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Proclamation

Proclamations, are binding upon the subject, where they do not either contradict the old laws, or tend to estaU blish new ones; but only enforce the execution of such Procopius. laws as are already in being, in such manner as the king shall judge necessary. Thus the established law is, that the king may prohibit any of his subjects from leaving the realm: a proclamation therefore forbidding this in general for three weeks, by laying an embargo upon all shipping in time of war, will be equally binding as an act of parliament, because founded upon a prior law. But a proclamation to lay an embargo in time of peace upon all vessels laden with wheat, (though in the time of a public scarcity), being contrary to law, and particularly to statute 22 Car. II. c. 13. the advisers of such a proclamation, and all persons acting under it, found it necessary to be indemnified by a special act of parliament, 7 Geo. III. c. 7. A proclamation for disarming Papists is also binding, being only in execution of what the legislature has first ordained : but a proclamation for allowing arms to Papists, or for disarming any Protestant subjects, will not bind; because the first would be to assume a dispensing power, the latter a legislative one; to the vesting of either of which in any single person the laws of England are absolutely strangers. Indeed, by the stat. 31 Hen. VIII. c. 8. it was enacted, that the king's proclamations should have the force of acts of parliament; a statute, which was calculated to introduce the most despotic tyranny and which must have proved fatal to the liberties of this kingdom, had it not been luckily repealed in the minority of his successor, about five years after. By a late act of parliament the king is empowered to raise regiments of Roman Catholics to serve in the pre

sent war.

PROCLUS, surnamed DIADOcus, a Greek philosopher and mathematician, was born in Lycia, and lived about the year 500. He was the disciple of Syrianus, and had a great share in the friendship of the emperor Anastasius. It is said, that when Vitalian laid siege to Constantinople, Proclus burnt his ships with large brazen speculums. This philosopher was a Pagan, and wrote against the Christian religion. There are still extant his Commentaries on some of Plato's books, and other of his works written in Greek.

PROCONSUL, a Roman magistrate, sent to govern a province with consular authority.

The proconsuls were appointed out of the body of the senate; and usually as the year of any one's consulate expired, he was sent proconsul into some province.

The proconsuls decided cases of equity and justice, either privately in their pretorium or palace, where they received petitions, heard complaints, granted writs under their seal, and the like; or else publicly, in the common hall, with the usual formalities observed in the court of judicature at Rome. They had besides, by virtue of their edicts, the power of ordering all things relating to the tributes, taxes, contributions, and provisions of corn and money, &c. Their office lasted only a year. See CONSUL.

PROCOPIUS, a famous Greek historian, born in Cæsaria, acquired great reputation by his works in the reign of Justinian, and was secretary to Belisarius during all the wars carried on by that general in Persia, Africa, and Italy. He at length became senator, ob

tained the title of illustrious, and was made pretor of Procopius Constantinople.

PROCREATION, the begetting and bringing forth young. See GENERATION and SEMEN.

PROCTOR, a person commissioned to manage another person's cause in any court of the civil or ecclesiastical law.

PROCTOR, in the English universities. See UNIVER

SITY.

PROCURATION, an act or instrument by which a RATI person is empowered to treat, transact, receive, &c. in another person's name.

PROCURATOR. See PROctor.

PROCYON, in Astronomy, a fixed star of the second magnitude, situated in canis minor, or the little dog. PRODIGALITY, means extravagance, profusion, waste, or excessive liberality, and is the opposite extreme to the vice of parsimony. By the Roman law, if a man by notorious prodigality was in danger of wasting his estate, he was looked upon as non compos, and committed to the care of curators, or tutors, by the prætor. And by the laws of Solon, such prodigals were branded with perpetual infamy.

PRODUCT, in Arithmetic and Geometry, the factum of two or more numbers, or lines, &c. into one another: thus 5X4 20 the product required.

PROEDRI, among the Athenians, were magistrates,. who had the first seats in the public assemblies, and whose office it was to propose at each assembly the things to be deliberated upon and determined. Their office always ended with the meeting. Their number was nine, so long as the tribes were ten in number.

