Statics, 1

In order, now follows, of Statike: to say something of what we mean by that name, and what benefit depends on such an art. Statike is a mathematical art, which demonstrates the causes of heaviness and lightness in all things, and of the motions and properties belonging to heaviness and lightness. And since by the bilanx, or balance (as the chief sensible instrument) experience of these demonstrations may be had, we call this art Statike: that is, the experiments of the balance. Oh, that men knew what profit, in every way, might grow from this art to the able examiner and diligent practitioner. "You alone know all things precisely (O God), who have made weight and balance your judgment; who have created all things in number, weight, and measure; who have weighed the mountains and hills in a balance, and poised in your hand both heaven and earth. We therefore, warned by the sacred word to consider your creatures, and by that consideration to win a glimpse, as it were, or a shadow of perception, that your wisdom, might, and goodness are infinite and unspeakable, declared in your creatures; and being further advised by your merciful goodness that three principal means were used in the creation of all your creatures, namely number, weight, and measure; and since the two arts of number and measure (ancient, famous, and to human uses most necessary) are already sufficiently known and available — this third key we beseech of you (through your accustomed goodness) that it may come to the needed and sufficient knowledge of such of your servants as would gladly find in your workmanship your true occasions (purposely designed) by which we should glorify your name, and show forth (to those weak in faith) your wondrous wisdom and goodness. Amen."

Do not marvel at this outburst, godly friend, you gentle and zealous student. Another day, perhaps, you will perceive what moved me to it. Here, for now, I will give you some foundation, and with it some show of the benefits arising from this art. And because this art is rare, my words and practices might be too dark, unless some light were held before the matter; and that will best be done by giving you, out of Archimedes' demonstrations, a few principal conclusions, as follows.

1 The surface of every liquid, standing by itself and at rest, is spherical; its center is the same as the center of the Earth.

2 If solid magnitudes of the same size or quantity as any liquid, and having the same weight, are lowered into that liquid, they will settle downward so that no part of them lies above the surface of the liquid; and yet for all that they will not sink utterly down, or drown.

3 If any solid magnitude lighter than a liquid is lowered into that same liquid, it will settle down into it so far that the quantity of that liquid equal to the part of the solid magnitude which has settled down into it is, in weight, equal to the weight of the whole solid magnitude.

4 Any solid magnitude lighter than a liquid, forced down into that same liquid, will move upward with a power equal to the amount by which the liquid of equal quantity to the whole magnitude is heavier than that same magnitude.

5 Any solid magnitude heavier than a liquid, lowered into that same liquid, will sink down utterly; and in that liquid it will be lighter by exactly the weight or heaviness of the liquid having size or quantity equal to the solid magnitude.

6 I. D. The cutting of a sphere according to any proportion assigned may by this proposition be done mechanically, by tempering a liquid to a certain weight in respect of the weight of the sphere swimming in it.

If any solid magnitude lighter than a liquid is lowered into that same liquid, the weight of that magnitude is to the weight of the liquid (which is equal in quantity to the whole magnitude) in the same proportion that the part of the magnitude settled down is to the whole magnitude.

By these truths, great errors may be corrected in the opinion of the natural motion of things, light and heavy. Which errors are allowed in natural philosophy by almost all men, from too much trust in authority and false assumptions; as, that of any two bodies, the heavier moves downward faster than the lighter.

A common error, noted.

This error was not first noted by me, but by one John Baptist de Benedictis. The chief of his propositions is this, which seems a paradox:

A paradox.

If there are two bodies of one form and of one kind, equal in quantity or unequal,

they will move through equal spaces in equal times: provided both their motions are in air, or both in water, or in any one medium.

Hereupon, in the practice of gunnery, certain good discourses may otherwise receive great amendment and furtherance.

The wonderful use of these propositions.

