The discourse made before the Royal Society the 26. of November, 1674, concerning the use of duplicate proportion in sundry important particulars together with a new hypothesis of springing or elastique motions / by Sir William Petty, Kt. ...

A DISCOURSE TO THE Royal Society.

FOrasmuch as this Society has been censured (though without much cause) for spending too much time in matters not directly tend∣ing to profit and palpable Advantages (as the Weigh∣ing of Air and the like)

I have therefore, to streigh∣ten this crooked stick, bent it and my present Dis∣course the quite contrary way, viz. to the Sails and Shapes of Ships; to Carpentry and Carriages; to Mills, Mill-dams, Bulwarks; to the Labour of Horses, and to several other particu∣lars: The which are not only gross enough of them∣selves, but are also as gros∣ly handled in this Exercise, to prevent the further im∣putation of needless Nici∣ty,

and to leave room for your own further thoughts upon the same.

And forasmuch as We have been also complained of for producing nothing New, I have together with my Instances and Applica∣tions, above and hereafter mentioned, presented you as an Appendix, to what is said of Springs and other Elastique bodies, with a new Theory (as I think) of Elasticity it self, and that mechanically explicated in

order to make a breach on this hard Rock in Philoso∣phy, and to chip off a little of that Block which has long lain thwart Us, in the way of Our Enquiries. Up∣on the whole matter I have followed the Example of Elderly Divines, who find∣ing their Flocks not to mend their lives by per∣plexed Discourses about Predestination, Transub∣stantiation, &c. betake themselves at last to preach Faith and Good Wooks,

Neighbourly Love and Cha∣rity, or Doing as we would be done unto, and the like. For I have in this Exercise declined all Speculations not tending to practice, and ventured at few new Hypotheses, but that of Elasticity; rather calling upon you to review your own former Observations, and to apply your Mathe∣maticks to Matter, so as both may be improved to the profitable purposes hereafter mentioned.

Wherefore the Title and Scope of this Exercise is, Several Instances, wherein the consideration of Duplicate & Subduplicate propoortion, or wherein the consideration of Sides and their Squares is of use in humane affairs. And the Instances which I have pitcht upon for this day are these following, viz.

• 1. In the Drawing or Driving powers, which force Ships or other bodies through the water, with reference to the respective

• ... Velocities caused thereby.
• 2. In the shapes or sharp∣ness of bodies, cutting or di∣viding the water, through which they are driven or drawn, and in the diffe∣rent Velocities arising from thence, where the Bodies and Forces are equal.
• 3. In the Strength of Timbers or other homoge∣neous materials applied to Buildings, to Carts, or any other Machinaments in∣tended for strength: And how by a Model to judg

• the sufficiency of such En∣gine as is represented by it.
• 4. In the effect of Oars upon equal and like Ves∣sels, according to their Numbers, Length, Blades, and Motions with or against the stream of smooth or un∣even waters.
• 5. In the Motion or Tra∣velling of Horses, on their several Paces, and with dif∣ferent Burthens on them.
• 6. In the Strength and Velocity of Mills and their Wheels.
• ...

• 7. In the Effects of Gun∣powder.
• 8. In the Distance at which Sounds may be heard.
• 9. In the Distances at which Odoriferous▪ matters may be smelt.
• 10. In the Distance at which the Objects of Sight may be seen.
• 11. In the time of the Returns made by vibrating Pendules.
• 12. In the Lives of men and their Duration.
• ...

• 13. In Musical & Sound∣ing Bodies, such as Strings and Bells.
• 14. In the Effects and Motions of Fire, and burn∣ing Spirits.
• 15. In the Rising and Falling of Bodies, but espe∣cially of Water in Pumps, Overshot Mills, Leaks in Ships, the Heights of Rivers at their head above their fall into the Sea.
• 16. In Bellows,
• 17. In the Prices of se∣veral Commodities, as Masts,

• ... Diamonds, large Timber, Amber, Loadstones, &c.
• 18. In Mill-dams, Sea∣banks, and in the Bulwarks or Walls of Fortresses.
• 19. In the Compression of Wooll, and other Elastick Bodies, and of the Air with∣in diving Vessels, as also in the Effects of Skrew-presses upon several Materials.

Having thus enumera∣ted my several Instances, wherein Duplicate, and Subduplicate proportion is of great importance; I might

now fall down-right upon the Application of those proportions to each of the respective matters above mentioned. But because Custome hath made it al∣most necessary to make a Preface to every Discourse, my Preface to this one Le∣cture shall be such, as may serve me for many more; that is, an Explication of what I my self (at least) un∣derstand by Matter, Body, Figure, Place, Motion, Quan∣tity, Quality, Habit, Time,

Proportion, Weight, Swift∣ness, Force, and Elasticity; which I shall do without imposing or scarce recom∣mending the same to any other. For I would be glad, when any man speaks to me in matters of impor∣tance, by words which he uses often, that he would first give me a Dictionary of such words, to contein what he himself meaneth by each of them. Wherefore I shall, as a Preface, prefix this Dictionary, wherein I

dare not define Matter by Ens, or Substance, because I think most men conceive Matter better than they do either of these two words, Ens, or Substance. Nor do I define the words, Think, Consider, or Conceive, by the words, Soul, Spirit, Act, or the like, for the same rea∣son. But presuming you all understand, conceive, imagine, or fancy the words Matter and Thought, as well as any other I can use, I venture to say as

followeth, and first, That

• 1. Place is the Image or Fancy of Matter, or Matter considered.
• 2. Quantity, the Fancy of Place.
• 3. Ratio, several Quan∣tities considered together.
• 4. Proportion, several like Rationes.
• 5. Situation, several Pla∣ces considered together.
• 6. Figure is Quantity and Situation considered together.
• ...

