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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #69116 (https://www.gutenberg.org/ebooks/69116)
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-The Project Gutenberg eBook of A discourse on the theory of gunnery,
-by John Pringle
-
-This eBook is for the use of anyone anywhere in the United States and
-most other parts of the world at no cost and with almost no restrictions
-whatsoever. You may copy it, give it away or re-use it under the terms
-of the Project Gutenberg License included with this eBook or online at
-www.gutenberg.org. If you are not located in the United States, you
-will have to check the laws of the country where you are located before
-using this eBook.
-
-Title: A discourse on the theory of gunnery
-       Delivered at the anniversary meeting of the Royal Society,
-       November 30, 1778
-
-Author: John Pringle
-
-Release Date: October 8, 2022 [eBook #69116]
-
-Language: English
-
-Produced by: deaurider and the Online Distributed Proofreading Team at
-             https://www.pgdp.net (This file was produced from images
-             generously made available by The Internet Archive)
-
-*** START OF THE PROJECT GUTENBERG EBOOK A DISCOURSE ON THE THEORY OF
-GUNNERY ***
-
-
-
-
-
-
-                                    A
-                                DISCOURSE
-                                  ON THE
-                           _Theory of Gunnery_.
-
-
-
-
-                                    A
-                                DISCOURSE
-                                  ON THE
-                           _Theory of Gunnery_.
-
-                             DELIVERED AT THE
-                Anniversary Meeting of the ROYAL SOCIETY,
-                            November 30, 1778.
-
-                       By Sir JOHN PRINGLE Baronet.
-
-                        PUBLISHED BY THEIR ORDER.
-
-                              [Illustration]
-
-                                 LONDON:
-                      PRINTED FOR THE ROYAL SOCIETY.
-                               MDCCLXXVIII.
-
-
-
-
-[Illustration]
-
-
-
-
-    GENTLEMEN,
-
-Among the several experiments communicated to the society, during the
-course of the preceding year, none seeming so much to engage your
-attention, as those contained in the Paper, intituled, _The force of
-fired gun-powder, and the initial velocity of cannon-balls, determined
-by experiments_: with much pleasure therefore I acquaint you, that, on
-account of the pre-eminence of that communication, your Council have
-judged the author, Mr. CHARLES HUTTON, worthy of the honour of the annual
-medal, instituted on the bequest of Sir GODFREY COPLEY Baronet, for
-raising a laudable emulation among men of genius, in making experimental
-inquiries. But, as on former occasions, so now, your Council, waving
-their privilege of determining the choice, have acted only as a select
-number deputed by you, to prepare matters for your final decision. I
-come then, on their part, briefly to lay before you the state of the
-_Theory of Gunnery_, from its rise to the time when its true foundation
-was laid, in order to evince how conducive those experiments may be
-to the improvement of an art of public concern, as well as to the
-advancement of _natural knowledge_, the great object of your institution.
-And if, upon a review of the subject, you shall entertain no less
-favourable an opinion of Mr. HUTTON’s performance, than what your Council
-have done, it is their earnest request that you would enhance the value
-of this prize, by authorizing your President to present it to our
-ingenious brother in your name.
-
-Artillery (in the large acceptation of the term) took place long before
-the invention of gun-powder. We trace the art to the remotest antiquity,
-since the Sacred Records acquaint us, that one of the kings of Judah,
-eight hundred years before the Christian æra, erected on the towers and
-bulwarks of Jerusalem engines of war, the contrivance of ingenious men,
-for shooting arrows and great stones for the defence of that city[1].
-Such machines were afterwards known to the Greeks and Romans by the
-names of _balista_, _catapulta_ and others, which had amazing powers,
-and were not less terrible in their effects than the cannon and mortars
-of the moderns. It appears that the _balista_ was contrived to shower
-volleys of darts and arrows of a very large size upon the enemy, whilst
-the _catapulta_ or _onagra_ (as it was otherwise called) was fitted
-not only for that purpose, but for discharging stones of an enormous
-weight; I might say _rocks_, since some of them are reported to have
-weighed several hundred pounds. Batteries composed of numerous pieces of
-that kind of artillery, nothing could withstand. Yet, if we are rightly
-informed, their sole principle of motion consisted in the spring of a
-strongly-twisted cordage, made of animal substances singularly tough and
-elastic. These warlike instruments continued, not only during the time
-of the Roman empire, but to the 12th and 13th centuries, as we find from
-history; nor indeed is it probable that they were totally laid aside,
-till gun-powder and the modern ordnance, attaining a good degree of
-perfection, superseded their use. The very intelligent commentator of
-POLYBIUS[2] is of opinion, that the military art rather lost than gained
-by the exchange of the _catapulta_ for the mortar: but however that
-point may be determined in speculation, it is not likely that the ancient
-_tormenta militaria_ will ever be revived; but that all nations will keep
-to the art of gunnery and study how to improve it; that is, they will
-adhere to a system of artillery, wherein the moving power depends on the
-expansive force of gun-powder, or of some other substance of a similar
-nature.
-
-Upon the first application of this principle to the purposes of war,
-nothing perhaps was less thought of than to assist so empirical a
-practice by scientific rules; for, however aiding in these matters the
-ancient mechanicians might have been, who, like ARCHIMEDES, had invented
-or perfected some of the _balistic_ machines, no praise seemed now due
-to the mathematicians for either the discovery or improvement of the new
-artillery. In fact, we find the practice of the art had subsisted about
-200 years, before any geometer considered it as one that admitted a
-theory, or at least such a theory as was grounded on geometry.
-
-It seems but just to trace and commemorate the inventors of the ingenious
-arts which furnish matter for discourses on these occasions; and not
-only the main inventors, but even those who first turned their thoughts
-upon the subject: for, though such men may not have produced any thing
-perfect, yet they may have suggested ideas to others of a less inventive,
-but of a more executive genius, and who, unprovided with those hints,
-would never have made any notable discovery. I must therefore observe,
-that the _Italians_ were the first who emerged out of those thick clouds
-of ignorance and barbarism which had so long overspread this quarter of
-the world. They profited by the unhappy fate of Constantinople; for by
-liberally receiving the learned emigrants on that distressful occasion,
-they were largely repaid by their arts and sciences, and still more
-abundantly by their language, whereby they were enabled to read and to
-translate those ancient manuscripts, which the Greeks had saved out of
-the wreck of their country. The art of printing, which was established
-soon after, was the means of quickly disseminating those treasures of
-knowledge, and concurred with the fall of the eastern empire to form an
-epoch for the advancement of learning, unparalleled in the annals of
-letters.
-
-The end of the 15th century, and the whole of the 16th, were chiefly
-employed by the Italians in the study and in the translation of the
-old Greek authors. The geometry of the ancient Greeks, as well as the
-arithmetic in numbers and species of the Arabians, were cultivated;
-but both remained, as it were, sciences by themselves, unassisting to,
-or at best but weak and reluctant auxiliaries to the philosophy of the
-schools: and indeed how could the abstracted doctrines of numbers and
-quantities be strained to co-operate with a system, in which neither
-the laws of motion, nor any but the superficial, and often delusive
-properties of matter, were to be met with? The genius of the Greeks, all
-acute and brilliant as it was, had never been properly directed to the
-interpretation of nature, and was indeed unfit (as Lord Bacon pronounced)
-for a study that made so slow and painful a progress, by re-iterated and
-varied experiments and observations. It was no wonder then, if the _mixed
-mathematics_, as they are called, descended to the moderns in a state
-no-wise corresponding to the elegance and certainty of those parts of the
-science which were elementary and pure; and that those mixed parts should
-have been found defective and erroneous, in proportion (if I may so
-express myself) to the physical considerations that were to be taken into
-the inquiry. The imperfection of the ancients, with regard to natural
-philosophy, was not perceived at that time; nay, at the period we are
-treating of, the learned were firmly persuaded of the contrary, and that
-all that was wanting to be known concerning the laws of nature, and the
-properties of matter, was to be taken either directly, or by deduction,
-from the physics of ARISTOTLE. It was not till the 17th century was
-somewhat advanced, that men of science began to listen to Lord BACON and
-GALILEO, the great founders of the experimental and the true philosophy.
-
-Mean while, in the beginning of the 16th century, unqualified as the
-Italians then were for entering upon physico-mathematical inquiries[3],
-they nevertheless made the attempt, and in particular took the theory
-of projectiles into consideration. Some imagined that a body impelled
-with violence, such as a ball discharged from a cannon, moved in a right
-line till the force was spent, and that then it fell in another right
-line perpendicularly to the earth. Upon this principle, absurd as it
-was, we find one of the earliest authors grounding his whole theory
-of gunnery[4]; whilst others, dissenting from his hypothesis, admitted
-only the straight line, in which the ball moved for some time after
-coming out of the piece, and that other straight line in which it fell
-to the ground; but asserted that these two were connected by a curve
-line, and that this curve was the segment of a circle. NICOLAS TARTAGLIA
-of Brescia, a mathematician of the first rank in those days, and still
-celebrated for his improvements in algebra, hath been supposed to be the
-author of this doctrine, no less erroneous than the former, and for which
-two of his books have been quoted[5]. Those I have never seen; but from
-another of his works, professedly written on this subject, and translated
-into English under the title of _Colloquies concerning the art of
-shooting in great and small pieces of artillery_[6], him I find, contrary
-to the opinion of his contemporaries, maintaining that no part of the
-track of a cannon-ball is in a right line, though the curvature in the
-first part of its flight be so small, that it needeth not to be attended
-to. But TARTAGLIA is far from supposing, that the line in question hath
-any relation to a _parabola_, or to any regular curve. It would seem
-then, that if this mathematician had at first been so far mistaken, as
-to fancy that some part of the course of a projectile was in a straight
-line, he had afterwards changed his opinion, and was perhaps singular in
-what he finally embraced.
