Tracts, mathematical and philosophical: By Charles Hutton, ... Vol.1.
[Page]
TRACTS, MATHEMATICAL AN [...] PHILOSOPHICAL.
F. R. S. OF LOND. AND EDINB. MEMB. OF THE SOCIETY OF SCIENCES OF HOLLAND, AND PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, WOOLWICH.
LONDON: PRINTED FOR G. G. J. AND J. ROBINSON, PATERNOSTER ROW.
TO HIS GRACE CHARLES, DUKE OF RICHMOND, LENNOX, AND AUBIGNY, &c. &c. &c.
[Page]MASTER GENERAL OF THE ORDNANCE, (UNDER WHOSE AUSPICES THE EXPERIMENTS IN GUNNERY WERE MADE)
THESE TRACTS ARE RESPECTFULLY INSCRIBED, BY
HIS GRACE'S MOST HUMBLE AND MOST OBEDIENT SERVANT, THE AUTHOR.
PREFACE.
[Page]THE Author preſumes to lay the following Tracts before the Public with the greater confidence, as he hopes theſe productions of his leiſure will be found to bear a due relation to the engagements of his official duty.
The preference given of late, even among profeſſed philoſophers, to ſtudies of a leſs abſtract kind, has too frequently diverted the purſuits of mathematicians into paths leſs ſuited to their talents, from the deſire of a vain and fleeting popularity, inſtead of the more laudable ambition of making real improvements in the ſciences which they had profeſſed to cultivate. The humble conſciouſneſs which the author has ever entertained of the limits of his own abilities, has, he hopes, preſerved him from this common and pernicious vanity. However ſolicitous to extend and diverſify his own acquirements, he can only hope to add, and that a little, to the public ſtock of knowledge, in thoſe parts of ſcience to which his early [Page vi] habits, and ſubſequent occupations, have led him peculiarly to conſecreate his ſtudies.
The very honourable diſtinction paid to the author by the Royal Society, for his former experiments in gunnery, as well as their general indulgence to his attempts in other mathematical ſubjects, would perhaps have given an obvious deſtination to theſe papers, had he not thought their publication in a collective form, better adapted, from the connection of their ſubject, to extend their utility.
The firſt ſix tracts in this volume will be found to have an obvious connection in reſpect to their ſubject; having all of them a tendency either to illuſtrate the hiſtory, or improve the theory of that ſpecies of mathematical quantities called Series. The particular ſubjects of theſe tracts are ſufficiently diſcuſſed in the introduction to each of them, reſpectively, to render any previous detail unneceſſary in this place. The author hopes, however, they will be thought to have ſome claim to the merit of invention; and that their utility will be readily recognized by thoſe who are converſant in theſe ſubjects.
The ſeventh and eighth tracts are rather detached in their nature, relating to ſubjects purely geometrical. It is hoped, however, that the former of them, being an inveſtigation of ſome new and curious properties of the ſphere and cone, which have always been a fruitful and favourite ſource of exerciſe to geometricians, will be both acceptable and uſeful [Page vii] to thoſe who are engaged in ſimilar ſpeculations. The latter problem, concerning the geometrical diviſion of a circle, having hitherto been deemed impracticable, the ſolution of it is here given, it is preſumed, for the firſt time.
The largeſt, and, in the author's opinion, the moſt important of theſe tracts, is the ninth, or laſt; the main purpoſe of which is altogether practical, though founded in a very ſubtle and complex theory.
Though the late excellent Mr. Robins firſt ſhewed the importance of this theory, and invented a very curious mechanical apparatus for the experiments which he made to verify it, the author is perſuaded that none have been hitherto made with cannon balls ſo completely, as thoſe here related and deſcribed; and that theſe are the firſt from which the Data for determining the reſiſtance of the medium can be accurately derived. It has been the author's great object, next to the accuracy of the experiments, and the full and preciſe deſcription of them, to ſimplify the theorems deduced from them, or from the theory itſelf; of which an example may be found in the new rule given for the velocity of the ball. And it is preſumed that the table of the correſponding Data, namely, of the Dimenſions and Elevation of the gun, and the Range, Velocity and Time of ſlight of the ball, is now ſo accurately framed, and ſo perſpicuous, that the ſeveral caſes of gunnery may be very certainly and eaſily referred to it; and rules of practice, adapted to the common purpoſes of the artilleriſt, may be very readily formed upon theſe principles.
1. MATHEMATICAL TRACTS, &c.
[Page]1.1. TRACT I. A Diſſertation on the Nature and Value of Infinite Series.
1. ABOUT five years ſince I diſcovered a very general and eaſy method of valuing ſeries whoſe terms are alternately poſitive and negative, which equally applies to ſuch ſeries, whether they be converging, or diverging, or their terms all equal; together with ſeveral other properties relating to certain ſeries: and as we ſhall have occaſion to deliver ſome of thoſe matters in the courſe of theſe tracts, I ſhall take this opportunity of premiſing a few ideas and remarks on the nature and valuation of ſome of the claſſes of ſeries which form the object of thoſe communications. This is done with a view to obviate [Page 2] any miſconceptions that might, perhaps, be made concerning the idea annexed to the term value of ſuch ſeries in thoſe intended tracts, and the ſenſe in which it is there always to be underſtood; which is the more neceſſary, as many controverſies have been warmly agitated concerning theſe matters, not only of late, by ſome of our own countrymen, but alſo by others among the ableſt mathematicians in Europe, at different periods in the courſe of the preſent century; and all this, it ſeems, through the want of ſpecifying in what ſenſe the term value or ſum was to be underſtood in their diſſertations. And in this diſcourſe, I ſhall follow, in a great meaſure, the ſentiments and manner of the late famous L. Euler, contained in a ſimilar memoir of his in the fifth volume of the New Peterſburgh Commentaries, adding and intermixing here and there other remarks and obſervations of my own.
3. Now concerning the ſums of theſe ſpecies of ſeries, there have been great diſſenſions among mathematicians; ſome affirming that they can be expreſſed by a certain ſum, while others deny it. In the firſt place, however, it is evident that the ſums of ſuch ſeries as come under the firſt of theſe ſpecies, will be really infinitely great, ſince by actually collecting the terms, we can arrive at a ſum greater than any propoſed number whatever: and hence there can be no doubt but that the ſums of this ſpecies of ſeries may be exhibited by expreſſions of this kind a/0. It is concerning the other ſpecies, therefore, that mathematicians have chiefly differed; and the arguments which both ſides allege in defence of their opinions, have been endued with ſuch force, that neither party could hitherto be brought to yield to the other.
4. As to the ſecond ſpecies, the famous Leibnitz was one of the firſt who treated of this ſeries 1 − 1 + 1 − 1 + 1 − 1 + &c. and he concluded the ſum of it to = ½, relying upon the following cogent reaſons. And firſt, that this ſeries ariſes by reſolving the fraction 1/1+a into the ſeries 1 − a + a2 − a3 + a4 − a5 + &c. by continual diviſion in the uſual way, and taking the value of a equal to unity. Secondly, for more confirmation, and for perſuading ſuch as are not accuſtomed to calculations, he reaſons in the following manner: If the ſeries terminate any where, and if the number of the terms be even, then its value will be = 0; [Page 4] but if the number of terms be odd, the value of the ſeries will be = 1: but becauſe the ſeries proceeds in infinitum, and that the number of the terms cannot be reckoned either odd or even, we may conclude that the ſum is neither = 0, nor = 1, but that it muſt obtain a certain middle value, equidifferent from both, and which is therefore = ½. And thus, he adds, nature adheres to the univerſal law of juſtice, giving no partial preference to either ſide.
5. Againſt theſe arguments the adverſe party make uſe of ſuch objections as the following. Firſt, that the fraction 1/1+a is not equal to the infinite ſeries 1 − a + a2 − a3 + &c. unleſs a be a fraction leſs than unity. For if the diviſion be any where broken off, and the quotient of the remainder be added, the cauſe of the paralogiſm will be manifeſt; for we ſhall then have [...]; and that although the number n ſhould be made infinite, yet the ſupplemental fraction [...] ought not to be omitted, unleſs it ſhould become evaneſcent, which happens only in thoſe caſes in which a is leſs than 1, and the terms of the ſeries converge to 0. But that in other caſes there ought always to be included this kind of ſupplement [...]; and although it be affected with the dubious ſign [...], namely − or + according as n ſhall be an even or an odd number, yet if n be infinite, it may not therefore be omitted, under the pretence that an infinite number is neither odd nor even, and that there is no reaſon why the one ſign ſhould be uſed rather than the other; for it is abſurd to ſuppoſe that there can be any integer number, even although it be infinite, which is neither odd nor even.
6. But this objection is rejected by thoſe who attribute determinate ſums to diverging ſeries, becauſe it conſiders an infinite number as a determinate number, and therefore either odd or even, when it is really [Page 5] indeterminate. For that it is contrary to the very idea of a ſeries, ſaid to proceed in infinitum, to conceive any term of it as the laſt, although infinite: and that therefore the objection above-mentioned, of the ſupplement to be added or ſubtracted, naturally falls of itſelf. Therefore, ſince an infinite ſeries never terminates, we never can arrive at the place where that ſupplement muſt be joined; and therefore that the ſupplement not only may, but indeed ought to be neglected, becauſe there is no place found for it.
And theſe arguments, adduced either for or againſt the ſums of ſuch ſeries as above, hold alſo in the fourth ſpecies, which is not otherwiſe embarraſſed with any further doubts peculiar to itſelf.
[Page 6] 8. The defenders therefore of the ſums of ſuch ſeries, in order to reconcile this ſtriking paradox, more ſubtle perhaps than true, make a diſtinction between negative quantities; for they argue that while ſome are leſs than nothing, there are others greater than infinite, or above infinity. Namely, that the one value of −1 ought to be underſtood, when it is conceived to ariſe from the ſubtraction of a greater number a + 1 from a leſs a; but the other value, when it is found equal to the ſeries 1 + 2 + 4 + 8 + &c. and ariſing from the diviſion of the number 1 by −1; for that in the former caſe it is leſs than nothing, but in the latter greater than infinite. For the more confirmation, they bring this example of fractions ¼, ⅓, ½, 1/1, 1/0, 1/−1, 1/−2, 1/−3, &c. which, evidently increaſing in the leading terms, it is inferred will continually increaſe; and hence they conclude that 1/−1 is greater than 1/0, and 1/−2 greater than 1/−1, and ſo on: and therefore as 1/−1 is expreſſed by −1, and 1/0 by ∼ or infinity, −1 will be greater than ∼, and much more will −½ be greater than ∼. And thus they ingeniouſly enough repelled that apparent abſurdity by itſelf.
9. But although this diſtinction ſeemed to be ingeniouſly deviſed, it gave but little ſatisfaction to the adverſaries; and beſides, it ſeemed to affect the truth of the rules of algebra. For if the two values of −1, namely 1 − 2 and 1/−1, be really different from each other, as we may not confound them, the certainty and the uſe of the rules, which we follow in making calculations, would be quite done away; which would be a greater abſurdity than that for whoſe ſake the diſtinction was deviſed: but if 1 − 2 = 1/−1, as the rules of algebra require, for by multiplication [...], the matter in debate is not ſettled; ſince the quantity −1, to which the ſeries 1 + 2 + 4 + 8 + &c. is made equal, is leſs than nothing, and therefore the ſame difficulty ſtill remains. In the mean time however, it ſeems but agreeable to truth, to ſay that the ſame quantities which are below nothing, may be [Page 7] taken as above infinite. For we know, not only from algebra, but from geometry alſo, that there are two ways, by which quantities paſs from poſitive to negative, the one through the cypher or nothing, and the other through infinity: and beſides that quantities, either by increaſing or decreaſing from the cypher, return again, and revert to the ſame term o; ſo that quantities more than infinite are the ſame with quantities leſs than nothing, like as quantities leſs than infinite agree with quantities greater than nothing.
10. But, farther, thoſe who deny the truth of the ſums that have been aſſigned to diverging ſeries, not only omit to aſſign other values for the ſums, but even ſet themſelves utterly to oppoſe all ſums belonging to ſuch ſeries, as things merely imaginary. For a converging ſeries, as ſuppoſe this 1 + ½ + ¼ + ⅛ + &c. will admit of a ſum = 2, becauſe the more terms of this ſeries we actually add, the nearer we come to the number 2: but in diverging ſeries the caſe is quite different; for the more terms we add, the more do the ſums which are produced differ from one another, neither do they ever tend to any certain determinate value. Hence they conclude that no idea of a ſum can be applied to diverging ſeries, and that the labour of thoſe perſons who employ themſelves in inveſtigating the ſums of ſuch ſeries, is manifeſtly uſeleſs, and indeed contrary to the very principles of analyſis.
11. But notwithſtanding this ſeemingly real difference, yet neither party could ever convict the other of any error, whenever the uſe of ſeries of this kind has occurred in analyſis; and for this good reaſon, that neither party is in an error, but that the whole difference conſiſts in words only. For if in any calculation I arrive at this ſeries 1 − 1 + 1 − 1 + &c. and that I ſubſtitute ½ inſtead of it; I ſhall ſurely not thereby commit any error; which however I ſhould certainly incur if I ſubſtitute any other number inſtead of that ſeries; and hence there remains no doubt but that the ſeries 1 − 1 + 1 − 1 + &c. and the [Page 8] fraction ½, are equivalent quantities, and that the one may always be ſubſtituted inſtead of the other without error. So that the whole matter in diſpute ſeems to be reduced to this only, namely, whether the fraction ½ can be properly called the ſum of the ſeries 1 − 1 + 1 − 1 + &c. Now if any perſons ſhould obſtinately deny this, ſince they will not however venture to deny the fraction to be equivalent to the ſeries, it is greatly to be feared they will fall into mere quarrelling about words.
12. But perhaps the whole diſpute will eaſily be compromiſed, by carefully attending to what follows. Whenever, in analyſis, we arrive at a complex function or expreſſion, either fractional or tranſcendental; it is uſual to convert it into a convenient ſeries, to which the remaining calculus may be more eaſily applied. And hence the occaſion and riſe of infinite ſeries. So far only then do infinite ſeries take place in analytics, as they ariſe from the evolution of ſome finite expreſſion; and therefore, inſtead of an infinite ſeries, in any calculus, we may ſubſtitute that formula, from whoſe evolution it aroſe. And hence, for performing calculations with more eaſe or more benefit, like as rules are uſually given for converting into infinite ſeries ſuch finite expreſſions as are endued with leſs proper forms; ſo, on the other hand, thoſe rules are to be eſteemed not leſs uſeful by the help of which we may inveſtigate the finite expreſſion from which a propoſed infinite ſeries would reſult, if that finite expreſſion ſhould be evolved by the proper rules: and ſince this expreſſion may always, without error, be ſubſtituted inſtead of the infinite ſeries, they muſt neceſſarily be of the ſame value: and hence no infinite ſeries can be propoſed, but a finite expreſſion may, at the ſame time, be conceived as equivalent to it.
13. If therefore, we only ſo far change the received notion of a ſum as to ſay, that the ſum of any ſeries, is the finite expreſſion by the evolution of which that ſeries may be produced, all the difficulties, [Page 9] which have been agitated on both ſides, vaniſh of themſelves. For, firſt, that expreſſion by whoſe evolution a converging ſeries is produced, exhibits at the ſame time its ſum, in the common acceptation of the term: neither, if the ſeries ſhould be divergent, could the inveſtigation be deemed at all more abſurd, or leſs proper, namely, the ſearching out a finite expreſſion which, being evolved according to the rules of algebra, ſhall produce that ſeries. And ſince that expreſſion may be ſubſtituted in the calculation inſtead of this ſeries, there can be no doubt but that it is equal to it. Which being the caſe, we need not neceſſarily deviate from the uſual mode of ſpeaking, but might be permitted to call that expreſſion alſo the ſum, which is equal to any ſeries whatever, provided however, that, in ſeries whoſe terms do not converge to o, we do not connect that notion with this idea of a ſum, namely, that the more terms of the ſeries are actually collected, the nearer we muſt approach to the value of the ſum.
14. But if any perſon ſhall ſtill think it improper to apply the term ſum, to the finite expreſſions by whoſe evolution all ſeries in general are produced; it will make no difference in the nature of the thing; and inſtead of the word ſum, for ſuch finite expreſſion, he may uſe the term value; or perhaps the term radix would be as proper as any other that could be employed for this purpoſe, as the ſeries may juſtly be conſidered as iſſuing or growing out of it, like as a plant ſprings from its root, or from its ſeed. The choice of terms being in a great meaſure arbitrary, every perſon is at liberty to employ them in whatever ſenſe he may think fit, or proper for the purpoſe in hand; provided always that he fix and determine the ſenſe in which he underſtands or employs them. And as I conſider any ſeries, and the finite expreſſion by whoſe evolution that ſeries may be produced, as no more than two different ways of expreſſing one and the ſame thing, whether that finite expreſſion be called the ſum, or value, or radix of the ſeries; ſo in the following paper, and in ſome others which may perhaps hereafter [Page 10] be produced, it is in this ſenſe I deſire to be underſtood when ſearching out the value of ſeries, namely, that the object of my enquiry, is the radix by whoſe evolution the ſeries may be produced, or elſe an approximation to the value of it in decimal numbers, &c.
Royal Military Acad. Woolwich, May 24, 1785.
1.2. TRACT II. A new Method for the Valuation of Numeral Infinite Series, whoſe Terms are alternately (+) Plus and (−) Minus; by taking continual Arithmetical Means between the Succeſſive Sums, and their Means.
[Page 11]Article 1. THE remarkable difference between the facility which mathematicians have found in their endeavours to determine the values of infinite ſeries whoſe terms are alternately affirmative and negative, and the difficulty of doing the ſame thing with reſpect to thoſe ſeries whoſe terms are all affirmative, is one of thoſe ſtriking appearances in ſcience which we can hardly perſuade ourſelves is true, even after we have ſeen many proofs of it, and which ſerve to put us ever after on our guard not to truſt to our firſt notions, or conjectures, on theſe ſubjects, till we have brought them to the teſt of demonſtration. For, at firſt ſight it is very natural to imagine, that thoſe infinite ſeries whoſe terms are all affirmative, or added to the firſt term, muſt be much ſimpler in their nature, and much eaſier to be ſummed, than thoſe whoſe terms are alternately affirmative and negative; which, nevertheleſs, we find, upon examination, to be directly contrary to the truth; the methods of finding the ſums of the latter ſeries being numerous and eaſy, and alſo very general, whereas thoſe that have been hitherto diſcovered for the ſummation of the former ſeries, are few and difficult, and confined to ſeries whoſe terms are generated from each other according to ſome particular laws, inſtead of extending, as the other methods do, to all ſorts of ſeries, whoſe terms are connected together by addition, by whatever law their terms are formed. Of this remarkable difference between theſe two ſorts of ſeries, the new method of finding the ſums of thoſe whoſe terms are [Page 12] alternately poſitive and negative, which is the ſubject of the preſent paper, will afford us a ſtriking inſtance, as it poſſeſſes the happy qualities of ſimplicity, eaſe, perſpicuity, and univerſality in a great degree; and yet, as the eſſence of it conſiſts in the alternation of the ſigns + and −, by which the terms are connected with the firſt term, it is of no uſe in the ſummation of thoſe other ſeries whoſe terms are all connected with each other by the ſign +.
It muſt be noted, however, that by the value of the ſeries, I always mean ſuch radix, or finite expreſſion, as, by evolution, would produce the ſeries in queſtion; according to the ſenſe I have ſtated in the former paper, on this ſubject; or an approximate value of ſuch radix; and which radix, as it may be ſubſtituted inſtead of the ſeries in any operation, I call the value of the ſeries.
4. Hence the value of every alternate ſeries s, is poſitive, and leſs than the firſt term a, the ſeries being always ſuppoſed to begin with a poſitive term a; and conſequently if the ſigns of all the terms be [Page 15] changed, or if the ſeries begin with a negative term, the value s will ſtill be the ſame, but negative, or the ſign of the ſum will be changed, and the value become −s = −a + b − c + d − &c. Alſo, becauſe the ſucceſſive ſums, in a converging ſeries, always approach nearer and nearer to the true value, while they recede always farther and farther from it in a diverging ſeries; it follows that, in a neutral ſeries, a − a + a − a + &c. which holds a middle place between the two former, the ſucceſſive ſums o, a, o, a, o, a &c. will neither converge nor diverge, but will be always at the ſame diſtance from the value of the propoſed ſeries a − a + a − a + &c. and conſequently that value will always be = ½a, which holds every where the middle place between o and a.
7. Hence the expreſſions 0, a/2, 3a−b/4, 7a−4b+c/8, 15a−11b+5c−a/16, 31a−26b+16c−6d+e/32, &c. are continual approximations to the value s of the converging ſeries a − b + c − d + e − &c. and are all below the truth. But we can eaſily expreſs all theſe ſeveral theorems by one general formula. For, ſince it is evident by the conſtruction, that whilſt the denominator in any one of them is ſome power (2^{n}) of 2 or 1 + 1, the numeral coefficients [Page 18] of a, b, c, &c. the terms in the numerator, are found by ſubtracting all the terms except the laſt term, one after another, from the ſaid power 2^{n} or [...] which is = 1 + n + n · n−1/2 + n · n−2/3 + &c. namely the coefficient of a equal to all the terms 2^{n} minus the firſt term 1; that of b equal to all except the firſt two terms 1 + n; that of c equal to all except the firſt three; and ſo on, till the coefficient of the laſt term be = 1 the laſt term of the power; it follows that the general expreſſion for the ſeveral theorems, or the general approximate value of the converging ſeries a − b + c − d + &c. will be [...] continued till the terms vaniſh and the ſeries break off, n being equal to o or any integer number. Or this general formula may be expreſſed by this ſeries, [...] where A, B, C, &c. denote the coefficients of the ſeveral preceding terms. And this expreſſion, which is always too little, is the nearer to the true value of the ſeries a − b + c − d + &c. as the number n is taken greater: always excepting however thoſe caſes in which the theorem is accurately true when n is ſome certain finite number. Alſo, with any value of n, the formula is nearer to the truth, as the terms a, b, c, &c. of the propoſed ſeries, are nearer to equality; ſo that the ſlower the propoſed ſeries converges, the more accurate is the formula, and the ſooner does it afford a near value of that ſeries: which is a very favourable circumſtance, as it is in caſes of very ſlow convergency that approximating formulae are chiefly wanted. And, like as the formula approaches nearer to the truth as the terms of the ſeries approach to an equality, ſo when the terms become quite equal, as in a neutral ſeries, the formula becomes quite accurate, and always gives the ſame value ½a for s or the ſeries, whatever integer number be taken for n. And farther, when the propoſed ſeries, from being converging, paſſes through [Page 19] neutrality, when its terms are equal, and becomes diverging, the formula will ſtill hold good, only it will then be alternately too great, and too little as long as the ſeries diverges, as we ſhall preſently ſhew more fully. So that in general the value s of the ſeries a − b + c − d + &c. whether it be converging, diverging, or neutral, is leſs than the firſt term a; when the ſeries converges, the value is above ½a; when it diverges, it is below ½a; and when neutral, it is equal to ½a.
Approx. Formulae. | Differences. |
a/2 | a−b/4 |
3a−b/4 | a−2b+c/8 |
7a−4b+c/8 | a−3b+3c−d/16 |
15a−11b+5c−d/16 | a−4b+6c−4d+e/32 |
31a−26b+16c−6d+e/32 | |
&c. | &c. |
9. Having, in Art. 5, 6, 7, 8, compleated the inveſtigations for the firſt or converging form of ſeries, the firſt four articles being introductory to both forms in common; we may now proceed to the diverging form of ſeries, for which we ſhall find the ſame method of arithmetical means, and the ſame general formula, as for the converging ſeries; as well as the mode of inveſtigation uſed in Art. 5 et ſeq. only changing ſometimes greater for leſs, or leſs for greater. Thus then, reaſoning from the table of ſucceſſive ſums in Art. 3, in which s is alternately above and below the expreſſions o, a, a − b, a − b + c, &c. becauſe o is below, and a above the value s of the ſeries a − b + c − d + &c. but o nearer than a to that value, it follows that s lies between [Page 21] o and ½a, and that ½a is greater than s, but nearer to s than a is. In like manner, becauſe a is above, and a − b below the value s, but a nearer to that value than a − b is, it follows that s lies between a and a − b, and that the arithmetical mean a − ½b is below s, but that it is nearer to s than a − b is. And thus, the ſame reaſoning holding in every pair of ſucceſſive ſums, the arithmetical means between them will form another ſeries of terms, which are alternately greater and leſs than s the value of the propoſed ſeries; but here greater and leſs in the contrary way to what they were for the converging ſeries, namely, thoſe ſteps greater here which were leſs there, and leſs here which before were greater. And this firſt ſet of arithmetical means, will either converge to the value of s, or will at leaſt diverge leſs from it than the progreſſion of ſucceſſive ſums. Again, the ſame reaſoning ſtill holding good, by taking the arithmetical means of thoſe firſt means, another ſet is found, which will either converge, or elſe diverge leſs than the former. And ſo on as far as we pleaſe, every new operation gradually checking the firſt or greateſt divergency, till a number of the firſt terms of a ſet converge ſufficiently faſt, to afford a near value of s the propoſed ſeries.
