Statics, 2
Original
Then, * set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, it hindreth nothyng.) So let thẽ most stedily be stayed. Now, if there be a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in water, 3. or 4. times: at the least: what stuff so euer they be made of. Let not your Solid angle, at the vertex, be to sharpe: but that the water may come with ease, to the very vertex, of your hollow Cone or Pyramis. Put one of your Solid Cubes in a Balance apt: take the waight therof exactly in water. Powre that water, (without losse) into the hollow Pyramis or Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take the waight of the same Cube againe: in the same kinde of water, which you had before:
(I. D. * Consider well whan you must put your waters togyther: and whan, you must empty your first water, out of your Pyramis or Cone. Els you will erre.)
put that* also, into the Pyramis or Cone, where you did put the first. Marke now againe, in what number or place of the lines, the water Cutteth them. Two wayes you may conclude your purpose: it is to wete, either by numbers or lines. By numbers: as, if you diuide the side of your Fundamentall Cube into so many æquall partes, as it is capable of, conueniently, with your ease, and precisenes of the diuision. For, as the number of your first and lesse line (in your hollow Pyramis or Cone,) is to the second or greater (both being counted from the vertex) so shall the number of the side of your Fundamentall Cube, be to the nũber belonging to the Radicall side, of the Cube, dubble to your Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it selfe, whether it be dubble or no, to the Cubik number of your Fundamentall Cube. By lines, thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is to the second or greater, so let the Radical side of your Fundamẽtall Cube, be to a fourth proportionall line, by the 12. proposition, of the sixth boke of Euclide. Which fourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired.
(☞ God be thanked for this Inuention, & the fruite ensuing.)
For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the holy and glorious Trinity: hauing greater cause therto, then
(* Vitruuius. Lib. 9. Cap. 3.)
* Archimedes had (for finding the fraude vsed in the Kinges Crowne, of Gold): as all men may easily Iudge: by the diuersitie of the frute following of the one, and the other. Where I spake before, of a hollow Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it with water, precisely full, and poure that water into your Pyramis or Cone. And here note the lines cutting in your Pyramis or Cone. Againe, fill your coffen, like as you did before. Put that Water, also, to the first. Marke the second cutting of your lines. Now, as you proceded before, so must you here procede.
(* Note.)
* And if the Cube, which you should Double, be neuer so great: you haue, thus, the proportion (in small) betwene your two litle Cubes: And then, the side, of that great Cube (to be doubled) being the third, will haue the fourth, found, to it proportionall: by the 12. of the sixth of Euclide.
(Note, as concerning the Sphæricall Superficies of the Water.)
Note, that all this while, I forget not my first Proposition Staticall, here rehearsed: that, the Superficies of the water, is Sphæricall. Wherein, vse your discretion: to the first line, adding a small heare breadth, more: and to the second, halfe a heare breadth more, to his length. For, you will easily perceaue, that the difference can be no greater, in any Pyramis or Cone, of you to be handled. Which you shall thus trye. For finding the swelling of the water aboue leuell.
(☞)
“Square the Semidiameter, from the Centre of the earth, to your first Waters Superficies. Square then, halfe the Subtendent of that watry Superficies (which Subtendent must haue the equall partes of his measure, all one, with those of the Semidiameter of the earth to your watry Superficies): Subtracte this square, from the first: Of the residue, take the Rote Square. That Rote, Subtracte from your first Semidiameter of the earth to your watry Superficies: that, which remaineth, is the heith of the water, in the middle, aboue the leuell.” Which, you will finde, to be a thing insensible. And though it were greatly sensible, *
(* Note.)
yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke, noted: you may reduce all, to a true Leuell. But, farther diligence, of you is to be vsed, against accidentall causes of the waters swelling: as by hauing (somwhat) with a moyst Sponge, before, made moyst your hollow Pyramis or Cone, will preuent an accidentall cause of Swelling, &c. Experience will teach you abundantly: with great ease, pleasure, and cõmoditie.