-
-From numerous instances one would imagine, that in those days, so far
-were men of science from making experiments themselves, that they even
-shut their eyes against what chance would have presented to their sight.
-For, whoever had minded the roving shot of an arrow, the flight of a
-stone from a sling, or had attended to a stream of water issuing from the
-spout of a cistern, might have been convinced, that the path of every
-projectile was in a continued curve, whatever little he otherwise knew
-concerning the properties of that one.
-
-But had the observation of the philosophers gone so far, they had
-still been at a distance from the truth. They might have perceived a
-likeness between the track of those bodies in motion and a parabola,
-and concluded, from analogy, that all projectiles delineated that curve
-in the air; but they could never have realized their conjectures by
-mathematical demonstration, without previously knowing the law of
-_acceleration_ in falling bodies: a discovery reserved for the next
-century, and for GALILEO[7], one of the greatest ornaments of it.
-
-It was he who first investigated the effects of _gravity_ on falling
-bodies, and upon that foundation demonstrated, that all projectiles
-would move in a parabola in a non-resisting medium. And as he made
-little account of the resistance of the air, whose properties were then
-imperfectly known, he proved that a ball shot horizontally would, in its
-flight, describe half a parabola; and when the piece had an elevation
-above the horizon, the ball would describe a whole parabola, supposing it
-to fall on the plane of the battery. By the same method of reasoning he
-shewed, that whatever the ranges of the projected body, or the elevations
-of the piece were, the ball would still trace that curve line, of a
-greater or lesser amplitude, by the time it descended to the level of the
-place from whence it came.
-
-Thus far went GALILEO, confining his projections to the horizontal plane
-of the battery; but TORRICELLI his disciple soon after carried the
-theory farther, by tracing the shot to its fall, whether that place was
-above or below the plane; and still found, by geometrical deductions,
-that it flew in a parabola of a larger or a smaller amplitude, according
-to the angle of elevation of the piece, and the strength of the powder.
-
-Various and numerous had been the disputes in Italy about the laws of
-motion in general, and especially about those of projectiles, from the
-time the mathematicians had begun the inquiry, till the publication of
-the dialogues of GALILEO on that subject (a space of upwards of a hundred
-years) but from that period, so evident did his demonstrations appear,
-that all contest ceased, and every man of science was convinced, that
-all projectiles moved in the track which he had discovered. For, as to
-the resistance of the air, which he had not passed unnoticed (as GALILEO
-himself had been the first, at least of the moderns, who started the
-notion of the weight of the air and the pressure of the atmosphere) yet
-so thin and so yielding did they esteem that fluid to be, that they were
-assured it could occasion no sensible, at least no material, deviation
-from that curve. As they had the principle from GALILEO, so they believed
-themselves warranted by that respectable author, not to fear from that
-cause any objection, which he himself had suggested, but had removed.
-_Among these projectiles_ (says he) _which we make use of, if they are of
-a heavy matter and a round form; nay if they are of a lighter matter, and
-have a cylindrical form, such as arrows shot from bows, their track or
-path will not sensibly decline from the curve of a parabola_[8].
-
-Here then was the theory of gunnery laid, in appearance, on the most
-solid foundation. And thus far the Italians having proceeded, they seemed
-to have taken leave, and to commit the subject to other nations, whose
-greater power, or greater ambition, was more likely to make them avail
-themselves of the perfection of a military art, than their instructors.
-We had reason therefore to expect, that a neighbouring state, intent upon
-the advancement of the arts and sciences in general, would not fail to
-give particular attention to those that should appear most subservient
-to its grandeur. Accordingly we find, that our sister-society of that
-kingdom had not been many years established, when an ingenious member of
-that illustrious Body, not questioning the soundness of the Galilean
-principle in regard to projectiles, in the year 1677, proposed to the
-academy, as a problem for the improvement of artillery, how to direct
-a piece (suppose a mortar) so as to make the shot fall where one had a
-mind; or in the common expression, _to hit a mark_, the strength of the
-powder being given[9]. This thought met with general approbation, and so
-far were the academy from raising any difficulty about the obstruction
-which the air might occasion to a body moving with so much velocity
-in it, that we do not find the making experiments on that head was
-considered by them as an essential step to the solution; but that their
-principal geometers straightway set about solving the problem as it had
-been announced to them, some following one method, some another, and all
-upon the supposition of a projectile moving in the line of a parabola.
-But M. BLONDEL, who had been the proposer, and who more particularly had
-studied the question, composed a large volume on the subject, which he
-published a few years after[10], under the title of _L’Art de jetter les
-Bombes_; a performance much celebrated at the time, and that continued in
-no small request long after, as containing, besides his own, the labours
-of several other members of that society of the most distinguished
-merit. So many, and such hands concurring in framing this work, it was
-no wonder that the learned throughout Europe were confirmed by it in
-the Galilean theory; and the more as M. BLONDEL had obviated the only
-objection they supposed could be made to it, the _resistance of the air_,
-which he had taken care expressly to mention, and so to combat as to
-persuade the reader, that the retardation arising from that cause was so
-inconsiderable as to be of no account in the practice.
-
-This illusion about the small or non-resistance of the air to bodies
-rapidly moving in it, was so prevalent at the end of the last century,
-and in the beginning of the present, that in the history of the Royal
-Academy for the year 1707, we find their worthy and most accomplished
-secretary, after taking notice of the joint labours of so many able
-mathematicians concerned in BLONDEL’s publication, venturing to say, _it
-did not appear that any thing was then wanting for the practice of the
-art_ [of Gunnery] _except perhaps perfecting the instruments for pointing
-a cannon or mortar ... but that geometry had done its part, so to speak,
-with regard to practice_. &c.[11]
-
-But far be it from our intention to relate the imperfections of others,
-in order to raise ourselves by the comparison. Candour requires of us not
-only to acknowledge, that in this country, as to the point in question,
-we did not surpass our neighbours; but ingenuously to own that, on the
-contrary, we were perhaps more liable to exception. For, some years
-before BLONDEL’s work appeared[12], a treatise was published by one of
-our own artillerists, ANDERSON (a person of eminence in his profession)
-intituled _The genuine use and effects of the gun_, in which the author
-strenuously supports the Galilean theory; nor do we learn he was ever
-contradicted among us, although he undertook to answer all those who
-should make objections to it. Nay, when he had an opportunity afterwards
-of making experiments on the ranges of bombs, and by those trials was
-assured that their flight was not in a parabola; yet so far was he from
-ascribing the deviation from that figure to the resistance of the air,
-that he had recourse to an hypothesis, repugnant to all the laws of
-motion, to salve appearances, and to reconcile those experiments with his
-former doctrine[13].
-
-And did not Dr. HALLEY, so long the ornament of this society, communicate
-in the year 1686 a Paper, which he calls _A discourse concerning
-gravity_, in which, treating of the motion of projectiles, he says, that
-being aware of the deflexion from the parabolic curve that might be
-occasioned by the resistance of the air, he had made some experiments,
-even with cannon-balls, to estimate the force of that resistance; yet
-conclude, _That in large shot of metal, whose weight many thousand times
-surpassed that of air, and whose force is very great, in proportion to
-the surface wherewith they press thereupon, this opposition was not
-discernible_. And again, _Though in small and light shot, the opposition
-of the air ought and must be accounted for; yet in shooting great and
-weighty bombs, there need be very tittle allowance made; and so these
-rules_ [those, to wit, grounded on the principle of GALILEO] _may be put
-in practice to all intents and purposes, as if this impediment_ [the
-resistance of the air] _were absolutely removed_[14]. Such conclusions,
-which we now find to be erroneous, were the less to be expected from so
-eminent a person, as they argued too much haste to finish a theory, that
-was to be made subservient to present use.
-
-It might indeed have been expected, that men of science applying
-themselves to this study, would have been sooner awakened to the
-consideration of the great opposition of the air, by the _Principia_ of
-NEWTON, published a little after this Paper of HALLEY’s[15]. For in that
-excellent work the illustrious author had demonstrated, that the curve
-described by a projectile, in a strongly resisting medium, differed much
-from a parabola, and that the resistance of the air was great enough to
-make the difference between the curve of projection of heavy bodies and a
-parabola far from being insensible, and therefore too considerable to be
-neglected.
-
-Have we not then less to plead for not attending to the _Principia_ of
-NEWTON in this article[16], than the mathematicians of other nations,
-who, as M. de FONTENELLE observes[17], partly from the difficulty
-of understanding that concise and profound work, and partly from a
-misapprehension of its tendency (which they fancied was to revive the
-exploded doctrine of _occult qualities_) were late in becoming acquainted
-with it? But it is not so easy to account for their inattention to
-HUYGENS, a known and even then a much esteemed author, and who indeed
-was second to NEWTON alone in science and in genius. For he in the year
-1690 had published a treatise on _Gravity_, written in a popular manner,
-wherein he gave an account of some experiments he had made at Paris, and
-in the academy, by which, as well as by mathematical investigations,
-he was convinced of the truth of NEWTON’s conclusions, in regard to
-the great opposition of the air to bodies moving swiftly in it; and,
-by consequence, believed that the track of all projectiles was very
-different from the line of a parabola[18].
