each Glasse alone, with the weight of the glasse 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. Then seing 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 shewed before; if we compare these two Gallon-weights of water, with severall experiments made by my self, 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 [Page 269] and magnitude together, which thing is most admira­ble and excellent use, as I shall shew more afterwards) we shal 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 forme 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 heighth, depth, or length, (which you wil) 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 powring in of water, and which was no bigger, then that he might guesse in the filling thereof to a drop or two of water, more or lesse: which Vessell therefore being precisely filled with fair setled Rain-water (as being the fittest, as I shal shew afterwards) and then as precisely weighed by Troy­weights, he found the water thereof alone to weigh 12 li. 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 equall in its dimensi­ons with the former, and that by the same Workman; which he filling with the like water, found it to agree 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 [Page 270] with the like water as aforesaid, found the weight of the said water alone to be 12 li. 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 chiefly from some difference of measure in the Inch, by which the two first Vessels, and the last were made, being done by two seve­rall 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 then made by the bye, 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 li. and 5 1/2 oun. Troy (or 149 1/2 ounces) and this to stand good.

And then after this, I got an exact cubical Vessel to be made of throughly seasoned wood, with all the accurat­nesse & precisenesse that could be, being 6 inches the in-side, (or the base thereof exactly 6 inches square) and so the whole Cube in its concave capacity, exactly 216 inches; and which then, 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 cubicall Vessell I then filling with all the exactnesse and precisenesse that [Page 271] might be, with fair setled 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 it selfe first of all had, from the weight of the vessel and water to­gether) to be upon the very point of 114 ounces, or 9 li. & 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 neer 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 severall 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 beween 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 severall Gallons of water aforesaid, be­ing compared severally with this experiment, the solid ca­pacity [Page 272] of the Wine-gallon, wil 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 trie the same thing over again, to see how nearly two severall trials would agree to con­firm the matter; knowing that two testimonies upon any thing are much better then one; I again moved the said M. W [...]gate and Mr. Sutton (whom I still desired as witnesses to what was done) for another triall of this thing, and that divers times; but could not accomplish my desires herein, til about two yeares after: And so in July 1647, I and Mr. Sutton went together a­gain 2 ^d. experiment upon the wine & ale-gallons. to the Guild-hall (Mr. Wingate ha­ving promised to go along with us, but was hindred by other occasions) carrying along with us two other great glasse-Vials like the for­mer, into wh [...]ch first powring the foresaid Standard-gal­lons of water exactly filled, and then 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 wel 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 li. 1 oun. 3 dr. avoirdupois (or 129 3/16 ounces avoirdupois) and of the Ale-gallon of wa­ter, to be 9 li. 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 Troy­weight before noted, viz. 14 to 17. and 56 to 51) 9 li. 9 oun. and 5.223 dr. or 13 p. w (which is 117.65 oun. troy) [Page 273] 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 neer the former proporti­on: And being compared with the foresaid experimentall 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 Ale­gallon, about 1 inch. Whith difference between these two experiments, especially in the Ale-gallon, though in the matter of Gauging, the same can breed no sensible errour or difference, as wil afterwards plainly appear, when we come to shew 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 triall: 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, ( [...]iffering [Page 274] much in form from the other before used, though indeed this be nothing materiall to the purpose,) which I first caused to be weighed severally by the standard- avoirdu­pois weights there; and then (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 accurate­nesse 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 Ballance, 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 lesse to the greater) as 100000 to 118699 ferè, which is very little lesse then 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 experimentall 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 streightway 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 Glasse toge­ther with its water, and afterwards each Glasse alone (being throughly drie) 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 lesse 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 Beer­gallon, 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 ballance, 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 ballance, only by 0.04 inch ferè, which is as much as nothing.