Thus, may you Double the Cube Mechanically, Treble it, and so forth, in any proportion.
(Note this Abridgement of Dubbling the Cube. &c.)
Now will I Abridge your paine, cost, and Care herein. Without all preparing of your Fundamentall Cubes: you may (alike) worke this Conclusion. For, that, was rather a kinde of Experimentall demõstration, then the shortest way: and all, vpon one Mathematicall Demonstration depending. “Take water (as much as conueniently will serue your turne: as I warned before of your Fundamentall Cubes bignes) Way it precisely. Put that water, into your Pyramis or Cone. Of the same kinde of water, then take againe, the same waight you had before: put that likewise into the Pyramis or Cone. For, in eche time, your marking of the lines, how the Water doth cut them, shall geue you the proportion betwen the Radicall sides, of any two Cubes, wherof the one is Double to the other: working as before I haue taught you:
(* ☞ Note.)
* sauing that for you Fundamentall Cube his Radicall side: here, you may take a right line, at pleasure.”
Yet farther proceding with our droppe of Naturall truth:
(To giue Cubes one to the other in any proportion, Rationall or Irrationall.)
you may (now) geue Cubes, one to the other, in any proportiõ geuẽ: Rationall or Irrationall: on this maner. Make a hollow Parallelipipedon of Copper or Tinne: with one Base wãting, or open: as in our Cubike Coffen. Frõ the bottome of that Parallelipipedon, raise vp, many perpendiculars, in euery of his fower sides. Now if any proportion be assigned you, in right lines: Cut one of your perpendiculars (or a line equall to it, or lesse then it) likewise: by the 10. of the sixth of Euclide. And those two partes, set in two sundry lines of those perpendiculars (or you may set them both, in one line) making their beginninges, to be, at the base: and so their lengthes to extend vpward. Now, set your hollow Parallelipipedon, vpright, perpendicularly, steadie. Poure in water, handsomly, to the heith of your shorter line. Poure that water, into the hollow Pyramis or Cone. Marke the place of the rising. Settle your hollow Parallelipipedon againe. Poure water into it: vnto the heith of the second line, exactly.
(* Emptying the first.)
Poure that water * duely into the hollow Pyramis or Cone: Marke now againe, where the water cutteth the same line which you marked before. For, there, as the first marked line, is to the second: So shall the two Radicall sides be, one to the other, of any two Cubes: which, in their Soliditie, shall haue the same proportion, which was at the first assigned: were it Rationall or Irrationall.
Thus, in sundry waies you may furnishe your selfe with such straunge and profitable matter: which, long hath bene wished for. And though it be Naturally done and Mechanically: yet hath it a good Demonstration Mathematicall.
(The demonstrations of this Dubbling of the Cube, and of the rest.)
Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones, in the proportions assigned: and like Pyramids or Cones, are in proportion, one to the other, in the proportion of their Homologall sides (or lines) tripled. Wherefore, if to the first, and second lines, found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in continuall proportion: that fourth line, shall be to the first, as the greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of Euclide. If Pyramis to Pyramis, or Cone to Cone, be double,
(I. D. * Hereby, helpe your self to become a præcise practiser. And so consider, how, nothing at all, you are hindred (sensibly) by the Conuexitie of the water.)
then shall * Line to Line, be also double, &c. But, as our first line, is to the second, so is the Radicall side of our Fundamentall Cube, to the Radicall side of the Cube to be made, or to be doubled: and therefore, to those twaine also, a third and a fourth line, in continuall proportion, ioyned: will geue the fourth line in that proportion to the first, as our fourth Pyramidall, or Conike line, was to his first: but that was double, or treble, &c. as the Pyramids or Cones were, one to an other (as we haue proued) therfore, this fourth, shalbe also double or treble to the first, as the Pyramids or Cones were one to an other: But our made Cube, is described of the second in proportion, of the fower proportionall lines:
(* By the 33. of the eleuenth booke of Euclide.)
therfore * as the fourth line, is to the first, so is that Cube, to the first Cube: and we haue proued the fourth line, to be to the first, as the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to the Cube, as Pyramis is to Pyramis, or Cone is to Cone.