PROFANATION, the acting disrespectfully to sa-cred things.

PROFANE, a term used in opposition to holy; and in general is applied to all persons who have not the sacred character, and to things which do not belong to the service of religion.

PROFESSION means a calling, vocation, or known employment. In Knox's Essays, vol. i. page 234, we find an excellent paper on the choice of a profession, which that elegant writer concludes thus: "All the occupations of life (says he) are found to have their advantages and disadvantages admirably adapted to preserve the just equilibrium of happiness. This we may confidently assert, that, whatever are the inconveniences of any of them, they are all preferable to a life of inaction; to that wretched listlessness, which is constrained to pursue pleasure as a business, and by rendering it the object of severe and unvaried attention, destroys its very

essence."

Among the Romanists profession denotes the entering into a religious order, whereby a person offers himself to God by a vow of inviolably observing obedience, chastity, and poverty.

PROFESSOR, in the universities, a person who teaches or reads public lectures in some art or science from a chair for that purpose.

PROFILE, in Architecture, the draught of a building, fortification, &c. wherein are expressed the several heights, widths, and thicknesses, such as they would ap- pear were the building cut down perpendicularly from the roof to the foundation. Whence the profile is also called the section, sometimes orthographical section, and i by Vitruvius also sciagraphy.

Profile,.

Profile.

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Profile, in this sense, amounts to the same with elevation; and stands opposed to plan or ichnography.

PROFILE is also used for the contour or outline of a figure, building, member of architecture, or the like; as a base, a cornice, &c. Hence profiling is sometimes used for designing, or describing the member with rule, compass, &c.

PROFILE, in sculpture and painting.-A head, a portrait, &c. are said to be in profile, when they are represented sidewise, or in a side-view; as, when in a portrait there is but one side of the face, one eye, one cheek, &c. shown, and nothing of the other.-On almost all medals, the faces are represented in profile.

PROFLUVIUM, in Medicine, denotes a flux, or liquid evacuation of any thing.

PROGNOSTIC, among physicians, signifies a judge

ment concerning the event of a disease; as whether it Progresóc shall end in life or death, be short or long, mild or malignant, &c.

PROGRAMMA, anciently signified a letter sealed with the king's seal.

Programma is also an university term for a billet or advertisement, posted up or given into the hand, by way of invitation to an oration, &c. containing the argument, or so much as is necessary for understanding thereof. PROGRESSION, in general, denotes a regular advancing, or going forwards, in the same course and man

ner.

PROGRESSION, in Mathematics, is either arithmetical or geometrical. Continued arithmetic proportion is, where the terms do increase and decrease by equal differences, and is called arithmetic progression : Thusa, a, a+d, a+2d, a+3d, &c. increasing by the difference d. a, a—d, a—2d, a-3d, &c. decreasing In numbers 2, 4, 6, 8, 10, &c. increasing 10, 8, 6, 4, 2, &c. decreasing S

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by the difference 2.

Geometric Progression, or Continued Geometric Proportion, is when the terms do increase or decrease by equal ratios: thus,

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Progres

Ston

I

Object of the science.

2

Effect of

gravity on projected bodies.

PROJECTILES.

THIS is the name for that part of mechanical philosophy which treats of the motion of bodies anyhow projected from the surface of this earth, and influenced by the action of terrestrial gravity.

It is demonstrated in the physical part of astronomy, that a body so projected must describe a conic section, having the centre of the earth in one focus; and that it will describe round that focus areas proportional to the times. And it follows from the principles of that sci>ence, that if the velocity of projection exceeds 36700 feet in a scond, the body (if not resisted by the air) would describe a hyperbola; if it be just 36700, it would describe a parabola; and if it be less than this, it would describe an ellipsis. If projected directly upwards, in the first case, it would never return, but proceed for ever; its velocity continually diminishing, but never becoming less than an assignable portion of the excess of the initial velocity above 36700 feet in a second; in the second case, it would never return, its velocity would diminish without end, but never be extinguished. In the third case, it would proceed till its velocity was reduced to an assignable portion of the difference between 36700 and its initial velocity; and would then return, regaining its velocity by the same degrees, and in the same places, as it lost it. These are necessary consequences of a gravity directed to the centre of the earth, and inversely proportional to the square of the distance. But in the greatest projections that we are able to make, the gravitations are so nearly equal, and in directions so nearly parallel,

that it would be ridiculous affectation to pay any regard to the deviations from equality and parallelism. A bul let rising a mile above the surface of the earth loses only Too of its weight, and a horizontal range of 4 miles makes only 4′ of deviation from parallelism.

