Lectiones Cutlerianæ, or, A collection of lectures, physical, mechanical, geographical, & astronomical made before the Royal Society on several occasions at Gresham Colledge : to which are added divers miscellaneous discourses / by Robert Hooke ...

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Potentia Restitutiva, OR SPRING.

THe Theory of Springs, though attempted by divers eminent Mathematicians of this Age has hitherto not been Published by any. It it now about eighteen years since I first found it out, but designing to apply it to some particular use, I omitted the publishing thereof.

About three years since His Majesty was pleased to see the Experiment that made out this Theory tried at White-Hall, as also my Spring Watch.

About two years since I printed this Theory in an Anagram at the end of my Book of the Descriptions of Helioscopes, viz. ceiiinosssttuu, id est, Ʋt tensio sic vis; That is, The Power of any Spring is in the same proportion with the Tension thereof: That is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the Theory is very short, so the way of try∣ing it is very easie.

Take then a quantity of even-drawn Wire, either Steel, Iron, or Brass, and coyl it on an even Cy∣linder into a Helix of what length or number of turns you please, then turn the ends of the Wire into Loops, by one of which suspend this coyl upon a nail, and by the other sustain the weight that you would have to extend it, and hanging on several Weights observe exactly to what length each of the weights do extend it beyond the length that its own weight doth stretch it to, and you shall find that if

one ounce, or one pound, or one certain weight doth lengthen it one line, or one inch, or one cer∣tain length, then two ounces, two pounds, or two weights will extend it two lines, two inches, or two lengths; and three ounces, pounds, or weights, three lines, inches, or lengths; and so forwards. And this is the Rule or Law of Nature, upon which all manner of Restituent or Springing motion doth pro∣ceed, whether it be of Rarefaction, or Extension, or Condensation and Compression.

Or take a Watch Spring, and coyl it into a Spiral, so as no part thereof may touch another, then pro∣vide a very light wheel of Brass, or the like, and fix it on an arbor that hath two small Pivots of Steel, upon which Pivot turn the edge of the said Wheel very even and smooth, so that a small silk may be coyled upon it; then put this Wheel into a Frame, so that the Wheel may move very freely on its Pivots; fasten the central end of the aforesaid Spring close to the Pivot hole or center of the frame in which the Arbor of the Wheel doth move, and the other end thereof to the Rim of the Wheel, then coyling a fine limber thread of silk upon the edge of the Wheel hang a small light scale at the end thereof fit to receive the weight that shall be put thereinto; then suffering the Wheel to stand in its own position by a little index fastned to the frame, and pointing to the Rim of the Wheel, make a mark with Ink, or the like, on that part of the Rim that the Index pointeth at; then put in a drachm weight into the scale, and suffer the Wheel to settle, and make another mark on the Rim where the Index doth point; then add a drachm more, and let the Wheel settle again, and note with Ink, as before, the place of the Rim pointed at by the In∣dex; then add a third drachm, and do as before, and so a fourth, fifth, sixth, seventh, eighth, &c. suffer∣ing the Wheel to settle, and marking the several places pointed at by the Index, then examine the

Distances of all those marks, and comparing them together you shall find that they will all be equal the one to the other, so that if a drachm doth move the Wheel ten degrees, two drachms will move it twenty, and three thirty, and four forty, and five fifty, and so forwards.

Or take a Wire string of twenty, or thirty, or forty foot long, and fasten the upper part thereof to a nail, and to the other end fasten a Scale to receive the weights: Then with a pair of Compasses take the distance of the bottom of the scale from the ground or floor underneath, and set down the said distance, then put in weights into the said scale in the same manner as in the former trials, and measure the several stretchings of the said string, and set them down. Then compare the several stretchings of the said string, and you will find that they will always bear the same proportions one to the other that the weights do that made them.

The same will be found, if trial be made, with a piece of dry wood that will bend and return, if one end thereof be fixt in a horizontal posture, and to the other end be hanged weights to make it bend downwards.

The manner of trying the same thing upon a body of Air, whether it be for the rarefaction or for the compression thereof I did about fourteen years since publish in my Micrographia, and therefore I shall not need to add any further description thereof.

Each of these ways will be more plainly under∣stood by the explanations of the annexed figures.