• 7. Body is Matter and Figure considered toge∣ther.
• 8. Motion is change of Place.
• 9. Time, the Image of Motion.
• 10. Quality, several Mo∣tions considered together.
• 11. Habit, the same Mo∣tions repeated.
• 12. Likeness, several Fi∣gures, or Qualities, and Proportions considered to∣gether.
• 13. Swiftness, Time and

• Place or Space considered together.
• 14. Force is Body and Swiftness considered toge∣ther.
• 15. Right is the Image of Possession, and is to it as Place to Body.
• 16. Elasticity I shall speak of hereafter.

In the next place, I sup∣pose all the First Matter of the World to be Atoms; that is, Matter Immutable in Magnitude and Figure. I suppose Corpuscles to be

as many Atoms joyned to∣gether, as make up a visible or sensible Object, and that all Iuncture of Atomes is made by their Innate moti∣ons. Moreover I suppose, That every Atom is like the Earths Globe or Mag∣net, wherein are three Points considerable, viz. two in the surface, called Poles, and one within the sub∣stance, called Center, or ra∣ther Byas, because in A∣toms we consider neither Magnitude nor Gravity.

These Atoms also may have each of them such Motions as Copernicus attributes to the Earth, or more. Lastly, Motion to or from a Point makes a streight Line, and, about it, a Circle. But from the Center to several Points in the Circle, is Angle. We further say, that the moti∣ons of Corpuscles are com∣pounded of the abovemen∣tioned motions of Atoms; and the motions of bigger and Tangible Bodies (viz. their qualities) are decom∣pounded

out of the Moti∣ons, Situation, Figure, and Magnitude of Corpuscles; and that out of, and by, the premisses all Phaenome∣na in nature must be sol∣ved. And this is all the Preface I shall trouble you with, being (as was said) the Dictionary wherein to find what I mean by every material word I intend to use in this ensuing Exer∣cise, which we thus begin, viz.

The First Instance, Wherein Duplicate, and Sub∣duplicate Ratio or Pro∣portion is considerable, Is

IN the Velocities of two equal and like Ships; which Velocities, I say, are the square Roots of the Powers which either drive or draw them; as, for ex∣ample, Such two Ships having sails near double

to each other, or as 49 to 25, the Velocity will be as 5, the square Root of 25 unto 7, the like Root of 49. Again, if the sails be near triple, or as 49 to 16, there the Velocity shall be as 7 (the Root of 49) to 4 (the Root of 16.) So as a quadruple Sail is re∣quisite to double swiftness, and noncuple to treble; that is, The sails must be in du∣plicate proportion to the swiftness of the Ship; or this, in subduplicate to that.

Again, let there be two Ships of Equal sails, but of unlike or unequal sharp∣ness, suppose the head of one extremely obtuse or quite flat, and the head of the other to be an Isosceles Triangle added thereunto; I say, the swiftness of these Bodies shall be as the Roots of the Perpendicular of that Triangle to the Root of half the Base, or half breadth of the same. Se∣condly, Or if the same Tri∣angular head be cyphered

away into an Angle from bottom to top; then, as the Root of the same Per∣pendicular is to the Root of the Depth or Thick∣ness, so are the Velocities. Thirdly, If the said head be cyphered both wayes together, then the Pro∣portion of Velocities shall be as half of one of the above mentioned Propor∣tions added to the other whole Proportion: Ex. gr. Suppose the Perpendicu∣lar of the triangle-head

be 36, the half breadth 9, and the whole depth be 4; then the one Proportion shall be as 6, the Root of 36, to 3, the Root of 9: The half of which Proportion is as 6 to 6; and the other Proportion is as 6, the Root of 36, to 2, the Root of 4. Now add the Pro∣portions of 6 to 6, to that of 6 to 2, the sum will be, as 36 to 12, or as 3 to 1.

Fifthly, Suppose two Pa∣ralellepipedons of unequal heads or resistances, Ex. gr.

as 8 to 5, or 64 to 40: And suppose the Sail on the big∣ger, to that on the lesser, to be as 9 to 4, or 72 to 32; then the Velocity of the bigger shall be to the Velo∣city of the lesser, as the Root of 45 is to the Root of 32. For if the Resistan∣ces be as 64 to 40; then, if the sail of the bigger to that of the less were pro∣portionable to the Resi∣stances, the sail of the less should be 45, whereas we suppose it but 32. Where∣fore

the Velocity shall be as the Root of 45, which is almost 7, to the Root of 32, which is about 5½, that is, as about 14 to 11.

Memorandum, That wet∣ting of Sails (by lessening the intersperst apertures between the threds of the Sail-cloth) doth make the Sail, as it were, bigger; which biggerness may be known and measured by the increase of the Ships velocity upon such wet∣ting. For, if the Ship should

move one tenth part quick∣er after wetting than be∣fore, we may conclude the Sails are swollen to the equivalent of about ⅕ part bigger; for 100 (whose Root is 10) exceeds 81, whose Root is 9, by about ⅕ of 100.

By these ways the diffe∣rent Velocities, arising from the different Trim of the same Ship, may be al∣so computed, the best Trim being that which makes least resistance, caeteris pa∣ribus.

Now, having said thus much of the Effects of Sharpness and Sails, (the two principal causes of Velocity in shipping, and unto which all others may be referred;) I shall add, That the want of these two Advantages are the chief cause, why short, bluff, un∣dermasted Vessels sail chea∣per than others.