10. Or, by taking the firſt terms of all the orders of means, we find the ſame ſet of theorems, namely [...], &c. or in general, [...] which will be alternately above and below s the value of the ſeries, till the divergency is overcome. Then this ſeries, which conſiſts of the firſt terms of the ſeveral orders of means, may be treated as the ſucceſſive ſums, taking ſeveral orders of means of theſe again. After which the firſt terms of theſe laſt orders may be treated again in the ſame manner; and ſo on as far as we pleaſe. Or the ſeries of ſecond terms, or third terms, &c. or ſometimes, the terms aſcending obliquely, may be treated in the ſame manner to advantage. And [Page 22] with a little practice and inſpection of the ſeveral ſeries, whether vertical, or horizontal, or oblique, (for they all tend to the detection of the ſame value s) we ſhall ſoon learn to diſtinguiſh whereabouts the required quantity s is, and which of the ſeries will ſooneſt approximate to it.
[Page 23] The conſtruction and continuation of this table, is a buſineſs of little labour. For the numbers in the firſt horizontal line next below the line of the powers of 2, are thoſe powers diminiſhed each by unity. The numbers in the next horizontal line, are made from the numbers in the firſt, by ſubtracting from each the index of that power of 2 which ſtands above it. And for the reſt of the table, the formation of it is obvious from this principle, which reigns through the whole, that every number in it is the ſum of two others, namely of the next to it on the left in the ſame horizontal line, and the next above that in the ſame vertical column. So that the whole table is formed from a few of its initial numbers, by eaſy operations of addition.
In converging ſeries, it will be farther uſeful, firſt to collect a few of the initial terms into one ſum, and then apply our method to the following terms, which will be ſooner valued becauſe they will converge ſlower.
12. For the firſt example, let us take the very ſlowly converging ſeries 1 − ½ + ⅓ − ¼ + ⅕ − ⅙ + &c. which is known to expreſs the hyp. log. of 2, which is = .69314718.
1. For the 10th Theorem. [...]
2. For the 20th Theorem. [...]
Again, to perform the operation by taking the ſucceſſive ſums, and the arithmetical means: let the terms ½, ⅓, ¼, &c. be reduced to decimal numbers, by dividing the common numerator 1 by the denominators 2, 3, 4, &c. or rather by taking theſe out of the table at the end of my Miſcellanea Mathematica, publiſhed in 1775, which contains a table of the ſquare roots and reciprocals of all the numbers, [Page 25] 1, 2, 3, 4, 5, 6, &c. to 1000, and which is of great uſe in ſuch calculations as theſe. Then the operation will ſtand thus: [...] Here, after collecting the firſt twelve terms, I begin at the bottom, and, aſcending upwards, take a very few arithmetical means between the ſucceſſive ſums, placing them on the right of them: it being unneceſſary to take the means of the whole, as any part of them will do the buſineſs, but the lower ones in a converging ſeries beſt, becauſe they are nearer the value ſought, and approach ſooner to it. I then take the means of the firſt means, and the means of theſe again, and ſo on, till the value is obtained as near as may be neceſſary. In this proceſs we ſoon diſtinguiſh whereabouts the value lies, the limits or means, which are alternately above and below it, gradually contracting, and approaching towards each other. And when the means are reduced to a ſingle one, and it is found neceſſary to get the value more exactly, I then go back to the columns of ſucceſſive ſums, and find another firſt mean, either next below or above thoſe before found, and continue it through the 2d, 3d, &c. means, which makes now a duplicate in the laſt column of means, and the mean between them gives another ſingle mean of the next order; and ſo on as far as we ſee it neceſſary. By ſuch a gradual progreſs we uſe no more terms nor labour than is quite requiſite for the degree of accuracy required.
Or, after having collected the ſum of any number of terms, we may apply any of the formulae to the following terms. So, having as above [Page 26] found .653211 for the ſum of the firſt 12 terms, and calling the next or 13th term .076923 = a, the 14th term .0714285 = b, the next, .06666 &c. = c, and ſo on: then the 2d theorem 3a−b/4 gives .039835, which added to .653211 the ſum of the firſt 12 terms, gives .693046, the value of the ſeries true in three places of figures; and the 3d theorem [...] gives .039927 for the following terms, and which added to .653211 the ſum of the firſt 12 terms, gives .693138, the value of the ſeries true in five places. And ſo on.
13. For a ſecond example, let us take the ſlowly converging ſeries 2/1 − 3/2 + 4/3 − 5/4 + 6/5 − 7/6 + &c. which is = ½ + hyp. log. of 2 = 1.19314718. Then [...]
Here, after the 3d column of means, the firſt four figures 1.193, which are common, are omitted, to ſave room and the trouble of writing them ſo often down; and in the laſt three columns, the proceſs is repeated with the laſt three figures of each number; and the laſt of theſe 147, joined to the firſt four, give 1.193147 for the value of the ſeries propoſed. And the ſame value is alſo obtained by the theorems uſed as in the former example.
14. For the third example let us take the converging ſeries 1 − ⅓ + ⅕ − 1/7 + 1/9 − 1/11 + &c. which is = .7853981 &c. or ¼ of the circumference of the circle whoſe diameter is 1. Here a = 1, b = ⅓, [Page 27] c = ⅕, &c. then turning the terms into decimals, and proceeding with the ſucceſſive ſums and means as before, we obtain the 5th mean true within a unit in the 6th place as here below: [...]
15. To find the value of the converging ſeries [...] which occurs in the expreſſion for determining the time of a body's deſcent down the arc of a circle:
The firſt terms of this ſeries I find ready computed by Mr. Baron Maſeres, pa. 219 Philoſ. Tranſ. 1777; theſe being arranged under one another, and the ſums collected, &c. as before, give .834625 for the value of that ſeries, being only 1 too little in the laſt figure.
16. To find the value of 1 − ¼ + ½ − 1/16 + 1/25 − &c. conſiſting of the reciprocals of the natural ſeries of ſquare numbers. [Page 28] [...] The laſt mean .822467 is true in the laſt figure, the more accurate value of the ſeries 1 − ¼ + 1/9 − 1/16 + &c. being .8224670 &c.
17. Let the diverging ſeries ½ − ⅔ + ¾ − ⅘ + &c. be propoſed; where the terms are the reciprocals of thoſe in Art. 13.
Here the ſucceſſive ſums are alternately + and −, as well as the terms themſelves of the propoſed ſeries, but all the arithmetical means are poſitive. The numbers in each column of means are alternately too great and too little, but ſo as viſibly to approach towards each other. The ſame mutual approximation is viſible in all the oblique lines from left to right, ſo that there is a general and mutual tendency, in all the columns, and in all the lines, to the limit of the value of the ſeries. But with this difference, that all the numbers in any line deſcending obliquely from left to right, are on one ſide of the limit, and [Page 29] thoſe in the next line in the ſame direction, all on the other ſide, the one line having its numbers all too great, while thoſe in the next line are all too little; but, on the contrary, the lines which aſcend from below obliquely towards the right, have their numbers alternately too great and too little, after the manner of thoſe in the columns, but approximating quicker than thoſe in the columns. So that, after having continued the columns of arithmetical means to any convenient extent, we may then ſelect the terms in the laſt, or any other line obliquely aſcending from left to right, or rather beginning with the laſt found mean on the right, and deſcending towards the left; then arrange thoſe terms below one another in a column, and take their continual arithmetical means, like as was done with the firſt ſucceſſive ſums, to ſuch extent as the caſe may require. And if neither theſe new columns, nor the oblique lines approach near enough to each other, a new ſet may be formed from one of their oblique lines which has its terms alternately too great and too little. And thus we may proceed as far as we pleaſe. Theſe repetitions will be more neceſſary in treating ſeries which diverge more; and having here once for all deſcribed the properties attending the ſeries, with the method of repetition, we ſhall only have to refer to them as occaſion ſhall offer. In the preſent inſtance, the laſt two or three means vary or differ ſo little, that the limit may be concluded to lie nearly in the middle between them, and therefore the mean between the two laſt 144 and 150, namely 147, may be concluded to be very near the truth, in the laſt three figures; for as to the firſt three figures 193, I dropt the repetition of them after the firſt three columns of means, both to ſave ſpace, and the trouble of writing them ſo often over again. So that the value of the ſeries in queſtion may be concluded to be .193147 very nearly, which is = − ½ + the hyp. log. of 2; or 1 leſs than its reciprocal ſeries in Art. 13.
Or, thus, taking the ſeveral orders of means, &c.
Here the ſucceſſive ſums are alternately + and −, but the arithmetical means are all +. After the ſecond column of means, the firſt two figures 56 are omitted, being common; and in the laſt three columns the firſt three figures 569, which are common, are omitted. Towards the end, all the numbers, both oblique and vertical, approach ſo near together, that we may conclude that the laſt three figures 035 are all true; and theſe being joined to the firſt three 569, we have .569035 for the value of the ſeries, which is otherwiſe found 2+√2/6 = .56903559 &c.
19. Let us take the diverging ſeries 2^{2}/1 − 3^{2}/2 + 4^{2}/3 − 5^{2}/4 + &c. or 4/1 − 9/2 + 16/3 − 25/4 + &c. or 4 − 4½ + 5⅓ − 6¼ + 7⅕ − 8⅙ + &c.
After the ſecond column of means, the firſt four figures 1.943 are omitted, being common to all the following columns; to theſe annexing the laſt three figures 147 of the laſt mean, we have 1.943147 for the ſum of the ſeries, which we otherwiſe know is equal to 5/4 + hyp. log. of 2. See Simp. Diſſert. Ex. 2. p. 75 and 76.
And the ſame value might be obtained by means of the formulae, uſing them as before.
20. Taking the diverging ſeries 1 − 2 + 3 − 4 + 5 − &c. the method of means gives us, [...]
Where the ſecond, and every ſucceeding column of means, gives ¼ for the value of the ſeries propoſed.
[Page 32] 21. Taking the ſeries 1 − 4 + 9 − 16 + 25 − 36 + &c. whoſe terms conſiſt of the ſquares of the natural ſeries of numbers, we have, by the arithmetical means, [...]
Where it is only in the ſecond column of means that the divergency is counteracted; after that the third and all the other orders of means give o for the value of the ſeries 1 − 4 + 9 − 16 + &c.
22. Taking the geometrical ſeries of terms 1 − 2 + 4 − 8 + &c. then [...]
[Page 33] Here the lower parts of all the columns of means, from the cipher 0 downwards, conſiſt of the ſame ſeries of terms + 1 − 1 + 3 − 5 + 11 − 21 + 43 − 85 + &c. and the other part of the columns, from the cipher upwards, as well as each line of oblique means, parallel to, and above the line of ciphers, forms a ſeries of terms ½, ¼, ⅜, 5/16.....⅓ · 2^{n} ± 1/2^{n}, alternately above and below the value of the ſeries, ⅓, and approaching continually nearer and nearer to it, and which, when infinitely continued, or when n is infinite, the term becomes ⅓ for the value of the geometrical ſeries, 1 − 2 + 4 − 8 + 16 − &c.
And the ſame ſet of terms would be given by each of the formulae.
23. Take the geometrical ſeries 1 − 3 + 9 − 27 + 81 − &c. Then [...] Here the column of ſucceſſive ſums, and every ſecond column of the arithmetical means, below the o, conſiſts of the ſame ſeries of terms 1, − 2, + 7, − 20, + &c. whilſt all the other columns of means conſiſt of this other ſet of terms ½, − ½, + 2½, − 6½, + &c. alſo the firſt oblique line of means, ½, 0, ½, 0, ½, 0, &c. conſiſts of the terms ½ and 0 alternately, which are all at equal diſtance from the value of the ſeries propoſed 1 − 3 + 9 − 27 + 81 − &c. as indeed are the terms of all the other oblique deſcending lines. And the mean between every two terms gives ¼ for that value. And the ſame terms would be given by the formulae, namely alternately ½ and 0.
And thus the value of any geometrical ſeries, whoſe ratio or ſecond term is r, will be found to be = 1/1+r.
[Page 34] 24. Finally, let there be taken the hypergeometrical ſeries 1 − 1 + 2 − 6 + 24 − 120 + &c. = 1 − 1 A + 2 B − 3 C + 4 D − 5 E + &c. which difficult ſeries has been honoured by a very conſiderable memoir written upon the valuation of it by the late famous L. Euler, in the New Peterſburg Commentaries, vol. v. where the value of it is at length determined to be .5963473 &c.
To ſimplify this ſeries, let us omit the firſt two terms 1 − 1 = 0, which will not alter the value, and divide the remaining terms by 2, and the quotients will give 1 − 3 + 12 − 60 + 360 − 2520 + &c. which, being half the propoſed ſeries, ought to have for its value the half of .596347 &c. namely .298174 nearly.
Now, ranging the terms in a column, and taking the ſums and means as uſual, we have [...] Where it is evident, that the diverging is ſomewhat diminiſhed, but not quite counteracted, in the columns and oblique deſcending lines from beginning to end, as the terms in thoſe directions ſtill increaſe, though not quite ſo faſt as the original ſeries; and that the ſigns of the ſame terms are alternately + and −, while thoſe of the terms in the other lines obliquely aſcending from left to right, are alternately one line all +, and another line all −, and theſe terms continually decreaſing. The terms in the oblique deſcending lines, being alternately too great and too little, are the fitteſt to proceed with again. Take therefore any one of thoſe lines, as ſuppoſe the firſt, and ranging it vertically, take the means as before, and they will approach nearer to the value of the ſeries, thus: [Page 35] [...] Here the ſame approximation in the lines and columns, towards the value of the ſeries, is obſervable again, only in a higher degree; alſo the terms in the columns and oblique deſcending lines, are again alternately too great and too little, but now within narrower limits, and the ſigns of the terms are more of them poſitive; alſo the terms in each oblique aſcending line, are ſtill either all above or all below the value of the ſeries, and that alternately one line after another as before. But the deſcending lines will again be the fitteſt to uſe, becauſe the terms in each are alternately above and below the value ſought. Taking therefore again the firſt of theſe oblique deſcending lines, treat it as before, and we ſhall obtain ſets of terms approaching ſtill nearer to the value, thus: [...] Here the approach to an equality, among all the lines and columns, is ſtill more viſible, and the deviations reſtricted within narrower limits, the terms in the oblique aſcending lines ſtill on one ſide of the value, and gradually increaſing, while the columns and the oblique deſcending lines, for the moſt part, have their terms alternately too great and too little, as is evident from their alternately becoming greater and leſs than each other: and from an inſpection of the whole, it is eaſy to pronounce that the firſt three figures of the number ſought, will be 298. Taking therefore the laſt ſour terms of the firſt deſcending line, and proceeding as before, we have [Page 36] [...]
And, finally, taking the loweſt aſcending line, becauſe it has moſt the appearance of being alternately too great and too little, proceed with it as before, thus: [...] where the numbers in the lines and columns gradually approach nearer together, till the laſt mean is true to the neareſt unit in the laſt figure, giving us .298174 for the value of the propoſed hypergeometrical ſeries 1 − 3 + 12 − 60 + 360 − 2520 + 20160 − &c.
And in like manner are we to proceed with any other ſeries whoſe terms have alternate ſigns.
POSTSCRIPT.Royal Military Acad. Woolwich, May, 1780.
SINCE the foregoing method was diſcovered, and made known to ſeveral friends, two paſſages have been offered to my conſideration, which I ſhall here mention, in juſtice to their authors, Sir Iſaac Newton, and the late learned Mr. Euler.
The firſt of theſe is in Sir Iſaac's letter to Mr. Oldenburg, dated October 24, 1676, and may be ſeen in Collins's Commercium Epiſtolicum, p. 177, the laſt paragraph near the bottom of the page, namely, Per [Page 37] ſeriem Leibnitii etiam, ſi ultimo loco dimidium termini adjiciatur, & alia quaedam ſimilia artificia adhibeantur, poteſt computum produci ad multas figuras. The ſeries here alluded to, is 1 − ⅓ + ⅕ − 1/7 + 1/9 − 1/11 + &c. denoting the area of the circle whoſe diameter is 1; and Sir Iſaac here directs to add in half the laſt term, after having collected all the foregoing, as the means of obtaining the ſum a little exacter. And this, indeed, is equivalent to taking one arithmetical mean between two ſucceſſive ſums, but it does not reach the idea contained in my method. It appears alſo, both by the other words, & alia quaedam ſimilia artificia adhibeantur, contained in the above extract, and by theſe, alias artes adhibuiſſem, a little higher up in the ſame page 177, that Sir Iſaac Newton had ſeveral other contrivances for obtaining the ſums of ſlowly converging ſeries; but what they were, it may perhaps be now impoſſible to determine.
The other is a paſſage in the Novi Comment. Petropol. tom. v. p. 226, where Mr. Euler gives an inſtance of taking one ſet of arithmetical means between a ſeries of quantities which are gradually too little and too great, to obtain a nearer value of the ſum of a ſeries in queſtion. But neither does this reach the idea contained in my method. However, I have thought it but juſtice to the characters of theſe two eminent men, to make this mention of their ideas, which have ſome relation to my own, though unknown to me at the time of my diſcovery.
1.3. TRACT III.
[Page 38]A Method of ſumming the Series a + bx + cx2 + dx3 + ex4 + &c. when it converges very ſlowly, namely, when x is nearly equal to 1, and the Coefficients a, b, c, d, &c. decreaſe very ſlowly: the Signs of all the Terms being poſitive.
Art. 1. WHEN we have occaſion to find the ſum of ſuch ſeries as that above-mentioned, having the coefficients a, b, c, d, &c. of the terms, decreaſing very ſlowly, and the converging quantity x pretty large; we can neither find the ſum by collecting the terms together, on account of the immenſe number of them which it would be neceſſary to collect; neither can it be ſummed by means of the differential ſeries, becauſe the powers of the quantity x/1−x will then diverge faſter than the differential coefficients converge. In ſuch caſe then we muſt have recourſe to ſome other method of tranſforming it into another ſeries which ſhall converge faſter. The following is a method by which the propoſed ſeries is changed into another, which converges ſo much the quicker as the original ſeries is ſlower.
2. The method is thus. Aſſume a2/D the given ſeries a + bx + cx2 + dx3 + &c. Then ſhall [...]; which, by actual diviſion, is [...] Conſequently a2 divided by this ſeries will be equal to the ſeries propoſed, and this new ſeries will be very eaſily ſummed, in compariſon with the original one, becauſe all the coefficients after the ſecond term are evidently very ſmall; and indeed they are ſo much the ſmaller, and fitter for ſummation, by how much the coefficients of the [Page 39] original ſeries are nearer to equality; ſo that when theſe a, b, c, d, &c. are quite equal, then the third, fourth, &c. coefficients of the new ſeries become equal to nothing, and the ſum accurately equal to [...]; which we alſo know to be true from other principles.
3. Although the firſt two terms, a − bx, of the new ſeries be very great in compariſon with each of the following terms, yet theſe latter may not always be ſmall enough to be entirely rejected where much accuracy is required in the ſummation. And in ſuch caſe it will be neceſſary to collect a great number of them, to obtain their ſum pretty near the truth; becauſe their rate of converging is but ſmall, it being indeed pretty much like to the rate of the original ſeries, but only the terms, each to each, are much ſmaller, and that commonly in a degree to the hundredth or thouſandth part.
5. But if theſe laſt terms be ſtill thought too large to be omitted, then find the ſum of them by the very ſame theorem: and thus proceed, by repeating the operation in the ſame manner, till the required degree of accuracy is obtained. Which it is evident, will happen after a ſmall number of repetitions, becauſe that, in each new denominator, the third, fourth, &c. terms are commonly depreſſed, in the ſcale of numbers, two or three places lower than the firſt and ſecond terms are. And the general theorem, denoting the ſum S when the proceſs is continually repeated, will be this, [...].
7. To exemplify now the uſe of this method, let it be propoſed to ſum the very ſlow ſeries x + ½x2 + ⅓x3 + ¼x4 + ⅕x5 + ⅙x6 + &c. when x = 9/10 = .9, which denotes the hyperbolic log. of 1/1−x, or in this caſe of 10.
I have written an unknown quantity z after the laſt denominator, to repreſent the ſmall quantity to be ſubtracted from the laſt denominator 344. Now, rejecting the ſmall quantity z, and beginning at the laſt fraction to calculate, their values will be as here ranged in the firſt annexed column.
[...] [Page 44] placing z below them for the next unknown fraction. Divide then every fraction by the next below it, placing the quotients or ratios in the next column. Then take the quotients or ratios of theſe; and ſo on till the laſt ratio [...]; which, from the nature of the ſeries of the firſt terms of every column, muſt be leſs than the next preceding one 2.39: conſequently z muſt be leſs than 1.68×187/63, or leſs than 5. But, from the nature of the ſeries in the vertical row or column of firſt ratios, 187/z muſt be leſs than 63; and conſequently z muſt be greater than 187/63, or greater than 3. Since then z is leſs than 5 and greater than 3, it is probable that the mean value 4 is near the truth: and accordingly taking 4 for z, or rather 4.3, as z appears to be nearer 5 than 3, and taking the continual ratios, as placed along the laſt line of the table, their values are found to accord very well with the next preceding numbers, both in the columns and oblique rows.
Then this value .518414 of the ſeries, being multiplied by x13 or .2541865828329, gives .1317738 for the ſum of all the terms of the original ſeries after the firſt 12 terms, to which therefore the ſum of the firſt 12 terms, or 2.17081162, being added, we have 2.30258542 for the ſum of the original ſeries x + ½x2 + ⅓x3 + ¼x4 + &c. Which value is true within about 3 in the 8th place of figures, the more accurate value being 2.30258509 &c. or the hyp. log. of 10.
1.4. TRACT IV. The Inveſtigation of certain eaſy and General Rules, for Extracting any Root of a given Number.
[Page 45]1. THE roots of given numbers are commonly to be found, with much eaſe and expedition, by means of logarithms, when the indices of ſuch roots are ſimple numbers, and the roots are not required to a great number of figures. And the ſquare or cubic roots of numbers, to a good practical degree of accuracy, may be obtained, almoſt by inſpection, by means of my tables of ſquares and cubes, publiſhed by order of the Commiſſioners of Longitude, in the year 1781. But when the indices of ſuch roots are certain complex or irrational numbers, or when the roots are required to be found to a great many places of figures, it is neceſſary to make uſe of certain approximating rules, by means of the ordinary arithmetical computations. Such rules as are here alluded to, have only been diſcovered ſince the great improvements in the modern algebra: and the perſons who have beſt ſucceeded in their enquiries after ſuch rules, have been ſucceſſively Sir Iſaac Newton, Mr. Raphſon, M. de Lagney, and Dr. Halley; who have ſhewn that the inveſtigation of ſuch theorems is alſo uſeful in diſcovering rules for approximating to the roots of all ſorts of affected algebraical equations, to which the former rules, for the roots of all ſimple equations, bear a conſiderable affinity. It is preſumed that the following ſhort tract contains ſome advantages over any other method that has hitherto been given, both as to the eaſe and univerſality of the concluſions, and the general way in which the inveſtigations are made: for here, a theorem is diſcovered, which, although it be general for all roots whatever, is at the ſame time very accurate, and ſo ſimple and [Page 46] eaſy to uſe and to keep in mind, that nothing more ſo can be deſired or hoped for; and farther, that inſtead of ſearching out rules ſeverally for each root, one after another, our inveſtigation is at once for any indefinite poſſible root, by whatever quantity the index is expreſſed, whether fractional, or irrational, or ſimple, or compound.