-
-But excepting NEWTON and HUYGENS, the learned seemed universally to
-acquiesce in the justness and sufficiency of the principles of gunnery
-invented by GALILEO, enlarged by TORRICELLI, confirmed and reduced to
-system by ANDERSON, BLONDEL, HALLEY and others; and so far were the
-theorists, in that branch of science, from suspecting any defect or
-fallacy in these principles, that they seemed rather to reproach the
-practical artillerists, for not profiting more by the instructions which
-they had so liberally imparted to them. Nor do we find that an apology
-was made for the empirical exercise of the art, by any author of note
-in that line, earlier than the sixteenth year of this century, when M.
-de RESSONS, a French officer of artillery, distinguished by the number
-of sieges at which he had served, by his high military rank, and by his
-abilities in his profession; when he, I say, thus qualified to bear
-testimony, presented a _memoire_ to the Royal Academy (of which he was
-a member) importing, that _although it was agreed that theory joined to
-practice did constitute the perfection of every art, yet experience had
-taught him, that theory was of very little service in the use of mortars.
-That the work of M. BLONDEL had justly enough described the several
-parabolic lines, according to the different degrees of the elevation of
-the piece; but that practice had convinced him there was no theory in the
-effects of gun-powder: for that having endeavoured, with the greatest
-precision, to point a mortar agreeably to those calculations, he had
-never been able to establish any solid foundation upon them[19]._
-
-Thus, after the theory of gunnery had exercised the genius of the
-learned for nearly two hundred years, and for almost fourscore of that
-time had rested on fundamentals which had never been contested, it was
-pronounced at once to be almost intirely useless, and that by one of
-the most competent judges. Now, whether it were owing to the deference
-due to the authority of that experienced artillerist, or to some other
-cause, I shall not determine, but observe, that it appears not from the
-history of the academy, that the sentiments of M. de RESSONS were at this
-time controverted, or any reason offered afterwards for the failure of
-the theory of projectiles when applied to use. Nor can I pass unnoticed
-the pause that ensued before any further attempts were made to improve
-the theory of the art, either upon the old principles or upon new ones,
-except by such authors as seemed ignorant of this transaction, and who of
-course were not sufficiently apprized of the inefficacy of the properties
-of the parabola for directing practice. Or by those who were employed in
-speculatively investigating the nature of the curve traced by a ball in
-the air; a curve which began at last to be considered as one deviating
-much from the line of a parabola. Or, finally, by such as, having taken
-notice that NEWTON’s ideas had not been duly attended to, endeavoured
-to avail themselves of them, and of some experiments that had been made
-by others, for proving the great opposition of the air to bodies of
-swift motion; but without ascertaining the degree of that resistance, or
-enriching the art by any practical rules[20].
-
-Such was the unhinged state of this part of the mixed mathematics,
-when within our memory Mr. BENJAMIN ROBINS took cognizance of it: nor
-could the subject have fallen into abler hands, endowed as he was by
-nature with a superior genius and unwearied application. Mr. ROBINS was
-deeply versed in geometry and the doctrine of numbers; but he knew the
-limits as well as the powers of both, and how insufficient they were for
-establishing any theory where matter was concerned, without preparing
-the way, by finding out the physical properties of that _matter_, by
-many and varied experiments and attentive observation. Those who had
-hitherto treated of the foundation of gunnery, by being too forward in
-the application of their mathematics, had in a manner hurt the credit
-of that admirable science. They ought to have seen the necessity of
-minutely examining every circumstance which could affect the course of
-a projectile, besides that of gravity. Mr. ROBINS perceived the error
-of his predecessors in that inquiry, and corrected it. Persuaded as he
-was from sir ISAAC NEWTON’s _Principia_ of the great resistance of the
-air to bodies moving in it, and also of the uncertainty of the force
-of gun-powder, and of the variations in the flight of shot, occasioned
-by the unavoidable varieties in the make of it, and in the make of the
-pieces of artillery which discharged it; apprized, I say, of so many
-causes of aberration, he justly concluded, that the foundation here was
-at least as much an affair of physics as of geometry, and that if the art
-of throwing bombs had not been advanced by theory, it was not because
-the art admitted of none, but because the theory which had hitherto been
-devised had been both defective and erroneous. He suspected that most of
-the writers on gunnery had been deceived, in supposing the resistance
-of the air to be inconsiderable, and thence asserting the track of all
-shot to be nearly in the curve of a parabola, by which means it came to
-pass that all their determinations, about the flight of projectiles of
-violent motion, had declined considerably from the truth. But in order
-to clear this point from every doubt, he found it necessary to ascertain
-the force of gun-powder, and by that step to estimate the velocity of
-the shot impelled by its explosion. That being done, he proceeded to
-measure the quickness of a musket-bullet, shot out of a given barrel,
-with a given quantity of powder; and to confirm the truth of his
-conclusions, he contrived a machine, by which the velocity of a bullet
-might be diminished in any given _ratio_, by being made to strike on a
-large body of a weight justly proportioned to it; whereby the swiftest
-motions, which otherwise would escape our examination, were to be exactly
-determined by these slower motions that had a given relation to them. The
-machine was a large wooden pendulum, which swung freely, but in so slow
-a manner, that its vibrations could easily be counted, whatever was the
-celerity of the bullet discharged against it. The thought was simple,
-ingenious, and incontestably his own.
-
-He next inquired into the resistance made by the air to projectiles of
-rapid motion, and which he discovered to be much greater than had been
-supposed by any writer on the subject; and indeed so great, that it
-was manifest the curve described by any shot was very different from a
-parabola, and consequently that all the applications of the properties
-of that conic section to gunnery were so erroneous as to be totally
-useless. For by means of this pendulum, placed at different distances
-from the mouth of the piece, he clearly demonstrated how much a bullet,
-flying with a given velocity, would gradually lose of that motion by the
-opposition of the air: therein furnishing to the learned a signal and
-instructive instance of the fallacy of the most specious theories, that
-do not proceed hand in hand with experiments.
-
-I should too much exceed the just bounds of a discourse of this kind,
-were I to enter more minutely into the system founded by Mr. ROBINS,
-confirmed and improved, as I find, by the labours of several of the
-learned in foreign parts of great celebrity[21]. I shall only add,
-that his performance well deserves the title he gives it of _The new
-principles of gunnery_, since the author may more properly be said to
-have invented a new science than to have added to an old one. And I
-believe I may venture to say, that no physico-mathematical disquisition
-hath done more honour to this country, or to the age, than the writings
-of Mr. ROBINS on this subject, which have been published, partly by this
-Society, partly by himself, and partly since his death (in the collection
-of his whole mathematical tracts) by his learned friend.
-
-But though our worthy brother will ever be celebrated for being the
-inventor of the true principles of gunnery, yet it would be too
-flattering to his memory, to say he had carried the theory of this art
-to perfection. He himself was far from entertaining so high an opinion
-of his labours; nay he expressly declared, that he left some material
-points to be inquired into at more leisure (which other occupations
-and his immature death deprived him of) and he much regretted that he
-wanted conveniency and opportunities for making experiments on balls of
-a greater weight, than what he had used for ascertaining the initial
-velocity of them.
-
-Much therefore are we indebted to Mr. HUTTON, who, treading in the
-footsteps of the deceased, hath resumed and prosecuted this last
-_desideratum_, and hath shewn himself not unequal to so difficult an
-enterprize.
-
-Mr. ROBINS, for determining the initial velocity of shot, arising from
-different quantities of powder, made use of balls of about an ounce
-weight; whereas Mr. HUTTON, for the same purpose, hath employed those of
-different weights, from one pound to nearly three; or, in other words,
-Mr. ROBINS made trial with musket-shot only, Mr. HUTTON with cannon-balls
-from 20 to about 50 times heavier. This was a considerable step gained
-in a disquisition of that part of the science, in which the resistance
-of the air and other circumstances were not concerned; and where neither
-analogy alone, nor mathematical deductions alone, nor the two combined,
-were sufficient for establishing principles applicable to the motion of
-cannon-balls, without making a new series of experiments: and with what
-labour and judgment these have been performed, you understood by the
-account which Mr. HUTTON gave of them in his Paper.
-
-But should it now be inquired, what advantages may be derived from Mr.
-HUTTON’s experiments, for the advancement of the art of gunnery, and of
-philosophy in general? I would reply, that as to the former it may be
-sufficient to observe, that though the improvements be only such as can
-be deduced from the force of fired gun-powder; yet they are in a higher,
-more certain, and in a more general manner, than what resulted from the
-labours of Mr. ROBINS; who indeed led the way, but who made, as it were
-in miniature, those experiments which Mr. HUTTON hath executed at large,
-and which ROBINS himself wished to have made, as well as others who have
-considered the subject since his time. Now these experiments, though made
-by Mr. HUTTON with cannon-balls of a small size, may nevertheless form
-just conclusions when applied to cannon-shot of the largest size. And
-such conclusions inform us of the real force of powder when fired, either
-in a cannon or a mortar, impelling a ball or bomb of a given weight;
-that is, they discover with what velocity a given quantity of powder
-drives those projectiles in a second, or in any other assigned portion
-of time. They also shew the law of variation in the velocity arising
-from different quantities of powder, with the same weight of metal,
-and likewise that law which takes place upon using balls of different
-weights. Further, they point out the advantage obtained by diminishing
-the windage in cannon, and teach us how we may increase the weight of
-the shot in the same piece, by making it of a cylindrical form, instead
-of a spherical: by this device, a smaller ship may be enabled to do the
-execution of a larger one. And experiments of the same kind will also
-determine the just length of cannon for shooting farthest with the same
-charge of powder.