The first whereof doth represent by A B the coyl or helix of Wire, C the end of it, by which it is su∣spended, D the other end thereof, by which a small Scale E is hanged, into which putting Weights as F G H I K L M N, singly and separately they being in proportion to one another as 1 2 3 4 5 6 7 8, the Spring will be thereby equally stretcht to o, p, q, r, s, t, u, w,

that is, if F stretch it so as the bottom of the Scale descend to o, then G will make it descend to p, H to q, I to r, K to s, L to t, M to u, and N to w, &c. So that x o shall be one space, x p, 2, x q, 3, x r, 4, x s, 5, x t, 6, x u, 7, x w, 8.

The second figure represents a Watch Spring coy∣led in a Spiral by C A B B B D, whose end C is fixed to a pin or Axis immovable, into the end of which the Axis of a small light Wheel is inserted, upon which it moves; the end D is fixed to a pin in the Rim of the Wheel y y y y, upon which is coyled a small silk, to the end of which is fixed a Scale to re∣ceive the weights. To the frame in which these are contained is fixed the hand or Index z; then trying with the former weights put into the Scale E, you will find that if F put into the Scale E sinks the bot∣tom of it x to o, then G will sink it to p, and H to q, I to r, K to s, L to t, and z will point at 1, 2, 3, 4, 5, 6, 7, 8 on the Wheel.

The trials with a straight wire, or a straight piece of wood laid Horizontal are so plain they need not an explication by figure, and the way of trying upon Air I have long since explained in my Micogra∣phia by figures.

From all which it is very evident that the Rule or Law of Nature in every springing body is, that the force or power thereof to restore it self to its na∣tural position is always proportionate to the Distance or space it is removed therefrom, whether it be by rarefaction, or separation of its parts the one from the other, or by a Condensation, or crowding of those parts nearer together. Nor is it observable in these bodys only, but in all other springy bodies whatso∣ever, whether Metal, Wood, Stones, baked Earths, Hair, Horns, Silk, Bones, Sinews, Glass, and the like. Respect being had to the particular figures of the bodies bended, and the advantagious or disadvan∣tagious ways of bending them.

From this Principle it will be easie to calculate the several strength of Bows, as of Long Bows or Cross-Bows, whether they be made of Wood, Steel, Horns, Sinews, or the like. As also of the Balistae or Catapultae used by the Ancients, which being once found, and Tables thereof calculated, I shall anon shew a way how to calculate the power they have in shooting or casting of Arrows, Bullets, Stones, Granadoes, or the like.

From these Principles also it will be easie to calculate the proportionate strength of the spring of a Watch upon the Fusey thereof, and consequently of adjust∣ing the Fusey to the Spring so as to make it draw or move the Watch always with an equal force.

From the same also it will be easie to give the rea∣son of the Isochrone motion of a Spring or extended string, and of the uniform sound produced by those whose Vibrations are quick enough to produce an audible sound, as likewise the reason of the sounds, and their variations in all manner of sonorous or springing Bodies, of which more on another occasion.

From this appears the reason, as I shall shew by and by, why a Spring applied to the balance of a Watch doth make the Vibrations thereof equal, whe∣ther they be greater or smaller, one of which kind I shewed to the right Honourable the Lord Viscount Brounker, the Honourable Robert Boyle Esq; and Sir Robert Morey in the year 1660. in order to have got∣ten Letters Patents for the use and benefit thereof.

From this it will be easie to make a Philosophical Scale to examine the weight of any body without putting in weights, which was that which I menti∣oned at the end of my description of Helioscopes, the ground of which was veiled under this Anagram, c e d i i n n o o p s s s t t u u, namely, Ʋt pondus sic ten∣sio. The fabrick of which see in the three first figures.

This Scale I contrived in order to examine the gra∣vitation of bodies towards the Center of the Earth,

viz. to examine whether bodies at a further distance from the Center of the Earth did not lose somewhat of their power or tendency towards it. And pro∣pounded it as one of the Experiments to be tried at the top of the Pike of Teneriff, and attempted the same at the top of the Tower of St. Pauls before the burning of it in the late great Fire; as also at the top and bottom of the Abby of St. Peters in Westminster though these being by but small distances removed from the Surface, I was not able certainly to perceive any manifest difference. I propounded the same also to be tried at the bottom and several stations of deep Mines; and D. Power did make some trials to that end, but his Instruments not being good, nothing could be certainly concluded from them.

These are the Phenomena of Springs and springy bodies, which as they have not hitherto been by any that I know reduced to Rules, so have all the attempts for the explications of the reason of their power, and of springiness in general, been very insufficient.

