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Proclamations, are binding upon the subject, where they tained the title of illustrious, and was made pretor of Procopius Proclama
ll do not either contradict the old laws, or tend to estation
blish new ones ; but only enforce the execution of such PROCREATION, the begetting and bringing forth Prolile. Procopius, laws as are already in being, in such manner as the young. See GENERATION and SEMEN.
king shall judge necessary. Thus the established law PROCTOR, a person commissioned to manage an-
another person's name.
PRODIGALITY, means extravagance, profusion,
service of religion.
vantages and disadvantages admirably adapted to preserve
dently assert, that, whatever are the inconveniencey of
tion; to that wretched listlessness, which is constrained
object of severe and unvaried attention, destroys its very
ing, fortification, &c. wherein are expressed the several
Profile, in this sense, amounts to the same with eleva ment concerning the event of a disease; as whether it progeste tion ; and stands opposed to plan or ichnography. shall end in life or death, be short or long, mild or ma
PROFILE is also used for the contour or outline of a lignant, &c. figure, building, member of architecture, or the like; PROGRAMMA, anciently signified a letter sealed as a base, a cornice, &c. Hence profiling is sometimes with the king's seal. used for designing, or describing the member with rule, Programma is also an university term for a billet or compass, &c.
advertisement, posted up or given into the band, by way PROFILE, in sculpture and painting.--A head, a por of invitation to an oration, &c. containing the trait, &c. are said to be in profile, when they are repre ment, or so much as is necessary for understanding thereof. sented sidewise, or in a side-view; as, when in a portrait PROGRESSION, in general, denotes a regular adthere is but one side of the face, one eye, one cheek, vancing, or going forwards, in the same course and man&c. shown, and nothing of the other.-On almost all medals, the faces are represented in profile.
PROGRESSION, in Mathematics, is either arithmetical PROFLUVIUM, in Medicine, denotes a flux, or li or geometrical. Continued arithmetic proportion is, quid evacuation of any thing.
where the terms do increase and decrease by eqnal difPROGNOSTIC, among physicians, signifies a judge- ferences, and is called arithmetic progression : Thusa, atd, a +2d, a+3d, &c. increasing 2
by the difference d. In numbers
§ 2, 4, 6, 8, 10, &c. increasing
710, 8, 6, 4, 2, &c. decreasing by the difference 2.
from a continual
2 division See the articles Fluxions, GEOMETRY, and SERIES.
a, =, &
Object of THIS is the name for that part of mechanical philo that it would be ridiculous affectation to pay any regard sophy which treats of the notion of bodies any to the deviations from equality and parallelism. A bul
. how projected from the surface of this earth, and influ let rising a mile above the surface of the earth loses only enced by the action of terrestrial gravity.
Toto of its weight, and a horizontal range of 4 miles
will describe round that focus areas proportional to the quite insensible in all the uses which can be made of this
It was the first
. These are necessary consequences of a gravity fects; and they have no measure but the effect, or what directed to the centre of the earth, and inversely propor measures the eflect; and every thing which we can distional to the square of the distance. But in the greatest cover with regard to those measures, we must aflirm with projections that we are able to make, the gravitations regard to the things of which we assume them as the are so nearly equal, and in directions so nearly parallel, measures. Therefore,
The motion of a falling body, or of a body projected half the length of the pendulum, as the circumference Conse. directly downwards, is uniformly accelerated, and that of a circle is to its diameter. The length of a penduquences of
of a body projected directly upwards is uniformly re lum can be ascertained with great precision ; and it can this fact.
tarded : ihat is, the acquired velocities are as the times be lengthened or shortened till it makes just 86,403
are proportional to the final velocities ; and the times of sured by comparing their accelerations with the accele-
all the later and best writeis on mechanical philosophy,
mark 3600; if it were then taken to the distance of the 3. The spaces described by falling bodies are propor moon, it would pull it out only to the mark 1. And tional to the souares of the times from the beginning of the we make this assertion on the authority of our having fall; and the spaces described by bodies projected directly observed that a body at the distance of the moon falls upsards are as the squares of the times of the ascents. from that distance zoro part of 16 feet in a second.
4. space described by a body falling from rest is We do not, therefore, compare the forces, which are one half of the space which the body would have uni- imperceptible things; we compare the accelerations, formly described in the same time, with the velocity ac which are their indications, effects, and measures. quired by the fall.--And the height to which a body This las made philosophers so anxious to determine Two modes will rise, in opposition to the action of gravity, is one with precision, the fall of beavy bodies, in order to bave of deterbalf of the space which it would uniformly describe in an exact value of the accelerating porser of terrestrial mining the the same time with the initial velocitv.
gravity. Now we must here observe, that this measure
vy bodies. In like manner the difierence of the spaces which a may be taken in two ways: we may take the space falling or rising body describes in any equal successive through which the heavy body falls in a second; or we parts of its fall or risc, is one half of the space which it may take the velocity which it acquires in consequence would uniformly describe in the same time with the of gravity having acted on it during a second. The difference of the initial and final velociues.