(I. D. * And your diligence in practise, can so (in waight of water) performe it: Therefore, now, you are able to geue good reason of your whole doing.)
But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the water Solide Parallelipipedons, are one to the other, as their heithes are, seing they haue one base. Wherfore the Pyramids or Cones, made of those water Parallelipipedons, are one to the other, as the lines are (one to the other) betwene which, our proportion was assigned. But the Cubes made of lines, after the proportiõ of the Pyramidal or Conik homologall lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before did proue) therfore, the Cubes made, shalbe one to the other, as the lines assigned, are one to the other: Which was to be demonstrated. Note.
(* Note this Corollary.)
* This, my Demonstratiõ is more generall, then onely in Square Pyramis or Cone: Consider well. Thus, haue I, both Mathematically and Mechanically, ben very long in wordes: yet (I trust) nothing tedious to them, who, to these thinges, are well affected. And verily I am forced (auoiding prolixitie) to omit sundry such things, easie to be practised: which to the Mathematicien, would be a great Threasure: and to the Mechanicien, no small gaine.
(* The great Commodities following of these new Inuentions.)
* Now may you, Betwene two lines giuen, finde two middle proportionals, in Continuall proportion: by the hollow Parallelipipedon, and the hollow Pyramis, or Cone. Now, any Parallelipipedon rectangle being giuen: thre right lines may be found, proportionall in any proportion assigned, of which, shal be produced a Parallelipipedon, æquall to the Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36. proposition, of the 11. boke of Euclide. Now, all those thinges, which Vitruuius in his Architecture, specified hable to be done, by dubbling of the Cube: Or, by finding of two middle proportionall lines, betwene two lines giuen, may easely be performed. Now, that Probleme, which I noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke of Euclide, is proued possible. Now, may any regular body, be Transformed into an other, &c. Now, any regular body: any Sphere, yea any Mixt Solid: and (that more is) Irregular Solides, may be made (in any proportiõ assigned) like vnto the body, first giuen. Thus, of a Manneken, (as the Dutch Painters terme it) in the same Symmetrie, may a Giant be made: and that, with any gesture, by the Manneken vsed: and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make one, of the same Mould (in any assigned proportion) bigger or lesser.
(* ☞)
Now, may you, of any * Gunne, or little peece of ordinaũce, make an other, with the same Symmetrie (in all pointes) as great, and as little, as you will. Marke that: and thinke on it. Infinitely, may you apply this, so long sought for, and now so easily concluded: and withall, so willingly and frankly communicated to such, as faithfully deale with vertuous studies.
(Such is the Fruite of the Mathematicall Sciences and Artes.)
Musean translation
Modern machine-assisted Musean edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.
Then set your cone or pyramid, with the vertex downward, perpendicularly with respect to the base. (Though it is placed otherwise, it hinders nothing.) So let them be most steadily supported. Now, if there is a cube which you would have doubled, make a neat cube of copper, silver, lead, tin, wood, stone, or bone; or else make a hollow cube, or cubic coffer, of copper, silver, tin, or wood, etc. You may so proportion these with respect to your pyramid or cone that the pyramid or cone will be able to contain their weight in water 3 or 4 times over, at the least, whatever stuff they are made of. Let not your solid angle at the vertex be too sharp, but let the water come with ease to the very vertex of your hollow cone or pyramid. Put one of your solid cubes in a suitable balance; take its weight exactly in water. Pour that water (without loss) into the hollow pyramid or cone, quietly. Mark in your lines what numbers the water cuts at. Take the weight of the same cube again, in the same kind of water which you had before:
I. D. Consider well when you must put your waters together, and when you must empty your first water out of your pyramid or cone. Else you will err.