For suppose two Ships▪ of equal burthens, but of unlike dimensions, the main Beam of the one be∣ing

scarse ⅓ of the Keels length, and in the other, a full ⅕^th; I say first, that the Hull of the latter shall cost ⅓ part more than that of the former, and the advantage as to sailing shall be scarce ⅙ part. Again, suppose, the sharper could carry ½ as much sail more as the bluf∣fer, whereof the advantage in sailing would be ⅙ part more, in all ⅓. Now, where the Sails are as 2 to 3, the Masts and Yards must be as 4 to 9 in substance;

and in value much more: And where the Masts and Yards are as 4 to 9 in weight and bulk, the Cord∣age and Rigging must be answerable: And where the Masts, Yards, Sails, and Rigging are great, the Wind-taught of the Ship will correspond, and will require proportionable Ca∣bles; and the weight of the Anchor must follow the size of the Cable, and the number of hands must be proportionable to all the

premisses: So as the one Ship will cost at least double as much as the o∣ther, and will sail at double charge of Wages and Vi∣ctuals, Ware and Tare, &c. Now if no trading Ship be (one time with another) above 1/10 of her whole reign under sail, or 6 days in 60, suppose the sharper and larger-sail'd Ship sail in 4 dayes what the other performs in 6; the diffe∣rence will be but 2 dayes in 60, or 1/30 part of the Wa∣ges,

and Victuals, and o∣ther charges; whereas the charges is supposed to be more than double. I say, this consideration is of great weight in Vessels of burden, especially such as carry gross and cheap bul∣ky Commodities, neither liable to damage or perish∣ing: Of which goods 7 parts of 10 of all Sea∣carriage do consist. But on the other hand, where safety against Enemies, speedy dispatch upon im∣portant

occasions, or pre∣occupation of a Market are in the case, there sharpness and great Sails may be ad∣mitted to the greatest pro∣portions practicable.

Having thus digressed, I mind you that we said, Velocities are the Roots of Resistances and Extent of Sails, &c. It may be well askt, How we know the same, since that very few Seamen or Shipwrights, ei∣ther in their writing or dis∣courses seem to understand

or own this important Po∣sition. To which I answer, that I have by many Ob∣servations, Calculations, and Comparisons, found the same to be praeter propter true, although there be ma∣ny circumstances which in∣termingle themselves in this Experiment, so as to disturb and confound it: As namely, The ill placing of Masts, The ill cutting and standing of Sails, The ill Trim of the Vessel, with the Cleanness or Foulness

of the same; The Sails more or less worn or wet; as also taught or slack Rig∣ging, &c. Wherefore not onely to avoid these last mentioned Intricacies, but also to make these Positions Examinable by every one that desires it; I say, that the different Velocity of Bodies (of several sharp∣nesses, and as drawn or dri∣ven by different Powers of knocks or falling weights,) have been by my self and others much experimented

in large Canales, or Troughs of water, fitted with a con∣venient Apparatus for that purpose, and by no man more, nor more judi∣ciously, than by the Right Honorable the Lord Brounc∣ker, President of this So∣ciety. For I do not think it hard to conceive, that Weights and Sails are pow∣ers of like Effect, and redu∣cible to the same Principle; so as if a Body have moved in double velocity, when drawn by a quadruple

weight; and in triple, when by a noncuple weight; I doubt not but the same will hold in Sails, or other impellent Powers of the same proportions.

And for the further clear∣ing or easier trying hereof, I offer two small Machina∣ments heretofore made in this Society: The one, to measure the Velocity of the Wind, and the other its Power or Equivalency to Weight; whereby it did and will appear, when the

wind is of double velocity, it will stir a quadruple weight; and the like in o∣ther cases according to the proportions of Roots and Squares above mentioned. The same may also be seen even in any good Turnspit-Jack, where a quadruple weight makes double Velo∣city (at the same distances of Time from the begin∣ning of the Motion) both in the time of the Weights descent, as also in the Revo∣lutions of the Fly, and each

intermediate Wheel. Now perhaps the reason of these Phaenomena may be here expected; to which I an∣swer, that the many parallel Instances above and here∣after mentioned, do, like concurrent witnesses, prove the premisses, at least as to any practical use. And as for giving other reasons (which I take to be Ex∣plaining this Subject from the very first Principles of Atomical Matter, and Moti∣on) I leave it to discourse,

as too long for this Exer∣cise.

The Second Instance Is in the Strength of Timber, &c.

LEt there be Square Rods or Pieces made of any Clean Timber, or other Ma∣terials, whose Ends let be supported with conve∣nient Blocks or Fulcra: These Rods in Experience will bear weight hung in

the middle of them, ac∣cording to the proportion of their lengths or distance, between the Fulcra; that is to say, a Rod A. being of double length to the Rod B. will bear ½ the weight which B can bear; and be∣ing of triple length, it will bear one third; & sic de caeteris. Again, let two of those equal and alike square Rods be placed one upon the other (so as to touch and sit,) then the two together shall bear 4 times

as much as one alone, and three of them, placed as a∣fore-said, shall bear nine times as much, and so on in proportion of Roots to Squares. Again, lay the same two Rods side by side, to each other, then they shall bear but double, three shall bear triple, and so forward, in Arithmetical proportion. From whence it follows, that four of them placed square, shall bear eight times as much as one alone. But if the same four

Rods taken as One, being of double length making an Octuple quantity to One, they shall bear but four times the weight of One alone. So as two like pie∣ces of Timber, that are in cubical or triplicate pro∣portion of their Sides, are strong but according to duplicate proportion, or the Squares of their respe∣ctive Sides; and conse∣quently, to have like Vessels (differing in Content as the Cubes of their like