Jast is the proper measure ; for the last is the immediate This proposition will be more conveniently expressed effect on the body. The action of gravity has changed for our purpose thus:
the state of the body--in what way? By giving it a deA body moving uniformly during the time of any termination to notion downwards, this both points out fall with the velocity acquired thereby, will in that time the kind and the degree or intensity of the force of describe a space double of that fall; and a body pro gravity. The space described in a second by falling, jected directly upwards will rise to a height which is is not an invariable measure; for, in the successive se. one balf of the space which it would, uniformly con conds, the body falls through 16, 48, 80, 112, &c. tinued, describe in the time of its ascent with the initial feet, but the changes of the body's state in each second velocity of projection.
is the same. At the beginning it bad no determination These theorems liave been already demonstrated in a to move with any appreciable velocity; at the end of popular way, in the article GUNNERY. But we would the first second it had a determination by which it recommend to our readers the 39th prop. of the first would have gone on for ever (had no subsequent force book of Newton's Principia, as giving the most general acted on it) it the rate of 32 feet per second. At the investigation of this subject ; equally easy with these end of the second second, it had a determination by more loose methods of demonstration, and infinitely su which it would have moved for ever, at the rate of 64 perior to them, by being equally applicable to every feet per second. At the end of the third second, it variation of the accelerating force. See an excellent bad a determination by which it wonld have moved application of this proposition by Mr Robins, for defin for ever, at the rate of g6 feet per second, &c. &c. ing the motion of a ball discharged from a cannon, in The difference of these determinations is a deiermination the article GUNNERY, N° 15.
to the rate of 32 feet per second. This is therefore The force
5. It is a matter of observation and experience, that constant, and the indication and proper measure of the of gravity a heavy body falls 16 feet and an inch English measure constant or invariable force of gravity. The space falin falling in a second of time ; and therefore acquires the velocity len through in the first second is of use only as it is Bodies can of 32 feet 2 inches per second. This cannot be ascer one half of the measure of this determination ; and as tained.
tained directly, with the precision that is necessary. A halves have the proportion of their wholes, different ac-
ticians on time of a vibration is to the time of falling through the accelerating force. Mathematicians of the first rank this subject Vol. XVII. Part I.
have committed great mistakes by not attending to this ; Leibnitz is one of the most obscure of his obscure witand it is necessary to notice it just now, because cases will ings, but deserves the attention of an intelligent and occur in the prosecution of this subject, where we shall be curious reader, and cannot fail of making an indelible very apt to confound our reasonings by a confusion in the impression on his mind, with relation to the modesty, use of those measures. Those mathematicians who are candour, and probity of the author. It is preceded accustomed to the geometrical consideration of curvili- by a dissertation on the subject which we are now enneal motions, are generally disposed to take the actual de- tering upon, the motion of projectiles in a resisting meflection from the tangent as the measure of the deflecting dium. Newton's Principiu had been published a few force; while those who treat the same subject algebrai- years before, and bad been reviewed in a manner shamecally, by the assistance of fluxions, take the change of fully slight, in the Leipsic Acts. Both these subjects
velocity, which is measured by twice the deflection. The make the capital articles of that immortal work. Mr Plate reason is this: when a body passes through the point B Leibnitz published these dissertations, without (says be) CCCCXLI. of a curve ABC, fig. 1. if the deflecting force were to having seen Newton's book, in order to show the world fig. 1.
cease at that instant, the body would describe the tan that he had, some years before, discovered the same
take by J.
Burnoulli but in consequence of the deflection, it is only CF: there it to be only an oversight, in drawing the tangent on fore he takes EC for the measure of the deflection, and of the wrong side of the ordinate. For in this very prothe deflecting force. Now EC is ultimately twice DC; position Newton exhibits, in the strictest and most beauand thus the measure of the algebraist (derived solely tiful manner, the difference between the geometrical from the nature of the differential method, and without and algebraical manner of considering the subject; and any regard to physical considerations) happens to coin- expressly warns the reader, that his algebraical symbol cide with the true physical measure. There is therefore expresses the deflection only, and not the variation of great danger of mixing these measures. Of this we can the increment of the ordinate. It is therefore in the Bat felse's
. Particular- not give a more remarkable instance than Leibnitz's at last degree improbable that he would make this misly of Leib- tempt to demonstrate the elliptical motion of the planets take. He most expressly does not; and as to the real
in the Leipsic Acts, 1689. He first considers the sub- mistake, which he corrected in the second edition, tbe
And he afterwards adds, that it appears by But he either had not observed where the paralogism some sheets being cancelled, and new ones substituted lay, or would not let himself down by acknowledging in this part of the work, that the mistake would have a mistake in what he wisbed the world to think his own continued, had be not corrected it. We would desire calculus (fluxions); he applied the correction where no our readers to consult this dissertation, which is ex. fault had been committed, for he had measured both tremely elegant, and will be of service to us in this artithe centrifugal force and the solicitation of gravity in the cle; and let them compare the civil things which is bere same way, but had applied the fluxionary expression to said of the vir incomparabilis, the omni laude major, the last and not to the first, and, by so doing, he com the summus Newtonus, with what the same author, in pletely destroyed all coincidence between his result and the same year, in the Leipsic Acts, but under a borthe planetary motions. We mention this instance, not rowed name, says of him. Our readers will have only as a caution to our mathematical readers, but also no hesitation in ascribing this letter to this author. as a very curious literary anecdote. This dissertation of For, after praising John Bernoulli as summus geometra,
v=56. =,=Vh.7°=h, =49 feet.