put that also into the pyramid or cone where you put the first. Mark now again in what number or place of the lines the water cuts them. Two ways you may reach your conclusion: that is, either by numbers or by lines. By numbers: as, if you divide the side of your fundamental cube into as many equal parts as it is capable of, conveniently, with your ease and the precision of the division. For, as the number of your first and lesser line (in your hollow pyramid or cone) is to the second or greater (both being counted from the vertex), so shall the number of the side of your fundamental cube be to the number belonging to the radical side of the cube double to your fundamental cube; which, being multiplied cubically, will soon show whether it is double the cubic number of your fundamental cube or not. By lines, thus: as your lesser and first line (in your hollow pyramid or cone) is to the second or greater, so let the radical side of your fundamental cube be to a fourth proportional line, by the 12th proposition of the sixth book of Euclid. Which fourth line shall be the cube root, or radical side, of the cube double to your fundamental cube; which is the thing we desired.
God be thanked for this invention and the fruit ensuing.
For this may I (with joy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, thanking the holy and glorious Trinity, having greater cause thereto than
Vitruvius, Book 9, Chapter 3.
Archimedes had, for finding the fraud used in the king's crown of gold; as all men may easily judge by the diversity of the fruit following from the one and the other.
Where I spoke before of a hollow cubic coffer, the like use is of it, and without weight. Thus: fill it with water precisely full, and pour that water into your pyramid or cone. And here note where the lines are cut in your pyramid or cone. Again fill your coffer, just as you did before. Put that water also to the first. Mark the second cutting of your lines. Now, as you proceeded before, so must you proceed here.
Note.
And if the cube which you should double be ever so great, you have thus the proportion (in small) between your two little cubes; and then the side of that great cube (to be doubled), being the third, will have the fourth found to it proportional, by the 12th of the sixth book of Euclid.
Note, concerning the spherical surface of the water.
Note that all this while I do not forget my first statical proposition, here rehearsed: that the surface of the water is spherical. In this use your discretion: to the first line adding a small hair's breadth more, and to the second half a hair's breadth more to its length. For you will easily perceive that the difference can be no greater in any pyramid or cone you are to handle. Which you shall thus try. For finding the swell of the water above level.
"Square the semidiameter from the center of the earth to the surface of your first water. Then square half the subtendent of that watery surface (which subtendent must have the equal parts of its measure all the same as those of the semidiameter of the earth to your watery surface). Subtract this square from the first; of the residue take the square root. Subtract that root from your first semidiameter of the earth to your watery surface: what remains is the height of the water in the middle above the level."
Which you will find to be a thing imperceptible. And though it were greatly perceptible,
Note.
yet, by help of my sixth theorem upon the last proposition of Euclid's twelfth book, you may reduce all to a true level. But further diligence is to be used by you against accidental causes of the water's swelling: as by having beforehand made moist your hollow pyramid or cone somewhat with a moist sponge, which will prevent an accidental cause of swelling, etc. Experience will teach you abundantly, with great ease, pleasure, and convenience.
Thus may you double the cube mechanically, treble it, and so forth, in any proportion.
Note this abridgment of doubling the cube, etc.
Now will I abridge your pain, cost, and care in this. Without all the preparing of your fundamental cubes, you may (alike) work this conclusion. For that was rather a kind of experimental demonstration than the shortest way, and all depending upon one mathematical demonstration. "Take water (as much as will conveniently serve your turn, as I warned before of your fundamental cube's size). Weigh it precisely. Put that water into your pyramid or cone. Of the same kind of water then take again the same weight you had before, and put that likewise into the pyramid or cone. For each time, your marking of the lines, how the water cuts them, shall give you the proportion between the radical sides of any two cubes, of which the one is double to the other; working as before I have taught you:
Note.
saving that, for your fundamental cube's radical side, here you may take a straight line at pleasure."