In the intended purpose also, allowing something to the imperfection of nature, which does not answer to the precision of demonstration. Moreover, by the foresaid propositions (wisely used) air, water, earth, and fire may be nearly known: how light or heavy they are (naturally) in their assigned parts, or in the whole. And then, turning your practice to elemental things, you may deal for the proportion of the elements in compounded things. Then, for the proportions of the humors in man: their weights, and the weight of his bones and flesh, etc. Then, by weight, to consider the force of a man in any manner, whole or in part. Then you may have pleasant consideration of the water-draft of ships, differing in the sea and in fresh water; and of weighing up anything sunk in sea or fresh water, etc. And (to lift your head aloft) by weight you may, as precisely as by any other instrument, measure the diameters of sun and moon, etc.

Friend, please weigh these things with the just balance of reason, and you will find wonders upon wonders; and esteem one drop of truth (yes, in natural philosophy) more worth than whole libraries of opinions undemonstrated, or not answering to nature's law and your experience. Leaving these things thus, I will give you two or three light practices to great purpose, and so finish my statical annotation. In mathematical matters, by the mechanician's aid, we will here see the benefit of weight.

The statical practice, to know the proportion between the cube and the sphere.

Make a cube of one uniform material throughout, of like heavy stuff; and of the same stuff make a sphere or globe, precisely, of a diameter equal to the radical side of the cube. Your stuff may be wood, copper, tin, lead, silver, etc. (being, as I said, of like nature, condition, and like weight throughout). And you may, by assay balance, have prepared a great number of the smallest weights, which by that balance can be discerned or tried; and so have proceeded to make a perfect pile, company and number of weights, up to the weight of six, eight, or twelve pounds: all most diligently tried. And of every one, the content known, in your least weight that is weighable.

[Those who cannot have weights of precision may, with uniform and well-sifted sand, make themselves a number of weights somewhat near precision, by always halving the sand; they will at length come to a least common weight. There I leave the further matter to the discretion of those whom necessity shall press.]

The Venetian reckoning of weight may seem precise enough: by eight successive descents, halving from a grain.

I. D. For so you have 256 parts of a grain.

Your cube, sphere, suitable balance, and convenient weights being ready, fall to work. First, weigh your cube. Note the number of the weights. Then weigh your sphere, and likewise note the number of the weights. If you now find the weight of your cube to be to the weight of the sphere as 21 is to 11, then you see how the mechanician and experimenter, without geometry and demonstration, are (as nearly in effect) taught the proportion of the cube to the sphere, as I have demonstrated it at the end of the twelfth book of Euclid. Often try with the same cube and sphere. Then change your sphere and cube to another material, or to another size, until you have made a perfect universal experience of it. It is possible that you may win to nearer terms in the proportion.

When you have found this one certain drop of natural truth, proceed on to infer, and duly to make trial of, the matter depending. For, since it is well demonstrated that a cylinder whose height and the diameter of its base are equal to the diameter of the sphere is sesquialter to that same sphere (that is, as 3 is to 2), to the number of the weight of the sphere add half as much again, and so you have the number of the weight of that cylinder. Which is also contained in our former cube: so that the base of that cylinder is a circle described in the square which is the base of our cube. But the cube and the cylinder, being both of one height, have their bases in the same proportion in which they are, one to another, in their bulk or solidity. But, beforehand, we have two numbers expressing their bulk, solidities, and quantities by weight; therefore,

The proportion of the square to the circle inscribed.

we have the proportion of the square to the circle inscribed in the same square. And so we have fallen into the sensible and experimental knowledge of Archimedes' great secret, sought and found by him through great travail of mind. Therefore, to any circle given you can give a square equal:

The squaring of the circle, mechanically.

as I have taught in my annotation upon the first proposition of the twelfth book; and likewise to any square given you may give a circle equal:

To any square given, to give a circle equal.

