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
A44320	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 ...	Hooke, Robert, 1635-1703.	1679 (1679)	Wing H2617; ESTC R4280	276,083	420

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 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 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 δ D d and let C D represent 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 imagining 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 return to A all those powers which are equal to the respective 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 returns 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 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 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 returning 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 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 Diameter 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 proportion 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 returning 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 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 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 calculate 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 impediment 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 deliver 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 second then shall the body touch the floor in a point very near at 32 foot from the aforesaid perpendicular point and the

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 quadruple 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 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 Root of 51. that is of 19+17 +15. At the end of the