Yet further proceeding with our drop of natural truth:
To give cubes one to another in any proportion, rational or irrational.
you may (now) give cubes one to another in any proportion given, rational or irrational, in this manner. Make a hollow parallelepiped of copper or tin, with one base wanting, or open, as in our cubic coffer. From the bottom of that parallelepiped raise up many perpendiculars in each of its four sides. Now if any proportion be assigned you in straight lines, cut one of your perpendiculars (or a line equal to it, or less than it) likewise, by the 10th of the sixth book of Euclid. And set those two parts in two separate lines of those perpendiculars (or you may set them both in one line), making their beginnings to be at the base, and so their lengths to extend upward. Now set your hollow parallelepiped upright, perpendicularly, steady. Pour in water to the height of your shorter line. Pour that water into the hollow pyramid or cone. Mark the place of the rise. Settle your hollow parallelepiped again. Pour water into it, to the height of the second line, exactly.
Emptying the first.
Pour that water duly into the hollow pyramid or cone; mark now again where the water cuts the same line which you marked before. For there, as the first marked line is to the second, so shall the two radical sides be, one to the other, of any two cubes which in their solidity shall have the same proportion which was at the first assigned, whether it be rational or irrational.
Thus, in sundry ways you may furnish yourself with such strange and profitable matter, which long has been wished for. And though it be done naturally and mechanically, yet it has a good mathematical demonstration.
The demonstrations of this doubling of the cube, and of the rest.
Which is this: always you have two like pyramids, or two like cones, in the proportions assigned; and like pyramids or cones are in proportion, one to another, in the proportion of their homologous sides (or lines) tripled. Therefore, if to the first and second lines found in your hollow pyramid or cone you join a third and a fourth in continual proportion, that fourth line shall be to the first as the greater pyramid or cone is to the lesser, by the 33rd of the eleventh book of Euclid. If pyramid to pyramid, or cone to cone, be double,
I. D. Hereby help yourself to become a precise practitioner. And so consider how you are hindered not at all (sensibly) by the convexity of the water.
then shall line to line be also double, etc. But, as our first line is to the second, so is the radical side of our fundamental cube to the radical side of the cube to be made or to be doubled; and therefore to those two also a third and a fourth line in continual proportion joined will give the fourth line in that proportion to the first as our fourth pyramidal or conic line was to its first; but that was double or triple, etc., as the pyramids or cones were one to another (as we have proved); therefore this fourth shall be also double or triple to the first, as the pyramids or cones were one to another. But our made cube is described of the second in proportion, of the four proportional lines:
By the 33rd of the eleventh book of Euclid.
therefore, as the fourth line is to the first, so is that cube to the first cube; and we have proved the fourth line to be to the first as the pyramid or cone is to the pyramid or cone. Therefore the cube is to the cube as pyramid is to pyramid, or cone is to cone.
I. D. And your diligence in practice can so (in weight of water) perform it; therefore, now, you are able to give good reason for your whole doing.
But we suppose pyramid to pyramid, or cone to cone, to be double or triple, etc. Therefore cube is to cube double, or triple, etc. Which was to be demonstrated. And of the parallelepiped it is evident that the solid water parallelepipeds are one to another as their heights are, seeing they have one base. Therefore the pyramids or cones made of those water parallelepipeds are one to another as the lines are (one to another) between which our proportion was assigned. But the cubes made of lines, after the proportion of the pyramidal or conic homologous lines, are one to another as the pyramids or cones are one to another (as we before did prove); therefore the cubes made shall be one to another as the lines assigned are one to another. Which was to be demonstrated. Note.
Note this corollary.
This my demonstration is more general than only in a square pyramid or cone; consider well. Thus have I, both mathematically and mechanically, been very long in words; yet (I trust) nothing tedious to those who are well affected toward these things. And truly I am forced (avoiding prolixity) to omit sundry such things easy to be practiced, which to the mathematician would be a great treasure, and to the mechanician no small gain.
The great benefits following from these new inventions.