And here having a fit occasion and op­portunity, I shal (by way of Digression) The foresaid severall Pro­portions be­tween the Troy and Avoirdu­pois Weights, cōpared again together, & ex­amined by the Ballance upon another Expe­riment. speak somwhat more concerning the Pro­portions between the Troy and Avoirdu­pois weights, for a further confirmation & verification of what was said forme [...]ly, 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 termes of proportion, viz. 60 to 73 for pound-weight, or 80 to 73 for ounce-weight, wil 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 Ballance 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 then by the other terms of proportion, viz. 14 to 17 for librall weight, or 56 to 51, for unciall-weight, the Wine-gallon wil 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 Ballance, (by way of excesse) 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. [Page 277] which differences are very inconsiderable, the greatest of them not amounting to a penny-weight, whereas the least 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 Ale­gallons 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 Brasse, of 4 inches the Side exactly, and so the whole Cube, 64 inches; 4th. experi­ment upon the Wine and Ale­gallon. into which (being set level) my selfe, and Mr. Sutton aforenamed, together pouring out the two Gallons of water in the Glas­ses which we had from the Standard-Vessels at the Guild­hall, as aforesaid, with all the accuratenesse and precisenesse that might be (at severall times) we found first the Wine­gallon to fill the Cube three times, and then halfe the Cube (as ne [...]rly as we could possibly measure it) which being computed, doe make 224 cube-inches exactly, for the solid content of the Wine-gallon; And then we found the Ale­gallon 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 neerly agree with the for­mer experiments (especially in the Wine-gallon) as that [Page 278] this experiment may sufficiently confirme the former; this being as plaine and demonstrative an experiment as [...]ay be. And if it had been performed by a cubicall Vessell 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 Cubicall Vessell of wood was made long after the triall of this Experiment) then the same might probably have yet come neerer the truth; for that the of [...]en filling of this small Cube, might cause some small errour, 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 heighth (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 my self, 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 str [...]ctnesse of Art, is (I may say) impossible, unlesse the Standard-Vessels were so narrow-mouthed, as that in the filling thereof, one might be able to guesse at a few drops of water, which in the Standard Vessels at the Guild-hall (and I thinke in-all others) cannot be done, they being so wide at the mouthes or tops, as that a spoonfull of water more or lesse in the filling thereof is hardly discernable, or so much more, as might breed the difference of halfe an ounce more or lesse, in the weight of the water, and so consequently of one inch more or lesse, in the solid con­tent; seing that one inch cubick or solid of water weigheth [Page 279] (as wee shall shew anon) halfe an ounce-troy, at least: And yet in all these severall experiments and ob­servations compared together, the greatest difference of so­lid measure is but about one inch and an halfe, 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 wil be but one inch solid, which wil breed no sensible errour in the matter of gauging, as I, said before. And which may shew 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 wel be gone.

But yet that there might no likely way be left unatrem­pted, for the discovery of this thing, I wil add to the former, one experiment more, 5 ^th. exper [...]i­ment in the Ale-gallon a­lone, by taking it's Dimensi­ons. being very demonstrative, which I made last of all in the foresaid brasse 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 internall or concave superficies from the top to the bottome, being very straight and smooth, aswell as the externall or convexe Supe [...]ficies, and also exactly circular throughout; only a little shelving or arching at its meeting or connexion with the bottome, and this not precisely plain, but rather a little hollowish, yet not so much as to make any sensible errour in giving the solid ca­pacity of the Vessel, as wil 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 [...]ore­cited [Page 280] place of his Circles of Proportion, concerning Gauging (as being shewed Circles of Pro­portion. Part 1 chap. 9. Sect. 4. unto him, and also the measures thereof first given him, by that great Antiqua [...]y 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 wel 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 bottome, and the height of that Vessell which they together measured, (and which you may see in the fore-mentioned 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 brasse Cube aforesaid, but little more then 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 least 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 Vessell, which might wel 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 Vessell (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 bottome, but 0.1 inch; and to want of that in the per­pendicular [Page 281] height (or the depth) 0.8 inch; by which I find the solid dimension of this Conicall 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 meane Proportionall (produced most exactly by Logarithmicall supputation) with a trient of the Altitude, thus,

[...]

Which comes very neer 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 Ale­gallon, 6th. Experi­ment in the Ale-gallon, ta­ken from the half-Peck: and that from the Bushell. may yet be further confirmed by another observation or experiment being taken from the half-Peck, w ^ch they hold at the Guild-hall, to be equall in Content to the Ale-gallon, as being taken therefrom (and so I find it to be) and which I shall [Page 282] deduce from the Bushell, according to the Dimensions thereof, established by an Act of the common-Councell of the City of London, and yearly published by authority of the Lord Major, which ordains the breadth or widenesse of a Bushell to be 19 inches, and the depth 7 2/2 inches; which being cast up according to a Cylindrical dimension, wil be fonnd 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 ballance) which gave the same, 265.6 inches; or somewhat nearlier, that from Troy-weight, 265.7 ferè. And if we shal mediate between that of the first observation, being the greatest (ex­cept that of the brasse Cube, which is too large) viz. 266.3 inches, and that of the second, being the least, viz. 265.3 inches, (as is usually done in such like cases, where severall 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, wil be just 265.8 inches, for the solid capacity of the Ale-gallon, exactly agreeing with that of the half-Peck.

And nearly agreing with the foresaid Bushell, 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 Brasse, in the last year but one of her reign viz. Anno 1601 having her Inscription or Title about it; and somwhat resembling a segment of a Cone, it being wider at the top then at the bottome, whose dimensions there fore I took with all the accuratnesse that [Page 283] 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 wil be found, 2122.165 inches; thus,

[...]