-
-Lastly, it is from these experiments, or from others that may be made
-after the like manner, we are instructed how to answer every question
-relative to military projectiles, except such as depend on the resistance
-of the air to bodies moving swiftly in it. This indeed is a consideration
-which leaves room for greater improvement in the art, and for conferring
-fresh honours on those, who, like Mr. HUTTON, shall have opportunities
-and abilities for continuing and perfecting this very curious and useful
-inquiry.
-
-As to the advantages accruing to philosophy from the labours both of
-Mr. ROBINS and Mr. HUTTON, speak they not for themselves? The sciences
-of motion and pneumatics are promoted by them; and of what avail their
-perfection would be for the farther interpretation of nature, you need
-not be informed. In fine, we have here before us, in these experiments,
-the surest test of our advancement in true knowledge, which is, the
-improvement of a liberal art, and the enlargement of the powers of man
-over the works of creation.
-
-Some however may think, that the objects of this society are the arts of
-peace alone, not those of war, and that considering how numerous and how
-keen the instruments of death already are, it would better become us to
-discourage than to countenance their farther improvement. These naturally
-will be the first thoughts of the best disposed minds. But when upon
-a closer examination we find, that since the invention of arms of the
-quickest execution, neither battles nor sieges have been more frequent
-nor more destructive, indeed apparently otherwise; may we not thence
-infer, that such means as have been employed to sharpen the sword, have
-tended more to diminish than to increase the number of its victims, by
-shortening contests and making them more decisive. I shall not however
-insist on maintaining so great a paradox; but only surmise, that
-whatever State would adopt the Utopian maxims, and proscribe the study of
-arms, would soon, I fear, become a prey to those who best knew how to use
-them. For yet, alas! far seem we to be removed from those promised times,
-_when nation shall not lift up sword against nation, neither shall they
-learn war any more_!
-
-       *       *       *       *       *
-
-_Here ended the President’s Discourse: after which he turned to Mr.
-HUTTON, and said_,
-
-You have heard, Sir, the account I have given of the rise and progress
-of the _theory of gunnery_, and of your improvement of it; a recital,
-which by no means would have done either you or the subject justice, had
-it been addressed to any other audience than to the present. But as my
-intention was only briefly to recall to the memory of these gentlemen
-what they knew of this subject, antecedently to your Paper, and to remind
-them of the result of your experiments, I flatter myself I have said
-what was sufficient on the occasion; being now authorized by them to
-deliver into your hand this medal, as the perpetual memorial of their
-approbation. And let me add, Sir, that they make you this present with
-the more cordial affection, as by your other ingenious and valuable
-communications they are assured, not only of your talents, but of your
-zeal, for promoting the interests and honour of their Institution.
-
-[Illustration]
-
-
-
-
-FOOTNOTES
-
-
-[1] 2 Chron. xxvi. 15.
-
-[2] M. FOLARD.
-
-[3] The chief exception that occurs to this general remark, is the rapid
-progress which in that age COPERNICUS made in astronomy; who was not
-indeed an Italian, but was supposed to have profited by his early travels
-into Italy, which he enlightened afterwards by his admirable discoveries.
-
-[4] See MONTUCLA, Hist. des Mathem. vol. I. p. 623.
-
-[5] Those were _La Nuova Scientia_, and _Quesiti ed Inventioni diverse_.
-
-[6] Published at London, A. 1588.
-
-[7] He was born in the year 1564; but few if any of his works were
-published till after the year 1600, and his dialogues on motion not
-before 1638.
-
-[8] See his 4th Dialogue on Motion.
-
-[9] See Hist. de l’Academ. Roy. des Sciences, A. 1707.
-
-[10] In the year 1683, see Hist. de l’Acad. R. des Sci. A. 1707.
-
-[11] Hist. de l’Acad. R. des Sc. A. 1707, under the article _Mechanique_.
-
-[12] Viz. in 1674.
-
-[13] See his treatise _To hit a Mark_, published in 1690.
-
-[14] Philos. Trans. No. 179, p. 20.
-
-[15] In the year 1687.
-
-[16] NEWTON, Princip. Mathem. lib. ii. sect. 7.
-
-[17] Eloge de NEWTON.
-
-[18] Discours de la Cause de la Pesanteur. Leide, 1690.
-
-[19] Mem. de l’Acad. R. des Sc. A. 1716.
-
-[20] DAN. BERNOULLI, Comment. Acad. Petropol. T. 2. & 3.
-
-[21] It is also much to the honour of Mr. ROBINS, that his writings on
-this subject have been translated into foreign languages by men that were
-the best judges of their merit. I need only name MM. EULER, and LE ROY.
-
-
-
-
-ERRATA.
-
-
-    Page 4. l. 5. _for_ this _read_ the
-        16.    9. _for_ combate _read_ combat
-        20.   17. _for_ tract _read_ track
-        26.   last line of the note, _for_ M. M. _read_ MM.
-
-Transcriber’s Note: The errata have been corrected.
-
-*** END OF THE PROJECT GUTENBERG EBOOK A DISCOURSE ON THE THEORY OF
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-<p style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of A discourse on the theory of gunnery, by John Pringle</p>
-<div style='display:block; margin:1em 0'>
-This eBook is for the use of anyone anywhere in the United States and
-most other parts of the world at no cost and with almost no restrictions
-whatsoever. You may copy it, give it away or re-use it under the terms
-of the Project Gutenberg License included with this eBook or online
-at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you
-are not located in the United States, you will have to check the laws of the
-country where you are located before using this eBook.
-</div>
-
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Title: A discourse on the theory of gunnery</p>
-<p style='display:block; margin-left:2em; text-indent:0; margin-top:0; margin-bottom:1em;'>Delivered at the anniversary meeting of the Royal Society, November 30, 1778</p>
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: John Pringle</p>
-<p style='display:block; text-indent:0; margin:1em 0'>Release Date: October 8, 2022 [eBook #69116]</p>
-<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p>
-  <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: deaurider and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</p>
-<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK A DISCOURSE ON THE THEORY OF GUNNERY ***</div>
-
-<p><span class="pagenum"><a id="Page_1"></a>[1]</span></p>
-
-<h1><span class="smaller">A</span><br />
-<span class="gesperrt">DISCOURSE</span><br />
-<span class="smaller gesperrt">ON THE</span><br />
-<i>Theory of Gunnery</i>.</h1>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<p><span class="pagenum"><a id="Page_2"></a>[2]</span></p>
-
-<p class="titlepage larger"><span class="smaller">A</span><br />
-<span class="gesperrt">DISCOURSE</span><br />
-<span class="smaller gesperrt">ON THE</span><br />
-<i>Theory of Gunnery</i>.</p>
-
-<p class="center"><span class="allsmcap">DELIVERED AT THE</span><br />
-Anniversary Meeting of the <span class="smcap">Royal Society</span>,<br />
-November 30, 1778.</p>
-
-<p class="titlepage">By Sir <span class="gesperrt">JOHN PRINGLE</span> Baronet.</p>
-
-<p class="titlepage allsmcap">PUBLISHED BY THEIR ORDER.</p>
-
-<div class="figcenter titlepage illowp100" style="max-width: 12.5em;">
-  <img class="w100" src="images/illus1.jpg" alt="" />
-</div>
-
-<p class="titlepage">LONDON:<br />
-PRINTED FOR THE ROYAL SOCIETY.<br />
-MDCCLXXVIII.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-
-<p><span class="pagenum"><a id="Page_3"></a>[3]</span></p>
-
-<div class="figcenter illowp100" style="max-width: 37.5em;">
-  <img class="w100" src="images/illus2.jpg" alt="" />
-</div>
-
-</div>
-
-<p class="indent4"><span class="smcap">Gentlemen</span>,</p>
-
-<p class="dropcap">Among the several experiments communicated to
-the society, during the course of the preceding
-year, none seeming so much to engage your attention,
-as those contained in the Paper, intituled, <i>The force of
-fired gun-powder, and the initial velocity of cannon-balls,
-determined by experiments</i>: with much pleasure therefore
-I acquaint you, that, on account of the pre-eminence
-of that communication, your Council have judged
-the author, Mr. <span class="smcap">Charles Hutton</span>, worthy of the honour
-of the annual medal, instituted on the bequest of Sir
-<span class="smcap">Godfrey Copley</span> Baronet, for raising a laudable emulation
-among men of genius, in making experimental
-inquiries. But, as on former occasions, so now, your
-Council, waving their privilege of determining the
-choice, have acted only as a select number deputed by<span class="pagenum"><a id="Page_4"></a>[4]</span>
-you, to prepare matters for your final decision. I come
-then, on their part, briefly to lay before you the state
-of the <i>Theory of Gunnery</i>, from its rise to the time when
-its true foundation was laid, in order to evince how
-conducive those experiments may be to the improvement
-of an art of public concern, as well as to the advancement
-of <i>natural knowledge</i>, the great object of
-your institution. And if, upon a review of the subject,
-you shall entertain no less favourable an opinion of Mr.