Now may you, between two lines given, find two middle proportionals in continual proportion, by the hollow parallelepiped and the hollow pyramid or cone. Now, any rectangular parallelepiped being given, three straight lines may be found proportional in any proportion assigned, of which shall be produced a parallelepiped equal to the parallelepiped given. Hereof I noted somewhat upon the 36th proposition of the 11th book of Euclid. Now, all those things which Vitruvius in his Architecture specified as able to be done by doubling of the cube, or by finding of two middle proportionals between two lines given, may easily be performed. Now, that problem which I noted to you at the end of my addition upon the 34th of the 11th book of Euclid is proved possible. Now may any regular body be transformed into another, etc. Now, any regular body: any sphere, yes, any mixed solid; and (what is more) irregular solids may be made (in any proportion assigned) like the body first given. Thus, of a mannequin (as the Dutch painters term it) in the same symmetry may a giant be made, and that with any gesture used by the mannequin, and conversely. Now may you, of any mold or model of a ship, make one of the same mold (in any assigned proportion) bigger or lesser.
Now may you, of any gun, or little piece of ordnance, make another, with the same symmetry (in all points) as great and as little as you will. Mark that, and think on it. Infinitely may you apply this, so long sought for and now so easily concluded, and withal so willingly and frankly communicated to such as faithfully deal with virtuous studies.
Such is the fruit of the mathematical sciences and arts.
Plain English translation
Modern machine-assisted Plain English edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.
Then set your cone or pyramid, with the vertex downward, perpendicular to the base. (It does not matter if it is the other way around; it hinders nothing.) So let them be kept as steady as possible. Now, if there is a cube that you want to double: make a small cube of copper, silver, lead, tin, wood, stone, or bone. Or else make a hollow cube, or a cubic box, of copper, silver, tin, or wood, etc. You may size these in respect of your pyramid or cone so that the pyramid or cone can hold 3 or 4 times their weight in water, at least, whatever material they are made of. Do not let your solid angle at the vertex be too sharp, so that the water can easily reach the very vertex of your hollow cone or pyramid. Put one of your solid cubes in a suitable balance and take its weight exactly in water. Pour that water (without loss) quietly into the hollow pyramid or cone. Mark in your lines what numbers the water cuts. Take the weight of the same cube again, in the same kind of water you had before:
John Dee. Consider well when you must put your waters together, and when you must empty your first water out of your pyramid or cone. Otherwise you will err.
Put that also into the pyramid or cone where you put the first. Mark again in what number or place of the lines the water cuts them. There are two ways to reach your aim: that is, either by numbers or by lines. By numbers: divide the side of your fundamental cube into as many equal parts as it can conveniently hold, with ease and precision in the division. For as the number of your first and lesser line (in your hollow pyramid or cone) is to the second and greater one, both counted from the vertex, so the number of the side of your fundamental cube is to the number belonging to the radical side of the cube that is double your fundamental cube. Multiply that number cube-wise and it will soon show whether it is double the cube number of your fundamental cube or not. By lines, thus: as your lesser and first line (in your hollow pyramid or cone) is to the second and greater, so let the radical side of your fundamental cube be to a fourth proportional line, by the 12th proposition of the sixth book of Euclid. That fourth line will be the cube root, or the radical side, of the cube that is double your fundamental cube, which is what we wanted.
God be thanked for this invention and the fruit that follows.
For this I may say with joy, "I have found it, I have found it, I have found it," thanking the holy and glorious Trinity, having greater cause for it than
Vitruvius, book 9, chapter 3.
Archimedes had for finding the fraud used in the king's crown of gold, as anyone can easily judge by the difference of the results that follow from the one and the other. Where I spoke before of a hollow cubic box: the same use is made of it, and without weighing. Thus: fill it with water precisely full, and pour that water into your pyramid or cone. And here note where the lines are cut in your pyramid or cone. Again, fill your box as you did before. Put that water also to the first. Mark the second cutting of your lines. Now, as you proceeded before, so you must proceed here.