In the year 1660. I printed a little Tract, which I called, An Attempt for the explication of the Phenome∣na, &c. of the rising of water in the pores of very small Pipes, Filtres, &c. And being unwilling then to publish this Theory, as supposing it might be preju∣dicial to my design of Watches, which I was then procuring a Patent for, I only hinted the principle which I supposed to be the cause of these Phaenomena of springs in the 31 page thereof in the English Edi∣tion, and in the 38 page of the Latine Edition, tran∣slated by M. Behem, and printed at Amsterdam, 1662. But referred the further explication thereof till some other opportunity.

The Principles I then mentioned I called by the names of Congruity and Incongruity of bodies. And promised a further explanation of what I thereby meant on some other occasion. I shall here only ex∣plain so much of it as concerns the explication of this present Phaenomenon.

By Congruity and Incongruity then I understand no∣thing else but an agreement or disagreement of Bo∣dys as to their Magnitudes and motions.

Those Bodies then I suppose congruous whose particles have the same Magnitude, and the same de∣gree of Velocity, or else an harmonical proportion of Magnitude, and harmonical degree of Velocity. And those I suppose incongruous which have neither the same Magnitude, nor the same degree of Velocity, nor an harmonical proportion of Magnitude nor of Velocity.

I suppose then the sensible Universe to consist of body and motion.

By Body I mean somewhat receptive and commu∣nicative of motion or progression. Nor can I have any other Idea thereof, for neither Extention nor Quan∣tity, hardness nor softness, fluidity nor fixedness, Rare∣faction nor Densation are the proprieties of Body, but of Motion or somewhat moved.

By Motion I understand nothing but a power or tendency progressive of Body according to several de∣grees of Velocity.

These two do always counterballance each other in all the effects, appearances, and operations of Na∣ture, and therefore it is not impossible but that they may be one and the same; for a little body with great motion is equivalent to a great body with little moti∣on as to all its sensible effects in Nature.

I do further suppose then that all things in the Uni∣verse that become the objects of our senses are com∣pounded of these two (which we will for the pre∣sent suppose distinct essences, though possibly they may be found hereafter to be only differing concepti∣ons of one and the same essence) namely, Body, and Moti∣on. And that there is no one sensible Particle of matter but owes the greatest part of its sensible Extension to Motion whatever part thereof it ows to Body accord∣ing to the common notion thereof: Which is, that

Body is somewhat that doth perfectly fill a determi∣nate quantity of space or extension so as necessarily to exclude all other bodies from being comprehended within the same Dimensions.

I do therefore define a sensible Body to be a de∣terminate Space or Extension defended from being penetrated by another, by a power from within.

To make this the more intelligible, Imagine a very thin plate of Iron, or the like, a foot square, to be moved with a Vibrative motion forwards and backwards the flat ways the length of a foot with so swift a motion as not to permit any other bo∣dy to enter into that space within which it Vi∣brates, this will compose such an essence as I call in my sense a Cubick foot of sensible Body, which dif∣fers from the common notion of Body as this space of a Cubick foot thus defended by this Vibrating plate doth from a Cubick foot of Iron, or the like, through∣out solid. The Particles therefore that compose all bodies I do suppose to owe the greatest part of their sensible or potential Extension to a Vibrative motion.

This Vibrative motion I do not suppose inherent or inseparable from the Particles of body, but communi∣cated by Impulses given from other bodies in the Uni∣verse. This only I suppose, that the Magnitude or bulk of the body doth make it receptive of this or-that peculiar motion that is communicated, and not of any other. That is, every Particle of matter ac∣cording to its determinate or present Magnitude is receptive of this or that peculiar motion and no other, so that Magnitude and receptivity of motion seems the same thing: To explain this by a similitude or example. Suppose a number of musical strings, as A B C D E, &c. tuned to certain tones, and a like number of other strings, as a, b, c, d, e, &c. tuned to the same sounds respectively, A shall be receptive of the motion of a, but not of that of b, c, nor d; in like manner B shall be receptive of the motion of b, but not of the motion

of a, c or d. And so of the rest. This is that which I call Congruity and Incongruity.

Now as we find that musical strings will be moved by Unisons and Eighths, and other harmonious chords, though not in the same degree; so do I suppose that the particles of matter will be moved principally by such motions as are Unisons, as I may call them, or of equal Velocity with their motions, and by other har∣monious motions in a less degree.