- N be the given number, whoſe root is ſought,
- n the index of that root,
- a its neareſt rational root, or a^{n}; the neareſt rational power to N, whether greater or leſs,
- x the remaining part of the root ſought, which may be either poſitive or negative, namely, poſitive when N is greater than a^{n}, otherwiſe negative.
3. Now, for the firſt rule, expand the quantity [...] by the binomial theorem, ſo ſhall we have [...] Subtract a^{n} from both ſides, ſo ſhall [...] Divide by [...], ſo ſhall [...] or [...] Here, on account of the ſmallneſs of the quantity x in reſpect of a, all the terms of this ſeries, after the firſt term, will be very ſmall, and may therefore be neglected without much error, which gives us [...] for a near value of x, being only a ſmall matter too great. And conſequently [...] is nearly = N^{1}/n the root ſought. And this may be accounted the firſt theorem.
[Page 47] 4. Again, let the equation [...] be multitiplied by n − 1, and a^{n} added to each ſide, ſo ſhall we have [...] for a diviſor: Alſo multiply the ſides of the ſame equation by a and ſubtract a^{n} + 1 from each, ſo ſhall we have [...] for a dividend: Divide now this dividend by the diviſor, ſo ſhall [...] Which will be nearly equal to x, for the ſame reaſon as before; and this expreſſion is nearly as much too little as the former expreſſion was too great. Conſequently, by adding a, we have a + x or N^{1}/n nearly [...] for a ſecond theorem, and which is nearly as much in defect as the former was in exceſs.
5. Now becauſe the two foregoing theorems differ from the truth by nearly equal ſmall quantities, if we add together the two numerators and the two denominators of the foregoing two fractional expreſſions, namely [...] and [...], the ſums will be the numerator and denominator of a new fraction, which will be much nearer than either of the former. The fraction ſo found is [...]; which will be very nearly equal to N^{1}/n or a + x the root ſought; for, by diviſion, it is found to be equal to a + x * − n−1/2 · n+1/6 · x3/a2 + &c. where the term is wanting which contains the ſquare of x, and the following terms are very ſmall. And this is the third theorem.
6. A fourth theorem might be found by taking the arithmetical mean [Page 48] between the firſt and ſecond, which would be [...]; which will be nearly of the ſame value, though not ſo ſimple, as the third theorem; for this arithmetical mean is found equal to a + x * + n−1/2 · n−2/3 · x3/a2 + &c.
From which we find p−q/p+q = 1/n and p ∶ q ∷ n + 1 ∶ n − 1. So that by ſubſtituting n + 1 and n − 1, or any quantities proportional to them, for p and q, we ſhall have [...] for the value of the aſſumed quantity [...], which is ſuppoſed nearly equal to a + x, the required root of the quantity N.
[Page 49] 8. Now this third theorem [...], which is general for roots, whatever be the value of n, and whether a^{n} be greater or leſs than N, includes all the rational formulas of De Lagney and Halley, which were ſeparately inveſtigated by them; and yet this general formula is perfectly ſimple and eaſy to apply, and eaſier kept in mind than any one of the ſaid particular formulas. For, in words at length, it is ſimply this: to n + 1 times N add n − 1 times a^{n}, and to n − 1 times N add n + 1 times a^{n}, then the former ſum multiplied by a and divided by the latter ſum, will give the root N^{1}/n nearly; or, as the latter ſum is to the former ſum, ſo is a, the aſſumed root, to the required root, nearly. Where it is to be obſerved that a^{n} may be taken either greater or leſs than N, and that the nearer it is to it, the better.
10. To exemplify now our formula, let it be firſt required to extract the ſquare root of 365. Here N = 365, n = 2; the neareſt ſquare is 361, whoſe root is 19.
[Page 50] Hence 3 N + a2 = 3 × 365 + 361 = 1456, and N + 3 a2 = 365 + 3 × 361 = 1448; then as 1448 ∶ 1456 ∷ 19 ∶ 19×182/181 = 19 19/181 = 19.10497 &c.
11. For a ſecond example, let it be propoſed to double the cube, or to find the cube root of the number 2.
Here N = 2, n = 3, the neareſt root a = 1, alſo a3 = 1. Hence 2 N + a3 = 4 + 1 = 5, and N + 2 a3 = 2 + 2 = 4; then as 4 ∶ 5 ∷ 1 ∶ 5/4 = 1.25 = the firſt approximation. Again, take a = 5/4, and conſequently a3 = 125/64; Hence 2N + a3 = 4 + 125/64 = 381/64, and N + 2a3 = 2 + 250/64 = 378/64; then as 378 : 381, or as 126 ∶ 127 ∷ 5/4 ∶ 5/4 × 127/126 = 635/504 = 1.259921, for the cube root of 2, which is true in the laſt figure.
And by taking 635/504 for the value of a, and repeating the proceſs, a great many more figures may be found.
[Page 51] 12. For a third example, let it be required to find the 5th root of 2.
Here N = 2, n = 5, the neareſt root a = 1.
Hence 3 N + 2 a5 = 6 + 2 = 8, and 2 N + 3 a5 = 4 + 3 = 7; then as 7 ∶ 8 ∷ 1 ∶ 8/7 = 1 1/7 for the firſt approximation.
Again, taking a = 8/7, we have 3 N + 2 a5 = 6 + 65536/16807 = 166378/16807, 2 N + 3 a5 = 4 + 98304/16807 = 165532/16807; then as 165532 ∶ 166378 ∷ 8/7 ∶ 8/7 × 83189/82766 = 4/7 × 83189/41383 = 332756/289681 = 1.148698 &c. for the 5th root of 2, and is true in the laſt figure.
13. To find the 7th root of 126⅓.
Here N = 126⅕, n = 7, the neareſt root a = 2, alſo a7 = 128.
Hence 4 N + 3 a7 = 504⅘ + 384 = 888⅘ = 4444/5, and 3 N + 4 a7 = 378⅗ + 512 = 890⅗ = 4453/5; then as 4453 ∶ 4444 ∷ 2 ∶ 8888/4453 = 1.995957, for the root very exact by one operation, being true to the neareſt unit in the laſt figure.
14. To find the 365th root of 1.05, or the amount of 1 pound for 1 day, at 5 per cent. per annum, compound intereſt.
Here N = 1.05, n = 365, a = 1 the neareſt root. Hence 366 N + 364 a = 748.3, and 364 N + 366 a = 748.2; then as 748.2 ∶ 784.3 ∷ 1 ∶ 7483/7482 = 1 1/7482 = 1.00013366, the root ſought very exact at one operation.
15. Let it be required to find the value of the quantity [...] or [...].
[Page 52] Now this may be done two ways; either by finding the ⅔ power or 3/2 root of 21/4 at once; or elſe by finding the 3d or cubic root of 21/4, and then ſquaring the reſult.
By the firſt way:—Here it is eaſy to ſee that a is nearly = 3, becauſe 3^{3}/2 = √27 = 5 + ſome ſmall fraction. Hence, to find nearly the ſquare root of 27, or √27, the neareſt power to which is 25 = a2 in this caſe: Hence 3 N + a2 = 3 × 27 + 25 = 106, and N + 3 a2 = 27 + 3 × 25 = 102; then as 102 : 106, or as 51 ∶ 53 ∷ 5 ∶ 5 × 53/51 = 265/51 = √ 27 nearly.
Then having N = 21/4, n = 3/2, a = 3, and a3/2 = 265/51 nearly; it will be 5/2 N + ½ a3/2 = 5/2 × 21/4 + ½ × 265/51 = 6415/408, and ½ N + 5/2 a5/2 = ½ × 21/4 + 5/2 × 265/51 = 6371/408; hence as 6371 ∶ 6415 ∷ 3 ∶ 19245/6371 = 3 134/6371 = 3.020719, for the value of the quantity ſought nearly, by this way.
Again, by the other method, in finding firſt the value of [...], or the cube root of 21/4. It is evident that 2 is the neareſt integer root, being the cube root of 8 = a3.
Hence 2 N + a3 = 21/2 + 8 = 74/4, and N + 2 a3 = 21/4 + 16 = 85/4; then as 85 ∶ 74 ∷ 2 ∶ 148/85 or = 7/4 nearly. Then taking 7/4 for a, we have 2 N + a3 = 21/2 + 343/64 = 1015,64, and N + 2 a3 = 21/4 + 2.343/64 = 1022/64; [Page 53] hence as 1022 : 1015, or as [...] nearly. Conſequently the ſquare of this, or [...] will be = 7^{2}/4^{2} × 145^{2}/146^{2} = 1030225/341056 = 3 7057/341056 = 3.020690, the quantity ſought more nearly, being true in the laſt figure.
1.5. TRACT V. A new Method of finding, in finite and general Terms, near Values of the Roots of Equations of this Form, [...]; namely, having the Terms alternately Plus and Minus.
[Page 54]1. THE following is one method more, to be added to the many we are already poſſeſſed of, for determining the roots of the higher equations. By means of it we readily find a root, which is ſometimes accurate; and when not ſo, it is at leaſt near the truth, and that by an eaſy finite formula, which is general for all equations of the above form, and of the ſame dimenſion, provided that root be a real one. This is of uſe for depreſſing the equation down to lower dimenſions, and thence for finding all the roots one after another, when the formula gives the root ſufficiently exact; and when not, it ſerves as a ready means of obtaining a near value of a root, by which to commence an approximation ſtill nearer, by the previouſly known methods of Newton, or Halley, or others. This method is farther uſeful in elucidating the nature of equations, and certain properties of numbers; as will appear in ſome of the following articles. We have already eaſy methods for finding the roots of ſimple and quadratic equations. I ſhall therefore begin with the cubic equation, and treat of each order of equations ſeparately, in aſcending gradually to the higher dimenſions.
2. Let then the cubic equation x3 − px2 + qx − r = o be propoſed. Aſſume the root x = a, either accurately or approximately, as it may happen, ſo that x − a = o, accurately or nearly. Raiſe this [Page 55] x − a = o to the third power, the ſame dimenſion with the propoſed equation, ſo ſhall x3 − 3 a x2 + 3 a2 x − a3 = o; but the propoſed equation is x3 − p x2 + q x − r = o; therefore the one of theſe is equal to the other. But the firſt term (x3) of each is the ſame; and hence, if we aſſume the ſecond terms equal between themſelves, it will follow that the ſum of the two remaining terms will alſo be equal, and give a ſimple equation by which the value of x is determined. Thus, 3a x2 being = px2, or a = ⅓p, we ſhall have 3a2 x − a3 = qx − r, and hence [...], by ſubſtituting ⅓p, the value of a, inſtead of it.
3. Now this value of x here found, will be the middle root of the propoſed cubic equation. For becauſe a is aſſumed nearly or accurately equal to x, and alſo equal to ⅓ p, therefore x is = ⅓ p nearly or accurately, that is, ⅓ of the ſum of the three roots, to which the coefficient p of the ſecond term of the equation, is always equal; and thus, being a medium among the three roots, it will be either nearly or accurately equal to the middle root of the propoſed equation, when that root is a real one.
4. Now this value of x will always be the middle root accurately, whenever the three roots are in arithmetical progreſſion; otherwiſe, only approximately. For when the three roots are in arithmetical progreſſion, ⅓ p or ⅓ of their ſum, it is well known, is equal to the middle term or root. In the other caſes, therefore, the above-found value of x is only near the middle root.
5. When the roots are in arithmetical progreſſion, becauſe the middle term or root is then = ⅓p, and alſo [...], therefore [...], or [...], an equation [Page 56] expreſſing the general relation of p, q, and r; where p is the ſum of any three terms in arithmetical progreſſion, q the ſum of their three rectangles, and r the product of all the three. For, in any equation, the coefficient p of the ſecond term, is the ſum of the roots; the coefficient q of the third term, is the ſum of the rectangles of the roots; and the coefficient r of the fourth term, is the ſum of the ſolids of the roots, which in the caſe of the cubic equation is only one:—Thus, if the roots, or arithmetical terms, be 1, 2, 3. Here p = 1 + 2 + 3 = 6, q = 1 × 2 + 1 × 3 + 2 × 3 = 2 + 3 + 6 = 11, r = 1 × 2 × 3 = 6; then 2 p3 = 2 × 6^{3} = 432, and [...] alſo.
6. To illuſtrate now the rule [...] by ſome examples; let us in the firſt place take the equation x3 − 6 x2 + 11 x − 6 = 0. Here p = 6, q = 11, and r = 6; conſequently [...]. This being ſubſtituted for x in the given equation, makes all the terms to vaniſh, and therefore it is an exact root, and the roots will be in arithmetical progreſſion. Dividing therefore the given equation by x − 2 = 0, the quotient is x2 − 4x + 3 = 0, the roots of which quadratic equation are 3 and 1, the other two roots of the propoſed equation x3 − 6 x2 + 11 x − 6 = 0.
7. If the equation be x3 − 39x2 + 479x − 1881 = 0; we ſhall have p = 39, q = 479, and r = 1881; then [...]. Then, ſubſtituting 11 2/7 for x in the propoſed equation, the negative terms are ſound to exceed the poſitive terms by 5, thereby ſhewing that 11 2/7 is very near, but ſomething above, the middle root, and that therefore the roots are not in arithmetical progreſſion. It is therefore probable [Page 57] that 11 may be the true value of the root, and on trial it is found to ſucceed.
Then dividing x3 − 39x2 + 479x − 1881 by x − 11, the quotient is x• − 28x + 171 = 0, the roots of which quadratic equation are 9 and 19, the two other roots of the propoſed equation.
8. If the equation be x2 − 6x2 + 9x − 2 = 0; we ſhall have p = 6, q = 9, and r = 2; then [...]. This value of x being ſubſtituted for it in the propoſed equation, cauſes all the terms to vaniſh, as it ought, thereby ſhewing that 2 is the middle root, and that the roots are in arithmetical progreſſion.
Accordingly, dividing the given quantity x3 − 6x2 + 9x − 2 by x − 2, the quotient is x• − 4x + 1 = 0, a quadratic equation, whoſe roots are 2 + √2 and 2 − √2, the two other roots of the equation propoſed.
9. If the equation be x3 − 5x2 + 5x − 1 = 0; we ſhall have p = 5, q = 5, and r = 1; then [...]. From which one might gueſs the root ought to be 1, and which being tried, is found to ſucceed.
But without ſuch trial, we may make uſe of this value 1 4/45, or 1 1/ [...] nearly, and approximate with it in the common way.
Having found the middle root to be 1, divide the given quantity x3 − 5x2 + 5x − 1 by x − 1, and the quotient is x2 − 4x + 1 = 0, the roots of which are 2 + √2 and 2 − √2, the two other roots, as in the laſt article.
[Page 58] 10. If the equation be x3 − 7x2 + 18x − 18 = 0; we ſhall have p = 7, q = 18, and r = 18; then [...] or 3 nearly. Then trying 3 for x, it is found to ſucceed. And dividing x3 − 7x2 + 18x − 18 by x − 3, the quotient is x• − 4x + 6 = 0, a quadratic equation whoſe roots are 2 + √−2 and 2 − √−2, the two other roots of the propoſed equation, which are both impoſſible or imaginary.
11. If the equation be x3 − 6x2 + 14x − 12 = 0; we ſhall have p = 6, q = 14, and r = 12; then [...]. Which being ſubſtituted for x, it is found to anſwer, the ſum of the terms coming out = 0. Therefore the roots are in arithmetical progreſſion. And, accordingly, by dividing x3 − 6x2 + 14x − 12 by x − 2, the quotient is x2 − 4x + 6 = 0, the roots of which quadratic equation are 2 + √−2 and 2 − √−2, the two other roots of the propoſed equation, and the common difference of the three roots is √−2.
12. But if the equation be x3 − 8x2 + 22x − 24 = 0; we ſhall have p = 8, q = 22, and r = 24; then [...]. Which being ſubſtituted for x in the propoſed equation, the ſum of the terms differs very widely from the truth, thereby ſhewing that the middle root of the equation is an imaginary one, as it is indeed, the three roots being 4, and 2 + √−2, and 2 − √−2.
13. In Art. 2 the value of x was determined by aſſuming the ſecond terms of the two equations equal to each other. But a like near value might be determined by aſſuming either the two third terms, or the two ſourth terms equal.
And if we aſſume the fourth terms equal, namely a3 = r, or 3√r, then the ſums of the ſecond and third terms will be equal, namely, 3ax − 3a2 = px − q; and hence [...], by ſubſtituting r⅓ inſtead of a. And either of theſe two formulas will give nearly the ſame value of the root as the firſt formula, at leaſt when the roots do not differ very greatly from one another.
But if they differ very much among themſelves, the firſt formula will not be ſo accurate as theſe two others, becauſe that in them the roots were more complexly mixed together; for the ſecond formula is drawn from the coefficient of the third term, which is the ſum of all the rectangles of the roots; and the third formula is drawn from the coefficient of the laſt term, which is equal to the continual product of all the roots; while the firſt formula is drawn from the coefficient of the ſecond term, which is ſimply the ſum of the roots. And indeed the laſt theorem is commonly the neareſt of all. So that when we ſuſpect the roots to be very wide of each other, let either the ſecond or third be uſed.
14. To proceed now, in like manner, to the biquadratic equation, which is of this general form x4 − px3 + qx2 − rx + s = 0.
Aſſume the root x = a, or x − a = 0, and raiſe this equation x − a = 0 to the fourth power, or the ſame height with the propoſed equation, which will give x4 − 4ax3 + 6a2 x2 − 4a3 x + a4 = 0; but the propoſed equation is x4 − px3 + qx2 − rx + s = 0; therefore theſe two are equal to each other. Now if we aſſume the ſecond terms equal, namely 4a = p, or a = ¼p, then the ſums of the three remaining terms will alſo be equal, namely, [...]; and hence [...], or [...] by ſubſtituting ¼p inſtead of a: then, reſolving this quadratic equation, we find its roots to be thus [...]; or if we put A = 3/2 p2 − 4q, B = p2 − 16r, C = p4 − 256s, the two roots will be [...].
15. It is evident that the ſame property is to be underſtood here, as for the cubic equation in Art. 3, namely, that the two roots above found, are the middle roots of the four which belong to the biquadratic equation, when thoſe roots are real ones; for otherwiſe the formulae are [Page 61] of no uſe. But however thoſe roots will not be accurate, when the ſum of the two middle roots, of the propoſed equation, is equal to the ſum of the greateſt and leaſt roots, or when the four roots are in arithmetical progreſſion; becauſe that, in this caſe, ¼ p, the aſſumed value of a, is neither of the middle roots exactly, but only a mean between them.
16. To exemplify this formula [...], let the propoſed equation be x4 − 12 x3 + 49 x2 − 78 x + 40 = 0. Then A = 3/2 p2 − 4 q = 12^{2} × 3/2 − 4 × 49 = 216 − 196 = 20, B = p3 − 16 r = 12^{3} − 16 × 78 = 1728 − 1248 = 480, C = p4 − 256s = 12^{4} − 256 × 40 = 20736 − 10240 = 10496. Hence [...] nearly, or 4¼ and 1¾ nearly, or nearly 4 and 2, whoſe ſum is 6. And trying 4 and 2, they are both found to anſwer, and therefore they are the two middle roots.
Then [...], by which dividing the given equation x4 − 12 x3 + 49 x2 − 78 x + 40 = 0, the quotient is x2 − 6 x + 5 = 0, the roots of which quadratic equation are 5 and 1, and which therefore are the greateſt and leaſt roots of the equation propoſed.
17. If the equation be x4 − 12 x3 + 47 x2 − 72 x + 36 = 0; then A = 3/2 p2 − 4 q = 12^{2} × 3/2 − 4 × 47 = 216 − 188 = 28, B = p3 − 16 r = 12^{3} − 16 × 72 = 1728 − 1152 = 576, C = p4 − 256 s = 12^{4} − 256 × 36 = 20736 − 9216 = 11520. Hence [...] and 2 1/7, or 3 and 2 nearly; both of which anſwer on trial; and therefore 3 and 2 are the two middle roots.
[Page 62] Then [...], by which dividing the given quantity x4 − 12 x3 + 47 x2 − 72 x + 36 = 0, the quotient is x2 − 7 x + 6 = 0, the roots of which quadratic equation are 6 and 1, which therefore are the greateſt and leaſt roots of the equation propoſed.
18. If the equation be x4 − 7 x3 + 15 x2 − 11 x + 3 = 0; we have A = 3/2 p2 − 4 q = 7^{2} × 3/2 − 4 × 15 = 73½ − 60 = 13½, B = p3 − 16 r = 7^{3} − 16 × 11 = 343 − 176 = 167, C = p4 − 256 s = 7^{4} − 256 × 3 = 2401 − 768 = 1633. Hence [...] or nearly 2 and 1; both which are found, on trial, to anſwer; and therefore 2 and 1 are the two middle roots ſought.
Then [...], by which dividing the given equation x4 − 7 x3 + 15 x2 − 11 x + 3 = 0, the quotient is x2 − 4 x + 1 = 0, the roots of which quadratic equation are 2 + √2 and 2 − √2, and which therefore are the greateſt and leaſt roots of the propoſed equation.
- A = 3/2p2 − 4 q = 9^{2} × 3/2 −4 × 30 = 121½ − 120 = 1½,
- B = p3 − 16 r = 9^{3} − 16 × 46 = 729 − 736 = − 7,
- C = p4 − 256 s = 9^{4} − 256 × 24 = 6561 − 6144 = 417.
20. And thus in other examples the two middle roots will be found when they are rational, or a near value when irrational, which in this [Page 63] caſe will ſerve for the foundation of a nearer approximation, to be made in the uſual way.
We might alſo find another formula for the biquadratic equation, by aſſuming the laſt terms as equal to each other; for then the ſum of the 2d, 3d, and 4th terms of each would be equal, and would form another quadratic equation, whoſe roots would be nearly the two middle roots of the biquadratic propoſed.
21. Or a root of the biquadratic equation may eaſily be found, by aſſuming it equal to the product of two ſquares, as [...]. For, comparing the terms of this with the terms of the equation propoſed, in this manner, namely, making the ſecond terms equal, then the third terms equal, and laſtly the ſums of the fourth and fifth terms equal, theſe equations will determine a near value of x by a ſimple equation. For thoſe equations are [...], [...], [...]. Then the values of ab and a + b, found from the firſt and ſecond of theſe equations, and ſubſtituted in the third, this gives [...], a general formula for one of the roots of the biquadratic equation x4 − px3 + qx2 − rx + s = 0.
22. To exemplify now this ſormula, let us take the ſame equation as in Art. 17, namely, x4 − 12 x3 + 47 x2 − 72 x + 36 = 0, the roots of which were there found to be 1, 2, 3, and 6. Then, by our laſt formula we ſhall have [...], or nearly 1, which is the leaſt root.
[Page 64] 23. Again, in the equation x4 − 7 x3 + 15 x − 11 x2 + 3 = 0, whoſe roots are 1, 2, 2 + √2, and 2 − √2, we have [...] nearly, which is nearly a mean between the two leaſt roots 1 and 2 − √2 or ⅗ nearly.
24. But if the equation be x4 − 9 x3 + 30 x2 − 46 x + 24 = 0, which has impoſſible roots, the four roots being 1, 2 + √−2, 2 − √−2, and 4; we ſhall have [...] nearly, which is of no uſe in this caſe of imaginary roots.
25. This formula will alſo ſometimes fail when the roots are all real. As if the equation be x4 − 12 x3 + 49 x2 − 78 x + 40 = 0, the roots of which are 1, 2, 4, and 5. For here [...], which is of no uſe.
26. For equations of higher dimenſions, as the 5th, the 6th, the 7th, &c. we might, in imitation of this laſt method, combine other forms of quantities together. Thus, for the 5th power, we might compare it either with [...], or with [...], or with [...], or with [...]. And ſo for the other powers.
1.6. TRACT VI. Of the Binomial Theorem. With a Demonſtration of the Truth of it in the General Caſe of Fractional Exponents.
[Page 65]1. IT is well known that this famous theorem is called binomial, becauſe it contains a propoſition of a quantity conſiſting of two terms, as a radix, to be expanded in a ſeries of equal value. It is alſo called emphatically the Newtonian theorem, or Newton's binomial theorem, becauſe he has commonly been reputed the author of it, as he was indeed for the caſe of fractional exponents, which is the moſt general of all, and includes all the other particular caſes, of powers, or diviſions, &c.