Note.
And if the cube you are to double is ever so great, you have in this way the proportion (on a small scale) between your two little cubes. And then the side of that great cube (to be doubled), being the third, will have a fourth found that is proportional to it, by the 12th of the sixth of Euclid.
Note concerning the spherical surface of the water.
Note that all this while I have not forgotten my first proposition in statics, restated here: that the surface of the water is spherical. In this use your judgment: to the first line add a small hairbreadth more, and to the second add half a hairbreadth more to its length. For you will easily see that the difference can be no greater in any pyramid or cone you are to handle. And you can try it this way, for finding how much the water swells above the level.
"Square the semidiameter from the center of the earth to the surface of your first water. Then square half the subtendent of that watery surface (the subtendent must have its measure divided into equal parts, the same as the semidiameter from the earth to your watery surface). Subtract that square from the first. Of the remainder, take the square root. Subtract that root from your first semidiameter from the earth to your watery surface: what remains is the height of the water in the middle, above the level." You will find this to be something too small to notice. And even if it were greatly noticeable,
Note.
Yet, by help of my sixth theorem on the last proposition of Euclid's twelfth book, noted, you may reduce everything to a true level. But you must take further care against accidental causes of the water swelling: for example, if you dampen your hollow pyramid or cone beforehand a little with a moist sponge, you will prevent an accidental cause of swelling, etc. Experience will teach you in abundance, with great ease, pleasure, and benefit.
Thus you can double the cube mechanically, treble it, and so on, in any proportion.
Note this abridgment of doubling the cube, etc.
Now I will shorten your effort, cost, and care in this. Without any preparation of your fundamental cubes, you may work this conclusion in the same way. For that was rather a kind of experimental demonstration than the shortest way, and it all depended on one mathematical demonstration: "Take water (as much as will conveniently serve your purpose, as I warned earlier about the size of your fundamental cubes). Weigh it precisely. Put that water into your pyramid or cone. Then take again the same kind of water, the same weight you had before, and put that also into your pyramid or cone. For each time, your marking of the lines, how the water cuts them, will give you the proportion between the radical sides of any two cubes, one of which is double the other, working as I taught you before:
Note.
Except that, for your fundamental cube's radical side, you may take any straight line you like here."
Yet a step further with our drop of natural truth:
To give cubes one to another in any proportion, rational or irrational.
You may now give cubes one to another in any proportion given, rational or irrational, in this way. Make a hollow parallelipipedon of copper or tin, with one base missing, or open, as in our cubic box. From the bottom of that parallelipipedon raise up many perpendiculars on each of its four sides. Now if any proportion is assigned to you in straight lines, cut one of your perpendiculars (or a line equal to it, or shorter than it) likewise, by the 10th proposition of the sixth book of Euclid. Set those two parts on two separate lines of those perpendiculars (or you may set them both on one line), making their beginnings at the base, so that their lengths extend upward.
Now set your hollow parallelipipedon upright, perpendicularly, and steady. Pour in water neatly to the height of your shorter line. Pour that water into the hollow pyramid or cone. Mark the place it rises to. Set your hollow parallelipipedon again. Pour water into it exactly to the height of the second line.
Empty out the first.
Pour that water properly into the hollow pyramid or cone. Mark now again where the water cuts the same line you marked before. For there, as the first marked line is to the second, so the two radical sides will be to each other, of any two cubes that in their solidity have the same proportion that was first assigned, whether rational or irrational.
Thus, in various ways, you may equip yourself with such strange and profitable matter, which has long been wished for. And though it is done naturally and mechanically, it still has a good mathematical demonstration.
The demonstrations of this doubling of the cube, and of the rest.