I do further suppose, A subtil matter that incom∣passeth and pervades all other bodies, which is the Menstruum in which they swim which maintains and continues all such bodies in their motion, and which is the medium that conveys all Homogenious or Har∣monical motions from body to body.

Further I suppose, that all such particles of matter as are of a like nature, when not separated by others of a differing nature will remain together, and strengthen the common Vibration of them all against the differing Vibrations of the ambient bodies.

According to this Notion I suppose the whole Universe and all the particles thereof to be in a con∣tinued motion, and every one to take its share of space or room in the same, according to the bulk of its body, or according to the particular power it hath to receive, and continue this or that peculiar motion.

Two or more of these particles joyned immediately together, and coalescing into one become of another nature, and receptive of another degree of motion and Vibration, and make a compounded particle differing in nature from each of the other par∣ticles.

All bulky and sensible bodies whatsoever I suppose to be made up or composed of such particles which have their peculiar and appropriate motions which are kept together by the differing or dissonant Vibrations of the ambient bodies or fluid.

According to the difference of these Vibrative motions of the Incompassing bulks. All bodies are more or less powerful in preserving their peculiar shapes.

All bodies neer the Earth are incompassed with a fluid subtil matter by the differing Velocity of whose parts all solid bodies are kept together in the peculiar shapes, they were left in when they were last fluid. And all fluid bodies whatsoever are mixed with this fluid, and which is not extruded from them till they become solid.

Fluid bulks differ from solids only in this, that all fluids consist of two sorts of particles, the one this common Menstruum near the Earth, which is inter∣spersed between the Vibrating particles appropriated to that bulk, and so participating of the motions and Vibrations thereof: And the other, by excluding wholly, or not participating of that motion.

Though the particles of solid bodies do by their Vibrative motions exclude this fluid from coming be∣tween them where their motions do immediately touch, yet are there certain spaces between them which are not defended by the motion of the par∣ticles from being pervaded by the Heterogeneous fluid menstruum.

These spaces so undefended by the bodies and Vi∣brative motion of the particles, and consequently pervaded by the subtil incompassing Heterogeneous fluid are those we call the insensible pores of bodies.

According to the bigness of the bodies the motions are, but in reciprocal proportion: That is, the big∣ger or more powerful the body is, the slower is its motion with which it compounds the particles; and the less the body is, the swifter is its motion.

The smaller the particles of bodies are, the nearer do they approach to the nature of the general fluid,

and the more easily do they mix and participate of its motion.

The Particles of all solid bodies do immediately touch each other; that is, the Vibrative motions of the bodies do every one touch each other at every Vibration. For explication, Let A B C represent three bodies, each of

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these bodies I sup∣pose to have a Vi∣brative motion on ei∣ther side of it, A be∣tween D and E, B be∣tween E and F, and C between F and G. I suppose then that B in every one of its Vibrations doth meet A at E, and C at F, and so the motions are continually interchanged: That is, B communicates its motion to A at E, and A at the same time and place communicates its motion to B, which returning to F meets there with C, and communicates its received motion to C, which at the same instant and place communicates its own motion to B, which returns it back to E: So that the Velocity of these bodies is always the same, and each body impresseth on the contiguous bodies such a determinate number of pulses within a certain space of time. Suppose for instance, in every second of time B communicates to A and to C one million of pulses, and hath received as many from each of them, by which means each of them doth pre∣serve its own space of Vibration, according to the power of its Vibration, that neither of the contigu∣ous bodies can enter into it. The extreme particles A and C are repercussed by the motion of the am∣bient Heterogeneous fluid, whereof though the bo∣dies are of differing magnitudes, yet the body and motion of the one are equivalent to the body and motion of the other, so that whatever the body be less, the motion is quicker; and where the body is bigger, the motion is less. But the Particles of fluid

bodies do not immediately touch each other, but permit the mixture of the other Heterogeneous fluid near the Earth, which serves to communicate the mo∣tion from particle to particle without the immediate contact of the Vibrations of the Particles.

All solid Bodies retain their solidity till by other extraordinary motions their natural or proper moti∣ons become intermixed with other differing motions, and so they become a bulk of compounded motions, which weaken each others Vibrative motions. So that though the similar parts do participate of each others motions, whereby they indeavour to joyn or keep together, yet do they also participate of an He∣terogeneous motion which endeavours to separate or keep them asunder. And according to the preva∣lency of the one or the other is the body more or less fluid or solid.

