Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	length_n	line_n	square_n	12,261	5	15.3418	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A44322	Lectures de potentia restitutiva, or, Of spring explaining the power of springing bodies : to which are added some collections viz. a description of Dr. Pappins wind-fountain and force-pump, Mr. Young's observation concerning natural fountains, some other considerations concerning that subject, Captain Sturmy's remarks of a subterraneous cave and cistern, Mr. G.T. observations made on the Pike of Teneriff, 1674, some reflections and conjectures occasioned thereupon, a relation of a late eruption in the Isle of Palma / by Robert Hooke ...	Hooke, Robert, 1635-1703.; Papin, Denis, 1647-1714.; Young, James.; Sturmy, Samuel, 1633-1669.; G. T.	1678 (1678)	Wing H2619; ESTC R38967	35,527	58

the first space returned 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 moved 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 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 √ 19 + 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 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 ABC and let it represent the way in which the end of the Spring by additional powers is to be moved draw to the end of it C at right Angles the Line C 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 D d and let CD represent the power that is sufficient to bend or move the end of the Spring A to C then draw the Line DA and from any point of the Line AC as BB. Draw Lines parallel to CD cutting the Line DA in E E the Lines BE BE will represent the respective powers requisite to bend the end of the Spring A to B which Lines BE BE CD will be in the same proportion with the length of the bent of the Spring AB AB AC And because the Spring hath in every point of the Line of bending AC a particular power therefore imagining infinite Lines drawn from every point of AC parallel to CD till they touch the Line AD they will all of them fill and compose the Triangle ACD The Triangle therefore ACD will represent the aggregate of the powers of the Spring bent from A to C and the lesser Triangles ABE ABE 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 AC and let go from thence will exert in its return to A all those powers which are equal to the respective ordinates BE BE in the Triangles the sum of all which make up the Triangles ABE ABE And the aggregate of the powers with which it returns from any point as from C to any point of the space CA as to BB is equal to the Trapezium CDEB CDEB 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 answer all Questions and Problems concerning them in the next place I come to represent the Velocities appropriated to the several powers The Velocities then being always in a subduplicate proportion of the powers that is as the Root of the powers impressed and the powers imprest being as the Trapezium 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 AC C being the point from which the Spring is supposed let go a Circle be described as CGGF and ordinates drawn from any point of CA the space to be past as from B B to the said Circle as BG BG these Lines BG BG will represent the Velocity of the Spring returning from C to B B c. the said ordinates being always in the same proportion with the Roots of the Trapeziums CDEB CDEB for putting AC = to a and AB = b BG will always be equal to √ aa bb 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 BG BG AF to find the times corresponding on the Diameter AC draw a Parabola CHF whose Vertex is C and which passeth through the point F. The Ordinates of this Parabola BH BH AF are in the same proportion with the Roots of the spaces CB CB CA then making GB to HB as HB to IB and through the points CIIF drawing the curve CIIIF the respective ordinates of this curve shall represent the proportionate time that the Spring spends in returning the spaces CB CB CA. If the powers or stiffness of the Spring be greater than what I before supposed and therefore must be expressed by the Triangle CdeA then the Velocities will be the Ordinates in an Ellipse as C 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 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 AF at a point as X within the point F. But if the powers of the Spring be weaker than I supposed then will C 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ee+A represent the powers and C 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 O the Ellipsis of Velocity whose Ordinates B 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 B 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 AO 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 proportionately 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 B2 and from thence let go B2EA is the Triangle of its powers the Ordinates of the Circle BgL are the Lines of its Velocity and the Ordinates of the S-like Line BiF are the Lines of time Having thus shewed you how the Velocity of a Spring may be computed it will be easie to calculate to what distance it will be

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 coalesce and joyn together and make up one solid body not perfectly every where contiguous and wholly 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 wholly 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 ambient whether fluid or solid These principles thus hinted I shall in the next place come to the particular explication of the manner how they serve to explain the Phaenomena of springing bodies whether solid or fluid Let AB represent a line of such a body compounded of eight Vibrating particles as 1 2 3 4 5 6 7 8 and suppose each of those Particles to perform a million of single Vibrations and consequently of occursions 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 proportionably extended and the number of Vibrations and consequently of occursions be reciprocally diminished and consequently their endeavour of receding from each other be reciprocally diminished also For supposing this second Dimension of Length be to the first as 3 to 2 the length of the Vibrations and consequently of occursions be reciprocally diminished For whereas I supposed 1000000 in a second of the former here can be but 666666 in this and consequently the Spring inward must be in proportion to the Extension beyond its natural length Secondly if by any external force the extreme particles 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 ballance to it in its natural state the length of the Vibrations 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. 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 conjoyned 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 common water and proportionably I suppose that its motion must be eight thousand times swifter than the former and above a thousand times swifter than the later If therefore a quantity of this body be inclosed by a solid body and that be so contrived as to compress it into less room the motion thereof supposing the heat the same will continue the same and consequently the Vibrations and Occursions will be increased 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 quarter the Vibrations and Occursions will be cuadruple 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 diminished 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 Occursions will be fewer by half and consequently its indeavours outward will be also weaker by half These Explanations will serve mutatis mutandis for explaining the Spring of any other Body whatsoever 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 degrees 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 posture 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 aggregate 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 Arithmetical 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 receiveth the whole aggregate of all the forces belonging 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