Let us therefore assume gravitation as equal and parallel. The errors arising from this assumption are quite insensible in all the uses which can be made of this theory.

The theory itself will ever be regarded with some veneration and affection by the learned. It was the first fruits of mathematical philosophy. Galileo was the first who applied mathematical knowledge to the motions of free bodies, and this was the subject on which he exercised his fine genius.

Gravity must be considered by us as a constant or u niform accelerating or retarding force, according as it produces the descent, or retards the ascent, of a body. A constant or invariable accelerating force is one which produces an uniform acceleration; that is, which in equal times produces equal increments of velocity, and therefore produces increments of velocity proportional to the times in which they are produced. Forces are of themselves imperceptible, and are seen only in their ef fects; and they have no measure but the effect, or what measures the effect; and every thing which we can dis cover with regard to those measures, we must affirm with regard to the things of which we assume them as the measures. Therefore,

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Cor. 1. If bodies simply fall, not being projected drawn from downwards by an external force, the times of the falls are proportional to the final velocities; and the times of ascents, which terminate by the action of gravity alone, are proportional to the initial velocities.

6

The force

bodies can be ascer tained.

2. The spaces described by a heavy body falling from rest are as the squares of the acquired velocities; and the differences of these spaces are as the differences of the squares of the acquired velocities: and, on the other hand, the heights to which bodies projected upwards will rise, before their motions be extinguished, are as the squares of the initial velocities.

3. The spaces described by falling bodies are proportional to the squares of the times from the beginning of the fall; and the spaces described by bodies projected directly upwards are as the squares of the times of the ascents.

4. The space described by a body falling from rest is one half of the space which the body would have uniformly described in the same time, with the velocity acquired by the fall.-And the height to which a body will rise, in opposition to the action of gravity, is one half of the space which it would uniformly describe in the same time with the initial velocity.

In like manner the difference of the spaces which a falling or rising body describes in any equal successive parts of its fall or rise, is one half of the space which it would uniformly describe in the same time with the difference of the initial and final velocities.

This proposition will be more conveniently expressed for our purpose thus:

A body moving uniformly during the time of any fall with the velocity acquired thereby, will in that time describe a space double of that fall; and a body projected directly upwards will rise to a height which is one half of the space which it would, uniformly continued, describe in the time of its ascent with the initial velocity of projection.

These theorems have been already demonstrated in a popular way, in the article GUNNERY. But we would recommend to our readers the 39th prop. of the first book of Newton's Principia, as giving the most general investigation of this subject; equally easy with these more loose methods of demonstration, and infinitely su perior to them, by being equally applicable to every variation of the accelerating force. See an excellent application of this proposition by Mr Robins, for defining the motion of a ball discharged from a cannon, in the article GUNNERY, N° 15.

5. It is a matter of observation and experience, that of gravity a heavy body falls 16 feet and an inch English measure in falling in a second of time; and therefore acquires the velocity of 32 feet 2 inches per second. This cannot be ascertained directly, with the precision that is necessary. A second is too small a portion of time to be exactly measured and compared with the space described; but it is done with the greatest accuracy by comparing the motion of a falling body with that of a pendulum. The time of a vibration is to the time of falling through VOL. XVII. Part I.

half the length of the pendulum, as the circumference of a circle is to its diameter. The length of a pendulum can be ascertained with great precision; and it can be lengthened or shortened till it makes just 86,400 vibrations in a day: and this is the way in which the space fallen through in a second has been accurately ascertained.