-<span class="smcap">Hutton</span>’s performance, than what your Council have
-done, it is their earnest request that you would enhance
-the value of this prize, by authorizing your President
-to present it to our ingenious brother in your name.</p>
-
-<p>Artillery (in the large acceptation of the
-term) took place long before the invention of gun-powder.
-We trace the art to the remotest antiquity,
-since the Sacred Records acquaint us, that one of the
-kings of Judah, eight hundred years before the Christian
-æra, erected on the towers and bulwarks of Jerusalem
-engines of war, the contrivance of ingenious
-men, for shooting arrows and great stones for the defence
-of that city<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a>. Such machines were afterwards<span class="pagenum"><a id="Page_5"></a>[5]</span>
-known to the Greeks and Romans by the names of
-<i>balista</i>, <i>catapulta</i> and others, which had amazing
-powers, and were not less terrible in their effects than
-the cannon and mortars of the moderns. It appears
-that the <i>balista</i> was contrived to shower volleys of darts
-and arrows of a very large size upon the enemy, whilst
-the <i>catapulta</i> or <i>onagra</i> (as it was otherwise called)
-was fitted not only for that purpose, but for discharging
-stones of an enormous weight; I might say <i>rocks</i>,
-since some of them are reported to have weighed several
-hundred pounds. Batteries composed of numerous
-pieces of that kind of artillery, nothing could withstand.
-Yet, if we are rightly informed, their sole principle of
-motion consisted in the spring of a strongly-twisted
-cordage, made of animal substances singularly tough
-and elastic. These warlike instruments continued, not
-only during the time of the Roman empire, but to the
-12th and 13th centuries, as we find from history; nor
-indeed is it probable that they were totally laid aside,
-till gun-powder and the modern ordnance, attaining a
-good degree of perfection, superseded their use. The very
-intelligent commentator of <span class="smcap">Polybius</span><a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> is of opinion,
-that the military art rather lost than gained by the exchange<span class="pagenum"><a id="Page_6"></a>[6]</span>
-of the <i>catapulta</i> for the mortar: but however
-that point may be determined in speculation, it is not
-likely that the ancient <i>tormenta militaria</i> will ever be
-revived; but that all nations will keep to the art of
-gunnery and study how to improve it; that is, they will
-adhere to a system of artillery, wherein the moving
-power depends on the expansive force of gun-powder,
-or of some other substance of a similar nature.</p>
-
-<p>Upon the first application of this principle to the purposes
-of war, nothing perhaps was less thought of
-than to assist so empirical a practice by scientific rules;
-for, however aiding in these matters the ancient mechanicians
-might have been, who, like <span class="smcap">Archimedes</span>, had
-invented or perfected some of the <i>balistic</i> machines, no
-praise seemed now due to the mathematicians for either
-the discovery or improvement of the new artillery. In
-fact, we find the practice of the art had subsisted about
-200 years, before any geometer considered it as one that
-admitted a theory, or at least such a theory as was
-grounded on geometry.</p>
-
-<p>It seems but just to trace and commemorate the inventors
-of the ingenious arts which furnish matter for<span class="pagenum"><a id="Page_7"></a>[7]</span>
-discourses on these occasions; and not only the main inventors,
-but even those who first turned their thoughts
-upon the subject: for, though such men may not have
-produced any thing perfect, yet they may have suggested
-ideas to others of a less inventive, but of a more
-executive genius, and who, unprovided with those
-hints, would never have made any notable discovery.
-I must therefore observe, that the <i>Italians</i> were the first
-who emerged out of those thick clouds of ignorance and
-barbarism which had so long overspread this quarter of
-the world. They profited by the unhappy fate of Constantinople;
-for by liberally receiving the learned emigrants
-on that distressful occasion, they were largely repaid
-by their arts and sciences, and still more abundantly
-by their language, whereby they were enabled to read and
-to translate those ancient manuscripts, which the Greeks
-had saved out of the wreck of their country. The art
-of printing, which was established soon after, was the
-means of quickly disseminating those treasures of knowledge,
-and concurred with the fall of the eastern empire
-to form an epoch for the advancement of learning, unparalleled
-in the annals of letters.</p>
-
-<p>The end of the 15th century, and the whole of the
-16th, were chiefly employed by the Italians in the study<span class="pagenum"><a id="Page_8"></a>[8]</span>
-and in the translation of the old Greek authors. The
-geometry of the ancient Greeks, as well as the arithmetic
-in numbers and species of the Arabians, were cultivated;
-but both remained, as it were, sciences by themselves,
-unassisting to, or at best but weak and reluctant
-auxiliaries to the philosophy of the schools: and indeed
-how could the abstracted doctrines of numbers and
-quantities be strained to co-operate with a system, in
-which neither the laws of motion, nor any but the superficial,
-and often delusive properties of matter, were to
-be met with? The genius of the Greeks, all acute and
-brilliant as it was, had never been properly directed to
-the interpretation of nature, and was indeed unfit (as
-Lord Bacon pronounced) for a study that made so slow
-and painful a progress, by re-iterated and varied experiments
-and observations. It was no wonder then, if the
-<i>mixed mathematics</i>, as they are called, descended to the
-moderns in a state no-wise corresponding to the elegance
-and certainty of those parts of the science which
-were elementary and pure; and that those mixed parts
-should have been found defective and erroneous, in proportion
-(if I may so express myself) to the physical considerations
-that were to be taken into the inquiry. The
-imperfection of the ancients, with regard to natural<span class="pagenum"><a id="Page_9"></a>[9]</span>
-philosophy, was not perceived at that time; nay, at the
-period we are treating of, the learned were firmly persuaded
-of the contrary, and that all that was wanting
-to be known concerning the laws of nature, and the
-properties of matter, was to be taken either directly, or
-by deduction, from the physics of <span class="smcap">Aristotle</span>. It was
-not till the 17th century was somewhat advanced, that
-men of science began to listen to Lord <span class="smcap">Bacon</span> and <span class="smcap">Galileo</span>,
-the great founders of the experimental and the
-true philosophy.</p>
-
-<p>Mean while, in the beginning of the 16th century,
-unqualified as the Italians then were for entering upon
-physico-mathematical inquiries<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a>, they nevertheless
-made the attempt, and in particular took the theory of
-projectiles into consideration. Some imagined that a
-body impelled with violence, such as a ball discharged
-from a cannon, moved in a right line till the force was
-spent, and that then it fell in another right line perpendicularly
-to the earth. Upon this principle, absurd
-as it was, we find one of the earliest authors grounding<span class="pagenum"><a id="Page_10"></a>[10]</span>
-his whole theory of gunnery<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a>; whilst others, dissenting
-from his hypothesis, admitted only the straight
-line, in which the ball moved for some time after coming
-out of the piece, and that other straight line in
-which it fell to the ground; but asserted that these two
-were connected by a curve line, and that this curve was
-the segment of a circle. <span class="smcap">Nicolas Tartaglia</span> of Brescia,
-a mathematician of the first rank in those days,
-and still celebrated for his improvements in algebra,
-hath been supposed to be the author of this doctrine, no
-less erroneous than the former, and for which two of his
-books have been quoted<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a>. Those I have never seen;
-but from another of his works, professedly written on
-this subject, and translated into English under the title
-of <i>Colloquies concerning the art of shooting in great and small
-pieces of artillery</i><a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a>, him I find, contrary to the opinion
-of his contemporaries, maintaining that no part of
-the track of a cannon-ball is in a right line, though the
-curvature in the first part of its flight be so small, that
-it needeth not to be attended to. But <span class="smcap">Tartaglia</span> is
-far from supposing, that the line in question hath any
-relation to a <i>parabola</i>, or to any regular curve. It<span class="pagenum"><a id="Page_11"></a>[11]</span>
-would seem then, that if this mathematician had at first
-been so far mistaken, as to fancy that some part of the
-course of a projectile was in a straight line, he had afterwards
-changed his opinion, and was perhaps singular
-in what he finally embraced.</p>
-
-<p>From numerous instances one would imagine, that
-in those days, so far were men of science from making
-experiments themselves, that they even shut their eyes
-against what chance would have presented to their sight.
-For, whoever had minded the roving shot of an arrow,
-the flight of a stone from a sling, or had attended to a
-stream of water issuing from the spout of a cistern,
-might have been convinced, that the path of every projectile
-was in a continued curve, whatever little he
-otherwise knew concerning the properties of that one.</p>
-
-<p>But had the observation of the philosophers gone so
-far, they had still been at a distance from the truth.