Sides) equally strong, the Timber of which they con∣sist must be Quadrato-qua∣dratic; that is to say, a Ship of 400 Tuns, equally strong with one of 50, must have not only 8 times as much Timber in it, but 16 times; which is seldom or never done. Which de∣fect is the true Reason, why great Shipping is both Dearer and Weaker than small Shipping, (no Ship in the world being so strong as a Nutshel;) I say,

Weaker, for what is here said; and Dearer, for what shall be said hereafter in the sixteenth Instance of Masts, Diamonds, &c. And on the other hand, if the Timbers were Quadrato∣quadratic, then the Ship of 400 Tuns would be loaden with her own Ma∣terials; if the Ship of 50 Tuns were not over-tim∣bered.

Now, for not well un∣derstanding these matters, many men designing En∣gines

of strength, do make Models of such Machina∣ments by a Scale (suppose wherein an inch represents a foot,) by which the Mo∣del is the 1/1728 part of the En∣gine intended: And there∣upon they conceive, that if the Model be strong e∣nough to bear 1/1728 part of what the great Machina∣ment is intended to bear, that then the said great Ma∣chinament will be strong enough. Whereas indeed the Model must bear the

full 1/144 of what is intended for the great Machinament; otherwise great mischiefs will appear in the Work. Wherefore the Square of the Linear Difference be∣tween the Model and En∣gin, is the measure and way of trying the strength and sufficiency sought for: The ignorance whereof hath made many a poor Proje∣ctor. Upon these Princi∣ples, a Cask which will hold a Tun, ought to have 16 times as much Timber in

it, as the Cask which holds onely a Barrel, or ⅛ of a Tun; provided one be as strong as the other (which is not usually seen.) For the bigger Vessels, Carts, &c. they are usually the weaker compar'd with the strength of the lesser; which appears also in Animals, whose strength is as the Square Roots of their weights and substance, viz. if 1728. Mice were equiponderate to one Horse, the said Horse is but 1/144 part as strong as

all the said Mice.

From these considerati∣ons the Scantlings of Tim∣ber in Buildings must be adjusted; as for example, Let the Walls of any Room be infinitely, that is, suffi∣ciently strong; let the length and the breadth of the Room be given: Next, suppose the Room is to be made so strong, as that eve∣ry foot and a half square shall bear a Man, and so, that 31½ square feet should bear a Tun weight, (rec∣koning

14 men to the Tun:) Lastly, let the strength of the Timber be also given. Now the Que∣stions are, to find the Scantlings of the Girders, Gise, &c. first in square pieces, and afterwards by altering the Squares into more advantageous ablong Sizes; as for example, Let the Room be supposed 26 foot long and 20 broad, viz. 520 foot in the Area, and able to receive about 250 men, and to bear a∣bout

16 Tuns. Suppose the Timber be such, as whereof a Rod of an inch square, and 20 foot long, will bear 1/20 part of an hun∣dred weight; or, that 20 such Rods, or a Board of 20 inches broad, and 20 foot long within the walls, an whole hundred weight; and so the whole Floor con∣sisting of about 16 such Boards, but 1600. Now if the same Board were planck of 4 inches thick, it would bear 16 times 1600 or 256

hundred weight: If 5 inch∣es, 400 hundred weight: But the whole weight de∣signed being but 325 hun∣dred, some size between 4 and 5 inches thick will suffice in this case, where we suppose the Floor to be of planck without Gise or Girder. Next, suppose in∣stead of this Planck there be used Gise of double thickness to the said Planck, and placed at qua∣druple distance; I say, the Effect and Strength will be

the same with half the stuff. And I also say, that one Girder alone of 18 inches square, and 20 foot long, is near Equivalent to the 17 Gises of 9 inches deep, and 4½ broad-abovemen∣tioned; which Girder has but half the stuff which the Gise had; as the Gise did contein but half the stuff, which the 4½ inch-Planck first mentioned did con∣tein. Which saving of stuff is the reason of divi∣ding Plank into Girders,

Gise, and Board. Where note, that these Proporti∣ons and Scantlings are not offered as exact and best for practice, but onely to intimate the method of in∣quiring into these matters so useful in the world.

The Third Instance▪ In the Oars of a Boat, &c.

TO determine or make a good estimate of the power of Oars, I first, for

easier calculation, suppose a Paralellipipedon-Boat or Vessel, of breadth fit for a pair of Skulls, viz. of a∣bout 5 foot broad, and of length sufficient for 9 such Skulls or Oars, viz. about 30 foot long, and one foot deep, and to draw but three inches water. Next, I suppose, that every Skul∣ler with his Skulls and Bench, &c. their weight to be equivalent to three Cu∣bical foot of water; so as every pair of Skulls (with

its appurtenances) depres∣ses or sinks the Vessel 1/50 of a foot, or about ¼ of an inch. Now, suppose also a smooth calm standing water, in which one Rower will row this Vessel 12000 foot, or above two miles in an hour or 3600 seconds; I say then, that, if one Remex or Skuller move 12 quarters or 3 inch.es draught, 12000 feet for∣ward in 3600 seconds; then 4 like Rowers shall move the same Vessel, drawing 15 quarters, or 3¾ inches

of water, the same 12000 feet, in 1800 seconds plus 360 seconds, or in all, 2160 seconds: And that 9 shall row the same Vessel, as the Root of 21 to the Root of 208, which is, as near 3 to 7; or in 3/7 of the time that one Rower alone could have done the same. Again, suppose each Oar lengthen∣ed from two to three, and that as many stroaks are made in the same time as before; then the Velocity shall increase proportiona∣bly.