natus ad summorum geometarum paralogismos corrigen- or the height through which a body must fall to acquire 16
In bolies dos, summi candoris ut et modestiæ, he betrays himself this velocity.
projected by an unguarded warmth, when defending J. B.'s de
upwards, monstration of the inverse problem of centripetal forces, by calling it MEAM demonstrationem. Let our readers now consider the scope and inten
or 56*=3136. tion of this dissertation on projectiles, and judge whether
64 the author's aim was to instruct the world, or to acquire
5. Suppose a body projected directly downwards with and directfame, by correcting Newton. The dissertation does
the velocity of 10 feet per second; what will be its ve-ly downnot contain one theorem, one corollary, nor one step
of argument, which is not to be found in Newton's first locity after four seconds ? In four seconds it will bave wards. edition ; nor has he gone farther than Newton's single acquired, by the action of gravity, the velocity of 4 x 32,
or 128 feet, and therefore its whole velocity will be 138 proposition the xth. To us it appears an exact com
feet per second.
6. To find how far it will have moved, compound its
motion of projection, which will be 40 feet in four se-
7. Suppose the body projected as already mentioned,
and that it is required to determine the time it will take We hope to be forgiven for this long digression. It
to go 296 feet downwards, and the velocity it will have is a very curious piece of literary history, and shows
acquired. tbe combination which envy and want of honourable
Find the height x, through which it must fall to acprinciple had formed against the reputation of our illu- quire the velocity of projection, 10 feet, and the time strious countryman ; and we think it our duty to em
y of falling from this height. Then find the time % of brace any opportunity of doing it justice.To return
falling through the height 296+x, and the velocity v to our subject : Accurate The accurate measure of the accelerative power of acquired by this fall. The time of describing the 296
zy, v measure of gravity, is the fall 161, feet, if we measure it by the From such examples, it is easy to see the way of an. the accele- space, or the velocity of 325 feet per second, if we take er or pow. the velocity. It will greatly facilitate calculation, and swering every question of the kind.
Writers on the higher parts of mechanics always More geneer of gravi-will be sufficiently exact for all our purposes, if we take compute the actions of other accelerating and retarding al forma16 and 32, supposing that a body falls 16 feet in a se
forces by comparing them with the acceleration of gra-læ cond, and acquires the velocity of 32 feet per second.vity, and in order to render their expressions more geneThen, because the heights are as the squares of the ral, use a symbol, such as g for gravity, leaving the reatimes, and as the squares of the acquired velocities, a
der to convert it into numbers. Agreeably to this body will fall one foot in one fourth of a second, and view, the general formulæ will stand thus : General
will acquire the velocity of eight feet per second. Now formula de let h express the height in feet, and call it the PRO I. v=v2gh, i. e. v2V&V h, =g ?, DUCING HEIGHT; v the velocity in feet per second, and
47 47 2h call it the PRODUCED VELOCITY, the velocity DUE ;
II. t= and t the time in seconds. We shall have the follow
28 ing formulæ, which are of easy recollection, and will
In all these equations, gravity, or its accelerating
power, is estimated, as it ought to be, by the cbange
of velocity which it generates in a particle of matter in
an unit of time. But many mathematicians, in tbeir
investigations of curvilineal and other varied motions, III. vh==4
measure it by the deflection which it produces in this
time from the tangent of the curve, or by the increIV. h=
ment by which the space described in an unit of time 64
exceeds the space described in the preceding unit. This To give some examples of their use, let it be requi- have produced, had the body moved throngh the whole
is but one balf of the increment which gravity would 1. To find the time of falling through 256 feet.
moment with the acquired addition of velocity. In this 16
sense of the symbol g, the equations stand thus :
IV.h> =gt, and wh=
15 Examples of their use red, in falling bodies.
3 D 2