-They might have perceived a likeness between the
-track of those bodies in motion and a parabola, and concluded,
-from analogy, that all projectiles delineated that
-curve in the air; but they could never have realized
-their conjectures by mathematical demonstration, without<span class="pagenum"><a id="Page_12"></a>[12]</span>
-previously knowing the law of <i>acceleration</i> in falling
-bodies: a discovery reserved for the next century,
-and for <span class="smcap">Galileo</span><a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a>, one of the greatest ornaments of it.</p>
-
-<p>It was he who first investigated the effects of <i>gravity</i>
-on falling bodies, and upon that foundation demonstrated,
-that all projectiles would move in a parabola in
-a non-resisting medium. And as he made little account
-of the resistance of the air, whose properties were then
-imperfectly known, he proved that a ball shot horizontally
-would, in its flight, describe half a parabola; and
-when the piece had an elevation above the horizon, the
-ball would describe a whole parabola, supposing it to
-fall on the plane of the battery. By the same method
-of reasoning he shewed, that whatever the ranges of the
-projected body, or the elevations of the piece were, the
-ball would still trace that curve line, of a greater or lesser
-amplitude, by the time it descended to the level of the
-place from whence it came.</p>
-
-<p>Thus far went <span class="smcap">Galileo</span>, confining his projections to
-the horizontal plane of the battery; but <span class="smcap">Torricelli<span class="pagenum"><a id="Page_13"></a>[13]</span></span>
-his disciple soon after carried the theory farther, by
-tracing the shot to its fall, whether that place was above
-or below the plane; and still found, by geometrical deductions,
-that it flew in a parabola of a larger or a
-smaller amplitude, according to the angle of elevation
-of the piece, and the strength of the powder.</p>
-
-<p>Various and numerous had been the disputes in Italy
-about the laws of motion in general, and especially
-about those of projectiles, from the time the mathematicians
-had begun the inquiry, till the publication of the
-dialogues of <span class="smcap">Galileo</span> on that subject (a space of upwards
-of a hundred years) but from that period, so
-evident did his demonstrations appear, that all contest
-ceased, and every man of science was convinced, that all
-projectiles moved in the track which he had discovered.
-For, as to the resistance of the air, which he had not
-passed unnoticed (as <span class="smcap">Galileo</span> himself had been the first,
-at least of the moderns, who started the notion of the
-weight of the air and the pressure of the atmosphere)
-yet so thin and so yielding did they esteem that fluid to
-be, that they were assured it could occasion no sensible,
-at least no material, deviation from that curve. As they
-had the principle from <span class="smcap">Galileo</span>, so they believed themselves<span class="pagenum"><a id="Page_14"></a>[14]</span>
-warranted by that respectable author, not to fear
-from that cause any objection, which he himself had
-suggested, but had removed. <i>Among these projectiles</i>
-(says he) <i>which we make use of, if they are of a heavy
-matter and a round form; nay if they are of a lighter
-matter, and have a cylindrical form, such as arrows shot
-from bows, their track or path will not sensibly decline from
-the curve of a parabola</i><a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a>.</p>
-
-<p>Here then was the theory of gunnery laid, in appearance,
-on the most solid foundation. And thus far
-the Italians having proceeded, they seemed to have taken
-leave, and to commit the subject to other nations, whose
-greater power, or greater ambition, was more likely to
-make them avail themselves of the perfection of a military
-art, than their instructors. We had reason
-therefore to expect, that a neighbouring state, intent
-upon the advancement of the arts and sciences in general,
-would not fail to give particular attention to
-those that should appear most subservient to its grandeur.
-Accordingly we find, that our sister-society of
-that kingdom had not been many years established,
-when an ingenious member of that illustrious Body, not<span class="pagenum"><a id="Page_15"></a>[15]</span>
-questioning the soundness of the Galilean principle in
-regard to projectiles, in the year 1677, proposed to the
-academy, as a problem for the improvement of artillery,
-how to direct a piece (suppose a mortar) so as to
-make the shot fall where one had a mind; or in the
-common expression, <i>to hit a mark</i>, the strength of the
-powder being given<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a>. This thought met with general
-approbation, and so far were the academy from raising
-any difficulty about the obstruction which the air might
-occasion to a body moving with so much velocity in it,
-that we do not find the making experiments on that
-head was considered by them as an essential step to the
-solution; but that their principal geometers straightway
-set about solving the problem as it had been announced
-to them, some following one method, some another,
-and all upon the supposition of a projectile moving in
-the line of a parabola. But <span class="smcap">M. Blondel</span>, who had
-been the proposer, and who more particularly had studied
-the question, composed a large volume on the subject,
-which he published a few years after<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a>, under the
-title of <i>L’Art de jetter les Bombes</i>; a performance much
-celebrated at the time, and that continued in no small<span class="pagenum"><a id="Page_16"></a>[16]</span>
-request long after, as containing, besides his own, the
-labours of several other members of that society of the
-most distinguished merit. So many, and such hands
-concurring in framing this work, it was no wonder that
-the learned throughout Europe were confirmed by it in
-the Galilean theory; and the more as <span class="smcap">M. Blondel</span> had
-obviated the only objection they supposed could be
-made to it, the <i>resistance of the air</i>, which he had taken
-care expressly to mention, and so to combat as to persuade
-the reader, that the retardation arising from
-that cause was so inconsiderable as to be of no account
-in the practice.</p>
-
-<p>This illusion about the small or non-resistance of the
-air to bodies rapidly moving in it, was so prevalent at
-the end of the last century, and in the beginning of the
-present, that in the history of the Royal Academy for
-the year 1707, we find their worthy and most accomplished
-secretary, after taking notice of the joint labours
-of so many able mathematicians concerned in <span class="smcap">Blondel</span>’s
-publication, venturing to say, <i>it did not appear that
-any thing was then wanting for the practice of the art</i> [of
-Gunnery] <i>except perhaps perfecting the instruments for
-pointing a cannon or mortar ... but that geometry<span class="pagenum"><a id="Page_17"></a>[17]</span>
-had done its part, so to speak, with regard to practice</i>.
-&amp;c.<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a></p>
-
-<p>But far be it from our intention to relate the imperfections
-of others, in order to raise ourselves by the comparison.
-Candour requires of us not only to acknowledge,
-that in this country, as to the point in question,
-we did not surpass our neighbours; but ingenuously to
-own that, on the contrary, we were perhaps more liable
-to exception. For, some years before <span class="smcap">Blondel</span>’s work
-appeared<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a>, a treatise was published by one of our
-own artillerists, <span class="smcap">Anderson</span> (a person of eminence in
-his profession) intituled <i>The genuine use and effects of
-the gun</i>, in which the author strenuously supports the
-Galilean theory; nor do we learn he was ever contradicted
-among us, although he undertook to answer all
-those who should make objections to it. Nay, when
-he had an opportunity afterwards of making experiments
-on the ranges of bombs, and by those trials was
-assured that their flight was not in a parabola; yet so
-far was he from ascribing the deviation from that figure
-to the resistance of the air, that he had recourse to an<span class="pagenum"><a id="Page_18"></a>[18]</span>
-hypothesis, repugnant to all the laws of motion, to salve
-appearances, and to reconcile those experiments with
-his former doctrine<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a>.</p>
-
-<p>And did not Dr. <span class="smcap">Halley</span>, so long the ornament of
-this society, communicate in the year 1686 a Paper,
-which he calls <i>A discourse concerning gravity</i>, in which,
-treating of the motion of projectiles, he says, that being
-aware of the deflexion from the parabolic curve that
-might be occasioned by the resistance of the air, he had
-made some experiments, even with cannon-balls, to estimate
-the force of that resistance; yet conclude, <i>That in
-large shot of metal, whose weight many thousand times surpassed
-that of air, and whose force is very great, in proportion
-to the surface wherewith they press thereupon, this
-opposition was not discernible</i>. And again, <i>Though in
-small and light shot, the opposition of the air ought and must
-be accounted for; yet in shooting great and weighty bombs,
-there need be very tittle allowance made; and so these rules</i>
-[those, to wit, grounded on the principle of <span class="smcap">Galileo</span>]
-<i>may be put in practice to all intents and purposes, as if
-this impediment</i> [the resistance of the air] <i>were absolutely<span class="pagenum"><a id="Page_19"></a>[19]</span>
-removed</i><a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a>. Such conclusions, which we now find to
-be erroneous, were the less to be expected from so
-eminent a person, as they argued too much haste to
-finish a theory, that was to be made subservient to
-present use.</p>
-
-<p>It might indeed have been expected, that men of
-science applying themselves to this study, would have
-been sooner awakened to the consideration of the great
-opposition of the air, by the <i>Principia</i> of <span class="smcap">Newton</span>, published
-a little after this Paper of <span class="smcap">Halley</span>’s<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a>. For in
-that excellent work the illustrious author had demonstrated,
-that the curve described by a projectile, in a
-strongly resisting medium, differed much from a parabola,
-and that the resistance of the air was great enough
-to make the difference between the curve of projection
-of heavy bodies and a parabola far from being insensible,
-and therefore too considerable to be neglected.</p>
-
-<p>Have we not then less to plead for not attending to
-the <i>Principia</i> of <span class="smcap">Newton</span> in this article<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a>, than the mathematicians
-of other nations, who, as M. de <span class="smcap">Fontenelle<span class="pagenum"><a id="Page_20"></a>[20]</span></span>
-observes<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">[17]</a>, partly from the difficulty of understanding
-that concise and profound work, and partly
-from a misapprehension of its tendency (which they
-fancied was to revive the exploded doctrine of <i>occult
-qualities</i>) were late in becoming acquainted with it?