If you describe a circle which shall be in that proportion to your inscribed circle as the square is to the same circle, this you may do by my annotations upon the second proposition of the twelfth book of Euclid, in my third problem there. Your diligence may come to a proportion of the square to the circle inscribed nearer the truth than is the proportion of 14 to 11. And consider that you may begin at the circle and square, and so come to conclude of the sphere and the cube what their proportion is, as now you came from the sphere to the circle. For, of silver or gold or brass sheets or plates (drawn through one hole, as is the manner) if you make a square figure and weigh it; and then, describing thereon the inscribed circle, cut off and file away precisely (to the circle) the surplus of the square, you shall then, weighing your circle, see whether the weight of the square be to your circle as 14 to 11. As I have noted at the beginning of Euclid's twelfth book, etc. After this, resort to my last proposition upon the last of the twelfth; and there help yourself to the end. And here note this by the way.

Note: squaring the circle without knowledge of the proportion between circumference and diameter.

That we may square the circle without having knowledge of the proportion of the circumference to the diameter, as you have here perceived. And otherwise also, I can demonstrate it. So that many have encumbered themselves superfluously, by laboring in that point first which was not of necessity first, and also very intricate. And easily you may (and that variously) come to the knowledge of the circumference, the circle's quantity being first known. Which thing I leave to your consideration, making haste to dispatch another masterly problem, and to bring it nearer to your knowledge and readier dealing with than the world (before this day) had it for you, that I can tell of. And that is, a mechanical doubling of the cube, etc. Which may thus be done:

To double the cube readily, by mechanical art, depending upon mathematical demonstration.

Make of copper plates or tin plates a four-square upright pyramid, or a cone, perfectly fashioned hollow within. In this let great diligence be used to approach (as near as may be) to the mathematical perfection of those figures. At their bases let them be all open; everywhere else most close and just. From the vertex to the circumference of the base of the cone, and to the sides of the base of the pyramid:

I. D. The 4 sides of this pyramid must be 4 isosceles triangles alike and equal.

Let 4 straight lines be drawn in the inside of the cone and pyramid, making at their fall upon the perimeters of the bases equal angles on both sides of themselves with those perimeters. These 4 lines (in the pyramid, and as many in the cone) divide: one into 12 equal parts, another into 24, another into 60, and another into 100 (counting up from the vertex). Or use other numbers of division, as experience shall teach you.

I. D. In all workings with this pyramid or cone, let their situations be in all points and conditions alike, or all the same, while you are about one work. Else you will err.

In order, now follows, of Statics: some words on what we mean by that name, and what benefit depends on such an art. Statics is a mathematical art. It shows the causes of heaviness and lightness in all things, and the motions and properties that belong to heaviness and lightness. And because the balance, or scales, is the chief instrument of the senses, and experience of these demonstrations can be had through it, we call this art statics: that is, experiments with the balance. If only people knew what profit, in every way, this art could bring to a capable examiner and a diligent practitioner.

"You alone know all things exactly (O God), who have made weight and balance your judgment, who have created all things in number, weight, and measure, and have weighed the mountains and hills in a balance, who have weighed both heaven and earth in your hand. We are therefore warned by your sacred word to consider your creatures, and by that consideration to gain a glimpse, as it were, or a shadow of understanding, that your wisdom, power, and goodness are infinite and unspeakable, declared in your creatures. And being further warned by your merciful goodness that three principal means were used in the creation of all your creatures, namely number, weight, and measure; and since the two arts of number and measure, ancient, famous, and most necessary for human uses, are already sufficiently known and available; we ask you, through your accustomed goodness, that this third key may come to the full and sufficient knowledge of such servants of yours as would gladly find in your workmanship your true designs (made purposely for that use), by which we should glorify your name and show your wondrous wisdom and goodness to those who are weak in faith. Amen."

Do not be surprised at this outburst, dear friend, you gentle and zealous student. Another day you may see what moved me to it. For now I will give you some basis, and also some show, of the benefits that come from this art. And because this art is rare, my words and practices might be too obscure, unless some light were held before the matter. That will best be done by giving you a few main conclusions out of Archimedes' demonstrations, as follows.

1 The surface of any liquid, standing by itself and at rest, is spherical. Its center is the same as the center of the Earth.