2. The binomial, as propoſed in its general form, was, by Newton, thus expreſſed [...]; where P is the firſt term of the binomial, Q the quotient of the ſecond term divided by the firſt, and conſequently PQ is the ſecond term itſelf; or PQ may repreſent all the terms of a multinomial, after the firſt term, and conſequently Q the quotient of all thoſe terms, except the firſt term, divided by that firſt term, and may be either poſitive or negative; alſo m/n repreſents the exponent of the binomial, and may denote any quantity, integral or fractional, poſitive or negative, rational or ſurd. When the exponent [Page 66] is integral, the denominator n is equal to 1, and the quantity then in this form [...], denotes a binomial to be raiſed to ſome power; the ſeries for which was fully determined before Newton's time, as I have ſhewn in the hiſtorical introduction to my Mathematical Tables, lately publiſhed. When the exponent is fractional, m and n may be any quantities whatever, m denoting the index of ſome power to which the binomial is to be raiſed, and n the index of the root to be extracted of that power: and to this caſe it was firſt extended and applied by Newton. When the exponent is negative, the reciprocal of the ſame quantity is meant; as [...] is equal to [...].
3. Now when the radical binomial is expanded in an equivalent ſeries, it is aſſerted that it will be in this general form, namely [...]. where the law of the progreſſion is viſible, and the quantities P, m, n, Q, include their ſigns + or −, the terms of the ſeries being all poſitive when Q is poſitive, and alternately poſitive and negative when Q is negative, independent however of the effect of the coefficients made up of m and n: alſo A, B, C, D, &c. in the latter form, denote each preceding term. This latter form is the eaſier in practice, when we want [Page 67] to collect the ſum of the terms of a ſeries; but the former is the fitter for ſhewing the law of the progreſſion of the terms.
4. The truth of this ſeries was not demonſtrated by Newton, but only inferred by way of induction. Since his time however, ſeveral attempts have been made to demonſtrate it, with various ſucceſs, and in various ways; of which however thoſe are juſtly preferred, which proceed by pure algebra, and without the help of fluxions. And ſuch has been eſteemed the difficulty of proving the general caſe independent of the doctrine of fluxions, that many eminent mathematicians to this day account the demonſtration not fully accompliſhed, and ſtill a thing greatly to be deſired. Such a demonſtration I think I have effected. But before I deliver it, it may not be improper to premiſe ſomewhat of the hiſtory of this theorem, its riſe, progreſs, extenſion, and demonſtrations.
5. Till very lately the prevailing opinion has been, that the theorem was not only invented by Newton, but firſt of all by him; that is, in that ſtate of perfection in which the terms of the ſeries for any aſſigned power whatever, can be found independently of the terms of the preceding powers; namely, the ſecond term from the firſt, the third term from the ſecond, the fourth term from the third, and ſo on, by a general rule. Upon this point I have already given an opinion in the hiſtory to my logarithms, above cited, and I ſhall here enlarge ſomewhat farther on the ſame head.
That Newton invented it himſelf, I make no doubt. But that he was not the firſt inventor, is at leaſt as certain. It was deſcribed by Briggs, in his Trigonometria Britannica, long before Newton was born; not indeed for fractional exponents, for that was the application of Newton, but for any integral power whatever, and that by the general law of the terms as laid down by Newton, independent of the terms of the powers preceding that which is required. For as to the generation of the coefficients of the terms of one power from thoſe of [Page 68] the preceding powers, ſucceſſively one after another, it was remarked by Vieta, Oughtred, and many others, and was not unknown to much more early writers on arithmetic and algebra, as will be manifeſt by a ſlight inſpection of their works, as well as the gradual advance the property made, both in extent and perſpicuity, under the hands of the ſucceſſive maſters in arithmetic, every one adding ſomewhat more towards the perfection of it.
6. Now the knowledge of this property of the coefficients of the terms in the powers of a binomial, is at leaſt as old as the practice of the extraction of roots; for this property was both the foundation, the principle, and the means of thoſe extractions. And as the writers on arithmetic became acquainted with the nature of the coefficients in powers ſtill higher, juſt ſo much higher did they extend the extraction of roots, ſtill making uſe of this property. At firſt it ſeems they were only acquainted with the nature of the ſquare, which conſiſts of theſe three terms, 1, 2, 1; and accordingly extracted the ſquare roots of numbers by means of them; but went no farther. The nature of the cube next preſented itſelf, which conſiſts of theſe four terms, 1, 3, 3, 1; and by means of theſe they extracted the cubic roots of numbers, in the ſame manner as we do at preſent. And this was the extent of their extractions in the time of Lucas de Burgo, an Italian, who, from 1470 to 1500, wrote ſeveral tracts on arithmetic, containing the ſum of what was then known of this ſcience, which chiefly conſiſted in the doctrine of the proportions of numbers, the nature of figurate numbers, and the extraction of roots, as far as the cubic root incluſively.
8. The contemplation of this table has probably been attended with the invention and extenſion of ſome of our moſt curious diſcoveries in mathematics, both in regard to the powers of a binomial, with the conſequent extraction of roots, the doctrine of angular ſections by Vieta, and the differential method by Briggs and others. For, one or two of the powers or ſections being once known, the table would be of excellent uſe in diſcovering and conſtructing the reſt. And accordingly we find this table uſed on many occaſions by Stifelius, Cardan, Stevin, Vieta, Briggs, Oughtred, Mercator, Paſcal, &c. &c.
9. On this occaſion I cannot help mentioning the ample manner in which I ſee Stifelius, at fol. 35, et ſeq. of the ſame book, treats of the nature and uſe of logarithms, though not under the ſame name, but under the idea of a ſeries of arithmeticals, adapted to a ſeries of geometricals. He there explains all their uſes; ſuch as that the addition of them, anſwers to the multiplication of their geometricals; ſubtraction to diviſion; multiplication of exponents, to involution; and dividing of exponents, to evolution. And he exemplifies the uſe of them in caſes of the Rule-of-Three, and in finding mean proportionals between given terms, and ſuch like, exactly as is done in logarithms. So that he ſeems to have been in the full poſſeſſion of the idea of logarithms, and wanted only the neceſſity of troubleſome calculations to induce him to make a table of ſuch numbers.
[Page 71] 10. But although the nature and conſtruction of this table, namely of figurate numbers, was thus early known, and employed in the raiſing of powers, and extracting of roots; yet it was only by raiſing the numbers one from another by continual additions, and then taking them from the table for uſe when wanted; till Briggs firſt pointed out the way of raiſing any horizontal line in the foregoing table by itſelf, without any of the preceding lines; and thus teaching to raiſe the terms of any power of a binomial, independent of any other powers; and ſo gave the ſubſtance of the binomial ſeries in words, wanting only the notation in ſymbols; as I have ſhewn at large at page 75 of the hiſtorical introduction to my Mathematical Tables.
11. Whatever was known however of this matter, related only to pure or integral powers, no one before Newton having thought of extracting roots by infinite ſeries. He happily diſcovered, that, by conſidering powers and roots in a continued ſeries, roots being as powers having fractional exponents, the ſame binomial ſeries would equally ſerve for them all, whether the index ſhould be fractional or integral, or the ſeries be finite or infinite.
12. The truth of this method however was long known only by trial in particular caſes, and by induction from analogy. Nor does it appear that even Newton himſelf ever attempted any direct proof of it. But various demonſtrations of this theorem have been ſince given by the more modern mathematicians, of which ſome are by means of the doctrine of fluxions, and others, more legally, from the pure principles of algebra only. Some of which I ſhall here give a ſhort account of.
13. One of the firſt was Mr. James Bernoulli. His demonſtration is, among ſeveral other curious things, contained in his little work called Ars Conjectandi, which has been improperly omitted in the collection of his works publiſhed by his nephew Nicholas Bernoulli. This is a ſtrict [Page 72] demonſtration of the binomial theorem in the caſe of integral and affirmative powers, and is to this effect. Suppoſing the theorem to be true in any one power, as for inſtance, in the cube, it muſt be true in the next higher power; which he demonſtrates. But it is true in the cube, in the fourth, fifth, ſixth, and ſeventh powers, as will eaſily appear by trial, that is by actually raiſing thoſe powers by continual multiplications. Therefore it is true in all higher powers. All this he ſhews in a regular and legitimate manner, from the principles of multiplication, and without the help of fluxions. But he could not extend his proof to the other caſes of the binomial theorem, in which the powers are fractional. And this demonſtration has been copied by Mr. John Stewart, in his commentary on Sir Iſaac Newton's quadrature of curves. To which he has added, from the principles of fluxions, a demonſtration of the other caſe, for roots or fractional exponents.
14. In No. 230 of the Philoſophical Tranſactions for the year 1697, is given a theorem, by Mr. De Moivre, in imitation of the binomial theorem, which is extended to any number of terms, and thence called the multinomial theorem; which is a general expreſſion in a ſeries, for raiſing any multinomial quantity to any power. His demonſtration of the truth of this theorem, is independent of the truth of the binomial theorem, and contains in it a demonſtration of the binomial theorem as a ſubordinate propoſition, or particular caſe of the other more general theorem. And this demonſtration may be conſidered as a legitimate one, for pure powers, founded on the principles of multiplication, that is, on the doctrine of combinations and permutations. And it proves that the law of the continuation of the terms, muſt be the ſame in the terms not computed, or not ſet down, as in thoſe that are written down.
15. The ingenious Mr. Landen has given an inveſtigation of the binomial theorem, in his Diſcourſe concerning the Reſidual Analyſis, printed in 1758, and in the Reſidual Analyſis itſelf, printed in 1764. [Page 73] The inveſtigation is deduced from this lemma, namely, if m and n be any integers, and q = v/x, then is [...] which theorem is made the principal baſis of his Reſidual Analyſis.
The inveſtigation is this: the binomial propoſed being [...], aſſume it equal to the following ſeries 1 + ax + bx2 + cx3 &c. with indeterminate coefficients. Then for the ſame reaſon as [...] will [...] Then, by ſubtraction, [...] And, dividing both ſides by x − y, and by the lemma, we have [...] Then, as this equation muſt hold true whatever be the value of y, take y = x, and it will become [...] Conſequently, multiplying by 1 + x, we have [...], or its equal by the aſſumption, viz. [...] [...] [Page 74] Then, by comparing the homologous terms, the value of the coefficients a, b, c, &c. are deduced for as many terms as you compare.
And a large account is given of this inveſtigation by the learned Dr. Hales, in his Analyſis Equationum, lately publiſhed at Dublin.
Mr. Landen then contraſts this inveſtigation with that by the method of fluxions, which is as follows. Aſſume as before; [...] Take the fluxion of each ſide, and we have [...] Divide by ẋ, or take it = 1, ſo ſhall [...]
Then multiply by 1 + x, and ſo on as above in the other way.
16. Beſides the above, which are the principal demonſtrations and inveſtigations that have been given of this important theorem, I have been ſhewn an ingenious attempt of Mr. Baron Maſeres, to demonſtrate this theorem in the caſe of roots or fractional exponents, by the help of De Moivre's multinomial theorem. But, not being quite ſatiſfied with his own demonſtration, as not expreſſing the law of continuation of the terms which are not actually ſet down, he was pleaſed to urge me to attempt a more complete and ſatisfactory demonſtration of the general caſe of roots, or fractional exponents. And he farther propoſed it in this form, namely, that if Q be the coefficient of one of the terms of the ſeries which is equal to [...], and P the coefficient of the next preceding term, and R the coefficient of the next followlowing term; then, if Q be [...], to prove that R will be [...]. This he obſerved would be quite perfect and ſatisfactory, [Page 75] as it would include all the terms of the ſeries, as well thoſe that are omitted, as thoſe that are actually ſet down. And I was, in my demonſtration, to ſuppoſe, if I pleaſed, the truth of the binomial and multinomial theorems for integral powers, as truths that had been previouſly and perfectly proved.
In conſequence I ſent him ſoon after the ſubſtance of the following demonſtration; with which he was quite ſatisfied, and which I now proceed to explain at large.
17. Now the binomial integral is [...]. where a, b, c, &c. denote the whole coefficients of the 2d, 3d, 4th, &c. terms, over which they are placed; and in which the law is this, namely, if P, Q, R, be the coefficients of any three terms in ſucceſſion, and if g/b P = Q, then is [...]; as is evident; and which, it is granted, has been proved.
18. And the binomial fractional is [...]. in which the law is this, namely, if P, Q, R be the coefficients of three terms in ſucceſſion; and if g/b P = Q, then is [...]. Which is the property to be proved.
[Page 76] 19. Again, the multinomial integral is [...] [Page 77] [...] &c. Or, if we put a, b, c, d, &c. for the coefficients of the 2d, 3d, 4th, 5th, &c. terms, the laſt ſeries, by ſubſtitution, will be transformed into this form, [...]
[Page 78] 20. Now, to find the ſeries in Art. 18, aſſume the propoſed binomial equal to a ſeries with indeterminate coefficients, as [...] Then raiſe each ſide to the n power, ſo ſhall [...]. But it is granted that the multinomial raiſed to any integral power is proved, and known to be, as in the laſt Art. [...] It follows then, that if this laſt ſeries be equal to 1 + x, by equating the homologous coefficients, all the terms after the ſecond muſt vaniſh, or all the coefficients b, c, d, &c. after the ſecond term, muſt be each = 0. Writing therefore, in this ſeries, 0 for each of the letters b, c, d, &c. it will become of this more ſimple form, [...]. Put now each of the coefficients, after the ſecond term, = 0, and we ſhall have theſe equations [...] [...] [...] [...] &c. [Page 79] The reſolution of which equations gives the following values of the aſſumed indeterminate coefficients, namely, [...], &c. which coefficients are according to the law propoſed, namely, when g/h P = Q, then g−n/h+n Q = R. Q. E. D.
21. Alſo, by equating the ſecond coefficients, namely, 1 = a = nA, we find A = 1/n. This being written for A in the above values of B, C, D, &c. will give the proper ſeries for the binomial in queſtion, namely [...].
1.6.1. Of the FORM of the ASSUMED SERIES.
22. In the demonſtrations or inveſtigations of the truth of the binomial theorem, the but or object has always been the law of the coefficients of the terms: the form of the ſeries, as to the powers of x, having never been diſputed, but taken for granted, either as incapable of receiving a demonſtration, or as too evident to need one. But ſince the demonſtration of the law of the coefficients has been accompliſhed, in which the main, if not the only, difficulty was ſuppoſed to conſiſt, we have extended our reſearches ſtill farther, and have even doubted or queried the very form of the terms themſelves, namely, 1 + Ax + Bx2 + Cx3 + Dx4 + &c. increaſing by the regular integral ſeries of the powers of x, as aſſumed to denote the quantity [...], or the n root of 1 + x. And in conſequence of theſe ſcruples, I have been required, by a learned friend, to vindicate the [Page 80] propriety of that aſſumption. Which I think is effectually done as follows.
23. To prove then, that any root of the binomial 1 + x can be repreſented by a ſeries of this form 1 + x + x2 + x3 + x4 &c. where the coefficients are omitted, our attention being now employed only on the powers of x; let the ſeries repreſenting the value of [...] be 1 + A + B + C + D + &c. where A, B, C, &c. now repreſent the whole of the 2d, 3d, 4th, &c. terms, both their coefficients and the powers of x, whatever they may be, only increaſing from the leſs to the greater, becauſe they increaſe in the terms 1 + x of the given binomial itſelf; and in which the firſt term muſt evidently be 1, the ſame as in the given binomial.
Raiſe now [...] and its equivalent ſeries 1 + A + B + C + &c. both to the n power by the multinomial theorem, and we ſhall have, as before, [...] Then equate the correſponding terms, and we have the firſt term 1 = 1.
Again, the ſecond term of the ſeries n/1 A, muſt be equal to the ſecond term x of the binomial. For none of the other terms of the ſeries are equipollent, or contain the ſame power of x, with the term n/1 A. Not any of the terms A^{2}, A^{3}, A^{4}, &c. for they are double, triple, quadruple, &c. in power to A. Nor yet any of the terms containing B, C, D, &c. becauſe, by the ſuppoſition, they contain all different and [Page 81] increaſing powers. It follows therefore, that n/1 A makes up the whole value of the ſecond term x of the given binomial. Conſequently the ſecond term A of the aſſumed ſeries, contains only the firſt power of x; and the whole value of that term A is = 1/nx.
But all the other equipollent terms of the expanded ſeries muſt be equal to nothing, which is the general value of the terms, after the ſecond, of the given quantity 1 + x or 1 + x + 0 + 0 + 0 + &c. Our buſineſs is therefore to find the ſeveral orders of equipollent terms of the expanded ſeries. And theſe I ſay will be as I have arranged them above, in which B is equipollent with A^{2}, C with A^{3}, D with A^{4}, and ſo on.
Now that B is equipollent with A^{2}, is thus proved. The value of the third term is 0. But [...] is a part of the third term. And it is only a part of that term: otherwiſe [...] would be = 0, which it is evident cannot happen in every value of n, as it ought; for indeed it happens only when n is = 1. Some other quantity then muſt be equipollent with n/1 · n−1/2 A^{2}, and muſt be joined with it, to make up the whole third term equal to 0. Now that ſupplemental quantity can be no other than n/1B: for all the other following terms are evidently plupollent than B. It follows therefore, that B is equipollent with A^{2}, and contains the ſecond power of x; or that [...], and conſequently [...].
Again, the fourth term muſt be = 0. But the quantities n/1 · n−1/2 · n−2/3 A^{3} + n/1 · n−1/2 AB are equipollent, and make up part of that fourth term. They are equipollent, or A^{3} equipollent with AB, becauſe A^{2} and B are equipollent. And they do not [Page 82] conſtitute the whole of that term; for if they did, then would n1 · n−1/2 · n−2/3 A^{3} + n/1 · n−1/2 AB be = 0 in all values of n, or n−2/3 A^{3} + B = 0: but it has been juſt ſhewn above, that n−1/2 A^{2} + B = 0; it would therefore follow that n−2/3 would be = n−1/2, a circumſtance which can only happen where n = −1, inſtead of taking place for every value of n. Some other quantity muſt therefore be joined with theſe to make up the whole of the fourth term. And this ſupplemental quantity can be no other than n/1 c, becauſe all the other following quantities are evidently plupollent than A^{3} or AB. It follows therefore, that C is equipollent with A^{3}, and therefore contains the 3d power of x. And the whole value of C is [...].
And the proceſs is the ſame for all the other following terms. Thus, then, we have proved the law of the whole ſeries, both with reſpect to the coefficients of its terms, and to the powers of the letter x.
1.7. TRACT VII. Of the Common Sections of the Sphere and Cone. Together with the Demonſtration of ſome other New Properties of the Sphere, which are ſimilar to certain Known Properties of the Circle.
[Page 83]THE ſtudy of the mathematical ſciences is uſeful and profitable, not only on account of the benefit derivable from them to the affairs of mankind in general; but are moſt eminently ſo, for the pleaſure and delight the human mind feels in the diſcovery and contemplation of the endleſs number of truths that are continually preſenting themſelves to our view. Theſe meditations are of a ſublimity far above all others, whether they be purely intellectual, or whether they reſpect the nature and properties of material objects: they methodiſe, ſtrengthen, and extend the reaſoning faculties in the moſt eminent degree, and ſo fit the mind the better for underſtanding and improving every other ſcience; but, above all, they furniſh us with the pureſt and moſt permanent delight, from the contemplation of truths peculiarly certain and immutable, and from the beautiful analogy which reigns through all the objects of ſimilar inquiry. In the mathematical ſciences, the diſcovery, often accidental, of a plain and ſimple property, is but the harbinger of a thouſand others of the moſt ſublime and beautiful nature, to which we are gradually led, delighted, from the more ſimple to the more compound and general, till the mind becomes quite enraptured at the full blaze of light burſting upon it from all directions.
[Page 84] Of theſe very pleaſing ſubjects, the ſtriking analogy that prevails among the properties of geometrical figures, or figured extenſion, is not one of the leaſt. Here we often find that a plain and obvious property of one of the ſimpleſt figures, leads us to, and forms only a particular caſe of, a property in ſome other figure, leſs ſimple; afterwards this again turns out to be no more than a particular caſe of another ſtill more general; and ſo on, till at laſt we often trace the tendency to end in a general property of all figures whatever.
The few properties which make a part of this paper, conſtitute a ſmall ſpecimen of the analogy, and even identity, of ſome of the more remarkable properties of the circle, with thoſe of the ſphere. To which are added ſome properties of the lines of ſection, and of contact, between the ſphere and cone. Both which may be farther extended as occaſions may offer: like as all of theſe properties have occurred from the circumſtance, mentioned near the end of the paper, of conſidering the inner ſurface of a hollow ſpherical veſſel, as viewed by an eye, or as illuminated by rays, from a given point.
1.7.1. PROPOSITION I.
All the tangents are equal, which are drawn, from a given point without a ſphere, to the ſurface of the ſphere quite around.
DEMONS. For, let PT be any tangent from the given point P; and draw PC to the center C, and join TC. Alſo let CTA be a great circle of the ſphere in the plane of the triangle TPC. Then, CP and CT, as well as the angle T, which is right (Eucl. iii. 18), being conſtant, in every poſition of the tangent, or of the point of contact T; the ſquare of PT will be every where equal to the difference of the ſquares of the conſtant lines CP, CT, and therefore conſtant; and conſequently the line or tangent PT itſelf of a conſtant length, in every poſition, quite round the ſurface of the ſphere.
1.7.2. PROP. II.
[Page 85]If a tangent be drawn to a ſphere, and a radius be drawn from the center to the point of contact, it will be perpendicular to the tangent; and a perpendicular to the tangent will paſs through the center.
DEMONS. For, let PT be the tangent, TC the radius, and CTA a great circle of the ſphere in the plane of the triangle TPC, as in the foregoing propoſition. Then, PT touching the circle in the point T, the radius TC is perpendicular to the tangent PT by Eucl. iii. 18, 19.
1.7.3. PROP. III.
If any line or chord be drawn in a ſphere, its extremes terminating in the circumference; then a perpendicular drawn to it from the center, will biſect it: and if the line drawn from the center, biſect it, it is perpendicular to it.
DEMONS. For, a plane may paſs through the given line and the center of the ſphere; and the ſection of that plane with the ſphere, will be a great circle (Theodoſ. i. 1), of which the given line will be a chord. Therefore (Eucl. iii. 3) the perpendicular biſects the chord, and the biſecting line is perpendicular.
COROL. A line drawn from the center of the ſphere, to the center of any leſſer circle, or circular ſection, is perpendicular to the plane of that circle. For, by the propoſition, it is perpendicular to all the diameters of that circle.
1.7.4. PROP. IV.
If from a given point, a right line be drawn in any poſition through a ſphere, cutting its ſurface always in two points; the rectangle contained under the whole line and the external part, that is the rectangle contained by the two diſtances between the given point, and the two points where the line meets the ſurface of the ſphere, will always be of [Page 86] the ſame conſtant magnitude, namely, equal to the ſquare of the tangent drawn from the ſame given point.
DEMONS. Let P be the given point, and AB the two points in which the line PAB meets the ſurface of the ſphere: through PAB and the center let a plane cut the ſphere in the great circle TAB, to which draw the tangent PT. Then the rectangle PA.PB is equal to the ſquare of PT (Eucl. iii. 36); but PT, and conſequently its ſquare, is conſtant by Prop. 1; therefore the rectangle PA.PB, which is always equal to this ſquare, is every where of the ſame conſtant magnitude.
1.7.5. PROP. V.
If any two lines interſect each other within a ſphere, and be terminated at the ſurface on both ſides; the rectangle of the parts of the one line, will be equal to the rectangle of the parts of the other. And, univerſally, the rectangles of the two parts of all lines paſſing through the point of interſection, are all of the ſame magnitude.
DEMONS. Through any one of the lines, as AB, conceive a plane to be drawn through the center C of the ſphere, cutting the ſphere in the great circle ABD; and draw its diameter DCPF through the point of interſection P of all the lines. Then the rectangle AP.PB is equal to the rectangle DP.PF (Eucl. iii. 35).