All bodies whatsoever would be fluid were it not for the external Heterogeneous motion of the Am∣bient.

And all fluid bodies whatsoever would be un∣bounded, and have their parts fly from each other were it not for some prevailing Heterogeneous mo∣tion from without them that drives them more power∣fully together.

Heterogeneous motions from without are propa∣gated within the solid in a direct line if they hit per∣pendicular to the superficies or bounds, but if ob∣liquely in ways not direct, but different and deflected, according to the particular inclination of the body striking, and according to the proportion of the Par∣ticles striking and being struck.

All springy bodies whatsoever consist of parts thus qualified, that is, of small bodies indued with ap∣propriate and peculiar motions, whence every one of these particles hath a particular Bulk, Extension, or Sphere of activity which it defends from the ingress of any other incompassing Heterogeneous body whilst

in its natural estate and balance in the Universe. Which particles being all of the same nature, that is, of equal bodies, and equal motions, they readily co∣alesce and joyn together, and make up one solid bo∣dy, not perfectly every where contiguous, and whol∣ly excluding the above mentioned ambient fluid, but permitting it in many places to pervade the same in a regular order, yet not so much but that they do whol∣ly exclude the same from passing between all the sides of the compounding particles.

The parts of all springy bodies would recede and fly from each other were they not kept together by the Heterogeneous compressing motions of the am∣bient whether fluid or solid.

These principles thus hinted, I shall in the next place come to the particular explication of the man∣ner how they serve to explain the Phaenomena of springing bodies whether solid or fluid.

First for solid bodies, as Steel, Glass, Wood, &c. which have a Spring both inwards and outwards, ac∣cording as they are either compressed or dilated be∣yond their natural state.

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Let A B represent a line of such a body compound∣ed of eight Vibrating particles, as 1, 2, 3, 4, 5, 6, 7, 8, and suppose each of those Particles to perform a mil∣lion of single Vibrations, and consequently of oc∣cursions with each other in a second minute of time,

their motion being of such a Velocity impressed from the Ambient on the two extreme Particles 1 and 8. First, if by any external power on the two extremes 1 and 8, they be removed further asunder, as to CD, then shall all the Vibrative Particles be proportiona∣bly extended, and the number of Vibrations, and con∣sequently of occursions be reciprocally diminished, and consequently their endeavour of receding from each other be reciprocally diminished also. For sup∣posing this second Dimension of Length be to the first as 3 to 2, the length of the Vibrations, and con∣sequently of occursions, be reciprocally diminished. For whereas I supposed 1000000 in a second of the former, here can be but 666666 in this, and conse∣quently the Spring inward must be in proportion to the Extension beyond its natural length.

Secondly, if by any external force the extreme par∣ticles be removed a third part nearer together than (the external natural force being alway the same both in this and the former instance, which is the bal∣lance to it in its natural state) the length of the Vi∣brations shall be proportionably diminished, and the number of them, and consequently of the occursions be reciprocally augmented, and instead of 1000000, there shall be 1500000.

Having thus explained the most simple way of springing in solid bodies, it will be very easie to ex∣plain the compound way of springing, that is, by flexure, supposing only two of these lines joyned

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together as at G H I K, which being by any external power bended into the form L N N O, L M will be extended, and N O will be diminished in proportion to the flexure, and consquently the same proportions and Rules for its endeavour of restoring it self will hold.

In the next place for fluid bodies, amongst which the greatest instance we have is air, though the same be in some proportion in all other fluid bodies.

The Air then is a body consisting of particles so small as to be almost equal to the particles of the Heterogeneous fluid medium incompassing the earth. It is bounded but on one side, namely, towards the earth, and is indefinitely extended upward, being only hindred from flying away that way by its own gravity, (the cause of which I shall some other time explain.) It consists of the same particles single and separated, of which water and other fluids do, con∣joyned and compounded, and being made of particles

exceeding small, its motion (to make its ballance with the rest of the earthy bodies) is exceeding swift, and its Vibrative Spaces exceeding large, comparative to the Vibrative Spaces of other terrestrial bodies. I suppose that of the Air next the Earth in its natural state may be 8000 times greater than that of Steel, and above a thousand times greater than that of com∣mon water, and proportionably I suppose that its mo∣tion must be eight thousand times swifter than the for∣mer, and above a thousand times swifter than the la∣ter. If therefore a quantity of this body be inclosed by a solid body, and that be so contrived as to com∣press it into less room, the motion thereof (supposing the heat the same) will continue the same, and con∣sequently the Vibrations and Occursions will be in∣creased in reciprocal proportion, that is, if it be Condensed into half the space the Vibrations and Occursions will be double in number: If into a quar∣ter the Vibrations and Occursions will be qua∣druple, &c.