-But it is not so easy to account for their inattention to
-<span class="smcap">Huygens</span>, a known and even then a much esteemed
-author, and who indeed was second to <span class="smcap">Newton</span> alone
-in science and in genius. For he in the year 1690
-had published a treatise on <i>Gravity</i>, written in a popular
-manner, wherein he gave an account of some experiments
-he had made at Paris, and in the academy, by
-which, as well as by mathematical investigations, he
-was convinced of the truth of <span class="smcap">Newton</span>’s conclusions, in
-regard to the great opposition of the air to bodies moving
-swiftly in it; and, by consequence, believed that the
-track of all projectiles was very different from the line
-of a parabola<a id="FNanchor_18" href="#Footnote_18" class="fnanchor">[18]</a>.</p>
-
-<p>But excepting <span class="smcap">Newton</span> and <span class="smcap">Huygens</span>, the learned
-seemed universally to acquiesce in the justness and sufficiency
-of the principles of gunnery invented by <span class="smcap">Galileo</span>,<span class="pagenum"><a id="Page_21"></a>[21]</span>
-enlarged by <span class="smcap">Torricelli</span>, confirmed and reduced
-to system by <span class="smcap">Anderson</span>, <span class="smcap">Blondel</span>, <span class="smcap">Halley</span> and others;
-and so far were the theorists, in that branch of science,
-from suspecting any defect or fallacy in these principles,
-that they seemed rather to reproach the practical artillerists,
-for not profiting more by the instructions which
-they had so liberally imparted to them. Nor do we
-find that an apology was made for the empirical exercise
-of the art, by any author of note in that line,
-earlier than the sixteenth year of this century, when M.
-de <span class="smcap">Ressons</span>, a French officer of artillery, distinguished
-by the number of sieges at which he had served, by his
-high military rank, and by his abilities in his profession;
-when he, I say, thus qualified to bear testimony, presented
-a <i>memoire</i> to the Royal Academy (of which he
-was a member) importing, that <i>although it was agreed
-that theory joined to practice did constitute the perfection
-of every art, yet experience had taught him, that theory
-was of very little service in the use of mortars. That
-the work of M. <span class="smcap">Blondel</span> had justly enough described the
-several parabolic lines, according to the different degrees
-of the elevation of the piece; but that practice had convinced
-him there was no theory in the effects of gun-powder:
-for that having endeavoured, with the greatest precision,<span class="pagenum"><a id="Page_22"></a>[22]</span>
-to point a mortar agreeably to those calculations, he had
-never been able to establish any solid foundation upon
-them<a id="FNanchor_19" href="#Footnote_19" class="fnanchor">[19]</a>.</i></p>
-
-<p>Thus, after the theory of gunnery had exercised the
-genius of the learned for nearly two hundred years, and
-for almost fourscore of that time had rested on fundamentals
-which had never been contested, it was pronounced
-at once to be almost intirely useless, and that
-by one of the most competent judges. Now, whether
-it were owing to the deference due to the authority of
-that experienced artillerist, or to some other cause, I shall
-not determine, but observe, that it appears not from the
-history of the academy, that the sentiments of M. de
-<span class="smcap">Ressons</span> were at this time controverted, or any reason
-offered afterwards for the failure of the theory of projectiles
-when applied to use. Nor can I pass unnoticed
-the pause that ensued before any further attempts were
-made to improve the theory of the art, either upon the
-old principles or upon new ones, except by such authors
-as seemed ignorant of this transaction, and who of
-course were not sufficiently apprized of the inefficacy of
-the properties of the parabola for directing practice.<span class="pagenum"><a id="Page_23"></a>[23]</span>
-Or by those who were employed in speculatively investigating
-the nature of the curve traced by a ball in the
-air; a curve which began at last to be considered as one
-deviating much from the line of a parabola. Or, finally,
-by such as, having taken notice that <span class="smcap">Newton</span>’s ideas
-had not been duly attended to, endeavoured to avail
-themselves of them, and of some experiments that had
-been made by others, for proving the great opposition
-of the air to bodies of swift motion; but without ascertaining
-the degree of that resistance, or enriching the
-art by any practical rules<a id="FNanchor_20" href="#Footnote_20" class="fnanchor">[20]</a>.</p>
-
-<p>Such was the unhinged state of this part of the
-mixed mathematics, when within our memory Mr.
-<span class="smcap">Benjamin Robins</span> took cognizance of it: nor could the
-subject have fallen into abler hands, endowed as he was
-by nature with a superior genius and unwearied application.
-Mr. <span class="smcap">Robins</span> was deeply versed in geometry
-and the doctrine of numbers; but he knew the limits
-as well as the powers of both, and how insufficient they
-were for establishing any theory where matter was concerned,
-without preparing the way, by finding out the
-physical properties of that <i>matter</i>, by many and varied<span class="pagenum"><a id="Page_24"></a>[24]</span>
-experiments and attentive observation. Those who had
-hitherto treated of the foundation of gunnery, by being
-too forward in the application of their mathematics, had
-in a manner hurt the credit of that admirable science.
-They ought to have seen the necessity of minutely examining
-every circumstance which could affect the
-course of a projectile, besides that of gravity. Mr. <span class="smcap">Robins</span>
-perceived the error of his predecessors in that inquiry,
-and corrected it. Persuaded as he was from sir <span class="smcap">Isaac
-Newton</span>’s <i>Principia</i> of the great resistance of the air to
-bodies moving in it, and also of the uncertainty of the
-force of gun-powder, and of the variations in the flight
-of shot, occasioned by the unavoidable varieties in the
-make of it, and in the make of the pieces of artillery
-which discharged it; apprized, I say, of so many causes
-of aberration, he justly concluded, that the foundation
-here was at least as much an affair of physics as of geometry,
-and that if the art of throwing bombs had not
-been advanced by theory, it was not because the art admitted
-of none, but because the theory which had
-hitherto been devised had been both defective and erroneous.
-He suspected that most of the writers on
-gunnery had been deceived, in supposing the resistance
-of the air to be inconsiderable, and thence asserting the<span class="pagenum"><a id="Page_25"></a>[25]</span>
-track of all shot to be nearly in the curve of a parabola,
-by which means it came to pass that all their determinations,
-about the flight of projectiles of violent motion,
-had declined considerably from the truth. But in order
-to clear this point from every doubt, he found it necessary
-to ascertain the force of gun-powder, and by that
-step to estimate the velocity of the shot impelled by its
-explosion. That being done, he proceeded to measure
-the quickness of a musket-bullet, shot out of a given
-barrel, with a given quantity of powder; and to confirm
-the truth of his conclusions, he contrived a machine,
-by which the velocity of a bullet might be diminished
-in any given <i>ratio</i>, by being made to strike on a large
-body of a weight justly proportioned to it; whereby
-the swiftest motions, which otherwise would escape our
-examination, were to be exactly determined by these
-slower motions that had a given relation to them.
-The machine was a large wooden pendulum, which
-swung freely, but in so slow a manner, that its vibrations
-could easily be counted, whatever was the celerity of
-the bullet discharged against it. The thought was
-simple, ingenious, and incontestably his own.</p>
-
-<p><span class="pagenum"><a id="Page_26"></a>[26]</span></p>
-
-<p>He next inquired into the resistance made by the air
-to projectiles of rapid motion, and which he discovered
-to be much greater than had been supposed by any
-writer on the subject; and indeed so great, that it was
-manifest the curve described by any shot was very different
-from a parabola, and consequently that all the
-applications of the properties of that conic section to
-gunnery were so erroneous as to be totally useless. For
-by means of this pendulum, placed at different distances
-from the mouth of the piece, he clearly demonstrated
-how much a bullet, flying with a given velocity, would
-gradually lose of that motion by the opposition of the
-air: therein furnishing to the learned a signal and instructive
-instance of the fallacy of the most specious
-theories, that do not proceed hand in hand with experiments.</p>
-
-<p>I should too much exceed the just bounds of a discourse
-of this kind, were I to enter more minutely into
-the system founded by Mr. <span class="smcap">Robins</span>, confirmed and improved,
-as I find, by the labours of several of the learned
-in foreign parts of great celebrity<a id="FNanchor_21" href="#Footnote_21" class="fnanchor">[21]</a>. I shall only add,<span class="pagenum"><a id="Page_27"></a>[27]</span>
-that his performance well deserves the title he gives it
-of <i>The new principles of gunnery</i>, since the author may
-more properly be said to have invented a new science
-than to have added to an old one. And I believe I may
-venture to say, that no physico-mathematical disquisition
-hath done more honour to this country, or to the age,
-than the writings of Mr. <span class="smcap">Robins</span> on this subject, which
-have been published, partly by this Society, partly by
-himself, and partly since his death (in the collection of
-his whole mathematical tracts) by his learned friend.</p>
-
-<p>But though our worthy brother will ever be celebrated
-for being the inventor of the true principles of
-gunnery, yet it would be too flattering to his memory,
-to say he had carried the theory of this art to perfection.