Which wanteth of the London-Bushell, (being 3126.465 inches) 4.3 inches: But indeed this Bushel was made a litle turning or winding outwards near the edg or top ther­of, (w ^ch was made very sharp or thin) which might give so much more in the solid content thereof, as to make it equall with the City-bushell; and so I suppose this London­bushell was first intended to be made equall 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 [Page 284] Bushell should be computed, the same would be found, 2134 inches, but that is too large: and the other comes nearer the truth: And the 8 ^th. part of this last, for the half-peck, and so for the Ale-gallon, wil 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 Brasse in the ve­ry same Yea [...]f her Raigne, ha­ving also her Epigraph about i [...]: & which, I went with an intent chiefly to have measured; but indeed before my comming 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 agree there with, without any conside­rable difference, both as one of the Founders men, and also the under-keeper of the Standards for the City, told me, who were present at the triall 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 fore-cited, 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 Vessell made of brasse, 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 geometricall 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 bot­tome then at the top, whereas that was wider at the top then at the bottome, 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 5 ^th.) most plainly and manifestly discovered in a geome­tricall manner, the dimensionall quantity of the Ale-gallon in inch-measure, as neerly 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 then by the 2 ^d. and 3 ^d. experiments the ponderall quantity thereof, in respect of the weight of the water exactly filling the same, (which two experiments also do not considerably differ; the difference being but 3dr. avoirdupois, as we there shewed, or about 1 1/3 dr. troy, being neer 1/6 of an ounce-troy) especially the 3 ^d. 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 [Page 286] me, from my large cubicall Vessell of 216 inches: and also confirme the same; and so consequently likewise confirme 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 neerly (by the 3 ^d. experiment) to be (in Troy-weight) 11 li. 8 oun. and neer about 4 p. w. or 140.2 ounces: the weight of the wa­ter exactly contained by our foresaid Cubicall Vessel, will be found thereby (according to geometricall 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 neerly confirmed by the Wine-Gallon, according to our 4 ^th. and most plain demonstrative expe­riment for the same (by the brasse 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 neer about 9 li. and 10 oun. Troy, or 118 oun. the weight of the water exactly contained in the fore­said large cubicall 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 shal 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, [Page 287] as aforesaid; we first weighed the said concave Cube by a small Ballance of his, with Troy-weights, and then again filled the same with fair water, as precisely as possibly we could; which we also weighing together, found the ad­ditional or differentiall weight (take it which way you wil) to be (as nearly as we could possibly guesse) 33.5 ounces, for the water alone; And afterwards I being de­sirous to make a further and exacter triall 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 then filling the Cube with fair setled wa­ter, as precisely as possibly I could, had the same weighed together, with the like accuratnesse; and so by collating these two severall 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 widenesse of the Vessel, to be such, (and that fully open on one side) as that some few drops of water more or lesse in the filling thereof, were not very discernable. Now according to our former experiments, the weight of this cubicall body of water, wil be found, 33.7 oun. troy; (or rather 33.8 ferè) which comes prety 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 experi­mentall 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 [Page 288] 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 aire and in the water, and the same compared together.

SECT. III. Containing the practice of Gauging, according to our artificiall way of measuring: together with the naturall Dimension (by way of comparison) for a confirma­tion of the same.

HAving by the severall waies 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 two 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 it self; 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 artificiall way of [Page 289] measuring, being here, as the solid dimension of a Cylin­der, before declared and demonstrated, (as I have already noted) by the like artificiall Lines of measure: though in­deed the reason thereof cannot be exactly referred to the 3 ^d. Theoreme before-going, which is for Cylindricall Di­mension in generall; nor yet to our other, or generall 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 (Cylindricall) bo­dy, expressed in some certain solid measure, according to the capacity thereof: whereas else the same being consi­dered either simply, absolutely, and intirely in it self, as such a liquid Measure, or in its Parts compounding, &c. (as being by themselves apart takē for other lesser or infe­rior liquid Measures) like as the Integer of measure (or of Weight) in generall, was formerly; then wil the Reason of this Dimension (as in reference to the quantities of mea­sure in the artificiall gauging-Lines) be the same with that of Measure (or Weight) in generall; especially according to our first or generall acception or consideration thereof; but wil exacty agree with that of weight, in both the res­pects of Reasons aforesaid. And thus the quantity of the ar­tificiall gauging-Line for Wine-measure (according to the Gallon of 224 inches) wil be 6.58 inches; and for Ale or Beer measure, (according to the Gallon of 266 inches) 6.97 inches. But yet considering the losse that happens in Wine by the dregs or lees 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 least 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 [Page 290] 270 inches; which we conceive to be the most neer and indifferent measures (in a generall 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 neer the same, the true Contents thereof do come: And therefore according to these two measures, we shall pro­ceed in the worke of Gauging. And the length of our ar­tificiall gauging-Line serving hereunto, will be for Wine­measure, 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 severall times upon a Rod or Rular of a convenient length) and then the Diameters and lengths of Vessels ta­ken thereby; their Contents shall be obtained in Gallon­measure immediatly, as the solid Contents of Cylinders were formerly had, according to any Measure appointed.

But seeing these Vessels are not exactly Cylindricall (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 bung-hole, and the head; Therefore must a way be first had for the finding out of such a mean Diameter, as may reduce this unequall Body into an absolute Cylinder, or so neer the same as pos­sibly may be; that so the true Contents thereof may be had, or very neer the same: By which mean Diameter, must not here be understood a Meane either arithmetical or geometricall, but a third sort of Meane differing from them both: the discovering of which, is the first and main thing in the practise of Gauging, next after the discove­ring of the Contents of the Wine and Ale-gallons in solid inch-measure. And which we shall shew, most easily and [Page 291] speedily how to performe, by this first Rule next ensu­ing.