But suppose, that the Oars remain of the same length, but that the Blade be doubled; then the Velo∣city shall increase but ac∣cording to the Roots of that doubling, or as 10 to 7, or 7 to 5, &c. supposing still the same number of stroaks, within the same time, in every Case or Ex∣periment.

Again, suppose these Experiments be made not in still water, but in water which runs 6000 foot an

hour; then, against the stream the Velocity will be lessened by one half, and accelerated answerably with it.

Lastly, if the said water be so rough, as that the Vessel heavs and sets, sup∣pose 20 degrees of the Qua∣drant in it; then, for as∣much as the Boats way will be encreased as much as the Tangent of 20 degrees ex∣ceeds the Radius, the way or Velocity of the Boat must abate proportionably.

The Fourth Instance. In the Motion of Horses.

SUppose an Horse can travel 5 miles an hour with 200 pound burthen on his back; then with half the said burthen he shall travel 7; and with double but three miles and a half. Again, suppose a Horse with 200 pound burthen can endure to tra∣vel 10 hours per diem; then

with half the same burthen he may endure 14 hours, and with double but 7 hours. Lastly, suppose a Horse (as Race-horses) can run after the rate of four miles in ⅛ of an hour, or 32 miles per hour, then they can run about 6 miles 1/28 in ¼, or after the rate of 24 1/7 miles per hour; and in one half an hour can run 8 miles, or after the rate of 16 miles per hour; and in a whole hour can run 12 1/14 miles; and in 2 hours

can run 16 miles, or 8 miles per hour; and in 4 hours can run 24 miles, at 6 miles per hour; and in 8 hours 32, or 4 miles per hour; and in 16 hours may go 48 miles, or 3 miles in an hour. All which agrees well enough with Experience.

The Fifth Instance, In Mills.

WHere the wind blows, suppose, on a Saw∣mill, in double Velocity, there the Saw-mill, which carried but one Saw, shall carry four; If treble, shall carry nine. And the like is true of water gush∣ing out upon the floats of Under-shot Mills; as may be seen in the Stampers of Paper-Mills, the Stocks of

Fulling-Mills; and other Works of the like nature.

The Sixth Instance, In Gunpowder.

THe way of a Bullet, shot out of a good Gun, is as the square Roots of the quantity of the Gun∣powder fired; I say, of Powder fired, because what goes out unburnt, goes ra∣ther as Shot than Powder; and the Length of Guns sig∣nifies only the constraining

of the Powder within the Lines of Direction, till it be all fired: The use of hard ramming and screw∣ing of Guns, being also the same; and the excellency of Powder being to fire quick, and before it goes out of the Gun. I say there∣fore, the Velocities caused by Gun-powder are as the Roots of the Powder fired, that is to say, 4 pound of Powder, all equally fired within the Piece, shall car∣ry a Bullet twice as far as

one pound shall do; and in Time, as 10 to 7; which last mentioned numbers are the Roots of the double distances afore-mentioned. Now, if the Capacity of the Concave of Guns ought to be, as the Weight of their Bullets or Powder; then, if the just length of any one Gun hath been well found by good Experimentation, then may also be known the length of every Gun for every Bullet respective∣ly. As, for example, sup∣pose

a Gun, that carries a Ball of 5 inches Diameter, be 10 foot long in the Con∣cave, then the Content of the said Concave will be 3000 Cylindrical inches. Now the question is, how long must the Piece be, which carries a Bullet of 7 inches Diameter? I say, that forasmuch as the Weight of the 5 inch Bul∣let, to that of 7, is as 125 to 343; the Concave of the greater Gun must be in the same proportion to

3000, viz. 8232 like inches, so as it may contein and fire a proportionable quan∣tity of powder: Which 8232 being divided by the Area of the Bullet, 49, the Quotient will be 168 inch∣es, or 14 foot; that is (to speak shortly and plainly) The Length of Guns must be measured by the Diameters of their respective Bullets. I cannot say, I have tried the effects of Gunpowder to be in the abovemention'd pro∣portion, but have credibly

heard it to be so; and be∣cause of the Similitude of Sails, Weights, Knocks, and the other points above de∣scribed, unto this of Gun∣powder, I believe it; and recommend it to your fur∣ther thoughts and experi∣ence.

The Seventh Instance. Of Sounds.

LEt there be many Equal Sounds; I say, that the Distances, at which they

may be heard, are the Roots of the Numbers of such Sounds. For, four Musquets will be heard twice as far as one, and nine thrice; and so of the rest. By which reckoning, the hearing of some of our Fleets Engagement with the Dutch even to S. Iames's Park near this City is ea∣sily solved; and the truth of that Observation doth reciprocally countenance this Doctrine. For sup∣pose both Fleets (consist∣ing

of two hundred Ships great and small) had a∣bout 12000 pieces of Ord∣nance on board them, which at a Medium suppose to be Demi-Culverins: Suppose also, that a Demi∣culverin, with the same cir∣cumstances of Wind and Air, may be as easily heard five miles, as the said En∣gagements were heard 160 miles. Then I say, that 1024 of the said 12000 Guns firing together, or very near the same time,

might (as they were) be well heard 160 miles; and that about 4000 such Guns might as well be heard 300- miles, as one Demi-Cul∣verin five miles; which last point I add, to prevent the unbelief of a probable mat∣ter, when it shall happen. Now what effect this had in the Popes Presage of the Battel of Lepanto, I know not.