-He himself was far from entertaining so high an opinion
-of his labours; nay he expressly declared, that he
-left some material points to be inquired into at more
-leisure (which other occupations and his immature death
-deprived him of) and he much regretted that he wanted
-conveniency and opportunities for making experiments
-on balls of a greater weight, than what he had used for
-ascertaining the initial velocity of them.</p>
-
-<p><span class="pagenum"><a id="Page_28"></a>[28]</span></p>
-
-<p>Much therefore are we indebted to Mr. <span class="smcap">Hutton</span>,
-who, treading in the footsteps of the deceased, hath resumed
-and prosecuted this last <i>desideratum</i>, and hath
-shewn himself not unequal to so difficult an enterprize.</p>
-
-<p>Mr. <span class="smcap">Robins</span>, for determining the initial velocity of shot,
-arising from different quantities of powder, made use of
-balls of about an ounce weight; whereas Mr. <span class="smcap">Hutton</span>,
-for the same purpose, hath employed those of different
-weights, from one pound to nearly three; or, in other
-words, Mr. <span class="smcap">Robins</span> made trial with musket-shot only,
-Mr. <span class="smcap">Hutton</span> with cannon-balls from 20 to about 50
-times heavier. This was a considerable step gained in
-a disquisition of that part of the science, in which the
-resistance of the air and other circumstances were not
-concerned; and where neither analogy alone, nor mathematical
-deductions alone, nor the two combined,
-were sufficient for establishing principles applicable to
-the motion of cannon-balls, without making a new series
-of experiments: and with what labour and judgment
-these have been performed, you understood by the account
-which Mr. <span class="smcap">Hutton</span> gave of them in his Paper.</p>
-
-<p><span class="pagenum"><a id="Page_29"></a>[29]</span></p>
-
-<p>But should it now be inquired, what advantages may
-be derived from Mr. <span class="smcap">Hutton</span>’s experiments, for the advancement
-of the art of gunnery, and of philosophy in
-general? I would reply, that as to the former it may be
-sufficient to observe, that though the improvements be
-only such as can be deduced from the force of fired
-gun-powder; yet they are in a higher, more certain, and
-in a more general manner, than what resulted from the
-labours of Mr. <span class="smcap">Robins</span>; who indeed led the way, but
-who made, as it were in miniature, those experiments
-which Mr. <span class="smcap">Hutton</span> hath executed at large, and which
-<span class="smcap">Robins</span> himself wished to have made, as well as others
-who have considered the subject since his time. Now
-these experiments, though made by Mr. <span class="smcap">Hutton</span> with
-cannon-balls of a small size, may nevertheless form just
-conclusions when applied to cannon-shot of the largest
-size. And such conclusions inform us of the real force
-of powder when fired, either in a cannon or a mortar,
-impelling a ball or bomb of a given weight; that is, they
-discover with what velocity a given quantity of powder
-drives those projectiles in a second, or in any other
-assigned portion of time. They also shew the law of
-variation in the velocity arising from different quantities
-of powder, with the same weight of metal, and likewise<span class="pagenum"><a id="Page_30"></a>[30]</span>
-that law which takes place upon using balls of different
-weights. Further, they point out the advantage
-obtained by diminishing the windage in cannon, and
-teach us how we may increase the weight of the shot
-in the same piece, by making it of a cylindrical form,
-instead of a spherical: by this device, a smaller ship
-may be enabled to do the execution of a larger one.
-And experiments of the same kind will also determine
-the just length of cannon for shooting farthest with
-the same charge of powder.</p>
-
-<p>Lastly, it is from these experiments, or from others
-that may be made after the like manner, we are instructed
-how to answer every question relative to military
-projectiles, except such as depend on the resistance
-of the air to bodies moving swiftly in it. This indeed
-is a consideration which leaves room for greater improvement
-in the art, and for conferring fresh honours
-on those, who, like Mr. <span class="smcap">Hutton</span>, shall have opportunities
-and abilities for continuing and perfecting this
-very curious and useful inquiry.</p>
-
-<p>As to the advantages accruing to philosophy from
-the labours both of Mr. <span class="smcap">Robins</span> and Mr. <span class="smcap">Hutton</span>, speak<span class="pagenum"><a id="Page_31"></a>[31]</span>
-they not for themselves? The sciences of motion and
-pneumatics are promoted by them; and of what avail
-their perfection would be for the farther interpretation
-of nature, you need not be informed. In fine, we have
-here before us, in these experiments, the surest test of
-our advancement in true knowledge, which is, the improvement
-of a liberal art, and the enlargement of the
-powers of man over the works of creation.</p>
-
-<p>Some however may think, that the objects of this
-society are the arts of peace alone, not those of war, and
-that considering how numerous and how keen the instruments
-of death already are, it would better become
-us to discourage than to countenance their farther improvement.
-These naturally will be the first thoughts
-of the best disposed minds. But when upon a closer
-examination we find, that since the invention of arms
-of the quickest execution, neither battles nor sieges have
-been more frequent nor more destructive, indeed apparently
-otherwise; may we not thence infer, that such
-means as have been employed to sharpen the sword,
-have tended more to diminish than to increase the number
-of its victims, by shortening contests and making
-them more decisive. I shall not however insist on<span class="pagenum"><a id="Page_32"></a>[32]</span>
-maintaining so great a paradox; but only surmise, that
-whatever State would adopt the Utopian maxims, and
-proscribe the study of arms, would soon, I fear, become
-a prey to those who best knew how to use them. For
-yet, alas! far seem we to be removed from those promised
-times, <i>when nation shall not lift up sword against
-nation, neither shall they learn war any more</i>!</p>
-
-<hr class="tb" />
-
-<p><i>Here ended the President’s Discourse: after which he
-turned to Mr. <span class="smcap">Hutton</span>, and said</i>,</p>
-
-<p>You have heard, Sir, the account I have given of
-the rise and progress of the <i>theory of gunnery</i>, and of
-your improvement of it; a recital, which by no means
-would have done either you or the subject justice, had
-it been addressed to any other audience than to the present.
-But as my intention was only briefly to recall to
-the memory of these gentlemen what they knew of this
-subject, antecedently to your Paper, and to remind them
-of the result of your experiments, I flatter myself I have
-said what was sufficient on the occasion; being now authorized<span class="pagenum"><a id="Page_33"></a>[33]</span>
-by them to deliver into your hand this medal,
-as the perpetual memorial of their approbation. And
-let me add, Sir, that they make you this present with
-the more cordial affection, as by your other ingenious
-and valuable communications they are assured, not only
-of your talents, but of your zeal, for promoting the interests
-and honour of their Institution.</p>
-
-<div class="figcenter illowp100" style="max-width: 15.625em;">
-  <img class="w100" src="images/illus3.jpg" alt="" />
-</div>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="footnotes">
-
-<div class="chapter">
-
-<h2 class="nobreak">FOOTNOTES</h2>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> 2 Chron. xxvi. 15.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> <span class="smcap">M. Folard.</span></p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> The chief exception that occurs to this general remark, is the rapid
-progress which in that age <span class="smcap">Copernicus</span> made in astronomy; who was not indeed
-an Italian, but was supposed to have profited by his early travels into Italy,
-which he enlightened afterwards by his admirable discoveries.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> See <span class="smcap">Montucla</span>, Hist. des Mathem. vol. I. p. 623.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> Those were <i>La Nuova Scientia</i>, and <i>Quesiti ed Inventioni diverse</i>.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> Published at London, A. 1588.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> He was born in the year 1564; but few if any of his works were published
-till after the year 1600, and his dialogues on motion not before 1638.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> See his 4th Dialogue on Motion.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> See Hist. de l’Academ. Roy. des Sciences, A. 1707.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> In the year 1683, see Hist. de l’Acad. R. des Sci. A. 1707.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> Hist. de l’Acad. R. des Sc. A. 1707, under the article <i>Mechanique</i>.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> Viz. in 1674.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> See his treatise <i>To hit a Mark</i>, published in 1690.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> Philos. Trans. No. 179, p. 20.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> In the year 1687.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> <span class="smcap">Newton</span>, Princip. Mathem. lib. ii. sect. 7.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_17" href="#FNanchor_17" class="label">[17]</a> Eloge de <span class="smcap">Newton</span>.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_18" href="#FNanchor_18" class="label">[18]</a> Discours de la Cause de la Pesanteur. Leide, 1690.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_19" href="#FNanchor_19" class="label">[19]</a> Mem. de l’Acad. R. des Sc. A. 1716.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_20" href="#FNanchor_20" class="label">[20]</a> <span class="smcap">Dan. Bernoulli</span>, Comment. Acad. Petropol. T. 2. &amp; 3.</p>
-
-</div>
-
-<div class="footnote">
-
-<p><a id="Footnote_21" href="#FNanchor_21" class="label">[21]</a> It is also much to the honour of Mr. <span class="smcap">Robins</span>, that his writings on this
-subject have been translated into foreign languages by men that were the best
-judges of their merit. I need only name <span class="allsmcap">MM.</span> <span class="smcap">Euler</span>, and <span class="smcap">le Roy</span>.</p>
-
-</div>
-
-</div>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-
-<h2 class="nobreak">ERRATA.</h2>
-
-</div>
-
-<table>
-  <tr>
-    <td>Page</td>
-    <td class="tdr"><a href="#Page_4">4.</a></td>
-    <td class="tdr">l.</td>
-    <td class="tdr">5.</td>
-    <td><i>for</i> this <i>read</i> the</td>
-  </tr>
-  <tr>
-    <td></td>
-    <td class="tdr"><a href="#Page_16">16.</a></td>
-    <td></td>
-    <td class="tdr">9.</td>
-    <td><i>for</i> combate <i>read</i> combat</td>
-  </tr>
-  <tr>
-    <td></td>
-    <td class="tdr"><a href="#Page_20">20.</a></td>
-    <td></td>
-    <td class="tdr">17.</td>
-    <td><i>for</i> tract <i>read</i> track</td>
-  </tr>
-  <tr>
-    <td></td>
-    <td class="tdr"><a href="#Page_26">26.</a></td>
-    <td colspan="3">last line of the note, <i>for</i> <span class="allsmcap">M. M.</span>
-    <i>read</i> <span class="allsmcap">MM.</span></td>
-  </tr>
-</table>
-
-<p class="transnote">Transcriber’s Note: The errata have been corrected.</p>
-
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