2 If solid bodies of the same size, or volume, as any liquid, and with the same weight, are lowered into that liquid, they will settle downward so that no part of them lies above the surface of the liquid, and yet they will not sink all the way down, or drown.

3 If any solid body lighter than a liquid is lowered into that liquid, it will settle down into it so far that the amount of liquid equal in volume to the part of the body settled into it weighs as much as the whole solid body.

4 Any solid body lighter than a liquid, pushed down into that liquid, will move upward with a force equal to how much heavier the liquid of the same volume as the whole body is than that body itself.

5 Any solid body heavier than a liquid, lowered into that liquid, will sink all the way down. In that liquid it will be lighter by exactly the weight of the liquid that has the same size, or volume, as the solid body.

6 (John Dee. A sphere can be cut to any assigned proportion by this proposition, mechanically, by mixing a liquid to a certain weight compared with the weight of the sphere floating in it.)

If any solid body lighter than a liquid is lowered into that liquid, the weight of that body is to the weight of the liquid (which is equal in volume to the whole body) in the same proportion as the part of the body that has settled down is to the whole body.

From these truths, great errors can be set right in people's opinions about the natural motion of light and heavy things. Almost everyone in natural philosophy allows these errors, trusting too much to authority and to false assumptions. For example, that of two bodies, the heavier moves downward faster than the lighter.

(A common error, noted.)

This error was not first pointed out by me, but by a man named John Baptist de Benedictis. His chief proposition is this, and it seems like a paradox:

If there are two bodies of one shape and one kind, equal in volume or unequal,

(A paradox.)

they move through equal spaces in equal times, whether both are moving in air, or both in water, or in any one medium.

Hereupon, in the art of gunnery,

certain good discussions, otherwise, may receive great improvement and advancement.

(The wonderful uses of these propositions.)

Also, in the intended purpose, allow something for nature's imperfection, which does not match the precision of a demonstration. Moreover, by the propositions above, wisely used, air, water, earth, and fire can be measured closely: how light or heavy they are naturally, in their assigned parts, or as a whole. Then, turning your practice to elemental things, you can work out the proportions of the elements in compound things. Then the proportions of the humors in a person: their weights, and the weight of their bones and flesh, and so on. Then, by weight, you can consider a person's strength in any way, whole or in part. Then you can consider, with pleasure, how differently ships draw water, in the sea and in fresh water, and how to weigh up anything sunk in the sea or in fresh water, and so on. And, to lift your head up high, by weight you can measure the diameters of the Sun and Moon as precisely as by any other instrument, and so on. Friend, please weigh these things with the fair balance of reason, and you will find wonders upon wonders. You will value one drop of truth, even in natural philosophy, more than whole libraries of opinions that are undemonstrated, or that do not answer to nature's law and your experience. Leaving these things there, I will give you two or three simple practices that serve a great purpose, and so finish my notes on statics. In mathematical matters, with the help of the mechanic, we will see here the value of weight.

(A practical exercise in statics, to find the proportion between the cube and the sphere.)

Make a cube of one uniform, evenly heavy material. From that same material, make a sphere, or globe, with a diameter exactly equal to the edge of the cube.

Your material may be wood, copper, tin, lead, silver, and so on (being, as I said, of the same nature and condition, and of even weight throughout).

And with an assay balance you may have prepared a great number of the smallest weights, weights that balance can tell apart or test. In that way you can go on to make up a complete set, a company and number of weights, up to six, eight, or twelve pounds in weight, every one of them most carefully tested. And know the amount of each one in your smallest weighable weight.

[Those who cannot get weights of precision can make a number of weights fairly close to precision from uniform, well-sifted sand, by always halving the sand; in time they will come to a smallest common weight. On that I leave the further matter to the judgment of those whom need compels.]

The Venetian way of reckoning weight may seem precise enough: by eight steps in a descending series, halving from a single grain.

(John Dee. For so you have 256 parts of a grain.)