The Eighth Instance Of Smells

I Say the same of Smells, viz. that the Distances at which they are perceived are the Roots of the Quan∣tity of the Matter out of which they are emitted; which Doctrin I apply to solve what I once did hard∣ly believe, viz. that Ships coming from America to∣wards Portugal, did smell the Rosemary and other

odoriferous herbs 60 miles off from the Land: The which seems not only cre∣dible, but very likely. For, if a foot square of a Rose∣mary-Field may be smelt one Perch or Rod (whereof 320 make a mile,) then a∣bout 8000 Acres of Land, whereon such sented Plants do grow (or a piece of Land about 4 miles long, and 3 miles broad; or 6 miles long, and 2 miles broad) may be smelt 64 miles: And 72000 Acres

of the like Land, or a par∣cel of such Land about 11 miles square, may be smelt as many leagues, or near 200 miles. And this Con∣sideration I pitch upon, as one of the grounds where∣upon I would build a Do∣ctrin concerning the Influ∣ence of the Stars, and other Celestial or remote Bodies upon the Globe of the Earth, and its Inhabitants, both Men and Brutes.

The Ninth Instance Concerns Visible Objects.

I Say also, that four equal and like Candles will give light but twice as far as one, and 9, thrice as far; and that 16 will also en∣lighten but 4 times as far as one, &c. And if a Flag or Ships-Vane of a yard square may be seen a league off at Sea, it must be 2 yards square, or 4 square yards

to be seen 2 leagues, and so forward. But whoever will make experiment here∣of, must first consider, how many miles in thickness of a Middling, Clear, and Di∣aphanous Air do make an Opaque. For we find, that although a very thin plate of clear Glass seems to hin∣der our sight of near Ob∣jects but very little; yet we also know, that great number of them (suppose one hundred) can scarce be seen through at all.

Hereunto also must be ad∣ded the Consideration of the Convexity of the Earth; and then I doubt not, but this Doctrin (of Roots and Squares) rectified and cor∣rected with the two addi∣tional Considerations last mentioned, will hold con∣cerning Visible and Lucid Bodies, as was above pro∣pounded.

The Tenth Instance, In the Time of the Vibration of Pendules.

THe times in which the Returns of a Vibra∣ting Pendulum are made, are the Roots of the Di∣stances between the Cen∣ter of the Pendulum, and the Center upon which it moves. I shall need to make no application of this Truth, since we all enjoy

the benefit of it in our more regulated Clocks and Mea∣sures of Time, which are now in common use, and from whose Improvements we may most hopefully ex∣pect a better measure of Longitude upon the Sur∣face of the Earth. The fur∣ther uses which may be made hereof, (it being a very simple and examina∣ble Experiment) is to wit∣ness and give evidence to other the more abstruse and complicate Positions, which

are of the like and parallel Nature.

The Eleventh Instance In the Life of Man, and its Duration.

IT is found by Experi∣ence, that there are more persons living of be∣tween 16 and 26 years old, than of any other Age or Decade of years in the whole life of Man (which David and Experience say

to be between 70 and 80 years:) The reasons where∣of are not abstruse, viz. be∣cause those of 16 have pas∣sed the danger of Teeth, Convulsions, Worms, Ric∣kets, Measles, and Small∣pox for the most part: And for that those of 26. are scarce come to the Gout, Stone, Dropsie, Pal∣sies, Lethargies, Apople∣xies, and other Infirmities of Old Age. Now whether these be sufficient reasons, is not the present Enquiry;

but taking the afore-menti∣oned Assertion to be true; I say, that the Roots of eve∣ry number of Mens Ages under 16 (whose Root is 4) compared with the said number 4, doth shew the proportion of the likely∣hood of such mens reach∣ing 70 years of Age. As for example; 'Tis 4 times more likely, that one of 16 years old should live to 70, then a new-born Babe. 'Tis three times more likely, that one of 9 years old

should attain the said age of 70, than the said Infant. Moreover, 'tis twice as like∣ly, that one of 16 should reach that Age, as that one of 4 years old should do it; and one third more likely, than for one of nine. On the other hand, 'tis 5 to 4, that one of 26 years old will die before one of 16; and 6 to 5, that one of 36 will die before one of 26; and 3 to 2, that the same person of 36 shall die before him of 16: And so forward ac∣cording

to the Roots of a∣ny other year of the decli∣ning Age compared with a number between 4 and 5, which is the Root of 21, the most hopeful year for Longaevity, as the mean between 16 and 26; and is the year of perfection, ac∣cording to the sense of Our Law, and the Age for whose life a Lease is most valua∣ble. To prove all which, I can produce the accompts of every Man, Woman, and Child, within a certain Pa∣rish

of above 330 Souls; all which particular Ages being cast up, and added together, and the Sum divi∣ded by the whole number of Souls, made the Quotient between 15 and 16; which I call (if it be Constant or Uniform) the Age of that Parish, or numerous Index or Longaevity there. Many of which Indexes for seve∣ral times and places, would make an useful Scale of Sa∣lubrity for those places; and a better Judg of Ayres than

the conjectural Notions we commonly read and talk of. And such a Scale the King might as easily make for all his Dominions, as I did this for this one Parish.

The Twelfth Instance In Musick.