Again, through any other of the interſecting lines GH, and the center, conceive another plane to paſs, cutting the ſphere in another great circle DGFH. Then, becauſe the points C and P are in this latter plane, the line CP, and conſequently the whole diameter DCPF, is in the ſame plane; and therefore it is a diameter of the circle DGFH, of which GPH is a chord. Therefore, again, the rectangle GP.PH is equal to the rectangle DP.PF (Eucl. iii. 35)
[Page 37] Conſequently all the rectangles AP.PB, GP.PH, &c. are equal, being each equal to the conſtant rectangle DP.PF.
COROL. The great circles paſſing through all the lines or chords which interſect in the point P, will all interſect in the common diameter DPF.
1.7.6. PROP. VI.
If a ſphere be placed within a cone, ſo as to touch it in two points; then ſhall the outſide of the ſphere, and the inſide of the cone, mutually touch quite around, and the line of contact will be a circle.
DEMONS. Let V be the vertex of the cone, C the center of the ſphere, T one of the two points of contact, and TV a ſide of the cone. Draw CT, CV. Then TVC is a triangle right-angled at T (Prop. 2). In like manner, t being another point of contact, and Ct being drawn, the triangle tVC will be right-angled at t. Theſe two triangles then, TVC, tVC, having the two ſides CT, TV, equal to the two Ct, tV (Prop. 1), and the included angle T equal to the included angle t, will be equal in all reſpects (Eucl. i. 4), and conſequently have the angle TVC equal to the angle tVC.
Again, let fall the perpendiculars TP, tP. Then the two triangles TVP, tVP, having the two angles TVP and TPV equal to the two tVP and tPV, and the ſide TV equal to the ſide tV (Prop. 1), will be equal in all reſpects (Eucl. i. 26); conſequently TP is equal to tP, and VP equal to VP. Hence PT, Pt are radii of a little circle of the ſphere, whoſe plane is perpendicular to the line CV, and its circumference every where equidiſtant from the point C or V. This circle is therefore a circular ſection both of the ſphere and of the cone, and is therefore the line of their mutual contact. Alſo CV is the axis of the cone.
[Page 88] COROL. 1. The axis of a cone, when produced, paſſes through the center of the inſcribed ſphere.
COROL. 2. Hence alſo, every cone circumſcribing a ſphere, ſo that their ſurfaces touch quite around, is a right cone; nor can any ſcalene or oblique cone touch a ſphere in that manner.
1.7.7. PROP. VII.
The two common ſections of the ſurfaces of a ſphere and a right cone, are the circumferences of circles if the axis of the cone paſs through the center of the ſphere.
DEMONS. Let V be the vertex of the cone, C the center of the ſphere, and S one point of the leſs or nearer ſection; draw the lines CS, CV. Then, in the triangle CSV, the two ſides CS, CV, and the included angle SCV, are conſtant for all poſitions of the ſide VS; and therefore the ſide VS is of a conſtant length for all poſitions, and is conſequently the ſide of a right cone having a circular baſe; therefore the locus of all the points S, is the circumference of a circle perpendicular to the axis CV, that is, the common ſection of the ſurfaces of the ſphere and cone, is that circumference.
In the ſame manner it is proved that, if A be any point in the farther or greater ſection, and CA be drawn; then VA is conſtant for all poſitions, and therefore, as before, is the ſide of a cone cut off by a circular ſection whoſe plane is perpendicular to the axis.
And theſe circles, being both perpendicular to the axis, are parallel to each other. Or, they are parallel becauſe they are both circular ſections of the cone.
COROL. 1. Hence SA = sa, becauſe VA = Va, and VS = Vs.
COROL. 2. All the intercepted equal parts SA, sa, &c. are equally diſtant from the center. For, all the ſides of the triangle SCA [Page 89] are conſtant, and therefore the perpendicular CP is conſtant alſo. And thus all the equal right lines or chords in a ſphere, are equally diſtant from the center.
COROL. 3. The ſections are not circles, and therefore not in planes, if the axis paſs not through the center. For then ſome of the points of ſection are farther from the vertex than others.
1.7.8. PROP. VIII.
Of the two common ſections of a ſphere and an oblique cone, if the one be a circle, the other will be a circle alſo.
DEMONS. Let SAas and ASVa be ſections of the ſphere and cone, made by a common plane paſſing through the axes of the cone and the ſphere; alſo Ss, Aa the diameters of the two ſections. Now, by the ſuppoſition, one of theſe, as Aa, is the diameter of a circle. But the angle VSs = the angle VaA (Eucl. i. 13, and iii. 22), therefore Ss cuts the cone in ſub-contrary poſition to Aa; and conſequently if a plane paſs through Ss, and perpendicular to the plane AVa, its ſection with the oblique cone will be a circle, whoſe diameter is the line Ss (Apol. i. 5). But the ſection of the ſame plane and the ſphere, is alſo a circle whoſe diameter is the ſame line Ss (Theod. i. 1). Conſequently the circumference of the ſame circle, whoſe diameter is Ss, is in the ſurface both of the cone and ſphere; and therefore that circle is the common ſection of the cone and ſphere.
In like manner, if the one ſection be a circle whoſe diameter is Sa, the other ſection will be a circle whoſe diameter is sA.
COROL. 1. Hence if the one ſection be not a circle, neither of them is a circle; and conſequently they are not in planes; for the ſection of a ſphere by a plane, is a circle.
COROL. 2. When the ſections of a ſphere and oblique cone are circles, the axis of the cone does not paſs through the center of the [Page 90] ſphere, (except when one of the ſections is a great circle, or paſſes through the center). For the axis paſſes through the center of the baſe, but not perpendicularly; whereas a line drawn from the center of the ſphere to the center of the baſe, is perpendicular to the baſe, by Cor. to Prop. 3.
COROL. 3. Hence, if the inſide of a bowl, which is a hemiſphere, or any ſegment of the ſphere, be viewed by an eye not ſituated in the axis produced, which is perpendicular to the ſection or brim; the lower, or extreme part of the internal ſurface which is viſible, will be bounded by a circle of the ſphere; and the part of the ſurface ſeen by the eye, will be included between the ſaid circle, and the border or brim, which it interſects in two points. For the eye is in the place of the vertex of the cone; and the rays from the eye to the brim of the bowl, and thence continued from the nearer part of the brim, to the oppoſite internal ſurface, form the ſides of the cone; which, by the propoſition, will form a circular arc on the ſaid internal ſurface; becauſe the brim, which is the one ſection, is a circle.
And hence, the place of the eye being given, the quantity of internal ſurface that can be ſeen, may be eaſily determined. For the diſtance and height of the eye, with reſpect to the brim, will give the greateſt diſtance of the ſection below the brim, together with its magnitude and inclination to the plane of the brim; which being known, common menfuration furniſhes us with the meaſure of the ſurface included between them. Thus, if AB be the diameter in the vertical plane paſſing through the eye at E, alſo AFB the ſection of the bowl by the ſame plane, and AIB the ſupplement of that arc. Draw EAF, EIB, cutting this vertical circle in F and I; and join IF. Then ſhall IF be the diameter of the ſection or extremity of the viſible ſurface, and BF its greateſt diſtance below the brim, an arc which meaſures an angle double the angle at A.
[Page 91] COROL. 4. Hence alſo, and from Propoſition 4, it follows, that if through every point in the circumference of a circle, lines be drawn to a given point E out of the plane of the circle, ſo that the rectangle contained under the parts between the point E and the circle, and between the ſame point E and ſome other point F, may always be of a certain given magnitude; then the locus of all the points F will alſo be a circle, cutting the former circle in the two points where the lines drawn from the given point E, to the ſeveral points in the circumference of the firſt circle, change from the convex to the concave ſide of the circumference. And the conſtant quantity, to which the rectangle of the parts is always equal, is equal to the ſquare of the line drawn from the given point E to either of the ſaid two points of interſection.
And thus the loci of the extremes of all ſuch lines, are circles.
1.7.9. PROP. IX. Prob.
To place a given ſphere, and a given oblique cone, in ſuch poſitions, that their mutual ſections ſhall be circles.
Let V be the vertex, VB the leaſt ſide, and VD the greateſt ſide of the cone. In the plane of the triangle VBD it is evident will be found the center of the ſphere. Parallel to BD draw Aa the diameter of a circular ſection of the cone, ſo that it be not greater than the diameter of the ſphere. Biſect Aa with the perpendicular EC; with the center A and radius of the ſphere, cut EC in C, which will be the center of the ſphere; from which therefore deſcribe a great circle of it cutting the ſides of the cone in the points S, s, A, a : ſo ſhall Ss and Aa be the diameters of circular ſections which are common to both the ſphere and cone.
[Page 92] NOTE. The ſubſtance of the above propoſitions was drawn up ſeveral years ago. And Mr. Bonnycaſtle and Mr. George Sanderſon have this day ſhewn me the ſolution of a queſtion in the London Magazine for April 1777, in which a ſimilar ſection of a ſphere with a cone, is proved to be a circle, and which I had never ſeen before. Nor do I know of any other writings on the ſame ſubject.
July 29, 1785.
1.8. TRACT VIII. Of the Geometrical Diviſion of Circles and Ellipſes into any Number of Parts, and in any propoſed Ratios.
[Page 93]ART. 1. IN the year 1774 was publiſhed a pamphlet in octavo, with this title, A Diſſertation on the Geometrical Analyſis of the Antients. With a Collection of Theorems and Problems, without Solutions, for the Exerciſe of Young Students. This pamphlet was anonymous; it was however well known to myſelf and ſeveral other perſons, that the author of it was the late Mr. John Lawſon, B. D. rector of Swanſcombe in Kent, an ingenious and learned geometrician, and, what is ſtill more eſtimable, a moſt worthy and good man; one in whoſe heart was found no guile, and whoſe pure integrity, joined to the moſt amiable ſimplicity of manners, and ſweetneſs of temper, gained him the affection and reſpect of all who had the happineſs to be acquainted with him. His collection of problems in that pamphlet concluded with this ſingular one, "To divide a circle into any number of parts, which ſhall be as well equal in area as in circumference.—N. B. This may ſeem a paradox, however it may be effected in a manner ſtrictly geometrical." The ſolution of this ſeeming paradox he reſerved to himſelf, as far as I know. I fell upon the diſcovery however ſoon after; and other perſons might do the ſame. My reſolution of it was publiſhed in an account which I gave of the pamphlet in the Critical Review for 1775, vol. xl. and which the author informed me was on [Page 94] the ſame principle as his own. This account is in page 21 of that volume, and in the following words:
2. "We have no doubt but that our mathematical readers will agree with us in allowing the truth of the author's remark concerning the ſeeming paradox of this problem; becauſe there is no geometrical method of dividing the circumference of a circle into any propoſed number of parts taken at pleaſure, and it does not readily appear that there can be any othermethod of reſolving the problem, than by drawing radii to the points of equal diviſion in the circumference. However another method there is, and that ſtrictly geometrical, which is as follows.
"Divide the diameter AB of the given circle into as many equal parts as the circle itſelf is to be divided into, at the points C, D, E, &c. Then on the diameters AC, AD, AE, &c. as alſo on BE, BD, BC, &c. deſcribe ſemicircles, as in the annexed figure: and they will divide the whole circle as required.
"For, the ſeveral diameters being in arithmetical progreſſion, of which the common difference is equal to the leaſt of them, and the diameters of circles being as their circumferences, theſe will alſo be in arithmetical progreſſion. But, in ſuch a progreſſion, the ſum of the extremes is equal to the ſum of each two terms equally diſtant from them; therefore the ſum of the circumferences on AC and CB, is equal to the ſum of thoſe on AD and DB, and of thoſe on AE and EB, &c. and each ſum equal to the ſemi-circumference of the given circle on the diameter AB. Therefore all the parts have equal perimeters, and each is equal to the circumference of the propoſed circle. Which ſatisfies one of the conditions in the problem.
[Page 95] "Again, the ſame diameters being as the numbers 1, 2, 3, 4, &c. and the areas of circles being as the ſquares of their diameters, the ſemicircles will be as the numbers 1, 4, 9, 16, &c. and conſequently the differences between all the adjacent ſemicircles are as the terms of the arithmetical progreſſion 1, 3, 5, 7, &c. and here again the ſums of the extremes, and of every two equidiſtant means, make up the ſeveral equal parts of the circle. Which is the other condition."
3. But this ſubject admits of a more geometrical form, and is capable of being rendered very general and extenſive, and is moreover very fruitful in curious conſequences. For firſt, in whatever ratio the whole diameter is divided, whether into equal or unequal parts, and whatever be the number of the parts, the perimeters of the ſpaces will ſtill be equal. For ſince circumferences of circles are always as their diameters, and becauſe AB and AD + DB and AC + CB are all equal, therefore the ſemi-circumferences c and b + d and a + e are all equal, and conſtant, whatever be the ratio of the parts AD, DC, CB, of the diameter. We ſhall preſently find too that the ſpaces TV, RS, and PQ, will be univerſally as the ſame parts AD, DC, CB, of the diameter.
4. The ſemicircles having been deſcribed as before mentioned, erect CE perpendicular to AB, and join BE. Then I ſay, the circle on the diameter BE, will be equal to the ſpace PQ. For, join AE. [Page 96] Now the ſpace P = ſemicircle on AB − ſemicircle on AC: but the ſemicir. on AB = ſemicir. on AE + ſemicir. on BE, and the ſemicir. on AC = ſemicir. on AE − ſemicir. on CE, theref. ſemic. AB − ſemic. AC = ſemic. BE + ſemicir. CE, that is the ſpace P is = ſemic. BE + ſemicir. CE; to each of theſe add the ſpace Q, or the ſemicircle on BC, then P + Q = ſemic. BE + ſemic. CE + ſemic. BC, that is P + Q = double the ſemic. BE, or = the whole circle on BE.
5. In like manner, the two ſpaces PQ and RS together, or the whole ſpace PQRS, is equal to the circle on the diameter BF. And therefore the ſpace RS alone, is equal to the difference, or the circle on BF minus the circle on BE.
6. But, circles being as the ſquares of their diameters, BE^{2}, BF^{2}, and theſe again being as the parts or lines BC, BD, therefore the ſpaces PQ, PQRS, RS, TV, are reſpectively as the lines BC, BD, CD, AD, And if BC be equal to CD, then will PQ be equal to RS, as in the firſt or ſimpleſt caſe.
7. Hence, to find a circle equal to the ſpace RS, where the points D and C are taken at random: From either end of the diameter, as A, take AG equal to DC, erect GH perpendicular to AB, and join AH; then the circle on AH will be equal to the ſpace RS. For, the ſpace PQ: the ſpace RS ∷ BC ∶ CD or AG, that is as BE^{2}: AH^{2} the ſquares of the diameters, or as the circle on BE to the circle on AH; but the circle on BE is equal to the ſpace PQ, and therefore the circle on AH is equal to the ſpace RS.
8. Hence, to divide a circle in this manner, into any number of parts, that ſhall be in any ratios to one another: Divide the diameter [Page 97] into as many parts, at the points D, C, &c. and in the ſame ratios as thoſe propoſed; then on the ſeveral diſtances of theſe points from the two ends A and B, as diameters, deſcribe the alternate ſemicircles on the different ſides of the whole diameter AB: and they will divide the whole circle in the manner propoſed. That is, the ſpaces TV, RS, PQ, will be as the lines AD, DC, CB.
9. But theſe properties are not confined to the circle alone, but are to be found alſo in the ellipſe, as the genus of which the circle is only a ſpecies. For if the annexed figure be an ellipſe deſcribed on the axis AB, the area of which is, in like manner, divided by ſimilar ſemiellipſes, deſcribed on AD, AC, BC, BD, as axes, all the ſemiperimeters f, ae, bd, c, will be equal to one another, for the ſame reaſon as before in Art. 3, namely, becauſe the peripheries of ellipſes are as their diameters. And the ſame property would ſtill hold good, if AB were any other diameter of the ellipſe, inſtead of the axis; deſcribing upon the parts of it ſemiellipſes which ſhall be ſimilar to thoſe into which the diameter AB divides the given ellipſe.
- PQ is equal to the ſimilar ellipſe on the diameter BE,
- PQRS is equal to the ſimilar ellipſe on the diameter BF,
- RS is equal to the ſimilar ellipſe on the diameter AH,
1.9. TRACT IX. New Experiments in Artillery; for determining the Force of fired Gunpowder, the Initial Velocity of Cannon Balls, the Ranges of Pieces of Cannon at different Elevations, the Reſiſtance of the Air to Projectiles, the Effect of different Lengths of Cannon, and of different Quantities of Powder, &c. &c.
[Page 99]Sect. 1. AT Woolwich in the year 1775, in conjunction with ſome able officers of the Royal Regiment of Artillery, and other ingenious gentlemen, I firſt inſtituted a courſe of experiments on fired gunpowder and cannon balls. My account of them was preſented to the Royal Society, who honoured it with the gift of the annual gold medal, and printed it in the Philoſophical Tranſactions for the year 1778. The object of thoſe experiments, was the determination of the actual velocities with which balls are impelled from given pieces of cannon, when fired with given charges of powder. They were made according to the method invented by the very ingenious Mr. Robins, and deſcribed in his treatiſe on the new principles of gunnery, of which an account was printed in the Philoſophical Tranſactions for the year 1743. Before the diſcoveries and inventions of that gentleman, very little progreſs had been made in the true theory of military projectiles. His book however contained ſuch important diſcoveries, that it was ſoon tranſlated into ſeveral of the languages on the continent, and the late ſamous Mr. L. Euler honoured it with a very learned and extenſive commentary, in his tranſlation of it into the German language. That [Page 100] part of Mr. Robins's book has always been much admired, which relates to the experimental method of aſcertaining the actual velocities of ſhot, and in imitation of which, but on a large ſcale, thoſe experiments were made which were deſcribed in my paper. Experiments in the manner of Mr. Robins were generally repeated by his commentators, and others, with univerſal ſatisfaction; the method being ſo juſt in theory, ſo ſimple in practice, and altogether ſo ingenious, that it immediately gave the fulleſt conviction of its excellence, and the eminent abilities of the inventor. The uſe which our author made of his invention, was to obtain the real velocities of bullets experimentally, that he might compare them with thoſe which he had computed a priori from a new theory of gunnery which he had invented, in order to verify the principles on which it was founded. The ſucceſs was fully anſwerable to his expectations, and left no doubt of the truth of his theory, at leaſt when applied to ſuch pieces and bullets as he had uſed. Theſe however were but ſmall, being only muſket balls of about an ounce weight: for, on account of the great ſize of the machinery neceſſary for ſuch experiments, Mr. Robins, and other ingenious gentlemen, have not ventured to extend their practice beyond bullets of that kind, but contented themſelves with ardently wiſhing for experiments to be made in a ſimilar manner with balls of a larger ſort. By the experiments deſcribed in my paper therefore I endeavoured, in ſome degree, to ſupply that defect, having uſed cannon balls of above twenty times the ſize, or from one pound to near three pounds weight. Thoſe are the only experiments, that I know of, which have been made in that way with cannon balls, although the concluſions to be deduced from ſuch a courſe, are of the greateſt importance in thoſe parts of natural philoſophy which are connected with the effects of fired gunpowder: nor do I know of any other practical method beſides that above, of aſcertaining the initial velocities of military projectiles within any tolerable degree of the truth; except that of the recoil of the gun, hung on an axis in the ſame manner as the pendulum; which was alſo firſt pointed out and uſed by Mr. Robins, and which has lately been practiſed alſo by Benjamin [Page 101] Thompſon, Eſq. in his very ingenious and accurate ſet of experiments with muſket balls, deſcribed in his paper in the Philoſophical Tranſactions for the year 1781. The knowledge of this velocity is of the greateſt conſequence in gunnery: by means of it, together with the law of the reſiſtance of the medium, every thing is determinable which relates to that buſineſs; for, as I remarked in the paper above-mentioned on my firſt experiments, it gives us the law relative to the different quantities of powder, to the different weights of balls, and to the different lengths and ſizes of guns, and it is alſo an excellent method of trying the ſtrength of different ſorts of powder. Beſide theſe, there does not ſeem to be any thing wanting to anſwer every inquiry that can be made concerning the flight and ranges of ſhot, except the effects ariſing from the reſiſtance of the medium.
2. In that courſe of experiments were compared the effects of different quantities of powder, from two to eight ounces; the effects of different weights of ſhot; and the effects of different ſizes of ſhot, or different degrees of windage, which is the difference between the diameter of the ſhot and the diameter of the bore; all of which were found to obſerve certain regular and conſtant laws, as far as the experiments were carried. And at the end of each day's experiments, the deductions and concluſions were made, and the reaſons clearly pointed out why ſome caſes of velocity differ from others, as they properly and regularly ought to do. So that I am ſurprized how they could be miſunderſtood by Mr. Templehof, captain in the Pruſſian artillery, when ſpeaking of the irregularities in ſuch experiments, he ſays, (page 126 of Le Bombardier Pruſſien, printed at Berlin, 1781) "La meme choſe arriva a Mr. Hutton, il la trouva de 626 pieds, & le jour ſuivant de 973 pieds, tout les circonſtances étant d'ailleurs égales:" which laſt words ſhew that Mr. T. had either miſunderſtood, or had not read the reaſon, which is a very ſufficient one, for this remarkable difference: it is expreſsly remarked in page 71 of my paper in the Philoſophical [Page 102] Tranſactions, that all the circumſtances were not the ſame, but that the one ball was much ſmaller than the other, and that it had the leſs degree of velocity, 626 feet, becauſe of the greater loſs of the elaſtic fluid by the windage in the caſe of the ſmaller ball. On the contrary, the velocities in thoſe experiments were even more uniform and ſimilar thancould be expected in ſuch large machinery, and in a firſt attempt of the kind too. And from the whole, the following important concluſions were fairly drawn and ſtated, viz.
"(1.) And firſt, it is made evident by theſe experiments, that powder fires almoſt inſtantaneouſly, ſeeing that almoſt the whole of the charge fires, though the time be much diminiſhed.
"(2.) The velocities communicated to ſhot of the ſame weight, with different quantities of powder, are nearly in the ſubduplicate ratio of thoſe quantities. A very ſmall variation, in defect, taking place when the quantities of powder become great.
"(3.) And when ſhot of different weights are fired with the ſame quantity of powder, the velocities communicated to them, are nearly in the reciprocal ſub-duplicate ratio of their weights.
"(4.) So that, univerſally, ſhot which are of different weights, and impelled by the firing of different quantities of powder, acquire velocities which are directly as the ſquare roots of the quantities of powder, and inverſely as the ſquare roots of the weights of the ſhot, nearly.
"(5.) It would therefore be a great improvement in artillery, to make uſe of ſhot of a long form, or of heavier matter; for thus the momentum of a ſhot, when fired with the ſame weight of powder, would be increaſed in the ratio of the ſquare root of the weight of the ſhot.
[Page 103] "(6.) It would alſo be an improvement, to diminiſh the windage: for, by ſo doing, one third or more of the quantity of powder might be ſaved.
"(7.) When the improvements mentioned in the laſt two articles are conſidered as both taking place, it is evident that about half the quantity of powder might be ſaved; which is a very conſiderable object. But important as this ſaving may be, it ſeems to be ſtill exceeded by that of the guns: for thus a ſmall gun may be made to have the effect and execution of one of two or three times its ſize in the preſent way, by diſcharging a long ſhot of two or three times the weight of its natural ball, or round ſhot: and thus a ſmall ſhip might diſcharge ſhot as heavy as thoſe of the greateſt now made uſe of.
"Finally, as the above experiments exhibit the regulations with regard to the weight of powder and balls, when fired from the ſame piece of ordnance; ſo by making ſimilar experiments with a gun, varied in its length, by cutting off from it a certain part before each courſe of experiments, the effects and general rules for the different lengths of guns, may be certainly determined by them. In ſhort, the principles on which theſe experiments were made, are ſo fruitful in conſequences, that, in conjunction with the effects of the reſiſtance of the medium, they ſeem to be ſufficient for anſwering all the inquiries of the ſpeculative philoſopher, as well as thoſe of the practical artilleriſt."