Again, If the conteining Vessel be so contrived as to leave it more space, the length of the Vibrations will be proportionably inlarged, and the number of Vibrations and Occursions will be reciprocally dimi∣nished, that is, if it be suffered to extend to twice its former dimensions, its Vibrations will be twice as long, and the number of its Vibrations and Occursi∣ons will be fewer by half, and consequently its indea∣vours outward will be also weaker by half.

These Explanations will serve mutatis mutandis for explaining the Spring of any other Body whatso∣ever.

It now remains, that I shew how the constitutions of springy bodies being such, the Vibrations of a Spring, or a Body moved by a Spring, equally and uniformly shall be of equal duration whether they be greater or less.

I have here already shewed then that the power of all Springs is proportionate to the degree of flexure, viz. one degree of flexure, or one space bended hath one power, two hath two, and three hath three, and so forward. And every point of the space of flexure hath a peculiar power, and consequently there being infinite points of the space, there must be infinite de∣grees of power.

And consequently all those powers beginning from nought, and ending at the last degree of tension or bending, added together into one sum, or aggregate, will be in duplicate proportion to the space bended or degree of flexure; that is, the aggregate of the powers of the Spring tended from its quiescent po∣sture by all the intermediate points to one space (be it what length you please) is equal, or in the same proportion to the square of one (supposing the said space infinitely divisible into the fractions of one;) to two, is equal, or in the same proportion to the square of two, that is four; to three is equal or in the same proportion to the square of three, that is nine, and so forward; and consequently the aggre∣gate of the first space will be one, of the second space will be three, of the third space will be five, of the fourth will be seven, and so onwards in an Arithme∣tical proportion, being the degrees or excesses by which these aggregates exceed one another.

The Spring therefore in returning from any degree of flexure, to which it hath been bent by any power receiveth at every point of the space returned an impulse equal to the power of the Spring in that point of Tension, and in returning the whole it re∣ceiveth the whole aggregate of all the forces belong∣ing to the greatest degree of that Tension from which it returned; so a Spring bent two spaces in its return receiveth four degrees of impulse, that is, three in the first space returning, and one in the second; so bent three spaces it receiveth in its whole return nine

degrees of impulse, that is, five in the first space re∣turned, three in the second, and one in the third.

So bent ten spaces it receives in its whole return one hundred degrees of impulse, to wit, nineteen in the first, seventeen in the second, fifteen in the third, thirteen in the fourth, eleven in the fifth, nine in the sixth, seven in the seventh, five in the eighth, three in the ninth, and one in the tenth.

Now the comparative Velocities of any body mo∣ved are in subduplicate proportion to the aggregates or sums of the powers by which it is moved, therefore the Velocities of the whole spaces returned are always in the same proportions with those spaces, they being both subduplicate to the powers, and consequently all the times shall be equal.

Next for the Velocities of the parts of the space returned they will be always proportionate to the roots of the aggregates of the powers impressed in every of these spaces; for in the last instance, where the Spring is supposed bent ten spaces, the Velocity at the end of the first space returned shall be as the root of 19. at the end of the second as the Root of 36. that is, of 19+17. at the end of the third as the Root of 51. that is of 19+17 +15. At the end of the fourth as the Root of 64. that is of 19+17+15+13. at the end of the tenth, or whole as the Root of 100. that is as 〈 math 〉〈 math 〉, equal to 100.

Now since the Velocity is in the same proportion to the root of the space, as the root of the space is to the time, it is easie to determine the particular time in which every one of these spaces are passed for dividing the spaces by the Velocities corresponding the quotients give the particular times.

To explain this more intelligibly, let A in the fourth figure represent the end of a Spring not bent, or at least

counterpoised in that posture by a power fixt to it, and movable with it, draw the line A B C, and let it repre∣sent the way in which the end of the Spring by addi∣tional powers is to be moved, draw to the end of it C at right Angles the Line C δ D d, and let C D re∣present the power that is sufficient to bend or move the end of the Spring A to C, then draw the Line D A, and from any point of the Line A C as B B. Draw Lines parallel to C D, cutting the Line D A in E, E, the Lines B E, B E, will represent the respective powers requisite to bend the end of the Spring A to B, which Lines B E, B E, C D will be in the same proportion with the length of the bent of the Spring A B, A B, A C.