TAke a Musical String, one end thereof be∣ing fastned; hang unto the other (over a convenient Bridg) any weight which may strain it to some grave Musical Tone or Note; then

set some other string of near the same length, Unisone thereunto. Lastly, instead of the first weight, hang to the first String the Quad∣ruple of the same weight; and it will appear, that the String with the quadruple weight shall yield a Tone of an 8^th or Diapason above it self, when singly charged. The reason is, because the quadruple weight doubles the number of Vibrations, (2 being the Root of 4:) And for that the Ratio For∣malis

of Tones lieth in the number of the Vibrations; and of the Diapasons, in the doublness of such num∣bers. By the same Method of hanging-on several weights at one end of the same String, all Tones may be produced, of which such String is capable. The Tones or Notes also of like Bells and Drums do follow the same proportions of their Tension and Mettal, so as able Artists can cast Bells in Tones assigned.

The Thirteenth Instance, of Fire and Spirits.

LEt a Cylindrical Flat∣bottom Vessel be filled with Water, and let it be tried, in what time one Lamp or Candle would make the water boyl through, or come up to its greatest heat: Then see, in how much lesser time, 2, 3, or 4 more like fires will hasten the same effect. I

cannot speak positively hereof, but know from se∣veral Observations, that the Acceleration abovesaid shall not be made in Arith∣metical Proportion; for∣asmuch as I know, that in Fire-works great Fires are more profitable than small; as in Brewers Coppers, and Iron-works may be seen; wherein double Fires pro∣duce more than double dis∣patch or advantage. I shall therefore suspend this mat∣ter, and pass to the measu∣ring

of the Spirituosity of Liquors, or in what pro∣portions several Liquors contein more or less of in∣flameable or ardent parts. Now in this case I conceive, the Consideration of Roots and Squares is also mate∣rial; for I understand by strength or multitude of Spirits, the Space, greater or lesser, into which such Liquors will be rarified, or will fill with Spirits: As for example, if a Pint of Water rarified into Va∣pour

will fill a Globe but of 3 foot Diameter; and a Pint of rectified Spirit of Wine will fill a Globe of six foot diameter, or 8 times as large as that of Water; I shall say, that there is 8 times as much Spirit or Va∣pour in one as in the other. But if these Liquors were put into open Lamps or Vessels, there the space in which the Spirits rise, are the Roots, whose Squares do shew the Spirituosity of those Liquors: Ex. gr. Let

there be a Lamplike Vessel of common Aquavitae; in which place a Week as high as the same will burn by the rising of the Spirit un∣to it, suppose an inch a∣bove the surface of the Li∣quor: Now, let there be a like Equal vessel with such a Spirit, as will rise up higher, suppose to a Week placed two inches above the Surface; in this case, I say, that the latter Liquor is quadruple in strength or extent of Spirit to the for∣mer;

for 'tis certain, that as the Spirit riseth double upwards, so also it emitteth or rarifieth it self double also sideways; and conse∣quently the quantity of the Spirit or Vapour must be quadruple; and so of other proportions.

The Fourteenth Instance, of Rising and Falling Bo∣dies; but particularly of Waters in Pumps and River-streams.

LEt it be observed in the Transparent Pipe of a Forcing Pump, at how ma∣ny stroaks the Water is for∣ced from the Bottom to the Top; and let as many marks be made at the seve∣ral places unto which the

Water mounted at every stroak (which stroaks we suppose to be all in Equal Times;) it will appear, that all the said Divisions will be according to the Pro∣portions or the Logarithms above-mentioned. As for the Descents and Accelera∣tions of falling Bodies, the Times are the Roots of these Spaces, which they fall in the said times respe∣ctively. The great effect whereof we see in Overshot-Mills, where a little water

falling upon a Wheel of a large Diameter, produceth wonderful Effects; the which may be well compu∣ted upon the Principles we hold forth.

Waters also have greater forces in the above-menti∣oned proportions, as the hole or place whereat they issue is lower from their Surface; as may be seen in all Breast▪ and Undershot-Mills; where it is pleasant to divide the Sinking of the water into Equal Spa∣ces,

and to count the Clacks, Revolutions or Stroaks made within the Time of the waters sinking every such equal Space; for therein the above-men∣tioned Logarithmes may also be observed.

Unto this head may be referred the Leakage of Ships. For let there be a hole in a Ship somewhere under water; then let it be seen, what water comes in at the said hole, within any space of Time; then let

the like hole be made at double the perpendicular distance from the top of the water, and there shall come in four times as much as at the upper hole; and let a third be at three distances, and that shall admit 9 times as much, &c. Again, let there be two E∣qual holes or Leaks in a Ship, the one at Head, and the other at Stern, and let the Ship be in motion; then the Leakage at the Head is composed of the pressure of

the water from the Surface, and of the Ships Motion together. Moreover, if the Ship make double way, the Leakage will be quadruple; if treble way, non cuple, &c. Wherefore to stop Leaks a∣fore, the Ship must stop its motion, lye by, or bear up to go with the Wind and Sea, &c.

Lastly, I shall add, that the Swiftnesses of Waters or River-streams, are the Roots of the Power that causes them; which

causes are Steepness or De∣scent in a sharper Angle from the Perpendicular. Wherefore knowing by ob∣servations, what degree of Steepness causeth any de∣gree of Swiftness; hereby, and by our Doctrin, the Height of ground where a∣ny River riseth above its fall into the Sea, may be computed.

The Fifteenth Instance, In the Blast of Bellows.

IN Iron-work Furnaces are the greatest and most regular moving Bellows that are any where used; the which are commonly turned by the evenest over∣shot Wheels. Now the Times wherein these Bel∣lows rise and fall, are Roots of the Strength of such Bellows-blast upon

the fire; for rising in double Quickness admits double air in the same Time; which being in like manner squeezed out a∣gain, double Quickness makes double Expulsion, and consequently double Swiftness; (the whole pas∣sing through the same Twire-pipe in half the time;) and double Swift∣ness makes quadruple ef∣fects upon the fire or Fur∣nace, as aforesaid.