3. Such then was the ſtate of the firſt ſet of experiments with cannon balls in the year 1775, and ſuch were the probable advantages to be derived from them. I do not however know that any uſe has hitherto been made of them by authority for the public ſervice; unleſs perhaps we are to except the inſtance of Carronades, a ſpecies of ordnance which hath ſince been invented, and in ſome degree adopted in the public [Page 104] ſervice; for in this inſtance the proprietors of thoſe pieces, by availing themſelves of the circumſtances of large balls, and very ſmall windage, with ſmall charges of powder, have been able to produce very conſiderable and uſeful effects with thoſe light pieces, at a very ſmall expence. Or perhaps thoſe experiments were too much limited, and of too private a nature, to merit a more general notice. Be that however as it may, the preſent additional courſe, which is to make the ſubject of this tract, will have very great advantages over the former, both in point of extent, variety, improvements in machinery, and in authority. His Grace the Duke of Richmond, the preſent maſter-general of the ordnance, in his indefatigable endeavours for the good of the public ſervice, was pleaſed to order this extenſive courſe of experiments, and to give directions for providing guns, and machinery, and every thing compleat and fitting for the proper execution of them.
4. This courſe of experiments has been carried on under the direction of Major Blomefield, inſpector of artillery, an officer of great profeſſional merit, and whoſe ingenious contrivances in the machinery do him great credit. It has been our employment for three ſucceſſive ſummers, namely, thoſe of the years 1783, 1784, and 1785; and indeed it might be continued ſtill much longer, either by extending it to more objects, or to more repetitions of experiments for the ſame object.
- (1.) The velocities with which balls are projected by equal charges of powder, from pieces of the ſame weight and calibre, but of different lengths.
- (2.) The velocities with different charges of powder, the weight and length of the gun being the ſame.
- [Page 105] (3.) The greateſt velocity due to the different lengths of guns, to be obtained by increaſing the charge as far as the reſiſtance of the piece is capable of ſuſtaining.
- (4.) The effect of varying the weight of the piece; every thing elſe being the ſame.
- (5.) The penetration of balls into blocks of wood.
- (6.) The ranges and times of flight of balls; to compare them with their initial velocities, for determining the reſiſtance of the medium.
- (7.) The effect of wads; of different degrees of ramming, or compreſſing the charge; of different degrees of windage; of different poſitions of the vent; of chambers, and trunnions, and every other circumſtance neceſſary to be known for the improvement of artillery.
1.9.1. Of the Nature of the Experiment, and of the Machinery uſed in it.
6. THE effects of moſt of the circumſtances laſt mentioned are determined by the actual velocity with which the ball is projected from the mouth of the piece. Therefore the primary object of the experiments is, to diſcover that velocity in all caſes, and eſpecially in ſuch as uſually occur in the common practice of artillery. This velocity is very great; from one thouſand to two thouſand feet or more, in a ſecond of time. For conveniently eſtimating ſo great a velocity, the firſt thing neceſſary is, to reduce it, in ſome known proportion, to a ſmall one. [Page 106] Which we may conceive to be effected in this manner: ſuppoſe the ball, projected with a great velocity, to ſtrike ſome very heavy body, ſuch as a large block of wood, from which it will not rebound, ſo that after the ſtroke they may both proceed forward together with a common velocity. By this means, it is obvious that the original velocity of the ball may be reduced in any proportion, or to any ſlow velocity which may conveniently be meaſured, by making the body ſtruck to be ſufficiently large: for it is well known that the common velocity, with which the ball and the block of wood would move on together after the ſtroke, bears to the original velocity of the ball before the ſtroke, the ſame ratio which the weight of the ball has to that of the ball and block together. Thus then velocities of one thouſand feet in a ſecond are eaſily reduced to thoſe of two or three feet only: which ſmall velocity being meaſured by any convenient means, let the number denoting it be increaſed in the ratio of the weight of the ball to the weight of the ball and block together, and the original velocity of the ball itſelf will thereby be obtained.
7. Now this reduced velocity is rendered eaſy to be meaſured by a very ſimple and curious contrivance, of Mr. Robins, which is this: the block of wood, which is ſtruck by the ball, inſtead of being left at liberty to move ſtraight forward in the direction of the motion of the ball, is ſuſpended, like the weight of the vibrating pendulum of a clock, by a ſtrong iron ſtem, having a horizontal axis at the top, on the ends of which it vibrates freely when ſtruck by the ball. The conſequence of this ſimple contrivance is evident: this large balliſtic pendulum, after being ſtruck by the ball, will be penetrated by it to a ſmall depth, and it will then ſwing round its axis, deſcribing an arch, which will be greater or leſs according to the force of the blow ſtruck; and from the magnitude of the arch deſcribed by the vibrating pendulum, the velocity of any point of the pendulum can be eaſily computed: for a body acquires the ſame velocity by falling from the ſame height, [Page 107] whether it deſcend perpendicularly down, or otherwiſe; therefore, having given the length of the arc deſcribed by the center of oſcillation, and its radius, the verſed ſine becomes known, which is the height perpendicularly deſcended by that point of the pendulum. The height deſcended being thus known, the velocity acquired in falling through that height becomes known alſo, from the common rules for the deſcent of bodies by the force of gravity. And the velocity of this center, thus obtained, is to be eſteemed the velocity of the whole pendulum itſelf: which being now given, that of the ball before the ſtroke becomes known, from the given weights of the ball and pendulum. Thus then the determination of the very great velocity of the ball is reduced to the menſuration of the magnitude of the arch deſcribed by the pendulum, in conſequence of the blow ſtruck.
8. Now this arch may be determined in various ways: in the following experiments it was aſcertained by meaſuring the length of its chord, which is the moſt uſeful line about it for making the calculation by; and this chord was meaſured ſometimes by means of a piece of tape or narrow ribbon, the one end of which was faſtened to the bottom of the pendulum, and the reſt of it made to ſlide through a ſmall machine contrived for the purpoſe; and ſometimes it was meaſured by the trace of the fine point of a ſtylette in the bottom of the pendulum, made in an arch concentric with the axis, and covered with a compoſition of a proper conſiſtence; which will be particularly deſcribed hereafter.
9. Another ſimilar method of meaſuring the great velocity of the ball is, by obſerving the arch of recoil of the gun, when it is hung alſo after the manner of a pendulum: for, by loading the gun with adventitious weight, it may be made ſo heavy as to ſwing any convenient extent of arch we pleaſe, which arch it is evident will be greater or leſs according to the velocity of the ball, or force of the inſlamed powder, ſince action and re-action are equal and contrary; that is, the velocity of the ball will [Page 108] be greater than the velocity of the center of oſcillation of the gun, in the ſame proportion as the weight of the gun exceeds the weight of the ball. And therefore, if the velocity of the center of oſcillation of the gun be computed, from the chord of the arc deſcribed by it in the recoil, the velocity of the ball will be found by this proportion; namely, as the weight of the ball is to the weight of the gun, ſo is the velocity of the gun to the velocity of the ball: that is, if the weight of powder had no effect on the recoil.
10. This deſcription may ſuffice to convey a general idea of the nature and principles of the experiment, for determining the velocity with which a ball is projected, by any charge of powder, from a piece of ordnance. But it is to be obſerved that, beſides the center of oſcillation, and the weights of the ball and pendulum, or gun, the effect of the blow depends alſo on the place of the center of gravity in the pendulum or gun, and that of the point ſtruck, or the place where the force is exerted; for it is evident that the arch of vibration will be greater or leſs according to the ſituation of theſe two points alſo. It will therefore be neceſſary now to give a more particular deſcription of the machinery, and of the methods of finding the aforeſaid requiſites; and then we ſhall inveſtigate our general rules for determining the velocity of the ball, in all caſes, from them and the chord of the arch of vibration, either of the pendulum or gun.
1.9.2. Of the Guns, Powder, Balls, and Machinery employed in theſe Experiments.
11. FIVE very fine braſs one-pounder guns were caſt and prepared, in Woolwich Warren, for theſe experiments, and bored as true as poſſible; the common diameter of their bore being 2 inches and 2/100 [Page 109] parts of an inch. Theſe five guns are exactly repreſented in plate 1, with the ſcale of their dimenſions, by which they were drawn. Three of theſe, namely, n^{o}. 1, 2, 3, are nearly of the ſame weight, but of the reſpective lengths of 15, 20, and 30 calibers; in order to aſcertain the effect of different lengths of bore, with the ſame weight of gun, powder, and ball. The other two, n^{o}. 4 and 5, were heavier, and of 40 calibers in length; to obtain the effects of the longeſt pieces. N^{o}. 5 was more expreſſly to ſhew the effect of different lengths of the ſame gun: and for this purpoſe, it was to be fired a ſufficient number of rounds with its whole length; and then to be ſucceſſively diminiſhed, by ſawing off it 6 or 12 inches at a time, till it ſhould be all cut away: firing a number of rounds with it at each length. And for the convenience of ſuſpending this gun near its center of gravity for all the different lengths of it, a long thin ſlip was caſt with it, extending along the under ſide of it, from the breech to almoſt the middle of its length. By perforating this ſlip through with holes immediately under the center of gravity for each length, after being cut, a bolt was to paſs through the hole, on which the gun might be ſuſpended. The other guns were ſlung by their trunnions.
The exact weight and dimenſions of all theſe guns are exhibited in the following table.
13. In theſe experiments, the velocity of the ball, by which the force of the powder is determined, was to be meaſured both by the balliſtic pendulum into which the ball was fired, and by the arch of recoil of the gun, which was hung on an axis by an iron ſtem, after the ſame [Page 111] manner as the pendulum itſelf, and the arcs vibrated in both caſes meaſured in the ſame way. Plates 11 and 111 contain general repreſentations of the machinery of both; namely, a ſide view and a front view of each, as they hung by their ſtem and axis on the wooden ſupports. In plate 11, fig. 1 is the ſide-view of the pendulum, and fig. 2 the ſideview of the gun, as ſlung in their frames. And in plate 111, fig. 1 and 2 are the front-views of the ſame.
14. In fig. 1, of both plates, A is the pendulous block of wood, into which the balls are fired, ſtrongly bound with thick bars of iron, and hung by a ſtrong iron ſtem, which is connected by an axis at top; the whole being firmly braced together by croſſing diagonal rods of iron. The cylindrical ends of the axis, both in the gun and pendulum, were at firſt placed to turn upon ſmooth flat plate-iron ſurfaces, having perpendicular pins put in before and behind the ſides of the axis, to keep it in its place, and prevent it from ſlipping backwards and forwards. But, this method being attended with too much friction, the ends of the axis were ſupported and made to roll upon curved pieces, having the convexity upwards, and the pins, before and behind the axis, ſet ſo as not quite to touch it; which left a ſmall degree of play to the axis, and made the friction leſs than before. But, ſtill farther to diminiſh the friction, the lower ſide of the ends of the axis was ſharpened off a little, ſomething like the axis of a ſcale beam, and made to turn in hollow grooves, which were rounded down at both ends, and ſtanding higher in the middle, like the curvature of a bent cylinder; by which means the edge of the axis touched the grooves, not in a line, but in one point only; when it vibrated with very great freedom, having an almoſt imperceptible degree of friction. The ſeveral times and occaſions when theſe, and other improvements, were introduced and uſed, will be more particularly noticed in the journal of the experiments.
[Page 112] 15. At firſt, the chord of the arc, of vibration and recoil, was meaſured by means of a prepared narrow tape, divided into inches and tenths, as before. A new contrivance of machinery was however made for it. From the bottom of the pendulum, or gun-frame, proceeded a tongue of iron, which was raiſed or lowered by means of a ſcrew at B; this was cloven at the bottom C, to receive the end of the tape, and the lips then pinched together by a ſcrew, which held the tape faſt. Immediately below this the tape was paſſed between two ſlips of iron, which could be brought to any degree of nearneſs by two ſcrews; theſe pieces were made to ſlide vertically up and down a groove in a heavy block of wood, and fixed at any height by a ſcrew D. One of theſe latter pieces was extended out a conſiderable length, to prevent the tape from getting over its ends, and entangling in the returns of the vibrations. The extent of tape drawn out in a vibration, it is evident, is the chord of the arc deſcribed, and counted in inches and tenths, to the radius meaſured from the middle of the axis to the bottom of the tongue.
16. This method however was found to be attended with much trouble, and many inconveniences, as well as doubts and uncertainty ſometimes. For which reaſons we afterwards changed this method of meaſuring the chord of vibration for another, which anſwered much better in every reſpect. This conſiſted in a block of wood, having its upper ſurface EF formed into a circular arc, whoſe center was in the middle of the axis, and conſequently its radius equal to the length from the axis to the upper ſurface of the block. In the middle of this arch was made a ſhallow groove of 3 or 4 inches broad, running along the middle, through the whole length of the arch. This groove was filled with a compoſition of ſoft-ſoap and wax, of about the conſiſtence of honey, or a little firmer, and its upper ſide ſmoothed off even with the general ſurface of the broad arch. A ſharp ſpear or ſtylette then proceeded from the bottom of the pendulum or gun-frame, and ſo low as juſt to enter and ſcratch along the ſurface of the compoſition in the groove, without [Page 113] having any ſenſible effect in retarding the motion of the body. The trace remaining, the extent of it could eaſily be meaſured. This meaſurement was effected in the following manner:—A line of chords was laid down upon the upper ſurface of the wooden arch, on each ſide of the groove, and the diviſions marked with lines on a ground of white paint: the edge of a ſtraight ruler being then laid acroſs by the correſponding diviſions, juſt to touch the fartheſt extent of the trace in the compoſition, gave the length of the chord as marked on the arch. To make the computations by the rule for the velocity eaſier, the diviſions on the chords were made exact thouſandth parts of the radius, which ſaved the trouble of dividing by the radius at every operation. The manner in which I conſtructed this line of chords on the face of the arch was this: The radius was made juſt 10 feet; I therefore prepared a ſmooth and ſtraight deal rod, upon which I ſet off 10 feet; I then divided each foot into 10 equal parts, and each of theſe into 10 parts again; by which means the whole rod or radius was divided into 1000 equal parts, being 100th parts of a foot. I then transferred the diviſions of the rod to the face of the arch in this manner, namely; the firſt diviſion of the rod was applied to the ſide of the arch at the beginning of it, and made to turn round there as a center; then, in that poſition, the rod, when turned vertically round that point, always touched the ſide of the arch, and the diviſions of it were marked on the edge of the arch, ſucceſſively as they came into a coincidence with it.
17. In fig. 2, plate 11, G ſhews the leaden weights placed about the trunnions; H a ſcrew for raiſing or depreſſing the breech of the gun, by means of the piece 1 embracing the caſcable, and moveable along the perpendicular arm KL, to ſuit the different lengths of guns, and held to it by a ſcrew paſſing through the ſlit made along it.
The machines and operations for finding the ranges will be deſcribed hereafter.
1.9.3. Of the Centers of Gravity and Oſcillation.
[Page 114]18. It being neceſſary to know the poſition of the centers of gravity and oſcillation, without which the velocity cannot be computed; theſe were commonly determined every day as follows:
The center of gravity was found by one or both of theſe two methods. Firſt, a triangular priſm of iron AB, being placed on the ground with one edge upwards, the pendulum or gun-frame was laid acroſs it, and moved backward or forward, on the ſtem or block, as the caſe required, till the two parts exactly balanced each other in a horizontal poſition. Then, as it lay, the diſtance was meaſured from the middle of the axis to the part which reſted on the edge of the priſm, or the place of the center of gravity, which is the diſtance g of that center below the axis.
- As p the weight of the pendulum:
- is to w the appended weight ∷
- ſo is d the whole length from the axis to the chord:
- to dw / p the diſtance from the axis to the center of gravity.
20. To find the center of oſcillation, the balliſtic pendulum, or the gun, was hung up by its axis in its place, and then made to vibrate in ſmall arcs, for 1 minute, or 2, or 5, or 10 minutes; the more the better; as determined either by a half ſecond pendulum, or a ſtop watch, or a peculiar time-piece, meaſuring the time to 40th parts of a ſecond; and the number of vibrations performed in that time carefully counted. Having thus obtained the time anſwering to a certain number of vibrations, the center of oſcillation is eaſily found: for if n denote the number of vibrations made in s ſeconds, and l the length of the ſecond pendulum, then it is well known that n2 ∶ s2 ∷ l ∶ s2l / n2 the diſtance from the axis of motion to the center of oſcillation. And here if s be 60 ſeconds, or one minute, and n the number of vibrations performed in 1 minute, as found by dividing the whole number of vibrations, actually performed, by the whole number of minutes; then is n2 ∶ 60^{2} ∷ l ∶ 3600l/nn the diſtance to the center of oſcillation. But, by the beſt obſervations on the vibration of pendulums, it is found that l = 39⅛ inches is the length of the ſecond pendulum for the latitude of London, or of Woolwich; and therefore [...] or 140850/nn = 0, will be the diſtance, in inches, or = 11737.5/nn in feet, of the center of oſcillation below the axis. And by this rule the place of that center was found for each day of the experiments.
1.9.4. Of the Rule for Computing the Velocity of the Ball.
21. Having deſcribed the methods of obtaining the neceſſary dimenſions and weights, I proceed now to the inveſtigation of the theorem by [Page 116] which the velocity of the ball is to be computed: and firſt by means of the pendulum.
The ſeveral weights and meaſures being found, let b denote the weight of the ball, p the weight of the pendulum, g the diſtance to its center of gravity, o the diſtance to its center of oſcillation, i the diſtance to the point of impact, or point ſtruck, c the chord of the arch deſcribed by the pendulum, r its radius, or diſtance to the tape or arch, v the initial or original velocity of the ball.
Then, from the nature of oſcillatory motion, bii will expreſs the ſum of the forces of the ball acting at the diſtance i from the axis, and pgo the ſum of the forces of the pendulum, and conſequently pgo + bii the ſum for both the ball and pendulum together; and if each be multiplied by its velocity, biiv will be the quantity of motion of the ball, and (pgo + bii) × z the quantity for the pendulum and ball together; where z is the velocity of the point of impact. But theſe quantities of motion, before and after the blow, muſt be equal to each other, therefore (pgo + bii) × z = biiv, and conſequently z = biiv/pgo+bii is the velocity of the point of impact. Now becauſe of the acceſſion of the ball to the pendulum, the place of the center of oſcillation will be changed; and the diſtance y of the new or compound center of oſcillation will be found by dividing pgo + bii the ſum of the forces, by pg + bi the ſum of the momenta, that is y = pgo+bii/pg+bi is the diſtance of the new or compound center of oſcillation below the axis. Then, becauſe biiv/pgo+bii is the velocity of the point whoſe diſtance is i, by ſimilar figures we ſhall have this proportion, as i ∶ pgo+bii/pg+bi (or y) ∷ biiv/pgo+bii ∶ biv/pg+bi the velocity of this compound center of oſcillation.
[Page 117] Again, by the property of the circle, 2r ∶ c ∷ c ∶ cc/2r, which will be the verſed ſine of the deſcribed arc, to the chord c and radius r; and hence, by ſimilar figures, r : y or pgo+bii/pg+bi ∷ cc/2r ∶ cc/2rr × pgo+bii/pg+bi the correſponding verſed ſine to the radius y, or the verſed ſine of the arc deſcribed by the compound center of oſcillation; which call v. Then, becauſe the velocity loſt in aſcending through the circular arc, or gained in deſcending through the ſame, is equal to the velocity acquired in deſcending freely by gravity through its verſed ſine, or perpendicular height, therefore the velocity of this center of oſcillation will alſo be equal to the velocity generated by gravity in deſcending through the ſpace v or cc2rr × pgo+bii/pg+bi. But the ſpace deſcribed by gravity in one ſecond of time, in the latitude of London, is 16.09 feet, and the velocity generated in that time 32.18; therefore, by the nature of free deſcents, √ 16.09 ∶ √v ∷ 32.18 ∶ 5.6727c/r √ pgo+bii/pg+bi, the velocity of the ſame center of oſcillation, as deduced from the chord of the arc which is actually deſcribed.
Having thus obtained two different expreſſions for the velocity of this center, independent of each other, let an equation be made of them, and it will expreſs the relation of the ſeveral quantities in the queſtion: thus then we have biv/pg+bi = 5.6727c/r √ pgo+bii/pg+bi. And from this equation we get [...] the true expreſſion for the original velocity of the ball the moment before it ſtrikes the pendulum. And this theorem agrees with thoſe of Meſſrs. Euler and Antoni, and alſo with that of Mr. Robins nearly, for the ſame purpoſe, when his rule is corrected by the paragraph which was by miſtake omitted in his book when firſt publiſhed; which correction he himſelf gave in a paper in the Philoſophical Tranſactions for April 1743, and where he informs us that all the velocities of balls, mentioned in his book, except the firſt only, [Page 118] were computed by the corrected rule. Though the editor of his works, publiſhed in 1761, has inadvertently neglected this correction, and printed his book without taking any notice of it. And that remark, had M. Euler obſerved it, might have ſaved him the trouble of many of his animadverſions on Mr. Robins's work.
22. But this theorem may be reduced to a form much more ſimple and fit for uſe, and yet be ſufficiently near the truth. Thus, let the root of the compound factor (pgo + bii) × (pg + bi) be extracted, and it will be equal to (pg + bi · o+i/20) × √0, within the 100000th part of the true value, in ſuch caſes as commonly happen in practice. But ſince bi · o+i/20, in our experiments, is uſually but about the 500th, or 600th, or 800th part of pg, and ſince bi differs from bi · o+i/20 only by about the 100th part of itſelf, therefore pg + bi is within the 50000th part of pg + bi · o+i/20. Conſequently v = 5.6727c · pg+bi/bir √o very nearly. Or, farther, if g be written for i in the laſt term bi, then finally v = 5.6727gc · p+b/bir √o; which is an eaſy theorem to be uſed on all occaſions; and being within the 5000th part of the true quantity, it will always give the velocity true within leſs than half a foot, even in the caſes of the greateſt velocity. Where it muſt be obſerved, that c, g, i, r, may be taken in any meaſures, either feet or inches, &c. provided they be but all of the ſame kind; but o muſt be in feet, becauſe the theorem is adapted to feet.
24. But as the diſtance of the center of oſcillation o, whoſe ſquare root is concerned in the theorem for the velocity v, is found from the number of vibrations n performed by the pendulum; it will be better to ſubſtitute, in that theorem, the value of o in terms of n. Now by Art. 20, the value of o is 11737.5/nn feet, and conſequently √o = 108.3398/n; which value of √o being ſubſtituted for it in the theorem v = 5.6727gc × p+b/bir √o, it becomes v = 614.58gc × p+b/birn, or 59000/96 × p+b/birn gc, the ſimpleſt and eaſieſt formula for the velocity of the ball in feet: where c, g, i, r may be taken in any one and the ſame meaſure, either all inches, or all feet, or any other meaſure.
25. It will be neceſſary here to add a correction for n inſtead of that for o in Art. 23. Now, the correction for o being [...], and the [Page 120] value of n = 375.3/√o inches, the correction for n will be [...] by ſubſtituting the value of o inſtead of it: Which correction is negative, or to be ſubtracted from the former value of n. The corrections for p and g being b and [...], as in Art. 23; which are both additive. But the ſigns of theſe quantities muſt be changed when b is negative.
26. Before we quit this rule, it may be neceſſary here to advert to three or four circumſtances which may ſeem to cauſe ſome ſmall error in the initial velocity, as determined by the formula in Art. 24. Theſe are the friction on the axis, the reſiſtance of the air to the back of the pendulum, the time which the ball employs in penetrating the wood of the pendulum, and the reſiſtance of the air to the ball in its paſſage between the gun and the pendulum.
As to the firſt of theſe, namely, the friction on the axis, by which the extent of its vibration is ſomewhat diminiſhed; it may be obſerved, that the effect of this cauſe can never amount to a quantity conſiderable enough to be brought into account in our experiments; for, beſides that care was taken to render this friction as ſmall as poſſible, the effect of the ſmall part which does remain is nearly balanced by the effect it has on the number n of vibrations performed in a minute; for the friction on the axis will a little retard its motion, and cauſe its vibrations to be ſlower, and ſewer; ſo that c the length of a vibration, and n the number of vibrations, being both diminiſhed by this cauſe, nearly in an equal degree, and c being a multiplier, and n a diviſor, in our formula, it is [Page 121] evident that the effect of the friction in the one caſe operates againſt that in the other, and that the difference of the two is the real diſturbing cauſe, and which therefore is either equal to nothing, or very nearly ſo.