As all other forces are ascertained by the accelerations which they produce, they are conveniently measured by comparing their accelerations with the acceleration of gravity. This therefore has been assumed by all the later and best writers on mechanical philosophy, as the unit by which every other force is measured. It gives us a perfectly distinct notion of the force which retains the moon in its orbit, when we say it is the 3600th part of the weight of the moon at the surface of the earth. We mean by this, that if a bullet were here weighed by a spring steelyard, and pulled it out to the mark 3600; if it were then taken to the distance of the moon, it would pull it out only to the mark 1. we make this assertion on the authority of our having observed that a body at the distance of the moon falls from that distance part of 16 feet in a second. We do not, therefore, compare the forces, which are imperceptible things; we compare the accelerations, which are their indications, effects, and measures.

And

7

fall of hea

This has made philosophers so anxious to determine Two modes with precision, the fall of heavy bodies, in order to have of deteran exact value of the accelerating power of terrestrial mining the gravity. Now we must here observe, that this measure vy bodies. may be taken in two ways: we may take the space through which the heavy body falls in a second; or we may take the velocity which it acquires in consequence of gravity having acted on it during a second. The last is the proper measure; for the last is the immediate effect on the body. The action of gravity has changed the state of the body-in what way? By giving it a determination to motion downwards, this both points out the kind and the degree or intensity of the force of gravity. The space described in a second by falling, is not an invariable measure; for, in the successive seconds, the body falls through 16, 48, 80, 112, &c. feet, but the changes of the body's state in each second is the same. At the beginning it had no determination to move with any appreciable velocity; at the end of the first second it had a determination by which it would have gone on for ever (had no subsequent force acted on it) at the rate of 32 feet per second. At the end of the second second, it had a determination by which it would have moved for ever, at the rate of 64 feet per second. At the end of the third second, it had a determination by which it would have moved for ever, at the rate of 96 feet per second, &c. &c. The difference of these determinations is a determination to the rate of 32 feet per second. This is therefore constant, and the indication and proper measure of the constant or invariable force of gravity. The space fallen through in the first second is of use only as it is one half of the measure of this determination; and as halves have the proportion of their wholes, different accelerating forces may be safely affirmed to be in the proportion of the spaces through which they uniformly impel bodies in the same time. But we should always recollect, that this is but one half of the true measure of mathema the accelerating force. Mathematicians of the first rank this subject. + 3 D have

8 Mistakes of

ticians on

Plate

:

Leibnitz is one of the most obscure of his obscure writings, but deserves the attention of an intelligent and curious reader, and cannot fail of making an indelible impression on his mind, with relation to the modesty, candour, and probity of the author. It is preceded by a dissertation on the subject which we are now entering upon, the motion of projectiles in a resisting medium. Newton's Principia had been published a few years before, and had been reviewed in a manner shamefully slight, in the Leipsic Acts. Both these subjects make the capital articles of that immortal work. Mr Leibnitz published these dissertations, without (says he) having seen Newton's book, in order to show the world that he had, some years before, discovered the same theorems. Mr Nicholas Fatio carried a copy of the Principia from the author to Hanover in 1686, where he expected to find Mr Leibnitz; he was then absent, but Fatio saw him often before his return to France in 1687, and does not say that the book was not given him. Read along with these dissertations Dr Keill's letter to John Bernoulli and others, published in the Journal Literaire de la Hayeć 1714, and to John Bernoulli in 1719.