And because the Spring hath in every point of the Line of bending A C, a particular power, therefore ima∣gining infinite Lines drawn from every point of A C parallel to C D till they touch the Line A D, they will all of them fill and compose the Triangle A C D. The Triangle therefore A C D will represent the aggregate of the powers of the Spring bent from A to C, and the lesser Triangles A B E, A B E will represent the aggregate of all the powers of the Spring bent from A to B, B, and the Spring bent to any point of the Line A C, and let go from thence will exert in its re∣turn to A all those powers which are equal to the re∣spective ordinates B E, B E, in the Triangles, the sum of all which make up the Triangles A B E, A B E. And the aggregate of the powers with which it re∣turns from any point, as from C to any point of the space C A as to B B, is equal to the Trapezium C D E B, C D E B, or the excesses of the greater Triangles above the less.

Having therefore shewn an Image to represent the flexure and the powers, so as plainly to solve and an∣swer all Questions and Problems concerning them, in the next place I come to represent the Velocities ap∣propriated to the several powers. The Velocities then being always in a subduplicate proportion of

the powers, that is, as the Root of the powers im∣pressed, and the powers imprest being as the Trapezi∣um or the excess of the Triangle or square of the whole space to be past above the square of the space yet unpassed; if upon the Center A, and space A C, (C being the point from which the Spring is supposed let go) a Circle be described as C G G F, and ordinates drawn from any point of C A the space to be past, as from B, B, to the said Circle, as B G, B G, these Lines B G, B G, will represent the Velocity of the Spring re∣turning from C to B, B, &c. the said ordinates being always in the same proportion with the Roots of the Trapeziums C D E B, C D E B for putting A C=to a, and A B=b, B G will always be equal to 〈 math 〉〈 math 〉, the square of the ordinate being always equal to the Rectangle of the intercepted parts of the Diameter.

Having thus found the Velocities, to wit, B G, B G, A F, to find the times corresponding, on the Diame∣ter A C draw a Parabola C H F whose Vertex is C, and which passeth through the point F. The Ordinates of this Parabola B H, B H, A F, are in the same pro∣portion with the Roots of the spaces C B, C B, C A, then making G B to H B as H B to I B, and through the points C I I F drawing the curve C I I I F, the respective ordinates of this curve shall represent the proportionate time that the Spring spends in re∣turning the spaces C B, C B, C A.

If the powers or stiffness of the Spring be greater than what I before supposed, and therefore must be expressed by the Triangle C de A. then the Velocities will be the Ordinates in an Ellipse as C γ γ N, greater than the Circle, as it will also if the power be the same, and the bulk moved by the Spring be less. Then will the S-like Line of times meet with the Line A F at a point as X within the point F. But if the powers of the Spring be weaker than I supposed, then will C δ e e A represent the powers, and C γ γ O the Ellipsis of

Velocity, whose Ordinates B γ, B γ, A O will give the particular Velocities, and the S-like Line of time will extend beyond N. The same will happen supposing the body (moved by the Spring) to be propor∣tionately heavy, and the powers of the Spring the same with the first.

And supposing the power of the Spring the same as at first, bended only to B 2, and from thence let go B 2 E A is the Triangle of its powers, the Ordinates of the Circle B g L are the Lines of its Velocity, and the Ordinates of the S-like Line B i F are the Liues of time.

Having thus shewed you how the Velocity of a Spring may be computed, it will be easie to calcu∣late to what distance it will be able to shoot or throw any body that is moved by it. And this must be done by comparing the Velocity of the ascent of a body thrown with the Velocity of the descent of Gravity, allowance being also made for the Resistance and im∣pediment of the medium through which it passes. For instance, suppose a Bow or Spring fixed at 16 foot above a Horizontal floor, which is near the space that a heavy body from rest will descend perpendicularly in a second of time. If a Spring de∣liver the body in the Horizontal line with a Velocity that moves it 16 foot in a second of time, then shall it fall at 16 foot from the perpendicular point on the floor over which it was delivered with such Velocity, and by its motion shall describe in the Air or space through which it passes, a Parabola. If the Spring be bent to twice the former Tension, so as to deliver the body with double the Velocity in a Horizontal Line, that is, with a Velocity that moves 32 foot in a se∣cond, then shall the body touch the floor in a point very near at 32 foot from the aforesaid perpen∣dicular point, and the Line of the motion of the body, so shot shall be moved in a Parabola, or a Line very near it, I say very near it, by reason that the

Impediment of the medium doth hinder the exactness of it. If it be delivered with treble, quadruple, quintuple, sextuple, &c. the first Velocity it shall touch the floor at almost treble, quadruple, quintu∣ple, sextuple, &c. the first distance. I shall not need to shew the reason why it is moved in a Parabola, it having been sufficiently demonstrated long since by many others.