27. The ſecond cauſe of error is the reſiſtance of the air againſt the back of the pendulum, by which its motion is ſomewhat impeded. This reſiſtance hinders the pendulum from vibrating ſo far, and deſcribing ſo large an arch, as it would do if there was no ſuch reſiſtance; therefore the chord of the arc which is actually deſcribed and meaſured, is leſs than it really ought to be; and conſequently the velocity of the ball, which is proportional to that chord, will be leſs than the real velocity of the ball at the moment it ſtrikes the pendulum. And although the pendulum be very heavy, and its motion but ſlow, and conſequently the reſiſtance of the air againſt it very ſmall, it will yet be proper to inveſtigate the real effect of it, that we may be ſure whether it may ſafely be neglected or not.
In order to this, let the annexed figure repreſent the back of the pendulum, moving on its axis; and put p = weight of the pendulum, a = DE its breadth, r = AB the diſtance to the bottom, e = AC the diſtance to the top, x = AF any variable diſtance, g = diſtance of the center of gravity, o = diſtance of the center of oſcillation, v = velocity of the center of oſcillation, in any part of the vibration, h = 16.09 feet, the deſcent of gravity in 1 ſecond, c = the chord of the arc actually deſcribed by the center of oſcillation, and c = the chord which would be deſcribed by it if the air had no reſiſtance.
[Page 122] Then o ∶ x ∷ v ∶ vx/o the velocity of the point F of the pendulum; and 4h2 ∶ h ∷ v2 x2/o2 ∶ v2 x2/4ho2 the height deſcended by gravity to generate the velocity vx/o. Now the reſiſtance of the air to the line DFE is equal to the preſſure of a column of air upon it, whoſe height is the ſame v2 x2/4ho2, and therefore that preſſure or weight is nav2 x2/4ho2, where n is the ſpecific gravity, or weight of one cubic meaſure of air, or n = 62½ / 850lb = 5/68lb. Hence then nav2 x2 x/4ho2 is the preſſure on DEed, and nav2 x3 x/4ho2 the momentum of the preſſure on the ſame De, or the fluxion of the momentum on the block of the pendulum; and the correct fluent gives [...] for the momentum of the air on the whole pendulum, ſuppoſing that on the ſtem AC to be nothing, as it is nearly, both on account of its narrowneſs, and the diminution of the momentum of the particles by their nearneſs to the axis. Put now A = the compound coefficient [...], ſo ſhall A v2 denote the momentum of the air on the back of the pendulum.
But the motion of the pendulum is alſo obſtructed by its own weight, as well as by the reſiſtance of the air; and that weight acts as if it were all concentered in the center of gravity, whoſe diſtance below the axis is g; therefore pg is its momentum in its natural or vertical direction, and pgs its momentum perpendicular to the motion of the pendulum, when s is the ſine of the angle which it makes at any time with the vertical poſition, to the radius 1. Hence pgs + Av2 is the momentum of both the reſiſtances together, namely that of the preſſure of the air, and of the weight of the pendulum. And conſequently pgs+Av2/pg = s + A / pg v2 is the real retarding force to the motion of the pendulum, at the center of oſcillation; which force call f.
[Page 123] Now if z denote the arc deſcribed by the center of oſcillation, when its velocity is v, or z/o the arc whoſe fine is s; we ſhall have [...], and, by the doctrine of forces, [...].
But cc/2o is the verſed ſine or height of the whole arc whoſe chord is c, and [...] is the verſed ſine or height of the part whoſe ſine is os, therefore [...] is their difference, or the height of the remaining part, and is nearly equal to the height due to the velocity v; therefore [...] nearly. Then by ſubſtituting this for v2 in the value of vv̇, we have [...]; and the fluents give [...]; where Q is a conſtant quantity by which the fluent is to be corrected. Now, ſubſtituting v^{2} for v2, and o for s, their correſponding values at the commencement of motion, the above fluent becomes v^{2} = 4ho + Q; from which the former ſubtracted, gives [...]. And when v = o, or the pendulum is at the full extent of its aſcent, then [...], at which point os is the ſine of the whole arc whoſe chord is c, and conſequently [...].
[Page 124] But the value of s being commonly ſmall in reſpect of c/o, we ſhall have theſe following values nearly true, namely, [...], [...], z = os + ⅙ os3, and 2o2−c2/2o2 z − os = − c2 s/2o + 2o2−c2/12o s3, which values, by ſubſtitution, give v2 = 2hc2/o + 16h2 oA / pq (c2 s/2o − 2o2−c2/12o s3).
But c^{2}/2o is the verſed ſine or height to the chord c, and v^{2} = 4h · c^{2}/2o = 2hc^{2}/o the ſquare of the velocity due to that height; therefore 2hc^{2}/o = 2hc2/o + 16h2 oA / pq (c2 s/2o − 2o2−c2/12o s3, and c^{2} = c2 + 8hoA / pq (c2 s/2 − 2o2−c2/12 s3), or c^{2} = c2 + 8hA / pq (c3/3 + c•/12o2), and c = c + 4c2 hA / 3pq nearly, or ſubſtituting for A, c = c + nac2/12pg · r4−c4/o2 = c (1 + nac/12pg · r4−e4/o2). So that the chord of the arc which is actually deſcribed, is to that which would be deſcribed if the air had no reſiſtance, as 1 is to 1 + nac/12pg · r4−e4/o2; and therefore nac/12pg · r4−e4/o2 is the part of the chord, and conſequently of the velocity, loſt by means of the reſiſtance of the air. And the proportion is the ſame for the chords deſcribed by the loweſt point, or any other point, of the pendulum.
And even this ſmall effect may be ſuppoſed to be balanced by the method of determining the center of oſcillation, or the number of vibrations made in a ſecond. So that the number of oſcillations, and the chord of the arc deſcribed, being both diminiſhed by the reſiſtance of the air; and the one of theſe quantities being a multiplier, and the other a diviſor, in the formula for the velocity; the one of thoſe ſmall effects will nearly balance the other; much in the ſame way as the effects of the firſt cauſe, or the friction on the axis. So that, theſe effects may both of them be ſafely neglected, as in no caſe amounting to any ſenſible quantity.
[Page 126] In the beginning of this inveſtigation, it is ſuppoſed that the reſiſtance of the medium is equal to the weight of a column of the medium, whoſe baſe is the moving ſurface, and its altitude equal to that from whence a heavy body muſt fall to acquire the velocity of that ſurface. But ſome philoſophers think the altitude ſhould be only one half of that, and conſequently the preſſure only one half: which would render the reſiſtance ſtill leſs conſiderable. But if the altitude and reſiſtance were even double of that above found, it might be ſtill ſafely neglected.
28. The third ſeeming cauſe of error in our rule is the time in which the ball communicates its motion to the pendulum, or the time employed in the penetration. The principle on which the rule is founded ſuppoſes the momentum of the ball to be communicated in an inſtant; but this is not accurately the caſe, becauſe this force is communicated during the time in which the ball makes the penetration. And although that time be evidently very ſmall, ſcarcely amounting to the 500th part of a ſecond, it will be proper to enquire what effect that circumſtance may have on the truth of our theorem, or on the velocity of the ball, as computed by it.
Then is R/b the retarding force of the ball, which is conſtant. Again, as the motion of the pendulum ariſes from the reſiſting force R of the [Page 127] wood, Ri will be its momentum; and as the ſum of the forces in the pendulum was found to be = pgo, the accelerating force of the point c will be Rii/pgo, which force is conſtant alſo. But in the action of forces that are conſtant, the time t is equal to the velocity divided by the force, and by 2h or 2 × 16.09 feet, and the ſpace is equal to the ſquare of the velocity divided by the force and by 4h; conſequently t = pgou/2hiiR, x = pgouu/4hiiR, and t = −bv / 2hR, x + z = −bvv / 4hR, or by correc. t = b/2hR × (v − v), x + z = b/4hR × (v2 − v^{2}). The two values of the time t being equated, we obtain pgou = bii(v − v), or pgou + biiv = biiv.
And when v = u, or the action of the ball on the pendulum ceaſes, this equation becomes pgoU + biiU = biiv, and hence u = biiv/pgo+bii is the greateſt velocity of the point C at the inſtant when the ball has penetrated to the greateſt depth, and ceaſes to urge the pendulum farther. So that this velocity is the ſame, whatever the reſiſting force of the wood is, and therefore to whatever depth the ball penetrates, and the ſame as if the wood were perfectly hard, or the ball made no penetration at all. And this velocity of the point of impact alſo agrees with that which was found in Art. 21. So that the velocity communicated to the point of impact is the ſame, whether the impulſe is made in an inſtant, or in ſome ſmall portion of time. And hence, in the uſual caſe of a penetration, becauſe the block will have moved ſome ſmall diſtance before it has attained its greateſt velocity, it might at firſt view ſeem as if it would ſwing or riſe higher than when that velocity is communicated in an inſtant, or when the pendulum is yet in its vertical poſition, and ſo might deſcribe a longer chord, and ſhew a greater velocity of the ball than it ought. But on the other hand it muſt be [Page 128] conſidered, that in the ſmall part of its ſwing, which the pendulum has made before the penetration is completed, or has attained its greateſt velocity, juſt as much velocity will be loſt by the oppoſing gravity or weight of the pendulum, as if it had ſet out from the vertical poſition with the ſaid greateſt velocity; and therefore the real velocity at that height will be the ſame in both caſes. Hence then it may ſafely be concluded, that the circumſtance of the ball's penetration cauſes no alteration in the velocity of it, as computed by our formula. And as it was before found that no ſenſible error is incurred by the two firſt circumſtances, namely, the friction on the axis, and the reſiſtance of the air to the back of the pendulum, we may be well aſſured that our formula brings out the true velocity with which the ball ſtrikes the pendulum, without any ſenſible error.
29. Since biiv/pgo+bii denotes the greateſt velocity which the point c of the pendulum acquires by the ſtroke, dividing by i, we ſhall have biv/pgo+bii for the angular velocity of the pendulum, or that of a radius 1. From which it appears that the vibration will be very ſmall when i or the diſtance AD is ſmall, and alſo when i is very great. And if we take this expreſſion a maximum, and make its fluxion = 0, i only being variable, we ſhall obtain pgo = bii, and i = √ pgo/b for the diſtance of the center of percuſſion, or the point where the ball muſt ſtrike ſo as to cauſe the greateſt vibration in the pendulum; which point, in this caſe, is neither the center of gravity nor the center of oſcillation; but will be at a great diſtance below the axis when p is great reſpect of b, as in the caſe of our experiments, in which p is 600 or 800 times b.
[Page 129] 30. It may not be improper here, by the way, to enquire a little into the time of the penetration, its extent or depth, and the meaſure of the reſiſting force of the wood.
It was found above that x = pgouu/4hiiR, and x + z = b/4hR × (v2 − v^{2}). Now ſubſtituting in theſe biiv/pgo+bii, the greateſt value of u, for u and v, we have [...], [...]. The latter of theſe being the greateſt depth penetrated by the ball into the wood, and the former the diſtance moved by the point C of the pendulum at the inſtant when the penetration is completed. Both of which, it is evident, are directly as the ſquare of the original velocity of the ball, and inverſely as the reſiſting force of the wood; the other quantities remaining conſtant.
Hence alſo it appears that, other things remaining, the penetration will be leſs, as i is greater, or as the point of impact is farther below the axis. It is farther evident that the penetration will diminiſh as the ſum of the forces pgo diminiſhes.
Here the value of v is 1500, and z = 14 inches or 7/6 feet. [Page 130] Hence [...] nearly, which is the value of R for a ball of that ſize and weight. Or the reſiſtance in this inſtance is 32000 times the force of gravity. Hence alſo [...] part of a foot, or 1/39 part of an inch, is the ſpace moved by the point C of the pendulum when the penetration is completed.
Alſo [...] part of a ſecond, is the time of completing the penetration of 14 inches deep.
31. Upon the whole then it appears, that our rule will give, without ſenſible error, the true velocity with which the ball ſtrikes the pendulum. But this is not, however, the ſame velocity with which the ball iſſues from the mouth of the gun, which will indeed be ſomething greater than the former, on account of the reſiſtance of the air which the ball paſſes through in its way from the gun to the pendulum. And although this ſpace of air be but ſmall, and although the elaſtic fluid of the powder purſue and urge the ball for ſome diſtance without the mouth of the piece, and ſo in ſome degree counteract the reſiſtance of the air, yet it will be proper to enquire into the effect of this reſiſtance, as it will probably cauſe a difference between the velocity of the ball, as computed from the vibration of the pendulum and the vibration of the gun; which difference will, by the bye, be no bad way of meaſuring the reſiſtance of the air, eſpecially if the gun be placed at a good diſtance from the pendulum; for the vibration of the gun will meaſure the velocity with which the ball iſſues from the mouth of it; and the vibration of the pendulum the velocity with which it is ſtruck by the ball.
32. To find therefore the reſiſtance of the air againſt the ball in any caſe: it is firſt to be conſidered that the reſiſtance to a plane moving [Page 131] perpendicularly through a fluid at reſt, is equal to the weight or preſſure of a column of the fluid whoſe altitude is the height through which the body muſt fall by the force of gravity to acquire the velocity with which it moves through the fluid, the baſe of the column being equal to the plane. So that, if a denote the area of the plane, v the velocity, n the ſpecific gravity of the fluid, and h = 16.09 feet; the altitude due to the velocity v being vv/4h, the whole reſiſtance or motive force m will be a × n × vv/4h = anvv/4h.
Now if d denote the diameter of the ball, and k = .7854, then ſhall a = kd2 be a great circle of the ball; and conſequently [...] the motive force on the ſurface of a circle equal to a great circle of the ball.
But the reſiſtance on the hemiſpherical ſurface of the ball is only one half of that on the flat circle of the ſame diameter; therefore [...] is the motive force on the ball; and if w denote its weight, [...] will be equal to f the retarding force.
Since ⅔kd3 is the magnitude of the ſphere, if N denote its denſity or ſpecific gravity, its weight w will be = ⅔kd3 N; conſequently the retarding force f or m / w will be [...].
But by the laws of forces vv̇ = 2hfẋ = −3nvv/8dN ẋ, and v̇/w = −3n/8dN ẋ = − eẋ, where x is the ſpace paſſed over, putting e = 3n/8dN, and making the value negative becauſe the velocity v is decreaſing. And the correct fluent of this is log. v − log. v or log. v / w = ex, where v is the firſt or greateſt velocity of projection. Or if A be = 2.718281828 &c. the number whoſe hyperbolic logarithm is 1, [Page 132] then is v / v = A^{ex}, and hence the velocity v = v / A^{ex} = VA^{−ex}. So that the firſt velocity is to the laſt velocity, as A^{ex} to 1. And the velocity loſt by the reſiſtance of the medium is (A^{ex} − 1) v or A^{ex}−1/A^{ex} V.
33. Now to adapt this to the caſe of our balls, which weighed on a medium 16¾ ounces when the diameter was 1.96 inches; we ſhall have 1.96^{3} × .5236 = the magnitude of the ball; and as 1 cubic foot, or 1728 cubic inches, of water, weighs 1000 ounces, therefore [...] is the ſpecific gravity of the iron ball; which is very juſtly ſomething leſs than the uſual ſpecific gravity of ſolid caſt iron, on account of the ſmall air bubble which is in all caſt metal balls. Alſo the mean ſpecific gravity of air is .0012, which is the value of n. Hence [...].
Now the common diſtance of the face of the pendulum from the trunnions of all the guns, was 35½ feet; and the diſtance of the muzzles of the four guns, was nearly 34¼ for the 1ſt or ſhorteſt gun, 34 for the 2d, 33 for the 3d, and 31½ for the 4th. But as the elaſtic fluid purſues and urges the ball for a few feet after it is out of the gun, it may be ſuppoſed to counter-balance the reſiſtance of the air for a few feet, the number of which cannot be certainly known, and therefore we ſhall ſuppoſe 32 feet to be the common diſtance, for each of the guns, which the ball paſſes through before it reach the pendulum. Hence then the diſtance x = 32; and conſequently ex = 32/2666 = 16/1333.
Then A^{ex} − 1 = .01207 = 1/83 nearly. That is, the ball loſes nearly the 83d part of its laſt velocity, or the 84th part of its firſt velocity, in paſſing from the gun to the pendulum, by the reſiſtance of the air. Or the velocity at the mouth of the gun, is to the velocity at the pendulum, as 84 to 83; ſo that the greater diminiſhed by its 84th part gives the leſs, and the leſs increaſed by its [Page 133] 83d part gives the greater. But if the reſiſtance to ſuch ſwift velocities as ours be about three times as great as that above, computed from the nature of perfect and infinitely compreſſed fluids, as Mr. Robins thinks he has found it to be, then ſhall the velocity at the gun loſe its 28th part, and the greater velocity will be to the leſs, as 28 to 27. This however is a circumſtance to be diſcovered from our experiments, or otherwiſe.
1.9.5. Of the Velocity of the Ball, as found from the Recoil of the Gun.
34. It has been ſaid by more than one writer on this ſubject, that the effect of the inflamed power on the recoil of the gun, is the ſame whether it be charged with a ball, or fired by itſelf alone; that is, that the exceſs of the recoil when charged with a ball, over the recoil when fired without a ball, is exactly that which is due to the motion and reſiſtance of the ball. And this they ſay they have found from repeated experiments. Now ſuppoſing thoſe experiments to be accurate, and the deductions from them juſtly drawn; yet as they have been made only with ſmall balls and ſmall charges of powder, it may ſtill be doubted whether the ſame law will hold good when applied to ſuch cannon balls, and large charges of powder, as thoſe uſed in our preſent experiments. Which is a circumſtance that remains to be determined from the reſults of them. And this determination will be eaſily made, by comparing the velocity of the ball as computed from this law, with that which is computed from the vibration of the balliſtic pendulum. For if the law hold good in ſuch caſes as theſe, then the velocity of the ball, as deduced from the vibration of the gun, will exceed that which is deduced from the vibration of the pendulum, by as much as the velocity is diminiſhed by the reſiſtance of the air between the gun and the pendulum.
[Page 134] 35. Taking this for granted then in the mean time, namely, that the effect of the charge of powder on the recoil of the gun, is the ſame either with or without a ball, it will be proper here to inveſtigate a formula for computing the velocity of the ball from the recoil of the gun. Now upon the foregoing principle, if the chord of vibration be found for any charge without a ball, and then for the ſame charge with a ball, the difference of thoſe chords will be equal to the chord which is due to the motion of the ball. This follows from the property of a circle and a body deſcending along it, namely, that the velocity is always as the chord of the arc deſcribed in a ſemivibration.
Let then c denote this difference of the two chords, that is c = the chord of arc due to the ball's velocity, G = weight of the gun and iron ſtem, &c. b = weight of the ball, g = diſtance of center of gravity of G, o = diſtance of its center of oſcillation, n = its N^{o}. of oſcillations per minute, i = diſtance of the gun's axis, or point of impact, r = radius of arc or chord c, v = velocity of the ball, v = velocity of the gun, or of the axis of its bore.
Then becauſe biiv is the ſum of the momenta of the ball, and Ggov the ſum of the momenta of the gun, and becauſe action and re-action are equal, theſe two muſt be equal to each other, that is biiv = Ggov: But becauſe v is the velocity of the diſtance i, therefore by ſimilar figures i ∶ o ∷ v ∶ DV / i the velocity of the center of oſcillation. And becauſe the velocity of this center, is equal to the velocity generated by gravity, in deſcending perpendicularly through the height or verſed ſine [Page 135] of the arc deſcribed by it, and becauſe 2r ∶ c ∷ c ∶ cc/2r = verſed ſine to radius r, and r ∶ o ∷ cc/2r ∶ cco/2rr = verſ. ſine to radius o, therefore √h ∶ √ cco/2rr ∷ 2h ∶ c/r √2ho, the velocity of the center of oſcillation as deduced from the chord c of the arc deſcribed, where h = 16.09 feet; which velocity was before found = ov / i.
Therefore oV / i = c/r √2ho, or oV = ci/r √2ho. Then this value of ov being ſubſtituted in the firſt equation biiv = Ggov, we have biiv = Ggci/r √2ho, and hence the velocity v = Ggc/bir √2ho = 5.6727Ggc/bir √o, being the formula by which the velocity of the ball will be found in terms of the diſtance of the center of oſcillation and the other quantities. Which is exactly ſimilar to the formula for the ſame velocity, by means of the pendulum in Art. 22, uſing only G, or the weight of the gun, for p + b or the ſum of the weights of the ball and pendulum.
And if, inſtead of √o be ſubſtituted its value √ 11737.5/nn or 108.3398/n, from Art. 20, it becomes v = 614.58 × Ggc/birn, or = 59000/96 × Ggc/birn, the formula for the velocity of the ball in terms of the number of vibrations which the gun will make in one minute, and the other quantities.
36. Farther, as the quantities G, g, b, i, r, n commonly remain the ſame, the velocity will be directly as the chord c. So that if we aſſume a caſe in which the chord ſhall be 1, and call its correſponding velocity u; then ſhall v = cu; or the velocity correſponding to any [Page 136] other chord c, will be found by multiplying that chord c by the firſt velocity u anſwering to the chord 1.
Now, by the following experiments, the uſual values of thoſe literal quantities were as follows: viz. G = 917 g = 80.47 b = 1.047, ſometimes a little more or leſs. i = 89.15 r = 1000 n = 40.0, for the gun n^{o} 2, (but the 400th part more for n^{o} 1, and the 400th part leſs for n^{o} 3, and the 200th part leſs for n^{o} 4.)
Then, writing theſe values in the theorem, inſtead of the letters, it becomes v = 12.15c. So that the number 12.15 multiplied by the difference between two chords deſcribed with any charge, the one with and the other without a ball, will give the velocity of the ball when the dimenſions are as ſtated above. And when the values of any of the letters vary from theſe, it is but increaſing or diminiſhing that product in the ſame proportion, according as the letter belongs to the numerator or denominator in the general formula 59000/96 × Ggc/birn. When ſuch variations happen, they will be mentioned in each day's experiments. And farther when only the values of G, g, i, n are as before ſpecified, the ſame formula will become 12718 × c/br.
But note that theſe rules are adapted to the gun n^{o} 2 only; therefore for n^{o} 1 we muſt ſubtract the 400th part, and add the 400th part for n^{o} 3, and add the 200th part for n^{o} 4.
1.9.6. OF THE EXPERIMENTS.
[Page 137]37. WE ſhall now proceed to ſtate the circumſtances of the experiments for each day ſeparately as they happened; by this means ſhewing all the proceſſes for each ſet of experiments, with the failure or ſucceſs of every trial and mode of operation; and from which alſo any perſon may recompute all the reſults, and otherwiſe combine and draw concluſions from them as occaſion may require. Making but a very few curſory remarks on each day's experiments, to explain them when neceſſary; and reſerving the chief philoſophical deductions, to be drawn and ſtated together, after the cloſe of the experiments, in a more connected and methodical way.
The machinery having been made as perfect as the circumſtances would permit, 20 barrels of government powder were procured, all by the beſt maker, and numbered from 1 to 20. A great number of iron balls were alſo caſt on purpoſe, very round, and their accidental aſperities ground off: they were a little varied in their ſize and weight, but moſt of them almoſt equal to the diameter of the bore, ſo as to have but little windage. The powder was uniformly mixed, and every day exactly weighed off by the ſame careful man, and put up in very thin flannel bags, of a ſize juſt to fit the bore of the gun; a thread was tied round cloſe by the powder, after being ſhaken down, and the flannel cut off cloſe by the thread, ſo as to leave as ſhort a neck as poſſible to the bag. The charge of powder was puſhed gently down to the lower or breech end of the bore, and the ſame quantity of powder always made to occupy nearly the ſame extent, by means of the diviſions of inches and tenths marked on the ramrod. The ball was then put in, without uſing any wads, and ſet cloſe to the charge of powder, and kept in its place by a fine thread croſſed two or three times about it, which by its friction gave it a hold of the ſides of the bore, as the windage was very ſmall. The gun was directed point blank, or horizontal, and [Page 138] perpendicular to the face of the pendulum block, 35½ feet diſtant from the trunnions, and was well wiped and cleaned out after each diſcharge, which was made by piercing the bottom of the charge through the vent, and firing it by means of a ſmall tube. An account was kept of the barometer and thermometer, placed within a houſe adjoining, and ſhaded from the ſun.
The machinery having been all prepared and ſet up in a convenient place in Woolwich Warren, Major Blomefield and I went out on the 6th of June 1783, with a ſufficient party of men, to try the effects of them for the firſt time, which were as follows.