have committed great mistakes by not attending to this ; and it is necessary to notice it just now, because cases will occur in the prosecution of this subject, where we shall be very apt to confound our reasonings by a confusion in the use of those measures. Those mathematicians who are accustomed to the geometrical consideration of curvilineal motions, are generally disposed to take the actual deflection from the tangent as the measure of the deflecting force; while those who treat the same subject algebraically, by the assistance of fluxions, take the change of velocity, which is measured by twice the deflection. The reason is this when a body passes through the point B CCCCXLI. of a curve ABC, fig. 1. if the deflecting force were to fig. 1. cease at that instant, the body would describe the tangent BD in the same time in which it describes the arch BC of the curve, and DC is the deflection, and is therefore taken for the measure of the deflecting force. But the algebraist is accustomed to consider the curve by means of an equation between the abscissæ H a, Hb, He, and their respective ordinates Aa, Bb, Cc; and he measures the deflections by the changes made on the increments of the ordinates. Thus the increment of the ordinate A a, while the body describes the arch AB of the curve, is BG. If the deflecting force were to cease when the body is at B, the next increment would have been equal to BG, that is, it would have been EF; but in consequence of the deflection, it is only CF: therefore he takes EC for the measure of the deflection, and of the deflecting force. Now EC is ultimately twice DC; and thus the measure of the algebraist (derived solely from the nature of the differential method, and without any regard to physical considerations) happens to coincide with the true physical measure. There is therefore great danger of mixing these measures. Of this we canParticular- not give a more remarkable instance than Leibnitz's atly of Leib- tempt to demonstrate the elliptical motion of the planets in the Leipsic Acts, 1689. He first considers the subject mechanically, and takes the deflection or DC for the measure of the deflecting force. He then introduces his differential calculus, where he takes the difference of the increments for the measure; and thus brings himself into a confusion, which luckily compensates for the false reasoning in the preceding part of his paper, and gives his result the appearance of a demonstration of Newton's great discovery, while, in fact, it is a confused jumble of assumptions, self-contradictory, and inconsistent with the very laws of mechanics which are used by him in the investigation. Seventeen years after this, in 1706, having been criticised for his bad reasoning, or rather accused of an envious and unsuccessful attempt to appropriate Newton's invention to himself, he gives a correction of his paralogism, which he calls a correction of language. But he either had not observed where the paralogism lay, or would not let himself down by acknowledging a mistake in what he wished the world to think his own calculus (fluxions); he applied the correction where no fault had been committed, for he had measured both the centrifugal force and the solicitation of gravity in the same way, but had applied the fluxionary expression to the last and not to the first, and, by so doing, he completely destroyed all coincidence between his result and the planetary motions. We mention this instance, not only as a caution to our mathematical readers, but also as a very curious literary anecdote. This dissertation of

nitz.

10

take by J.

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Newton has been accused of a similar oversight by Newton John Bernoulli, (who indeed calls it a mistake in prin- cused of a ciple) in his Proposition x. book 2. on the very sub- similar mis ject we are now considering. But Dr Keill has shown Bernoulli it to be only an oversight, in drawing the tangent on the wrong side of the ordinate. For in this very proposition Newton exhibits, in the strictest and most beautiful manner, the difference between the geometrical and algebraical manner of considering the subject; and expressly warns the reader, that his algebraical symbol expresses the deflection only, and not the variation of the increment of the ordinate. It is therefore in the But falsely. last degree improbable that he would make this mistake. He most expressly does not; and as to the real mistake, which he corrected in the second edition, the writer of this article has in his possession a manuscript copy of notes and illustrations on the whole Principia, written in 1693 by Dr David Gregory, Savilian professor of astronomy at Oxford, at the desire of Mr Newton, as preparatory for a new edition, where he has rectified this and several other mistakes in that work, and says that Mr Newton had seen and approved of the amendments. We mention these particulars, because Mr Insincerity Bernoulli published an elegant dissertation on this sub- of Bernou ject in the Leipsic Acts in 1713; in which he charges with reNewton (though with many protestations of admiration et ta and respect) with this mistake in principle; and says, that he communicated his correction to Mr Newton, by his nephew Nicholas Bernoulli, that it might be corrected in the new edition, which he heard was in the press. And he afterwards adds, that it appears by some sheets being cancelled, and new ones substituted in this part of the work, that the mistake would have continued, had be not corrected it. We would desire our readers to consult this dissertation, which is extremely elegant, and will be of service to us in this article; and let them compare the civil things which is bere said of the vir incomparabilis, the omni laude major, the summus Newtonus, with what the same author, in the same year, in the Leipsic Acts, but under a borrowed name, says of him. Our readers will have no hesitation in ascribing this letter to this author. For, after praising John Bernoulli as summus geometra,

natus

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Newton.

or the height through which a body must fall to acquire

natus ad summorum geometarum paralogismos corrigen-
dos, summi candoris ut et modestia, he betrays himself this velocity.
by an unguarded warmth, when defending J. B.'s de-
monstration of the inverse problem of centripetal forces,
by calling it MEAM demonstrationem.