If the body be delivered by the Spring at the floor, but shot by some Angle upwards, knowing withwhat Velocity the same is moved when delivered, and with what Inclination to the Perpendicular the same is di∣rected, and the true Velocity of a falling body, you may easily know the length of the Jactus or shot, and the time it will spend in passing that length.

This is found by comparing the time of its ascent with the time of the descent of heavy bodies. The as∣cent of any body is easily known by comparing its Velocity with the Angle of Inclination.

Let a b then in the fifth Figure represent 16 foot, or the space descended by a heavy body in a second minute of time. If a body be shot from b, in the Line b f with a Velocity as much swifter than that equal motion of 16 foot in a second, as this Line b f is longer than a b the body shall fall at e; for in the same space of time that the oblique equal motion would make it ascend from b d to a c, will the accelerated direct motion downward move it from a c to b d, and there∣fore at the end of the space of one second, when the motions do equal and balance each other, the body must be in the same Horizontal Line in which it was at first, but removed asunder by the space b e, and for the points it passeth through in all the intermediate spaces this method will determine it.

Let the Parallelogram a b p q then represent the whole Velocity of the ascent of a body by an equal motion of 16 foot in a second, and the Triangle p q r represent the whole Velocity

of the accelerated descending motion, p b is then the Velocity with which the body is shot, and p is the point of rest where the power of Gravity begins to work on the body and make it descend. Now draw∣ing Lines parallel to a q r, as s t u, s t gives the Velo∣city of the point t ascending, and t u the Velocity of the same point t descending.

Again, p b s t signifies the space ascended, and p t u the space descended, so that subtracting the descent from the ascent you have the height above the Line b d, the consideration of this, and the equal progress for∣wards will give the intermediate Velocities, and de∣termine the points of the Parabola.

Now having the Jactus given by this Scheme or Scale, appropriated to the particular Velocity, where∣with any body is moved in this or that line of Incli∣nation, it will be easie to find what Velocity in any Inclination will throw it to any length; for in any Inclination as the square of the Velocity thus found in this Scale for any inclination is to the square of any other Velocity, so is the distance found by this Scale to the distance answering to the second Velo∣city.

I have not now time to inlarge upon this speculati∣on, which would afford matter enough to fill a Vo∣lume, by which all the difficulties about impressed and received motions, and the Velocities and effects resulting would be easily resolved.

Nor have I now time to mention the great number of uses that are and may be made of Springs in Me∣chanick contrivances, but shall only add, that of all springy bodies there is none comparable to the Air for the vastness of its power of extention and contracti∣on. Upon this Principle I remember to have seen long since in Wadham Colledge, in the Garden of the learned Dr. Wilkins, late Bishop of Chester, a Foun∣tain so contrived as by the Spring of the included Air to throw up to a great height a large and lasting

stream of water: Which water was first forced into the Leaden Cistern thereof by two force Pumps which did alternately work, and so condense the Air included into a small Room. The contrivance of which Engine was not unknown to the Ancients, as Hero in his Spiritalia does sufficiently manifest, nor were they wanting in applying it to very good uses, namely, for Engines for quenching fire: As Vitruvius (by the help of the Ingenious Monsieur Clande Per∣raults interpretation) hath acquainted us in the Twelfth Chapter of his Tenth Book, where he en∣deavours to describe Ctesibius his Engine for quench∣ing fire. Not long since a German here in England hath added a further improvement thereof by con∣veying the constant stream of water through Pipes made of well tanned and liquored Leather, joyned together to any convenient length by the help of brazen Screws. By which the stream of water may be conveyed to any convenient place through narrow and otherwise inaccessible passages.

The ingenious Dr. Denys Pappin hath added a fur∣ther improvement that may be made to this Ctesibian Engine by a new and excellent contrivance of his own for making of the forcing Syringe or Pump, which at my desire he is pleased to communicate to the Pub∣lique by this following Description, which he sent me some time since.

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