1.9.6.1. 38. Friday, June 6, 1783; from 10 till 12 A. M.
The weather was warm, dry, and clear.
The barometer at 30.17, and thermometer at 60°.
The intention of this day's experiments, was to try and adjuſt the apparatus; to aſcertain the proper diſtance of the pendulum; as alſo the comparative ſtrength of the different barrels of powder, by firing ſeveral charges of it, without balls or wads. Out of the 20 barrels of powder, were ſelected the 6 which had been found to be moſt uniform, and neareſt alike, by the different eprouvettes at Purfleet, which were n^{o•} 2, 5, 13, 15, 18, 19; of which the firſt two only were tried this day, as below. The gun was the ſhort one, n^{o} 1, and weighed this day, with leaden weights and iron ſtem, 906 lb: the diſtance of the tape, by which the chord of its recoil was meaſured, was not taken, and it was probably a little more than the uſual length, 110 inches, employed in moſt of the experiments of this year.
[Page 139] Here it appears that the quantity of recoil increaſed in a higher ratio than the quantity of powder.
The pendulum was not moved by the blaſt of the powder in theſe experiments.
1.9.6.2. 39. Saturday, June 7, 1783; from 9½ A. M. till 12.
The weather cloudy or hazy, but it did not rain.
Barometer 30.25, Thermometer 60°.
To try all the 6 ſorts of powder, and the effect of the blaſt on the pendulum, when high charges are uſed.
The firſt 14 rounds were with the ſame apparatus and gun n^{o} 1, as the former day.
The other four rounds with the gun n^{o} 4, but without the leaden weights; it weighed with the iron 561 lb.
[Page 140] Theſe recoils are very uniform, and there appears to be but little difference in the quality of the powder among the ſeveral ſorts.
All the charges were in flannel bags, except n^{o•} 14 and 18, of 16 oz each, for want of bags large enough provided to put it in. Each charge was rammed with two or three ſlight ſtrokes. A conſiderable quantity of the powder of n^{o} 14 was blown out unfired; many of the grains were found on the ground, and on the top of the pendulum block, and many were found ſticking in the face of it. By the force of theſe ſtriking it, and by the blaſt of the powder, or motion of the air, the pendulum was obſerved viſibly to vibrate a little: but the meaſuring tape had not been put to it. This was therefore now added, to meaſure the vibration by. And, to try to what degree the pendulum would be affected by the exploſion of the powder, the 7 feet amuſette was ſuſpended, and pointed oppoſite the center of the pendulum for the laſt 4 rounds. The pendulum was accordingly obſerved to move with the 8 ounces, but more with the 16 ounces, as appears at the bottom of the laſt column of the table above. The pendulum being thus much affected, we were convinced of the neceſſity of making a paper ſcreen to place between the gun and the pendulum; which we accordingly did, and uſed it in the whole courſe of experiments, at leaſt in the larger charges. At the laſt charge, which was 16 ounces of looſe powder, much ſewer grains were blown out than with the like charge at n^{o} 14 with the ſhort gun. The recoil at n^{o} 14 is evidently leſs than it ought to be; [Page 141] owing to the quantity of unfired powder that was blown out. It is remarkable that the recoil of the two guns, with the ſame charge, both for 2 ounces and 8 ounces, are nearly in the reciprocal ratio of the weights of the guns; a ſmall exceſs only, over that proportion, taking place in favour of the long gun, as due to its ſuperior length. The recoils are each viſibly in a higher proportion than the charges of powder: for, in the laſt four experiments, the charges of 2, 4, 8, 16 ounces, are in the continued proportion of 1 to 2; which their recoils 4.5, 10.8, 24.7, 53.3, are all in a higher ratio than that of 1 to 2; for, dividing the 2d by the 1ſt, the 3d by the 2d, and the 4th by the 3d, the three ſucceſſive quotients are 2.40, 2.29, 2.16, which are all above the double ratio, but approximating, however, towards it as the charge is increaſed. And farther, if we divide theſe quotients ſucceſſively one by another, the two new ratios or quotients will be nearly equal. So that, ranging thoſe recoils in a column under each other, and their two ſucceſſive orders of ratios in the adjacent columns, we ſhall have in one view the law which they obſerve, as here below, where they always tend to equality.
1.9.6.3. 40. Friday, June 13, 1783; from 11 till 1 o'clock.
[Page 142]The air moiſt, with ſmall rain at intervals.
The gun n^{o} 2 was mounted, and loaded with all the leaden weights: it was charged with the following quantities of powder; ſometimes with a ball, and ſometimes without one, as denoted by the cipher o, in the columns of weight and diameter of ball. The radius to the tape was —. As theſe experiments were made only to diſcover if the leaden weights would render the gun ſufficiently heavy, that the recoil might not be too large with the high charges of powder and ball, the pendulum block was removed, to let the balls enter and lodge in the bank which was behind it
Weight of powder | oz 2 | 4 | 8 | 16 |
Recoils with a ball | 8.9 | 16.2 | 26.5 | 34.7 |
Recoils without | 2.5 | 5.2 | 13.5 | 28.0 |
Differences | 6.4 | 11.0 | 13.0 | 6.7 |
There muſt have been ſome miſtake in the 10th round, as the recoil, which is 41.75 inches, is greater than can well be expected with that charge of powder. Probably the tape had entangled, and been drawn farther out in the return of the gun from the recoil.
1.9.6.4. 41. Monday, June 23, 1783.
We went with the workmen, and took the weight and dimenſions of the ſeveral parts of the machinery, both of the pendulum with its ſtem, and of the guns with their frame or iron ſtem, and the leaden weights to fit on about the trunnions.
That is, breadth of the face 18, height of the face 24, and length from front to back 22.
- A being the point through which the axis paſſes,
- G the point in the ſtem where it reſts in equilibrio, ſhewing the diſtance AG of the center of gravity below the axis,
- G g C perpendicular to A G,
- A P a plumb-line cutting G C in g,
- g the center of gravity of the iron work,
- B D a fixed perpendicular arm,
- E F a ſliding piece to ſupport the gun,
- T the center of the trunnions,
- t the place of the tape or loweſt point.
1.9.6.5.
[Page 145]42. The following are alſo the meaſures taken to ſettle the poſition of the compound center of gravity of the gun with its leaden weights and iron ſtem all together.
The numbers in the laſt column of this table, are the values of the letter g, in the formula for the velocity by means of the recoil of the gun. This letter may always be ſuppoſed to have the value 80.47 inches, as the two laſt numbers of the column differ from it but .03 only, or about the 2700 part of the whole, inducing an error of only about half a foot in the velocity of the ball.
The values of g, in this laſt column of the table, were computed in the following manner.
1.9.6.6.
1.9.6.7.
Of the great number of theſe meaſures that were taken, the variations among them would be ſometimes in exceſs and ſometimes in defect; and therefore the above numbers, which are the means among the whole, as long as the iron work remains the ſame, will probably be very near the truth. And by uſing always theſe, with proportional alterations in g and n for any alteration in the weight p, the computations of the velocity of the ball will be made by a rule that is uniform, and not ſubject at leaſt to accidental ſingle errors. When the weight of the pendulum varies by the wood alone of the block, or the ſtraps about it, the alteration is to be made at the center of the block, which is exactly 88.3 inches below the axis; that is, in that caſe the value of i is 88.3 in the formula [...], or the correction for g; and in [...], the correction [Page 148] of n. But when the alteration of the weight p ariſes from the balls and plugs lodged in the ſame block, then the value of i in thoſe corrections is the medium among the diſtances of the point ſtruck. And when the iron work is altered, the middle of the place altered gives the value of i in the ſame theorems.
In theſe corrections too p denotes 660, g 77.3, n 40.2, and b the difference between 660 and any other given weight of the pendulum; which value of b will be negative when this weight is below 660, otherwiſe poſitive; ſo that p + b is always equal to this weight of pendulum.
And if theſe values of p, g, n be ſubſtituted for them in thoſe corrections, they will become [...] or i−77.3/p b, the correction for g, and [...] the correction for n.
And farther, when i = 88.3, the ſame become 11b/660+b or 11 − 7260/p the correction for g, and b/1263+2.2b or .4545 − 261/p−86 the correction for n, as adapted to an alteration at the center of the pendulum.
And in that caſe G = 88.3 − 7260/p is the new value of g, and N = 39.7454 + 261/p−86 is the new value of n. But thoſe corrections will have contrary ſigns when b is negative, as well as the ſecond term in each of the denominators.
1.9.6.8. 45.Monday, June 30, 1783; from 9½ A. M. to 2½ P. M.
[Page 149]Barometer 30.34, and Thermometer 74.
We began this day for the firſt time to fire with balls againſt the pendulum block. The powder of the ſix barrels before-mentioned, had been all well mixed together for the uſe of our experiments, that they might be as uniform as poſſible, in that, as well as in other reſpects.
The GUN was n^{o} 1, with the leaden weights.
Its weight and the diſtance of its center of gravity, were as beforementioned; the diſtance of the tape it was forgotten this day to meaſure, but from circumſtances judged to be 106½.
The firſt 4 rounds were with powder only; the other 6 with balls, all of the ſame ſize and weight.
The two oaken plugs which were driven in, to fill up the holes, after the 8th and 9th rounds, weighed about 1¼ oz. to each inch of their length. The whole weights of theſe plugs, and the weights of the balls lodged in the block, were continually added to the weight of the pendulum, to compleat the numbers for the values of p in the 9th column; and from theſe numbers the correſpondent values of g and n, in the next two columns, are computed by their proper corrections in Art. 23 and 25. After which the velocities contained in the laſt column are computed by the formula in Art. 24. And the medium among all theſe velocities, as well as that of the recoils of the gun, are placed at the bottom of their reſpective columns.
Then, having b = 1.051, and r = 106.5, by the rule 12718 × c/br in Art. 36, we have only 1315 feet, for the velocity of the ball as deduced from the recoil of the gun; which is 141 leſs than the velocity found by the vibration of the pendulum, or about 1/10th of the whole velocity.
The powder blown out unfired was not much. The apparatus performed all very well, except only that the wood of the pendulum ſeemed not to be very ſound, as it was pierced quite through by the end of this day's experiments; though the ſheet lead with which the back was covered, as well as the face, juſt prevented the balls and pieces of the wood from falling out at the back of the pendulum.
1.9.6.9. 46. Saturday, July 5, 1783; from 9 till 2 o'clock.
[Page 151]The weather clear, dry, and hot. Barometer 30.27, and Thermometer 74.
A large piece had been cut out of the middle part of the pendulum, from the face almoſt to the back, to clear away the damaged part of the wood; and the vacuity was run full of lead, from an idea that the pendulum would not ſo ſoon be ſpoiled, and conſequently that it would need leſs repairs. But this did not ſucceed at all; for the only ſhot we diſcharged, namely, n^{o} 9, would not lodge in the lead, but broke into a thouſand ſmall pieces, many of which ſtuck in the lead, and formed a curious appearance; but the greater number rebounded back again, to the great danger of the by-ſtanders. The ball made a large round excavation in the face of the lead, of 5 inches diameter in the front, and 3½ inches deep in the center of the hole.
1.9.6.10. 47. Friday, July 11, 1783; from 9 A. M. till.
[Page 152]Length of the charge of 16 oz was 11.2 inches.
The pendulum had been altered ſince the former day. The core of lead being taken out, ſome layers of rope were laid at the bottom of the hole, then the remainder up to the front filled with a piece of ſound elm, and the face covered with ſheet lead.
At the laſt round, or that with ball, the iron tongue which held the tape of the pendulum, having ſlipped down by the looſening of a ſcrew, was ſtrained and bent. Which ſtopped the experiments till it could be repaired.
1.9.6.11. 48. Saturday, July 12, 1783; from 9 A. M. till.
[Page 153]The pendulum, gun n^{o} 3, and apparatus, were in every reſpect the ſame as in the laſt day's experiments, excepting that the radius of the tape, in the gun, was 110.2 inches inſtead of 110.
The mean length of the charge of 16 oz was 11.7 inches. But this height was always taken when the cartridge was uncompreſſed: ſo that the powder lay looſer than in former experiments. By a ſmall preſſure it occupied about ¼ of an inch leſs ſpace.
The value of p at beginning this day is made a little leſs than the pendulum weighed at firſt, for reaſons to be mentioned hereafter.
The mean recoil with a ball is 44.0, and without a ball 28.5, the difference of which is 15.5 = c. Alſo, in the formula for the velocity by means of the gun, we have b = 1.051, and r = 110.2. Conſequently v = 401/400 × 12718 × c / br = 1706 for the velocity by that method. But the mean velocity by the pendulum is 2030, which exceeds the former by 324, or almoſt ⅙ of the whole velocity.
1.9.6.12. 49. Thurſday, July 17, 1783; from 12 till 3 P. M.
[Page 154]Fine, clear, hot weather. Barometer 30.23, Thermometer 72° at 9 o'clock.
It ſwung very freely, and would have continued its vibrations a long time; owing to the ends of the axis being made to turn or roll upon a convex iron ſupport, and kept from going backward and forward, with the vibrations, by two upright iron pins, placed ſo as not quite to touch the axis, but at a very ſmall or hair-breadth diſtance from it.
The pendulum would not vibrate longer than 1 minute before the arcs became imperceptible, owing to the friction of the upright pins, which touched and bore hard againſt the ſides of the axis, unlike thoſe of the gun, although they had the ſame kind of round ſupport to roll upon. The pendulum had been well repaired, and ſtrengthened with iron bars, and ſtraps going round it in ſeveral places, except over the face. Alſo thick iron plates were let into, and acroſs it, near the back part, then over them was laid a firm covering of rope, after which the reſt of the hole was filled up with a block of elm, and ſinally the face covered over with ſheet lead.
The mean length of the charge of 8 oz was 5.9.
The pendulum, having been ſo well ſecured, ſuffered but little by this day's firing, only bulging or ſwelling out a little at the back part. All the balls were left in it, and all the holes were ſucceſſively plugged up with oaken pins of near 2 inches diameter, which weighed 11 oz to every 10 inches in length.—The arcs deſcribed, both by the gun and pendulum, are pretty regular. And the whole forms a good ſet of experiments.
Then v = 199/200 × 12718 × c/br = 399/400 × 12718 × 13.71/1.051×110.2 = 1501, the velocity of the ball as deduced from the recoil of the gun; which exceeds that deduced from the pendulum by 30, or nearly 1/49th part of this latter.
1.9.6.13. 50. Friday, July 18, 1783; from 9 A. M. till 12.
[Page 156]A freſh barrel of the mixed powder was opened for uſe this morning; and in the firſt 7 rounds, which were with powder only, that of the old and new barrel were uſed alternately, but no difference was obſerved.— The length of the charge of 4 oz was 3.2, and that of 8 oz was 5.9 inches.
It had remained hanging ſince the laſt day's experiments, with all the balls and plugs in it, which increaſed its weight by 10 lb, except an allowance for evaporation, and increaſed the diſtance of the center of gravity by little more than 1/10th of an inch. It vibrated with great freedom; for it had this day been made to turn very freely on its axis, by placing the upright pins, which confine it ſide-ways, ſo as not quite to touch the axis, like thoſe of the gun yeſterday; and the effect was very great indeed, for it appeared as if it would have vibrated for a great length of time; whereas on the former days it ſtopped motion in about 1 minute, or at leaſt after that the arcs ſoon became too ſmall to be counted.—By this day's firing the pendulum ſeemed not to be much injured, the back part not appearing to be altered, and the fore part only a little ſwelled out, the piece of wood, that had been fitted in there, ſtarting a little forward, and bulging out the facing of lead.
This appears to be a good ſet, being very uniform, except the 13th round, which has been omitted, as evidently defective in the arc deſcribed both by the gun and pendulum, from ſome undiſcovered and unaccountable cauſe.
1.9.6.14. 51. Saturday, July 19, 1783; from 9 till 3.
[Page 158]Of the plugs every 10 inches weighed 11 ounces.
Length of the charge of 2 oz was 1.7; and that of 4 oz was 3.2.
The GUN was n^{o}; 1 for the firſt 12 n^{o•};, and n^{o}; 3 for the reſt; in order to complete the compariſon between theſe two guns with 2, 4, 8, and [Page 159] 16 oz of powder. The radius to the tape 110 inches, and the other circumſtances as before.
The PENDULUM had been left hanging ſince yeſterday, and the radius to the tape was 117.8 as before. It became however ſo full of balls and plugs to-day, that no more plugs could be driven in, all the iron ſtraps being bent and forced out to their utmoſt ſtretch. It was therefore ordered to be gutted and repaired.
This is a good ſet of experiments; all the apparatus having performed well; and the arcs deſcribed, both by the gun and pendulum, being very uniform.
1.9.6.15. 52. Wedneſday, July 23, 1783; from 10 till 3.
[Page 160][Page 161] Of the plugs every 10 inches weighed 12 ounces.
It heated very little by firing.
It had been gutted, and repaired, by placing a ſtratum of lead, of 2 inches thick, before the iron plate, then the lead was covered with a block of wood, and the whole faced with ſheet lead.
So that the recoil gives the velocity with 2, 4 and 8 ounces of powder greater, but with 16 ounces much leſs, than the velocity ſhewn by the pendulum.
1.9.6.16. 53. Monday, July 28, 1783; from 10 till 2.
[Page 162][Page 163] The gun was very hot before firing, with the heat of the ſun. But heated little more with firing. It was hotteſt at the muzzle, where the hand could not long bear the heat of it.
The PENDULUM had been gutted and repaired ſince the laſt day.
1.9.6.17. 54. Tueſday, July 29, 1783; from 12 till 3.
[Page 164] Of the plugs every 10 inches weighed 13½ ounces.
The GUN n^{o} 4.—Its weight and other circumſtances as uſual. It did not become near ſo hot as yeſterday.
The PENDULUM was as weighed and meaſured yeſterday, having hung unuſed.
The tape drawn out in the laſt three rounds, both of the gun and pendulum, was rather doubtful, owing to the wind blowing and entangling it.
1.9.6.18. 55. Wedneſday, July 30, 1783; from 10 till 12.
[Page 165]The GUN was again n^{o} 4, and every circumſtance about it as before.
The PENDULUM the ſame as left hanging ſince yeſterday, with the addition of the balls and plugs in it.
1.9.6.19. 56. Thurſday, July 31, 1783; from 10 till 12.
[Page 166]The GUN n^{o} 1.—Weight and every thing elſe as uſual.—The annular leaden weights, which fit on about the trunnions, have gradually been knocked much out of form by the ſhocks of the ſudden recoils; ſo that, not fitting cloſely, they are ſubject to ſhake, a circumſtance which probably has occaſioned the irregularities in the recoils of this day.
The PENDULUM continued hanging ſtill. It is ſuſpected that its vibrations are not to be ſtrictly depended on with the high charges of powder; owing to the ſtriking of the balls againſt the iron plate within the block, and ſo perhaps cauſing them to rebound within it, and diſturb [Page 167] the vibrations, which are not regular this day. After it was taken down, the pendulum was found to weigh 726lb. But, from the weight of the balls and plugs lodged in it, it ought to have weighed 732 lb. It is therefore likely that the 6 lb had been loſt, by evaporation of the moiſture, in the 4 days, which is 1½lb per day. At the beginning of each day's experiments therefore 1½lb is deducted from the weight of the pendulum, or 2lb before each of the laſt three days. And the like was done on ſome former days, for the ſame reaſon, when it appeared neceſſary.
1.9.6.21. 57. Tueſday, Auguſt 12, 1783; from 10 till 2½.
[Page 168]1.9.6.22. The GUN was n^{o} 1 in the firſt 8 rounds; and n^{o} 2 in the reſt to the end. The weight, &c. as before.
At the 7th and 15th rounds the balls ſtruck both in firm and ſolid wood, when their penetrations, to the hinder part of the ball, meaſured 10½ and 11 inches; ſo that the fore part penetrated 12½ inches in the firſt caſe, and 13 inches in the latter.
1.9.6.23. 59. Wedneſday, Auguſt 13, 1783; from 10 till 2.
[Page 171] The GUN was n^{o} 3. In the 5, 6, 7, 8, and 12th rounds, the gun had from 15′ to 20′ elevation. At the 6th round an uncommon large quantity of powder came out unfired, ſo as to ſcatter a great way over the ground, and beſpatter the face of the ſcreen and pendulum very much; which was not the caſe in any other round. And this may account for the ſmaller arcs deſcribed at that number.
The PENDULUM was in the ſame condition as it had been left hanging after the laſt day's experiments, with all the balls and plugs in it. After this day's experiments, its weight was found to be 681 lb, including all the balls and plugs, except one which flew out behind the pendulum at the 7th round, occaſioned by this ball ſtriking in the ſame hole as n^{o} 6, and knocking it out. This ball, which came out, was quite whole and perfect; it was black on the hinder part with the powder, but rubbed bright before with the friction in paſſing through the wood. The tape of the pendulum alſo broke at this round, ſo that the vibration could not be meaſured.
The value of i, or the mean among the diſtances of the point ſtruck this day and the laſt is 88.
Of the plugs, this day and the laſt, 10 inches weighed 9 oz.
1.9.6.24. 60. Monday, September 8, 1783; from 10 till 1½ P. M.
[Page 172]The GUN n^{o} 3, with every circumſtance as uſual; except that in the laſt four rounds it had 15′ elevation.
The PENDULUM had been repaired, the balls and plugs taken out, a ſquare hole cut quite through, and a ſound piece fitted in; and the face covered with ſheet lead as before.
[Page 173] The vibration at n^{o} 8 a little doubtful, as the tape broke.
The plugs weighed 1 oz per inch.
The value of i, or the mean diſtance of the points ſtruck, 87.3.
1.9.6.25. 61. Wedneſday, September 10, 1783; from 10 till 12.
[Page 174]The GUN, n^{o} 1. Weight and other circumſtances as uſual.
The PENDULUM as left hanging ſince Monday. Its radius, &c. as uſual.—The value of i, or the mean diſtance among the points ſtruck this day and the former, is 88.0.
The plugs weighed 1 oz per inch.
1.9.6.26. 62. Thurſday, September 11, 1783; from 10 till 12.
[Page 176] The GUN n^{o} 2. In the laſt 5 rounds it had about 10′ depreſſion.
The PENDULUM the ſame as left hanging ſince yeſterday. After the experiments were concluded to-day, it weighed 694 lb.—The plugs weighed 1 oz per inch.
The weight of balls and plugs lodged in the block, theſe laſt three days, was 36 lb; which added to 663, the weight at the beginning, makes 699: but it weighed at the end only 694; ſo that it loſt 5 lb of its weight in the 4 days, or 1¼ lb per day on a medium.
The value of i, or the mean among the diſtance of the points ſtruck theſe three days, is 88.3.
1.9.6.27. 63. Tueſday, September 16, 1783; from 12 till 2.
[Page 177]The laſt n^{o} very uncertain; the tape, being very wet, twiſted, and was entangled.
1.9.6.28. 64. Thurſday, September 18, 1783; from 10 till 3 P. M.
[Page 178][Page 179] The GUN n^{o} 1. The charge of powder was gradually increaſed till the gun became quite full at n^{o} 10, when there was juſt room for half the ball to lie within the muzzle; which being too ſhort a length to give a direction to the ball, it miſſed the pendulum, going over and juſt ſtriking the top of the ſcreen frame, about 21½ inches above the line of direction, which, though a very ſlender piece of wood, turned the ball up into a ſtill higher direction, in which it ſtruck the bank over the pendulum, and entered it ſloping, though but a little way: all which circumſtances ſhew that the force of the ball was but ſmall. And even at the 9th round, when the center of the ball was about 3 inches within the gun, the ball ſtruck the pendulum 5 inches out of the line of direction. The gun was ſcarce ever ſenſibly heated.
The diameter of the balls 1.96 inches.
The PENDULUM had been gutted, and had received a new core. It was hung up in the morning of the day before yeſterday, when it weighed 659 lb. And when taken down this evening it weighed only 686 lb, which is near 4 lb leſs than the balls and plugs ought to make it; and which 4 pounds muſt have evaporated in the 3 days.
The plugs weighed ⅞ of an ounce to the inch.
The value of i, or mean point ſtruck, 89.7.
All the three rounds with 16 oz are very doubtful, and ſeem to be too low, from ſome unknown cauſe.