Your cube, sphere, suitable balance, and suitable weights being ready, get to work. First, weigh your cube. Note the number of weights. Then weigh your sphere, and note the number of weights again. If you now find the weight of the cube to be to the weight of the sphere as 21 is to 11, then you see how the mechanic and the experimenter, without geometry and demonstration, are taught in effect the proportion of the cube to the sphere, as I have demonstrated at the end of the twelfth book of Euclid. Try often with the same cube and sphere. Then change the sphere and the cube to another material, or another size, until you have made a complete general test of it. You may well come closer to the true terms of the proportion.

When you have found this one certain drop of natural truth, go on to draw conclusions, and properly test the matters that depend on it. For it is well demonstrated that a cylinder whose height and the diameter of its base equal the diameter of the sphere is sesquialter to that sphere, that is, as 3 to 2. To the weight of the sphere, add half as much again, and you have the weight of that cylinder. That cylinder is also contained in our earlier cube, so that the base of that cylinder is a circle drawn inside the square that is the base of our cube. But the cube and the cylinder, being the same height, have their bases in the same proportion as they bear to each other in bulk, or solidity. But before that, we have two numbers expressing their bulk, solidity, and quantity by weight; therefore,

(The proportion of the square to the inscribed circle.)

we have the proportion of the square to the circle inscribed in that same square. And so we come to a sensible and practical knowledge of Archimedes' great secret, sought and found by him through great labor of mind. Therefore, to any given circle, you can give an equal square:

(Squaring the circle mechanically.)

as I have taught in my annotation on the first proposition of the twelfth book. Likewise, to any given square, you can give an equal circle:

(To any given square, to give an equal circle.)

If you draw a circle that stands in the same proportion to your inscribed circle as the square does to that same circle, you can do this by my annotations on the second proposition of the twelfth book of Euclid, in my third problem there. Your diligence may bring you to a proportion of the square to the inscribed circle nearer the truth than 14 to 11. And notice that you can begin with the circle and the square, and so work out what proportion the sphere and the cube bear, just as you have now come from the sphere to the circle. For, if you make a square figure from silver, or gold, or brass sheets or plates (drawn through one hole, as is the custom), and weigh it; then describe the inscribed circle on it, and cut off and file away, precisely down to the circle, the surplus of the square; then, weighing your circle, you will see whether the weight of the square is to your circle as 14 to 11. As I have noted at the beginning of Euclid's twelfth book, and so on. After that, go to my last proposition, on the last of the twelfth, and help yourself through to the end. And note this in passing.

(Note: squaring the circle without knowing the proportion between the circumference and the diameter.)

That we can square the circle without knowing the proportion of the circumference to the diameter, as you have just seen. And I can demonstrate it in other ways too. So many people have burdened themselves needlessly by working first on that point, which did not need to come first, and is also very intricate. And you can easily, in several different ways, come to know the circumference, once the circle's size is known first. I leave that to your consideration, and hurry on to dispatch another masterly problem, and to bring it nearer to your knowledge, and easier for you to handle, than the world has had it before this day, so far as I can tell. And that is, a mechanical doubling of the cube, and so on. It can be done like this:

(To double the cube readily, by mechanical art, depending on mathematical demonstration.)

Make a square upright pyramid, or a cone, of copper plates or tin plates, perfectly shaped hollow on the inside. In this, take great care to approach, as near as possible, the mathematical perfection of those figures. Let their bases be open all around; everywhere else, as closed and exact as possible. From the vertex to the circumference of the base of the cone, and to the sides of the base of the pyramid:

(John Dee. The 4 sides of this pyramid must be 4 matching equal isosceles triangles.)

let 4 straight lines be drawn on the inside of the cone and the pyramid, so that where they fall on the perimeters of the bases they make equal angles on both sides of themselves with those perimeters. Divide these 4 lines (in the pyramid, and as many in the cone): one into 12 equal parts, another into 24, another into 60, and another into 100, counting up from the vertex. Or use other numbers of divisions, as experience teaches you.

(John Dee. In all work with this pyramid or cone, keep their positions alike in every point and condition while you are doing one job. Otherwise you will go wrong.)