Which is this: you always have two similar pyramids, or two similar cones, in the proportions assigned; and similar pyramids or cones stand to one another in the proportion of their homologous sides (or lines) tripled. Therefore, if to the first and second lines found in your hollow pyramid or cone you join a third and a fourth in continued proportion, that fourth line will be to the first as the greater pyramid or cone is to the lesser, by the 33rd of the eleventh of Euclid. If pyramid to pyramid, or cone to cone, is double,
John Dee. By this, help yourself become a precise practitioner. And so consider how the convexity of the water does not hinder you at all, in any way you can notice.
Then line to line will also be double, etc. But as our first line is to the second, so is the radical side of our fundamental cube to the radical side of the cube to be made or to be doubled. And therefore, joining a third and a fourth line in continued proportion to those two as well will give a fourth line in that proportion to the first as our fourth pyramidal or conic line was to its first. But that was double, or treble, etc., as the pyramids or cones were to one another (as we have proved). Therefore this fourth will also be double or treble to the first, as the pyramids or cones were to one another. But the cube we make is generated from the second of the four proportional lines:
By the 33rd of the eleventh book of Euclid.
Therefore, as the fourth line is to the first, so that cube is to the first cube. And we have proved the fourth line to be to the first as the pyramid or cone is to the pyramid or cone. Therefore the cube is to the cube as pyramid is to pyramid, or cone is to cone.
John Dee. And your diligence in practice can perform it in this way (by the weight of water). Therefore you are now able to give a good reason for all you do.
But we suppose pyramid to pyramid, or cone to cone, to be double or treble, etc. Therefore cube is to cube double, or treble, etc. Which was to be demonstrated. And for the parallelipipedon it is evident that the solid parallelipipedons of water stand to one another as their heights do, since they have one base. Therefore the pyramids or cones made from those water parallelipipedons stand to one another as the lines stand to one another between which our proportion was assigned. But the cubes made from lines, in the proportion of the pyramidal or conic homologous lines, stand to one another as the pyramids or cones stand to one another (as we proved before). Therefore the cubes made will stand to one another as the assigned lines stand to one another. Which was to be demonstrated. Note.
Note this corollary.
This demonstration of mine is more general than only in a square pyramid or cone: consider it well. Thus I have been very long-winded, both mathematically and mechanically; yet, I trust, nothing tedious to those who are well disposed toward these things. And in truth I am forced (to avoid prolixity) to omit several such things, easy to practice: they would be a great treasure to the mathematician, and no small gain to the mechanic.
The great benefits that follow from these new inventions.
Now, between two given lines, you may find two mean proportionals in continued proportion, by the hollow parallelipipedon and the hollow pyramid or cone. Now, any rectangular parallelipipedon being given, three straight lines may be found, proportional in any proportion assigned, out of which a parallelipipedon equal to the given parallelipipedon will be produced. Of this I noted something on the 36th proposition of the 11th book of Euclid. Now, all those things that Vitruvius in his Architecture specifies as able to be done by doubling the cube, or by finding two mean proportional lines between two given lines, can easily be performed. Now, that problem which I noted to you at the end of my addition on the 34th of the 11th book of Euclid is proved possible. Now, any regular body may be transformed into another, etc. Now, any regular body, any sphere, yes, any mixed solid, and what is more, irregular solids, may be made (in any proportion assigned) like the body first given. Thus, from a little figure (as the Dutch painters call it) in the same symmetry, a giant can be made, and that with any gesture used by the little figure, and the other way around. Now, from any mold or model of a ship you may make one, of the same mold (in any assigned proportion), bigger or smaller.
Now, you may, of any gun, or small piece of ordnance, make another with the same symmetry (in all points) as big and as small as you want. Mark that and think on it. You may apply this infinitely, something so long sought for and now so easily worked out, and at the same time so willingly and freely shared with those who deal faithfully with virtuous studies.
Such is the fruit of the mathematical sciences and arts.
About this section
Statics, 2 is part of The Mathematicall Praeface to the Elements of Geometrie by John Dee. This page sets the 16th-century English original beside 2 direct-source translations, so any line can be checked against the source. It is part 18 of 34 in the work, which is kept in the European tradition.
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