Let our readers now consider the scope and inten-
tion of this dissertation on projectiles, and judge whether
the author's aim was to instruct the world, or to acquire
fame, by correcting Newton. The dissertation does
not contain one theorem, one corollary, nor one step of
argument, which is not to be found in Newton's first
edition; nor has he gone farther than Newton's single
proposition the xth. To us it appears an exact com-
panion to his proposition on centripetal forces, which he
boasts of having first demonstrated, although it is in
every step a transcript of the 42d of the first Book of
Newton's Principia, the geometrical language of New-
ton being changed into algebraic, as he has in the pre-
sent case changed Newton's algebraic analysis into a
very elegant geometrical one.

We hope to be forgiven for this long digression. It is a very curious piece of literary history, and shows the combination which envy and want of honourable principle had formed against the reputation of our illustrious countryman; and we think it our duty to embrace any opportunity of doing it justice. To return to our subject: 13

Accurate

ty.

The accurate measure of the accelerative power of measure of gravity, is the fall 16 feet, if we measure it by the the accele- space, or the velocity of 323 feet per second, if we take rative pow- the velocity. It will greatly facilitate calculation, and er of gravi will be sufficiently exact for all our purposes, if we take 16 and 32, supposing that a body falls 16 feet in a second, and acquires the velocity of 32 feet per second. Then, because the heights are as the squares of the times, and as the squares of the acquired velocities, a body will fall one foot in one fourth of a second, and will acquire the velocity of eight feet per second. Now formule de- let h express the height in feet, and call it the PROduced. DUCING HEIGHT; v the velocity in feet per second, and call it the PRODUCED VELOCITY, the velocity DUE; and t the time in seconds.-We shall have the following formula, which are of easy recollection, and will serve, without tables, to answer all questions relative to projectiles.

14 General

15

I. v=8/h, 8×41,320

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Examples
of their use red,

in falling
bodies.

1. To find the time of falling through 256 feet.
16

Here h=256, 256-16, and -=4. Answer 4".

4

2. To find the velocity acquired by falling four seconds. 4.32X4=128 feet per second.

3. To find the velocity acquired by falling 625 feet. h=625. √h=25.8√/h-200 feet per second.

4. To find the height to which a body will rise when projected with the velocity of 56 feet per second,

56

v=56. ===7,= √/h • 7a=h, =49 feet.

or 56*=3136.

16 In bodies projected upwards,

17

3136 49 feet. 64 5. Suppose a body projected directly downwards with and directthe velocity of 10 feet per second; what will be its ve-ly downlocity after four seconds? In four seconds it will have wards. acquired, by the action of gravity, the velocity of 4 X 32, or 128 feet, and therefore its whole velocity will be 138 feet per second.

6. To find how far it will have moved, compound its motion of projection, which will be 40 feet in four seconds, with the motion which gravity alone would have given it in that time, which is 256 feet; and the whole

motion will be 296 feet.

7. Suppose the body projected as already mentioned, to go 296 feet downwards, and the velocity it will have and that it is required to determine the time it will take

acquired.

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In all these equations, gravity, or its accelerating
power, is estimated, as it ought to be, by the change
of velocity which it generates in a particle of matter in
an unit of time. But many mathematicians, in their
investigations of curvilineal and other varied motions,
measure it by the deflection which it produces in this
time from the tangent of the curve, or by the incre-
ment by which the space described in an unit of time
exceeds the space described in the preceding unit. This
is but one half of the increment which gravity would
have produced, had the body moved through the whole
moment with the acquired addition of velocity. In this
sense of the symbol g, the equations stand thus:
I. v=2√gh=25t

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