VELOCITY. 



In the doftrine of fluxions, it is ufual to confider the ve- 

 locity with which magnitudes flow, or are generated. Thus 

 the velocity with which a line flows, is the fame as that of 

 the point which is fuppofed to defcribe or generate the line. 

 The velocity with which a furface flows, is the fame as the 

 velocity of a given right line, that, by moving parallel to 

 itfelf, is fuppofed to generate a reftangle, always equal to 

 the furface. The velocity with which a folid flows may be 

 meafured by the velocity of a given plane furface, that, by 

 moving parallel to itfelf, is fuppofed to generate an ereft 

 prifm, or cylinder, always equal to the folid. The velocity 

 with which an angle flows, is meafured by the velocity of a 

 point, fuppofed to defcribe the arc of a given circle, which 

 fubtends die angle, and meafures it. See Macl. Fluxions, 

 book i. chap. i. 



All thefe velocities are meafured at any term of the time 

 of the motion, by the fpaces which would be defcrlbed in a 

 given point of time, by thefe points, lines, or furfaces, with 

 their motions continued uniformly from that term. 



The velocity with which a quantity flows at any term of 

 the time, while it is fuppofed to be generated, is called its 

 fluxion. See Fluxion. 



Velocities of Bodies moving in Curves. According to 

 Galileo's fyftem of the fall of heavy bodies, which is now 

 admitted by all philofophers, the velocities of a body falling 

 vertically are, each moment of its fall, as the roots of the 

 heights from whence it has fallen ; reckoning from the be- 

 ginning of the fall. Hence that author inferred, that if a 

 body fall along an inclined plane, the velocities it has, at the 

 different times, will be in the fame ratio ; for fince its velo- 

 city is altogether owing to its fall, { and it only falls as much 

 as there is perpendicular height in the inclined plane, ) the 

 velocity Ihould be meafured by that height as if it were ver- 

 'tical. See Inclined Plane. 



The fame principle, likewife, led him to conclude, that 

 if a body fall through two contiguous inclined planes, 

 making an angle between them, much like a ftick when 

 broken, the velocity would be regulated after the fame 

 manner, by the vertical height of the two planes taken to- 

 gether ; for it is only this height that it falls ; and from its 

 fall it has all its velocity. 



This conclufion was univerfally admitted till the year 

 1693, when M. Varignon demonftrated it to be falfe. From 

 his demonftration it fhould feem to follow, that the velo- 

 cities of a body falling along the cavity of a curve, for in- 

 itance, of a cycloid, ought not to be as the roots of the 

 heights, fmce a curve is only a feries of an infinity of infi- 

 nitely little contiguous planes, inclined towards one another ; 

 fo that Galileo's propofition would feem to fail in this cafe 

 too ; and yet it holds good, only with fome reftriftion. 



All this mixture of truth and error, fo near akin to each 

 other, (hewed that they had not got hold of the firft prin- 

 ciple ; M. Varignon, therefore, undertook to clear what re- 

 lated to the velocities of falling bodies, and to fet the whole 

 matter in a new light. He ftili fuppofes Galileo's firft 

 fyftem, that the velocities, at the different times of a vertical 

 fall, are as the roots of the correfponding heights. The 

 great principle he makes ufe of to attain his end, is that of 

 compound motion. 



If a body fall along two contiguous inclined planes, 

 making an obtufe angle, or a kind of concavity between 

 them J M. Varignon fhews, from the compofition of thofe 

 motions, that the body, as it meets the fecond plane, lofes 

 fomewhat of its velocity, and, of confequence, that it is not 

 the fame at the end of the fall, as it would be, had it fallen 

 through the firft plane prolonged ; fo that the proportion of 



the roots of the heights aflerted by Galileo does not here 

 obtain. 



The reafon of this lofs of velocity is, that the motion, 

 which was parallel to the firft plane, becomes oblique to the 

 fecond, fince they make an angle : this motion, which is 

 oblique to the fecond plane, being conceived as compounded, 

 that part perpendicular to the plane is loft, by the oppofi- 

 tion thereof, and part of the velocity along with it ; confe- 

 quently, the lefs of the perpendicular there is in the oblique 

 motion, or, which is the fame thing, the lefs the two planes 

 are from being one, i. c. the more obtufe the angle is, the 

 lefs velocity does the body lofe. 



Now all the infinitely little, contiguous, inclined planes 

 of which a curve confifts, making infinitely obtufe angles 

 among themfelves ; a body falUng along the concavity of 

 a curve, the lofs of velocity it undergoes each in ft ant is in- 

 finitely httle ; but a finite portion of any curve, how little 

 foever, confifting of an infinity of infinitely httle planes, a 

 body moving through it lofes an infinite number of infi- 

 nitely little parts of its velocity ; and an infinity of infi- 

 nitely little parts makes an infinity of a higher order, i. e. an 

 infinity of infinitely httle parts makes a finite magnitude, if 

 they be of the firft order or kind ; and an infinitely little 

 quantity of the firft order, if they be of the fecond, and fo 

 in infinitum. Therefore, if the lodes of velocity of a body, 

 faUing along a curve, be of the firft order, they will amount 

 to a finite quantity in any finite part of the curve, &c. 



The nature of every curve is abundantly determined by 

 the ratio of the ordinates to the correfpondent portions of 

 the axis ; and the effence of curves in general may be con- 

 ceived a» confifling in this ratio, which is variable in a 

 thoufand ways. Now this fame ratio will be, hkewife, 

 that of two fimple velocities, by whofe concurrence a body 

 will defcribe any curve ; and, of confequence, the ef- 

 fence of all curves, in the general, is the fame thing as the 

 concourfe or combination of all the forces, wliich, taken two 

 by two, may move the fame body. Thus we have a moft 

 fimple and general equation of all poffible curves, and of all 

 pofiible velocities. 



By means of this equation, as foon as the two fimple 

 velocities of a body are known, the curve refulting from 

 them is immediately determined. It is obfervable that, ac- 

 cording to this equation, an uniform velocity, and a velocity 

 that always varies according to the roots of the heights, 

 produce a parabola, independent of the angle made by the 

 two projeftile forces that give the velocities ; and, confe- 

 quently, a cannon-ball, ftiot either horizontally or obhquely 

 to the horizon, m'.ift always defcribe a parabola. The heft 

 mathematicians, hitherto, had laboured much to prove, 

 that oblique projeftions formed parabolas as well as hori- 

 zontal ones. 



To have fome meafure of velocity, the fpace is to be 

 divided into as many equal parts as the time is conceived 

 to be divided into ; for the quantity of fpace corre- 

 fponding to that divifion of time, is the meafure of the ve- 

 locity. For an inftance : fuppofe the moveable A paftes 

 through a fpace of 80 feet in 40 feconds of time : dividing 

 80 by 40, the quotient 2 fhews the velocity of the moveable 

 to be fuch, as that it paftes over an interval of two feet in 

 one fecond ; the velocity, therefore, is rightly exprefled by 

 ■1-0 ; that is, by 2. Suppofe, again, another moveable B, 

 which in 30 feconds of time travels 90 feet ; the index of 

 the celerity will be 3. Wherefore, fince in each cafe the 

 meafure of the fpace is a foot, which is fuppofed every 

 where of the fame length, and tlie meafure of time a fecond, 

 which is conceived every where of the lame deration ; 



the 