The Sixteenth Instance, In the Price of several Com∣modities.

SUppose a Mast for a small Ship be of 10 inches Diameter, and as is usual, of 70 foot in heighth, and be worth 40 s; then a Mast of 20 inches through, and double length also, shall not onely cost eight times as much, according to the Octuple quantity of

Timber it conteins, but shall cost 16 times as much, or 32 l. And by the same Rule, a Mast of 40 inches through shall cost 16 times 32 l. or 516 l. Of which last Case there have been some instances. But where∣as it may be objected, That there are no Masts of four times 70, or 280 foot long, I still say, that the Rule holds in common pra∣ctice and dealing. For, if a Mast of 10 inches thick, and 60 foot long, be worth

30 s; a Mast of 20 inches throughout, and 80 foot long, shall be worth 15 l. And a Mast of 40 inches thorough, and 100 foot long (not 280 foot) shall be worth near 100 l.

Moreover, suppose Dia∣monds or Pearls be equal and like in their Figures, Waters, Colours, and Even∣ness, and differ onely in their Weights and Magni∣tudes; I say, the Weights are but the Roots of their Prices, as in the Case a∣foregoing.

So a Diamond of Decuple weight, is of Centuple value. The same may be said of Looking∣glass-Plates. I might add, that the Loadstone A, if it take up 10 times more than the Loadstone B, may be also of Centuple value.

Lastly, A Tun of ex∣treme large Timber may be worth two Tuns of ordi∣nary dimensions; which is the cause of the dear∣ness of great Shipping a∣bove small; for the Hull

of a Vessel of 40 Tuns may be worth but 3 l. per Tun, whereas the Hull of a Ves∣sel of 1000 Tuns may be worth near 15 l. per Tun. From whence arises a Rule, how by any Ships Burthen to know her worth by the Tun, with the Number and Size of her Ordnance, &c.

The Seventeenth Instance, In Mill-Dams, Sea-Bancks, and Bulwarks of For∣tresses.

SUppose any Wall, Dam, or Banck, to be just sufficient to keep out or resist the Sea, or other Stream against the appulse of its waters, being of a cer∣tain force; I say, that to make this Wall or Damm strong enough against a

double swiftness of ap∣pulse, it must be augmen∣ted by quadruple thic∣kness; and if it must be made sufficient against the greatest violence which e∣ver was observed, then that violence being known, is the Root of the number by which the Walls thickness must be augmented.

So Cannon-Bullets do Execution or batter in du∣plicatâ ratione of their swiftness; and therefore Ramperts must be strong

and thick in duplicatâ ra∣tione of the said swiftness, which depends upon the Distance of the Battery, and the degrees of Tardati∣on, which Bullets make in every part of their way be∣tween the Gun and the Rampert, which they are to batter. Where note, that Bullets commonly beat out a Cone of Wall, whose Vertex is in the Bullets En∣try, and like the Conical Fovea to be seen in the Sand of an Hourglass.

The Eighteenth Instance, In the Compression of Yield∣ing and Elastic Bodies, as Wooll, &c.

SUppose some Cylindri∣cal or other parallell'd sided Vessel, fill'd with Wool, or Down, or Fea∣thers, or other Elastic Ma∣terials; let the same be covered with a moveable Head (such as in pressing of Pilchards they call a

Buckler;) then first ob∣serve, how low the Buck∣ler descendeth by its own weight; and then upon this Head or Buckler lay a triple weight, to make the whole quadruple, and it will appear, that the Buckler will sink but just as much lower; and being Noncuple, another like Space lower: So as the se∣veral Spaces of Depressi∣ons are the Roots of the depressing Powers. From hence may be seen, how

the Force must be increas∣ed at every Turn or Thred of a Screw-Press; which being done according to the proportions here un∣derstood, I doubt not, but a Light Substance with a convenient Apparatus, might be compressed unto the Density and Weight e∣ven of Gold. But, that Sil∣ver might be so condens'd, I made no question, till I heard of some Anomaly in the practice, which I must better consider of. The

further Truth whereof doth appear in the Vnder-water∣Air within the Vessels of Water-Divers, who the low∣er they go, do find their stock of Air more and more to shrink; and that according to the Roots of the Quantities of the super-incumbent Water or Weight. In like manner take a Bow, and hang any weight to the middle of its string, and observe how low it draweth the said string. Now, if you shall

quadruple the same weight, it will draw down double the first distance, and non∣cuple will draw it down treble, &c. So as in a drawn Bow, let the Arrow be divided into quotcunque partes, each equal part of the Tension carrieth the Arrow to an Equal Di∣stance, notwithstanding each equal part of the Ten∣sion was made by Unequal power, and that each equal Space or Part also of the Arrows first flight requires

Unequal Force, viz. least strength at first, and most at last; and that, in the proportion first mention∣ed. So in the Fuze of a Watch, the greatest strength of the Spring is made to work upon the shortest Ve∣ctis; and the least upon the longest, so as to equalize the whole. The like also happens in the Traction of Muscles upon two Bones with a turning Joynt be∣tween them; which Bones and Muscles make a Tri∣angle,

whereof the Muscle is the Base, subtending the Angle-Joynt. Now in the working, the Muscle is strongest, when the Vectis is smallest, as lying most obliquely; and vice versâ, when the Muscle and mo∣ving Bone come to make a right Angle.